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## Book.MathNotes istorijaPaslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus 2020 gegužės 30 d., 16:03
atliko -
Pakeistos 90-92 eilutės iš
The onesome defines order, whereas the twosome defines entropy, and so the two are related in that way. į:
The onesome defines order, whereas the twosome defines entropy, and so the two are related in that way. The reversal of entropy is the reversal of the twosome. 2020 gegužės 30 d., 15:14
atliko -
Pakeistos 88-90 eilutės iš
The twosome is an expression of the nature of entropy, the second law of thermodynamics. The mind shifts from a state of greater ambiguity (where opposites coexist) to a state of lesser ambiguity (where all is the same). Correspondingly, the mind shifts from a state of higher energy to a state of lower energy. į:
The twosome is an expression of the nature of entropy, the second law of thermodynamics. The mind shifts from a state of greater ambiguity (where opposites coexist) to a state of lesser ambiguity (where all is the same). Correspondingly, the mind shifts from a state of higher energy to a state of lower energy. The onesome defines order, whereas the twosome defines entropy, and so the two are related in that way. 2020 gegužės 30 d., 13:18
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Pakeistos 86-88 eilutės iš
Could particles be extremal black holes? į:
Could particles be extremal black holes? The twosome is an expression of the nature of entropy, the second law of thermodynamics. The mind shifts from a state of greater ambiguity (where opposites coexist) to a state of lesser ambiguity (where all is the same). Correspondingly, the mind shifts from a state of higher energy to a state of lower energy. 2020 gegužės 29 d., 19:27
atliko -
Pakeistos 84-86 eilutės iš
House of knowledge: Pushdown automata: Every question has an answer. Their two wings, entering the game and leaving the game, are linked by the three-cycle. į:
House of knowledge: Pushdown automata: Every question has an answer. Their two wings, entering the game and leaving the game, are linked by the three-cycle. Could particles be extremal black holes? 2020 gegužės 29 d., 12:52
atliko -
Pakeista 84 eilutė iš:
į:
House of knowledge: Pushdown automata: Every question has an answer. Their two wings, entering the game and leaving the game, are linked by the three-cycle. 2020 gegužės 29 d., 12:33
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Pakeistos 82-84 eilutės iš
What is the relationship between circumstances (12 topologies) and the meaning of context in automata theory. į:
What is the relationship between circumstances (12 topologies) and the meaning of context in automata theory. Pushdown automata: Every question has an answer. 2020 gegužės 27 d., 17:03
atliko -
Pakeistos 80-82 eilutės iš
Yoneda Lemma: Covariant version and contravariant version are two PDAs that come together to form a Turing machine. į:
Yoneda Lemma: Covariant version and contravariant version are two PDAs that come together to form a Turing machine. What is the relationship between circumstances (12 topologies) and the meaning of context in automata theory. 2020 gegužės 27 d., 17:02
atliko -
Pakeistos 78-80 eilutės iš
Algebraic thinking is step-by-step, as with a PDA, state-by-state. į:
Algebraic thinking is step-by-step, as with a PDA, state-by-state. Yoneda Lemma: Covariant version and contravariant version are two PDAs that come together to form a Turing machine. 2020 gegužės 27 d., 17:01
atliko -
Pakeistos 76-78 eilutės iš
* Splits in two the validation, "division into two". The subroutine validates the changes in the internal step and the external step, but assumes that the incoming natural transformation has already been validated. į:
* Splits in two the validation, "division into two". The subroutine validates the changes in the internal step and the external step, but assumes that the incoming natural transformation has already been validated. Algebraic thinking is step-by-step, as with a PDA, state-by-state. 2020 gegužės 27 d., 17:01
atliko -
Pakeistos 73-76 eilutės iš
* Actual entropy does not require any partitioning. į:
* Actual entropy does not require any partitioning. Yoneda Lemma * Splits in two the validation, "division into two". The subroutine validates the changes in the internal step and the external step, but assumes that the incoming natural transformation has already been validated. 2020 gegužės 26 d., 14:25
atliko -
Pakeistos 67-73 eilutės iš
* the elements in F(A) are just dummies, they don't matter - they don't have meaning - double check, what could they possibly mean? į:
* the elements in F(A) are just dummies, they don't matter - they don't have meaning - double check, what could they possibly mean? Entropy * Should distinguish between interpretations of apparent entropy based on probability (as in physics) and actual entropy based on irreversibility (as in automata theory). * Apparent entropy depends on having a particular partitioning that depends on the observer. * The observer is defined by interpretation of the choice frameworks. * Actual entropy does not require any partitioning. 2020 gegužės 24 d., 23:40
atliko -
Pakeistos 66-67 eilutės iš
* alpha (f) <-> Hom (Hom(f,_),alpha) į:
* alpha (f) <-> Hom (Hom(f,_),alpha) * the elements in F(A) are just dummies, they don't matter - they don't have meaning - double check, what could they possibly mean? 2020 gegužės 24 d., 23:37
atliko -
Pakeistos 65-66 eilutės iš
* The extension happens on the inside and the outside, the validations balance each other, like left and right parentheses. į:
* The extension happens on the inside and the outside, the validations balance each other, like left and right parentheses. * alpha (f) <-> Hom (Hom(f,_),alpha) 2020 gegužės 24 d., 23:34
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Pakeistos 60-65 eilutės iš
When you have an object, then you ignore the other objects, the "non-objects", thus in the binomial theorem you don't have to deal with their relationships. į:
When you have an object, then you ignore the other objects, the "non-objects", thus in the binomial theorem you don't have to deal with their relationships. Yoneda Lemma: * Validating a single step, in a generic way, is equivalent to validating everything. * The theta was validated by somebody else, and we just validate the extension. * The extension happens on the inside and the outside, the validations balance each other, like left and right parentheses. 2020 gegužės 17 d., 18:24
atliko -
Pakeistos 58-60 eilutės iš
Thus 34 or 37 particles. į:
Thus 34 or 37 particles. When you have an object, then you ignore the other objects, the "non-objects", thus in the binomial theorem you don't have to deal with their relationships. 2020 gegužės 16 d., 11:59
atliko -
Pakeista 54 eilutė iš:
* 6 į:
* 6 quarks and 6 antiquarks Pakeista 58 eilutė iš:
Thus į:
Thus 34 or 37 particles. 2020 gegužės 16 d., 11:25
atliko -
Pakeistos 46-58 eilutės iš
Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. Relate these paths to a matrix and to symmetric functions on the eigenvalues of a matrix. į:
Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. Relate these paths to a matrix and to symmetric functions on the eigenvalues of a matrix. Standard model: leptons mediate forces * photon - electromagnetic force * 8 gluons - strong force * Z boson - weak force * W (+,-) boson - weak force Also: * 6 quarks * 6 leptons * 3 or 6 neutrinos * 1 Higgs Thus 28 or 31 particles. 2020 gegužės 16 d., 09:34
atliko -
Pakeista 46 eilutė iš:
Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. į:
Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. Relate these paths to a matrix and to symmetric functions on the eigenvalues of a matrix. 2020 gegužės 16 d., 09:33
atliko -
Pakeistos 44-46 eilutės iš
With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries and explore. į:
With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries and explore. Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. 2020 gegužės 15 d., 22:48
atliko -
Pakeista 44 eilutė iš:
With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries. į:
With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries and explore. 2020 gegužės 15 d., 22:48
atliko -
Pakeistos 42-44 eilutės iš
Functors preserve all į:
Functors preserve all paths With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries. 2020 gegužės 15 d., 22:38
atliko -
Pakeistos 40-42 eilutės iš
* Problem -> Contentualize -> Contextualize -> Reformulate problem -> Find relevant categorical requirements -> Solve į:
* Problem -> Contentualize -> Contextualize -> Reformulate problem -> Find relevant categorical requirements -> Solve problem Functors preserve all paths 2020 gegužės 15 d., 22:37
atliko -
Pakeistos 39-40 eilutės iš
Relate content and context. į:
Relate content and context. * Problem -> Contentualize -> Contextualize -> Reformulate problem -> Find relevant categorical requirements -> Solve problem 2020 gegužės 15 d., 22:36
atliko -
Pakeistos 37-39 eilutės iš
Functor is a contextualization. Is Z a group or an abelian group? "Form follows function". į:
Functor is a contextualization. Is Z a group or an abelian group? "Form follows function". Relate content and context. 2020 gegužės 15 d., 22:35
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Pakeistos 35-37 eilutės iš
[[https://books.google.lt/books/about/The_Logical_Foundations_of_Cognition.html?id=_A1nDAAAQBAJ&redir_esc=y | The Logical Foundations of Cognition]] John Mcnamara, Gonzalo E. Reyes, 1994. į:
[[https://books.google.lt/books/about/The_Logical_Foundations_of_Cognition.html?id=_A1nDAAAQBAJ&redir_esc=y | The Logical Foundations of Cognition]] John Mcnamara, Gonzalo E. Reyes, 1994. Functor is a contextualization. Is Z a group or an abelian group? "Form follows function". 2020 gegužės 15 d., 21:36
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Pakeistos 33-35 eilutės iš
Is every computation system of sufficient sophistication equivalent to a house of knowledge? And are lesser systems equivalent to a part of a house of knowledge? į:
Is every computation system of sufficient sophistication equivalent to a house of knowledge? And are lesser systems equivalent to a part of a house of knowledge? [[https://books.google.lt/books/about/The_Logical_Foundations_of_Cognition.html?id=_A1nDAAAQBAJ&redir_esc=y | The Logical Foundations of Cognition]] John Mcnamara, Gonzalo E. Reyes, 1994. 2020 gegužės 15 d., 21:03
atliko -
Pakeista 33 eilutė iš:
Is every computation system of sufficient sophistication equivalent to a house of knowledge? į:
Is every computation system of sufficient sophistication equivalent to a house of knowledge? And are lesser systems equivalent to a part of a house of knowledge? 2020 gegužės 15 d., 21:03
atliko -
Pakeistos 31-33 eilutės iš
Is there a kind of mathematics that is behind every science, every house of knowledge, every person? į:
Is there a kind of mathematics that is behind every science, every house of knowledge, every person? Is every computation system of sufficient sophistication equivalent to a house of knowledge? 2020 gegužės 15 d., 21:02
atliko -
Pakeistos 29-31 eilutės iš
* "Grothendieck thought about this very hard and invented his concept of topos, which is roughly a category that serves as a place in which one can do mathematics." A place for figuring things out? What would that mean? į:
* "Grothendieck thought about this very hard and invented his concept of topos, which is roughly a category that serves as a place in which one can do mathematics." A place for figuring things out? What would that mean? Ways of extending the mind by leveraging basic ways of figuring things out and organizing them around a particular observer? Is there a kind of mathematics that is behind every science, every house of knowledge, every person? 2020 gegužės 15 d., 21:01
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Pakeistos 27-29 eilutės iš
[[http://math.ucr.edu/home/baez/topos.html | John Baez about toposes and Lawvere]]: "In algebraic geometry we are often interested not just in whether or not something is true, but in where it is true." Relate this to scopes: truths about everything, anything, something, nothing. į:
[[http://math.ucr.edu/home/baez/topos.html | John Baez about toposes and Lawvere]]: * "In algebraic geometry we are often interested not just in whether or not something is true, but in where it is true." Relate this to scopes: truths about everything, anything, something, nothing. * "Grothendieck thought about this very hard and invented his concept of topos, which is roughly a category that serves as a place in which one can do mathematics." A place for figuring things out? What would that mean? 2020 gegužės 15 d., 20:54
atliko -
Pakeistos 25-27 eilutės iš
How do position-momentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without a notion of momentum? į:
How do position-momentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without a notion of momentum? [[http://math.ucr.edu/home/baez/topos.html | John Baez about toposes and Lawvere]]: "In algebraic geometry we are often interested not just in whether or not something is true, but in where it is true." Relate this to scopes: truths about everything, anything, something, nothing. 2020 gegužės 15 d., 17:51
atliko -
Pakeista 25 eilutė iš:
How do position-momentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without momentum? į:
How do position-momentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without a notion of momentum? 2020 gegužės 15 d., 17:51
atliko -
Pakeista 25 eilutė iš:
How do position-momentum relate symplectic geometry and the entropy of phase space? į:
How do position-momentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without momentum? 2020 gegužės 15 d., 17:51
atliko -
Pakeistos 23-25 eilutės iš
Mass is an indicator of subsystems. į:
Mass is an indicator of subsystems. How do position-momentum relate symplectic geometry and the entropy of phase space? 2020 gegužės 15 d., 14:00
atliko -
Pakeistos 21-23 eilutės iš
Looking at the Standard Model: 12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome. į:
Looking at the Standard Model: 12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome. Mass is an indicator of subsystems. 2020 gegužės 15 d., 11:29
atliko -
Pakeista 21 eilutė iš:
12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome. į:
Looking at the Standard Model: 12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome. 2020 gegužės 15 d., 11:29
atliko -
Pakeistos 19-21 eilutės iš
Yoneda Lemma. The set function {$\theta \rightarrow \alpha \theta \textrm{Hom}(f,\_)$} is a rule for a pushdown automaton. The {$\alpha$} comes from the finite automaton (the input) and the {$\textrm{Hom}(f,\_)$} should describe the stack of memory. This all, on the left-hand side, is compared to a finite automaton on the right-hand side. į:
Yoneda Lemma. The set function {$\theta \rightarrow \alpha \theta \textrm{Hom}(f,\_)$} is a rule for a pushdown automaton. The {$\alpha$} comes from the finite automaton (the input) and the {$\textrm{Hom}(f,\_)$} should describe the stack of memory. This all, on the left-hand side, is compared to a finite automaton on the right-hand side. 12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome. 2020 gegužės 14 d., 12:27
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Pakeistos 17-19 eilutės iš
Study categories with a single initial state and a single final state. What does Yoneda Lemma mean for them? į:
Study categories with a single initial state and a single final state. What does Yoneda Lemma mean for them? Yoneda Lemma. The set function {$\theta \rightarrow \alpha \theta \textrm{Hom}(f,\_)$} is a rule for a pushdown automaton. The {$\alpha$} comes from the finite automaton (the input) and the {$\textrm{Hom}(f,\_)$} should describe the stack of memory. This all, on the left-hand side, is compared to a finite automaton on the right-hand side. 2020 gegužės 14 d., 12:23
atliko -
Pakeistos 15-17 eilutės iš
Automata are important for the dynamics of the three languages. į:
Automata are important for the dynamics of the three languages. Study categories with a single initial state and a single final state. What does Yoneda Lemma mean for them? 2020 gegužės 13 d., 17:24
atliko -
Pakeistos 13-15 eilutės iš
Pushdown automata have a stack of priorities. In general, automata deal with concerns - rūpesčiai. į:
Pushdown automata have a stack of priorities. In general, automata deal with concerns - rūpesčiai. Automata are important for the dynamics of the three languages. 2020 gegužės 13 d., 17:24
atliko -
Pakeistos 11-13 eilutės iš
Anyons are composite particles in two-dimensions that have statistics in between fermion (object) and boson (arrow) statistics. How can they be understood in terms of category theory? į:
Anyons are composite particles in two-dimensions that have statistics in between fermion (object) and boson (arrow) statistics. How can they be understood in terms of category theory? Pushdown automata have a stack of priorities. In general, automata deal with concerns - rūpesčiai. 2020 gegužės 12 d., 22:50
atliko -
Pakeistos 9-11 eilutės iš
David Corfield: [[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere. į:
David Corfield: [[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere. Anyons are composite particles in two-dimensions that have statistics in between fermion (object) and boson (arrow) statistics. How can they be understood in terms of category theory? 2020 gegužės 11 d., 21:48
atliko -
Pakeista 9 eilutė iš:
[[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere. į:
David Corfield: [[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere. 2020 gegužės 11 d., 21:48
atliko -
Pakeista 9 eilutė iš:
[[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components į:
[[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere. 2020 gegužės 11 d., 21:47
atliko -
Pakeistos 7-9 eilutės iš
Study [[https://www.math3ma.com/blog/at-the-interface-of-algebra-and-statistics | Tai-Danae Bradley's thesis]] to learn about the transition between classical and quantum probabilities, between sets (F1) and vector spaces (Fq), and look for a connection with N and N2 as per the Yoneda lemma. į:
Study [[https://www.math3ma.com/blog/at-the-interface-of-algebra-and-statistics | Tai-Danae Bradley's thesis]] to learn about the transition between classical and quantum probabilities, between sets (F1) and vector spaces (Fq), and look for a connection with N and N2 as per the Yoneda lemma. [[https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | Spatial notions of cohesion]] as the basis for geometry. A fourfold adjunction: components ⊣\dashv discrete ⊣\dashv points ⊣\dashv codiscrete. and a threefold adjunction of modalities based on that, originally due to Lawvere. 2020 gegužės 11 d., 21:42
atliko -
Pakeistos 5-7 eilutės iš
Binomial theorem. Derivative. There is no volume, just the faces, as with the cross-polytopes. Also, the derivative has the boundary conditions as in homology. į:
Binomial theorem. Derivative. There is no volume, just the faces, as with the cross-polytopes. Also, the derivative has the boundary conditions as in homology. Study [[https://www.math3ma.com/blog/at-the-interface-of-algebra-and-statistics | Tai-Danae Bradley's thesis]] to learn about the transition between classical and quantum probabilities, between sets (F1) and vector spaces (Fq), and look for a connection with N and N2 as per the Yoneda lemma. 2020 gegužės 08 d., 10:00
atliko -
Pakeistos 3-5 eilutės iš
All systems are the same: The Principle of Computational Equivalence [1, 2], due to S. Wolfram, is the heuristic statement that almost all processes (involving classical computations) that are not obviously simple are of equivalent sophistication. į:
All systems are the same: The Principle of Computational Equivalence [1, 2], due to S. Wolfram, is the heuristic statement that almost all processes (involving classical computations) that are not obviously simple are of equivalent sophistication. Binomial theorem. Derivative. There is no volume, just the faces, as with the cross-polytopes. Also, the derivative has the boundary conditions as in homology. 2020 gegužės 07 d., 19:27
atliko -
Pakeistos 1-3 eilutės iš
[++++数学笔记++++] į:
[++++数学笔记++++] All systems are the same: The Principle of Computational Equivalence [1, 2], due to S. Wolfram, is the heuristic statement that almost all processes (involving classical computations) that are not obviously simple are of equivalent sophistication. 2020 gegužės 07 d., 13:02
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Pakeistos 1-173 eilutės iš
[++++数学笔记++++] Duality * Adjunction is a form of duality. * Equality and equivalence, in general, are forms of duality. * Duality is an extension of equivalence where the two sides of the equation are somehow different. For example, one side may be a variable and the other side a value that it is set to. * Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such. * Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome. Math Discovery * Kan extensions are examples of "extending the domain". So consider what they say about the three-cycle in the house of knowledge, how they relate to arguments by continuity and by self-superimposition. Think of monads as describing models, that is, simplified accounts of a system. Note that monads are built on adjoint functors, which express least upper bounds and greatest lower bounds. So this suggests that the four levels of logic/geometry are related to the four levels of algebra/analysis. But how, perhaps inversely? Because the bounds are third level, but the models are second level. Thus: * center / induction = variables * balance-parity / maximum-minimum = working backwards, implication * sets / bounds = models * vector spaces / limits = contradictions What this suggests is that the richest structures are inherently contradictory. But if we pull back to simpler structures then they become safer. Category theory * In a category, if two objects have morphisms into each other, than they are isomorphic. So if we are only interested in nonisomorphic objects, then we can consider equivalence classes. And we can see that the maps only go in one way, so they are naturally partially ordered. * Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less. * Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D. * In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable? Divisions * The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | 维基百科: Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? Universality is unconditionality is how God thinks. I am interested not in how the various mathematicians think but how God thinks. * The identity morphism falls out as a special morphism among the possible morphisms from B to B. It is the basis for Graziano's awareness schema. * So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others. * Why I feel strange that Set is not definite: "every function should have a definite class as domain and a definite class as range". Riehl quotes Eilenberg and Maclane: ". . . the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation . . . . The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether and adopt an even more intuitive standpoint, in which a functor such as “Hom” is not defined over the category of “all” groups, but for each particular pair of groups which may be given. [EM45]" * Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity. * The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, we * John Baez: "Every sufficiently good analogy is yearning to become a functor." Compare with metaphors, blends. * Can a functor or a function add information? Investigate: A function can add context in the codomain. For example, a loop can be understood as a circle or as a complicated closed curve. It is a loop to itself, but in another context it is a complicated curve, perhaps in the complex numbers, perhaps in a multi-dimensional space. Walks * What is the connection between random walks (all walks) on a graph and homotopy loops on a surface? Universality. The tendency of the eigenvalues of random matrices to space themselves out uniformly. Similarly, in nature, physical phenomena space themselves out in different orders of magnitude. Similarly, we have orders of scale in Alexander's theory. * [[https://www.quantamagazine.org/in-mysterious-pattern-math-and-nature-converge-20130205 | Natalie Wolchover. In Mysterious Pattern, Math and Nature Converge.]] * [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | 维基百科: Tracy-Widom distribution]] There are Tracy-Widom distributions for orthogonal ($F_1$}, unitary {$F_2$} and symplectic {$F_4$} random matrices. * [[https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents | Painlevé transcendents]] * [[https://www.quantamagazine.org/beyond-the-bell-curve-a-new-universal-law-20141015/ | At the far ends of a new universal law.]] * [[https://www.quantamagazine.org/tag/universality/ | Quanta Magazine articles on universality]] * [[https://www.youtube.com/watch?v=HrtJ3SRQF4E | V: What is Universality?]] * Does the number {$\sqrt{2n}$} relate to the root systems with {$2n$} simple roots? Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? Can the Yoneda lemma help model such a phase transition? * There may be a shift from relations (n2) to objects (n) when n grows large so as to reduce the overhead. This would break a system down into subsystems, which can operate independently, line by line. So there should be a pressure for a system to break down into subsystems. * The Yoneda Lemma establishes the underlying isomorphism. But how can such naturally isomorphic structures be meaningfully different? It is because the interpreting perspective is different. As with entropy, the perspective matters, the coordinate system we choose. This perspective is inherent in whether we have a system or a subsystem. * Think also of the orders of magnitude in the universe. And orders of scale in Alexander's principles of life. * Switching over from a wave (or boson) point of view to a particle (or fermion) point of view. Note that multiple morphisms can have the same location, but multiple objects cannot. * Consider how the identity morphism becomes distinguished. Note that the Yoneda lemma can be explaining how an identity morphism gets inferred. Because perhaps there aren't identity morphisms in the beginning. * Consider how a system of morphisms unfolds from a single morphism, consider how a new morphism arises and consider the conditions that it has to satisfy to be consistent with the system, especially with composition. * Consider how symmetric functions of eigenvalues are relevant here. Morally, the phase transition can model good and bad behavior. Bad behavior - such as slowing down your bus so that you are followed by another bus and take all of its customers - is behavior that complicates the phase transition. Also, the overhead of words: if you think in words (based on the world) rather than concepts (rooted in the features of your mind) then you will be taxed for that and at a certain point it won't be sustainable. In general, this is modeling nonsustainability. Random matrices are related to random walks and other symmetric functions of the eigenvalues of matrices. If we think of a hook as a walk, within a rim hook or a special rim hook, then we can think of these tableaux as ways of assembling walks, and thus of assembling systems from relationships. [[https://arxiv.org/abs/1001.0722 | Martin R. Zirnbauer. Symmetry Classes.]] Dyson's Threefold Way: * F.J. Dyson, “The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics”, J. Math. Phys.3(1962) 1199-1215 * "the most general matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of whichbelongs to one of the three known types": Complex Hermitian, real symmetric, or quaternion self-dual. Bott periodicity and symmetry classes are related in the [[https://topocondmat.org/w8_general/classification.html | periodic table of topological insulators]]. [[http://www.ft.uam.es/personal/rubio/Random_Matrix_symmetries.pdf | Rubio. Random Matrix Symmetries]] Energy levels are eigenvalues. Thus we consider the distribution of energy levels and their separation. On the analytic wing of the house of knowledge, we can think of these as the energy levels, the eigenvalues, that must be kept separate, kept distinguished, like fermions. And in the case of a long tail we have a low energy extremes. [[https://www.youtube.com/channel/UCVAmzfNGfC6zOnUojEC2yag | V: Course on topology in condensed matter]] Zero energy excitations - "do nothing" - whether or not they exist - basis for topology in condensed matter. [[https://www.youtube.com/watch?v=5ysdSoorJz4 | Topology and symmetry intro (by Anton Akhmerov)]] Without zero energy excitations, one can't transform certain systems into other systems. Thus we can group these systems into classes. Quantum dots - zero dimensional systems. [[https://en.wikipedia.org/wiki/Chern_class | 维基百科: Chern class]] [[https://en.wikipedia.org/wiki/Majorana_fermion | 维基百科: Majorana fermion]] - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? [[https://nbviewer.jupyter.org/github/topocm/topocm_content/blob/edx_2015/w1_topointro/1D.ipynb | Bulk-edge correspondence outlook (by Jay Sau)]] [[https://www.youtube.com/watch?v=wiHPQlEha6g | Video]] domain walls between Strong coupling limit - Kitaev chain, Dirac limit - produces delocalized Majorana modes in weakly gapped system. Different sides of the same coin. Bulk topological invariant connects topology in condensed matter with mathematics. Block Hamiltonian associated with crystal momentum K constrained on a circle. Difference between cylinder and a Mobius strip. K-theory worked this out. Rimhook - is the growth of a tableaux by a continuous set of sells: a rimhook, as opposed to a row or a special rim hook, etc. [[https://www.youtube.com/watch?v=Je4bU3g_QGk | V: P. Vivo. Random Matrices: Theory and Practice.]] [[https://en.wikipedia.org/wiki/Random_matrix | 维基百科: Random matrix]] [[https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents | 维基百科: Painlevé transcendents]] Calculate and interpret symmetric functions of the eigenvalues of random matrices, starting with the determinant and the trace. * [[https://terrytao.files.wordpress.com/2008/03/determinant.pdf | Terence Tao. Singularity and determinant of random matrices]] * [[https://mathoverflow.net/questions/13008/expected-determinant-of-a-random-nxn-matrix | Math StackExchange: Expected determinant of a random nxn matrix]] * [[https://arxiv.org/pdf/1112.0752.pdf | Nguyen, Vu. Random matrices: Law of the determinants.]] * [[https://mathoverflow.net/questions/317135/expected-determinant-of-random-symmetric-matrix-with-different-gaussian-distribu | Math StackExchange: Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements]] There are always dual categories {$C$} and {$C^{op}$}. [[https://hackr.io/blog/programming-paradigms | Four programming paradigms]]: * Procedural (imperative, top-down) - Fortran, C, Cobol. Statements are structured into procedures, subroutines, functions, they make procedure calls. * Logical (declarative) - Prolog. Statements express facts (assertions), rules (inferences) and queries about problems within a system. Rules are logical clauses with a head and a body. Expresses knowledge independently of implementation. Knowledge separated from use. Programs can be more flexible, compressed, understandable. * Functional - Haskell, Lisp. Computation is an evaluation of mathematical functions. Pure functions are those that take an argument list as input and output a return value. They do not depend on global data or class member's data. No side effects. Input not affected. A function can recursively call itself. Referentially transparent expressions can be replaced by their value without changing the program's behavior. Functions are first-class, they can be treated as any other variable: they can be used as input, returned as output, or be assigned to a variable. Variables are immutable, they can't be modified after initialization. * Object-oriented - Ruby, Java, C++, Python. Real world entities are represented by Classes. Objects are instances of classes such that each object encapsulates a state (fields, attributes) and behavior (methods, what you do with the object). Objects interact with each other by passing messages. ** Encapsulation: Classes bundle the data and methods, hide the internal representation, and provide a simple and clear interface. ** Inheritance: A hierarchy of classes by which one class derives from another class. ** Data abstraction: Interface shows essential information, hides details. ** Polymorphism: A variable, function or object may take on multiple forms. [[https://projecteuclid.org/euclid.bia/1403013939#toc | Robert Goldblatt. Topoi: The Categorial Analysis of Logic]] Yoneda lemma. * Executing an entire subroutine is in addition to executing individual steps. It is the use of a new parser at a second level. What would it mean on a further level? * Moving from How (arrows) to What (objects). * The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. * How does the "null activity" relate to the "do nothing action"? * In the Yoneda lemma, the natural transformation on the left hand side seems to coordinate vertical and horizontal composition. * The crucial point of the Yoneda lemma is that the set function is on an object A and not on a morphism. * The four levels Whether, What, How, Why differ in how they focus on objects or morphisms. * alpha theta Hom(f,_) takes as input the entire morphism: what value it goes to X and where that goes to in Y * Does the Yoneda lemma suppose commutativity of summation? as with multiplication? * Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. The divisions of everything. What would the foursome look like as actions taking place on doing nothing, on the nullsome? Or is it the fact that the foursome is consciousness of the onesome? * Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces the object d to a set of the arrows from c to d. It reduces an arrow f:a to b to a set function from (c,a) to (c,b). Every functor is about forgetting, and the least forgetful functor is simply an isomorphism. But the category Set has very little explicit information, so in some sense it is the space of greatest forgetting, it is a blank slate. * Yoneda lemma models determinism: If you know where the root of the tree goes, then you know where everything goes. * The left side of the Yoneda lemma relates the concepts of forgetful and free. * [[https://math.stackexchange.com/questions/2030418/basic-example-of-yoneda-lemma | Math Stack Exchange: Basic example of Yoneda lemma]] mentions subroutines. * [[https://en.wikipedia.org/wiki/Continuation-passing_style | 维基百科: Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. * [[https://bartoszmilewski.com/2015/09/01/the-yoneda-lemma/ | Bartosz Milewski. The Yoneda Lemma.]] * [[https://bartoszmilewski.com/2015/10/28/yoneda-embedding/ | Bartosz Milewski. Yoneda Embedding.]] * [https://www.cs.ox.ac.uk/jeremy.gibbons/publications/proyo.pdf | What You Needa Know about Yoneda. Profunctor Optics and the Yoneda Lemma. (Functional Pearl)]] Guillaume Boisseau, Jeremy Gibbons. Mention the preorder example as proofs by indirect equality. * [[https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-16/ | Category theory notes 16: Yoneda lemma (Part 3)]] Linguist Chenchen (Julio) Song works through the details in his own way. * Yoneda lemma: A set of actions becomes an object. * Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f). * "Does nothing" means that it has never has any effect on other actions upon composition with them. An action does something if it takes one action into another action. (Not whether it takes an object into another object.) Thus actions are maps on actions. * Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. This relates a system and a pre-system. Thus it sets up the point where the pre-system becomes a system (a subsystem). And it can do this by distinguishing objects - those identity morphisms that do nothing - they get segregated into the subsystem and understood as such. * Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another. * Analyze the Yoneda Lemma in terms of the arithmetical hierarchy. Express its statements in terms of existential and universal quantifiers. Natural transformation has four levels: * Why: f:x->y in C * How: F(f):F(x)->F(y) in F(C) * What: G(f):G(x)->G(y) in G(C) * Whether: F(x)->G(y) Objectification of morphisms * center - whether - identity morphism * balance - what - "from and to" * set - how - *elements* in a set * list - why - *list* of items Are these all internal structures? But what about external relationships? Are they given by analysis? Choice frameworks * The binomial theorem (x+h)^n can be interpreted as the context for a derivative of a polynomial power, distinguishing a volume and its boundary, where the second term is the derivative. How does this express the coordinate system? And how may taking the limit h->0 be considered as shifting to the asymmetric interpretation in terms of a simplex? And what does that say about the continuity and differentiability of nature, how it relates symmetry and asymmetry? Topos theory * [[https://klevas.mif.vu.lt/garunkstis/lietuviskas.htm | Ramūnas Garunkštis]] Studying topos theory. Modular forms * [[https://klevas.mif.vu.lt/garunkstis/preprintai/ModFormos.pdf | Ramūnas Garunkštis. Modulinių formų įvadas.]] Does the Law of Forms define the sevensome? į:
[++++数学笔记++++] 2020 gegužės 07 d., 12:39
atliko -
Pakeistos 12-13 eilutės iš
į:
* Kan extensions are examples of "extending the domain". So consider what they say about the three-cycle in the house of knowledge, how they relate to arguments by continuity and by self-superimposition. Pakeistos 24-26 eilutės iš
į:
* Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D. * In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable? Ištrintos 29-35 eilutės:
Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D. In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable? Kan extensions are examples of "extending the domain". So consider what they say about the three-cycle in the house of knowledge, how they relate to arguments by continuity and by self-superimposition. Pakeistos 32-42 eilutės iš
The identity morphism falls out as a special morphism among the possible morphisms from B to B. It is the basis for Graziano's awareness schema. So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others. Objectification of morphisms * center - whether - identity morphism * balance - what - "from and to" * set - how - *elements* in a set * list - why - *list* of items Are these all internal structures? But what about external relationships? Are they given by analysis? į:
* The identity morphism falls out as a special morphism among the possible morphisms from B to B. It is the basis for Graziano's awareness schema. * So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others. Ištrintos 36-38 eilutės:
Pakeistos 40-41 eilutės iš
į:
* Can a functor or a function add information? Investigate: A function can add context in the codomain. For example, a loop can be understood as a circle or as a complicated closed curve. It is a loop to itself, but in another context it is a complicated curve, perhaps in the complex numbers, perhaps in a multi-dimensional space. Ištrintos 61-66 eilutės:
* Why: f:x->y in C * How: F(f):F(x)->F(y) in F(C) * What: G(f):G(x)->G(y) in G(C) * Whether: F(x)->G(y) Pakeistos 151-157 eilutės iš
Can a functor or [[https://klevas.mif.vu.lt/garunkstis/lietuviskas.htm | Ramūnas Garunkštis]] Studying topos į:
Natural transformation has four levels: * Why: f:x->y in C * How: F(f):F(x)->F(y) in F(C) * What: G(f):G(x)->G(y) in G(C) * Whether: F(x)->G(y) Objectification of morphisms * center - whether - identity morphism * balance - what - "from and to" * set - how - *elements* in a set * list - why - *list* of items Are these all internal structures? But what about external relationships? Are they given by analysis? Choice frameworks * The binomial theorem (x+h)^n can be interpreted as the context for a derivative of a polynomial power, distinguishing a volume and its boundary, where the second term is the derivative. How does this express the coordinate system? And how may taking the limit h->0 be considered as shifting to the asymmetric interpretation in terms of a simplex? And what does that say about the continuity and differentiability of nature, how it relates symmetry and asymmetry? Topos theory * [[https://klevas.mif.vu.lt/garunkstis/lietuviskas.htm | Ramūnas Garunkštis]] Studying topos theory. Modular forms * [[https://klevas.mif.vu.lt/garunkstis/preprintai/ModFormos.pdf | Ramūnas Garunkštis. Modulinių formų įvadas.]] 2020 gegužės 07 d., 12:33
atliko -
Pakeista 171 eilutė iš:
The binomial į:
The binomial theorem (x+h)^n can be interpreted as the context for a derivative of a polynomial power, distinguishing a volume and its boundary, where the second term is the derivative. How does this express the coordinate system? And how may taking the limit h->0 be considered as shifting to the asymmetric interpretation in terms of a simplex? And what does that say about the continuity and differentiability of nature, how it relates symmetry and asymmetry? 2020 gegužės 07 d., 11:48
atliko -
Pridėtos 169-171 eilutės:
* Analyze the Yoneda Lemma in terms of the arithmetical hierarchy. Express its statements in terms of existential and universal quantifiers. The binomial theorem 2020 gegužės 07 d., 11:47
atliko -
Pakeistos 3-12 eilutės iš
Adjunction is a form of duality. Equality and equivalence, in general, are forms of duality. Duality is an extension of equivalence where the two sides of the equation are somehow different. For example, one side may be a variable and the other side a value that it is set to. Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such. Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome. į:
Duality * Adjunction is a form of duality. * Equality and equivalence, in general, are forms of duality. * Duality is an extension of equivalence where the two sides of the equation are somehow different. For example, one side may be a variable and the other side a value that it is set to. * Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such. * Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome. Math Discovery Pakeistos 20-31 eilutės iš
The crucial point of the Yoneda lemma is that the set function is on an object A The four levels Whether, After Yoneda lemma, study the Yates Index Theorem. Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less. į:
Category theory * In a category, if two objects have morphisms into each other, than they are isomorphic. So if we are only interested in nonisomorphic objects, then we can consider equivalence classes. And we can see that the maps only go in one way, so they are naturally partially ordered. * Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less. Divisions * The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | 维基百科: Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? Pakeistos 47-95 eilutės iš
alpha theta Hom(f,_) takes Does the Yoneda lemma suppose commutativity of summation? as with multiplication? Yoneda Lemma: Acting on the identity is acting on doing nothing Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces Yoneda lemma models determinism: If you know where the root of the tree goes, then you know where everything goes. Why I feel strange that Set is not definite: "every function should have a definite class as domain and a definite class as range our basic concepts are essentially those of a functor and of a natural transformation . . . . The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether and adopt an even more intuitive standpoint, in which a functor such as “Hom” is not defined over the category of “all” groups, but for each particular pair of groups which may be given. [EM45]" The left side of the Yoneda lemma relates the concepts of forgetful and free. Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity. The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, we John Baez: "Every sufficiently good analogy is yearning to become a functor." Compare with metaphors, blends. [[https://math.stackexchange.com/questions/2030418/basic-example-of-yoneda-lemma | Math Stack Exchange: Basic example of Yoneda lemma]] mentions subroutines. [[https://en.wikipedia.org/wiki/Continuation-passing_style | 维基百科: Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. [[https://bartoszmilewski.com/2015/09/01/the-yoneda-lemma/ | Bartosz Milewski. The Yoneda Lemma.]] [[https://bartoszmilewski.com/2015/10/28/yoneda-embedding/ | Bartosz Milewski. Yoneda Embedding.]] [https://www.cs.ox.ac.uk/jeremy.gibbons/publications/proyo.pdf | What You Needa Know about Yoneda. Profunctor Optics and the Yoneda Lemma. (Functional Pearl)]] Guillaume Boisseau, Jeremy Gibbons. Mention the preorder example as proofs by indirect equality. [[https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-16/ | Category theory notes 16: Yoneda lemma (Part 3)]] Linguist Chenchen (Julio) Song works through the details in his own way. Yoneda lemma: A set of actions becomes an object. Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f). What is the connection between random walks (all walks) on a graph and homotopy loops on a surface? į:
* Why I feel strange that Set is not definite: "every function should have a definite class as domain and a definite class as range". Riehl quotes Eilenberg and Maclane: ". . . the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation . . . . The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether and adopt an even more intuitive standpoint, in which a functor such as “Hom” is not defined over the category of “all” groups, but for each particular pair of groups which may be given. [EM45]" * Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity. * The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, we * John Baez: "Every sufficiently good analogy is yearning to become a functor." Compare with metaphors, blends. Walks * What is the connection between random walks (all walks) on a graph and homotopy loops on a surface? Ištrintos 81-86 eilutės:
Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. This relates a system and a pre-system. Thus it sets up the point where the pre-system becomes a system (a subsystem). And it can do this by distinguishing objects - those identity morphisms that do nothing - they get segregated into the subsystem and understood as such. Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another. Pridėtos 149-168 eilutės:
* In the Yoneda lemma, the natural transformation on the left hand side seems to coordinate vertical and horizontal composition. * The crucial point of the Yoneda lemma is that the set function is on an object A and not on a morphism. * The four levels Whether, What, How, Why differ in how they focus on objects or morphisms. * alpha theta Hom(f,_) takes as input the entire morphism: what value it goes to X and where that goes to in Y * Does the Yoneda lemma suppose commutativity of summation? as with multiplication? * Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. The divisions of everything. What would the foursome look like as actions taking place on doing nothing, on the nullsome? Or is it the fact that the foursome is consciousness of the onesome? * Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces the object d to a set of the arrows from c to d. It reduces an arrow f:a to b to a set function from (c,a) to (c,b). Every functor is about forgetting, and the least forgetful functor is simply an isomorphism. But the category Set has very little explicit information, so in some sense it is the space of greatest forgetting, it is a blank slate. * Yoneda lemma models determinism: If you know where the root of the tree goes, then you know where everything goes. * The left side of the Yoneda lemma relates the concepts of forgetful and free. * [[https://math.stackexchange.com/questions/2030418/basic-example-of-yoneda-lemma | Math Stack Exchange: Basic example of Yoneda lemma]] mentions subroutines. * [[https://en.wikipedia.org/wiki/Continuation-passing_style | 维基百科: Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. * [[https://bartoszmilewski.com/2015/09/01/the-yoneda-lemma/ | Bartosz Milewski. The Yoneda Lemma.]] * [[https://bartoszmilewski.com/2015/10/28/yoneda-embedding/ | Bartosz Milewski. Yoneda Embedding.]] * [https://www.cs.ox.ac.uk/jeremy.gibbons/publications/proyo.pdf | What You Needa Know about Yoneda. Profunctor Optics and the Yoneda Lemma. (Functional Pearl)]] Guillaume Boisseau, Jeremy Gibbons. Mention the preorder example as proofs by indirect equality. * [[https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-16/ | Category theory notes 16: Yoneda lemma (Part 3)]] Linguist Chenchen (Julio) Song works through the details in his own way. * Yoneda lemma: A set of actions becomes an object. * Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f). * "Does nothing" means that it has never has any effect on other actions upon composition with them. An action does something if it takes one action into another action. (Not whether it takes an object into another object.) Thus actions are maps on actions. * Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. This relates a system and a pre-system. Thus it sets up the point where the pre-system becomes a system (a subsystem). And it can do this by distinguishing objects - those identity morphisms that do nothing - they get segregated into the subsystem and understood as such. * Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another. 2020 gegužės 06 d., 15:35
atliko -
Pridėta 195 eilutė:
* How does the "null activity" relate to the "do nothing action"? 2020 gegužės 06 d., 13:19
atliko -
Pakeistos 200-202 eilutės iš
[[https://klevas.mif.vu.lt/garunkstis/preprintai/ModFormos.pdf | Ramūnas Garunkštis. Modulinių formų įvadas.]] į:
[[https://klevas.mif.vu.lt/garunkstis/preprintai/ModFormos.pdf | Ramūnas Garunkštis. Modulinių formų įvadas.]] Does the Law of Forms define the sevensome? 2020 gegužės 06 d., 13:00
atliko -
Pakeistos 196-200 eilutės iš
Can a functor or a function add information? Investigate: A function can add context in the codomain. For example, a loop can be understood as a circle or as a complicated closed curve. It is a loop to itself, but in another context it is a complicated curve, perhaps in the complex numbers, perhaps in a multi-dimensional space. į:
Can a functor or a function add information? Investigate: A function can add context in the codomain. For example, a loop can be understood as a circle or as a complicated closed curve. It is a loop to itself, but in another context it is a complicated curve, perhaps in the complex numbers, perhaps in a multi-dimensional space. [[https://klevas.mif.vu.lt/garunkstis/lietuviskas.htm | Ramūnas Garunkštis]] Studying topos theory. [[https://klevas.mif.vu.lt/garunkstis/preprintai/ModFormos.pdf | Ramūnas Garunkštis. Modulinių formų įvadas.]] 2020 gegužės 06 d., 12:36
atliko -
Pakeistos 194-196 eilutės iš
* The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. į:
* The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. Can a functor or a function add information? Investigate: A function can add context in the codomain. For example, a loop can be understood as a circle or as a complicated closed curve. It is a loop to itself, but in another context it is a complicated curve, perhaps in the complex numbers, perhaps in a multi-dimensional space. 2020 gegužės 05 d., 12:14
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Pakeistos 189-191 eilutės iš
[[https://projecteuclid.org/euclid.bia/1403013939#toc | Robert Goldblatt. Topoi: The Categorial Analysis of Logic]] į:
[[https://projecteuclid.org/euclid.bia/1403013939#toc | Robert Goldblatt. Topoi: The Categorial Analysis of Logic]] Yoneda lemma. * Executing an entire subroutine is in addition to executing individual steps. It is the use of a new parser at a second level. What would it mean on a further level? * Moving from How (arrows) to What (objects). * The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. 2020 gegužės 04 d., 22:29
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Pakeistos 189-191 eilutės iš
Yoneda Lemma: The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. į:
Yoneda Lemma: The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. [[https://projecteuclid.org/euclid.bia/1403013939#toc | Robert Goldblatt. Topoi: The Categorial Analysis of Logic]] 2020 gegužės 04 d., 20:57
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Pakeistos 187-189 eilutės iš
** Polymorphism: A variable, function or object may take on multiple forms. į:
** Polymorphism: A variable, function or object may take on multiple forms. Yoneda Lemma: The movement of time: where the entire system moves forwards in that the final state becomes a new initial state. Whereas the other perspective you can have an opposite category. 2020 gegužės 02 d., 12:37
atliko -
Pakeista 102 eilutė iš:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | 维基百科: Tracy-Widom distribution]] There are Tracy-Widom distributions for orthogonal ($F_1$}, unitary {$F_2$} and symplectic {$F_ į:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | 维基百科: Tracy-Widom distribution]] There are Tracy-Widom distributions for orthogonal ($F_1$}, unitary {$F_2$} and symplectic {$F_4$} random matrices. 2020 gegužės 02 d., 12:36
atliko -
Pakeistos 177-187 eilutės iš
There are always dual categories {$C$} and {$C^{op}$}. į:
There are always dual categories {$C$} and {$C^{op}$}. [[https://hackr.io/blog/programming-paradigms | Four programming paradigms]]: * Procedural (imperative, top-down) - Fortran, C, Cobol. Statements are structured into procedures, subroutines, functions, they make procedure calls. * Logical (declarative) - Prolog. Statements express facts (assertions), rules (inferences) and queries about problems within a system. Rules are logical clauses with a head and a body. Expresses knowledge independently of implementation. Knowledge separated from use. Programs can be more flexible, compressed, understandable. * Functional - Haskell, Lisp. Computation is an evaluation of mathematical functions. Pure functions are those that take an argument list as input and output a return value. They do not depend on global data or class member's data. No side effects. Input not affected. A function can recursively call itself. Referentially transparent expressions can be replaced by their value without changing the program's behavior. Functions are first-class, they can be treated as any other variable: they can be used as input, returned as output, or be assigned to a variable. Variables are immutable, they can't be modified after initialization. * Object-oriented - Ruby, Java, C++, Python. Real world entities are represented by Classes. Objects are instances of classes such that each object encapsulates a state (fields, attributes) and behavior (methods, what you do with the object). Objects interact with each other by passing messages. ** Encapsulation: Classes bundle the data and methods, hide the internal representation, and provide a simple and clear interface. ** Inheritance: A hierarchy of classes by which one class derives from another class. ** Data abstraction: Interface shows essential information, hides details. ** Polymorphism: A variable, function or object may take on multiple forms. 2020 balandžio 30 d., 18:15
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Pridėtos 176-177 eilutės:
There are always dual categories {$C$} and {$C^{op}$}. 2020 balandžio 29 d., 12:13
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Pakeistos 172-173 eilutės iš
* [[https:// į:
* [[https://terrytao.files.wordpress.com/2008/03/determinant.pdf | Terence Tao. Singularity and determinant of random matrices]] * [[https://mathoverflow.net/questions/13008/expected-determinant-of-a-random-nxn-matrix | Math StackExchange: Expected determinant of a random nxn matrix]] Pakeista 175 eilutė iš:
* į:
* [[https://mathoverflow.net/questions/317135/expected-determinant-of-random-symmetric-matrix-with-different-gaussian-distribu | Math StackExchange: Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements]] 2020 balandžio 29 d., 12:10
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Pakeistos 171-174 eilutės iš
į:
Calculate and interpret symmetric functions of the eigenvalues of random matrices, starting with the determinant and the trace. * [[https://mathoverflow.net/questions/13008/expected-determinant-of-a-random-nxn-matrix | Math StackOverflow: Expected determinant of a random nxn matrix]] * [[https://arxiv.org/pdf/1112.0752.pdf | Nguyen, Vu. Random matrices: Law of the determinants.]] * 2020 balandžio 28 d., 21:35
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Pridėtos 167-171 eilutės:
[[https://en.wikipedia.org/wiki/Random_matrix | 维基百科: Random matrix]] [[https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents | 维基百科: Painlevé transcendents]] 2020 balandžio 28 d., 21:34
atliko -
Pakeistos 22-23 eilutės iš
The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? į:
The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | 维基百科: Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? Pakeistos 84-85 eilutės iš
[[https://en.wikipedia.org/wiki/Continuation-passing_style | Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. į:
[[https://en.wikipedia.org/wiki/Continuation-passing_style | 维基百科: Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. Pakeista 102 eilutė iš:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | į:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | 维基百科: Tracy-Widom distribution]] There are Tracy-Widom distributions for orthogonal ($F_1$}, unitary {$F_2$} and symplectic {$F_$} random matrices. Pakeistos 157-159 eilutės iš
[[https://en.wikipedia.org/wiki/Chern_class | [[https://en.wikipedia.org/wiki/Majorana_fermion | Majorana fermion]] - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? [[https://nbviewer.jupyter.org/github/topocm/topocm_content/blob/edx_2015/w1_topointro/1D.ipynb | Bulk-edge correspondence outlook (by Jay Sau)]] [[https://www.youtube.com/watch?v=wiHPQlEha6g | Video]] į:
[[https://en.wikipedia.org/wiki/Chern_class | 维基百科: Chern class]] [[https://en.wikipedia.org/wiki/Majorana_fermion | 维基百科: Majorana fermion]] - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? [[https://nbviewer.jupyter.org/github/topocm/topocm_content/blob/edx_2015/w1_topointro/1D.ipynb | Bulk-edge correspondence outlook (by Jay Sau)]] [[https://www.youtube.com/watch?v=wiHPQlEha6g | Video]] 2020 balandžio 28 d., 21:32
atliko -
Pakeistos 165-168 eilutės iš
* Dvejonės * Jauduliai: išsako sąmonės ir pasąmonės santykį su Dievu * Aštuongubas kelias: išsako sąmoningumo, sąmonės ir pasąmonės santykį su Dievu į:
[[https://www.youtube.com/watch?v=Je4bU3g_QGk | V: P. Vivo. Random Matrices: Theory and Practice.]] 2020 balandžio 28 d., 21:28
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Pakeistos 163-168 eilutės iš
Rimhook - is the growth of a tableaux by a continuous set of sells: a rimhook, as opposed to a row or a special rim hook, etc. į:
Rimhook - is the growth of a tableaux by a continuous set of sells: a rimhook, as opposed to a row or a special rim hook, etc. * Poreikių tenkinimai: išsako kieno santykį su Dievu? * Dvejonės: išsako pasąmonės santykį su Dievu (troškimu) * Jauduliai: išsako sąmonės ir pasąmonės santykį su Dievu * Aštuongubas kelias: išsako sąmoningumo, sąmonės ir pasąmonės santykį su Dievu 2020 balandžio 28 d., 20:43
atliko -
Pakeistos 161-163 eilutės iš
Strong coupling limit - Kitaev chain, Dirac limit - produces delocalized Majorana modes in weakly gapped system. Different sides of the same coin. Bulk topological invariant connects topology in condensed matter with mathematics. Block Hamiltonian associated with crystal momentum K constrained on a circle. Difference between cylinder and a Mobius strip. K-theory worked this out. į:
Strong coupling limit - Kitaev chain, Dirac limit - produces delocalized Majorana modes in weakly gapped system. Different sides of the same coin. Bulk topological invariant connects topology in condensed matter with mathematics. Block Hamiltonian associated with crystal momentum K constrained on a circle. Difference between cylinder and a Mobius strip. K-theory worked this out. Rimhook - is the growth of a tableaux by a continuous set of sells: a rimhook, as opposed to a row or a special rim hook, etc. 2020 balandžio 28 d., 20:18
atliko -
Pakeistos 159-161 eilutės iš
[[https://en.wikipedia.org/wiki/Majorana_fermion | Majorana fermion]] - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? [[https://nbviewer.jupyter.org/github/topocm/topocm_content/blob/edx_2015/w1_topointro/1D.ipynb | Bulk-edge correspondence outlook (by Jay Sau)]] [[https://www.youtube.com/watch?v=wiHPQlEha6g | Video]] į:
[[https://en.wikipedia.org/wiki/Majorana_fermion | Majorana fermion]] - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? [[https://nbviewer.jupyter.org/github/topocm/topocm_content/blob/edx_2015/w1_topointro/1D.ipynb | Bulk-edge correspondence outlook (by Jay Sau)]] [[https://www.youtube.com/watch?v=wiHPQlEha6g | Video]] domain walls between Strong coupling limit - Kitaev chain, Dirac limit - produces delocalized Majorana modes in weakly gapped system. Different sides of the same coin. Bulk topological invariant connects topology in condensed matter with mathematics. Block Hamiltonian associated with crystal momentum K constrained on a circle. Difference between cylinder and a Mobius strip. K-theory worked this out. 2020 balandžio 28 d., 20:11
atliko -
Pakeistos 157-159 eilutės iš
[[https://en.wikipedia.org/wiki/Chern_class | V: Chern class]] į:
[[https://en.wikipedia.org/wiki/Chern_class | V: Chern class]] [[https://en.wikipedia.org/wiki/Majorana_fermion | Majorana fermion]] - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? [[https://nbviewer.jupyter.org/github/topocm/topocm_content/blob/edx_2015/w1_topointro/1D.ipynb | Bulk-edge correspondence outlook (by Jay Sau)]] [[https://www.youtube.com/watch?v=wiHPQlEha6g | Video]] 2020 balandžio 28 d., 20:07
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Pakeistos 155-157 eilutės iš
Quantum dots - zero dimensional systems. į:
Quantum dots - zero dimensional systems. [[https://en.wikipedia.org/wiki/Chern_class | V: Chern class]] 2020 balandžio 28 d., 20:00
atliko -
Pakeistos 153-155 eilutės iš
Zero energy excitations - "do nothing" - į:
Zero energy excitations - "do nothing" - whether or not they exist - basis for topology in condensed matter. [[https://www.youtube.com/watch?v=5ysdSoorJz4 | Topology and symmetry intro (by Anton Akhmerov)]] Without zero energy excitations, one can't transform certain systems into other systems. Thus we can group these systems into classes. Quantum dots - zero dimensional systems. 2020 balandžio 28 d., 19:59
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Pakeista 153 eilutė iš:
Zero energy excitations - "do nothing" - basis for topology in condensed matter. [[https://www.youtube.com/watch?v=5ysdSoorJz4 | Topology and symmetry intro (by Anton Akhmerov)]] į:
Zero energy excitations - "do nothing" - basis for topology in condensed matter. [[https://www.youtube.com/watch?v=5ysdSoorJz4 | Topology and symmetry intro (by Anton Akhmerov)]] Without zero energy excitations, one can't transform certain systems into other systems. 2020 balandžio 28 d., 19:58
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Pakeistos 151-153 eilutės iš
[[https://www.youtube.com/channel/UCVAmzfNGfC6zOnUojEC2yag | V: Course on topology in condensed matter]] į:
[[https://www.youtube.com/channel/UCVAmzfNGfC6zOnUojEC2yag | V: Course on topology in condensed matter]] Zero energy excitations - "do nothing" - basis for topology in condensed matter. [[https://www.youtube.com/watch?v=5ysdSoorJz4 | Topology and symmetry intro (by Anton Akhmerov)]] 2020 balandžio 28 d., 19:55
atliko -
Pridėta 151 eilutė:
[[https://www.youtube.com/channel/UCVAmzfNGfC6zOnUojEC2yag | V: Course on topology in condensed matter]] 2020 balandžio 28 d., 19:48
atliko -
Pakeistos 145-150 eilutės iš
[[http://www.ft.uam.es/personal/rubio/Random_Matrix_symmetries.pdf | Rubio. Random Matrix Symmetries]] į:
[[http://www.ft.uam.es/personal/rubio/Random_Matrix_symmetries.pdf | Rubio. Random Matrix Symmetries]] Energy levels are eigenvalues. Thus we consider the distribution of energy levels and their separation. On the analytic wing of the house of knowledge, we can think of these as the energy levels, the eigenvalues, that must be kept separate, kept distinguished, like fermions. And in the case of a long tail we have a low energy extremes. 2020 balandžio 28 d., 19:35
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Pridėta 145 eilutė:
[[http://www.ft.uam.es/personal/rubio/Random_Matrix_symmetries.pdf | Rubio. Random Matrix Symmetries]] 2020 balandžio 28 d., 19:33
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Pridėta 143 eilutė:
Bott periodicity and symmetry classes are related in the [[https://topocondmat.org/w8_general/classification.html | periodic table of topological insulators]]. 2020 balandžio 28 d., 19:26
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Pakeistos 139-141 eilutės iš
Dyson's Threefold Way: į:
Dyson's Threefold Way: * F.J. Dyson, “The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics”, J. Math. Phys.3(1962) 1199-1215 * "the most general matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of whichbelongs to one of the three known types": Complex Hermitian, real symmetric, or quaternion self-dual. 2020 balandžio 28 d., 19:25
atliko -
Pakeistos 135-141 eilutės iš
If we think of a hook as a walk, within a rim hook or a special rim hook, then we can think of these tableaux as ways of assembling walks, and thus of assembling systems from relationships. į:
If we think of a hook as a walk, within a rim hook or a special rim hook, then we can think of these tableaux as ways of assembling walks, and thus of assembling systems from relationships. [[https://arxiv.org/abs/1001.0722 | Martin R. Zirnbauer. Symmetry Classes.]] Dyson's Threefold Way: 1962 paper. The Threefold Way: algebraic structure of symmetry groups and ensembles in quantum mechanics. "the most general matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of whichbelongs to one of the three known types": Complex Hermitian, real symmetric, or quaternion self-dual. 2020 balandžio 28 d., 19:17
atliko -
Pakeistos 133-135 eilutės iš
Random matrices are related to random walks and other symmetric functions of the eigenvalues of matrices. į:
Random matrices are related to random walks and other symmetric functions of the eigenvalues of matrices. If we think of a hook as a walk, within a rim hook or a special rim hook, then we can think of these tableaux as ways of assembling walks, and thus of assembling systems from relationships. 2020 balandžio 28 d., 19:16
atliko -
Pakeistos 131-133 eilutės iš
Also, the overhead of words: if you think in words (based on the world) rather than concepts (rooted in the features of your mind) then you will be taxed for that and at a certain point it won't be sustainable. In general, this is modeling nonsustainability. į:
Also, the overhead of words: if you think in words (based on the world) rather than concepts (rooted in the features of your mind) then you will be taxed for that and at a certain point it won't be sustainable. In general, this is modeling nonsustainability. Random matrices are related to random walks and other symmetric functions of the eigenvalues of matrices. 2020 balandžio 28 d., 19:05
atliko -
Pridėta 103 eilutė:
* [[https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents | Painlevé transcendents]] 2020 balandžio 28 d., 19:03
atliko -
Pakeista 102 eilutė iš:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | W: Tracy-Widom distribution]] į:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | W: Tracy-Widom distribution]] There are Tracy-Widom distributions for orthogonal ($F_1$}, unitary {$F_2$} and symplectic {$F_$} random matrices. 2020 balandžio 27 d., 18:47
atliko -
Pakeistos 128-130 eilutės iš
Morally, the phase transition can model good and bad behavior. Bad behavior - such as slowing down your bus so that you are followed by another bus and take all of its customers - is behavior that complicates the phase transition. į:
Morally, the phase transition can model good and bad behavior. Bad behavior - such as slowing down your bus so that you are followed by another bus and take all of its customers - is behavior that complicates the phase transition. Also, the overhead of words: if you think in words (based on the world) rather than concepts (rooted in the features of your mind) then you will be taxed for that and at a certain point it won't be sustainable. In general, this is modeling nonsustainability. 2020 balandžio 27 d., 18:23
atliko -
Pakeistos 126-128 eilutės iš
Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another. į:
Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another. Morally, the phase transition can model good and bad behavior. Bad behavior - such as slowing down your bus so that you are followed by another bus and take all of its customers - is behavior that complicates the phase transition. 2020 balandžio 27 d., 17:26
atliko -
Pakeista 124 eilutė iš:
Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. į:
Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. This relates a system and a pre-system. Thus it sets up the point where the pre-system becomes a system (a subsystem). And it can do this by distinguishing objects - those identity morphisms that do nothing - they get segregated into the subsystem and understood as such. 2020 balandžio 27 d., 17:11
atliko -
Pakeistos 124-126 eilutės iš
Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. į:
Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. Yoneda Lemma explains what happens in the gap between consciousness and the unconscious, the tipping point at which the conscious gets dealt with, one way or another. 2020 balandžio 27 d., 16:19
atliko -
Pakeistos 122-124 eilutės iš
"Does nothing" means that it has never has any effect on other actions upon composition with them. An action does something if it takes one action into another action. (Not whether it takes an object into another object.) Thus actions are maps on actions. į:
"Does nothing" means that it has never has any effect on other actions upon composition with them. An action does something if it takes one action into another action. (Not whether it takes an object into another object.) Thus actions are maps on actions. Yoneda Lemma expresses the "symmetry group" at the heart of the ways of figuring things out in math. 2020 balandžio 27 d., 13:01
atliko -
Pakeistos 114-122 eilutės iš
* Consider how symmetric functions of eigenvalues are relevant here. į:
* Consider how symmetric functions of eigenvalues are relevant here. Natural transformation has four levels: * Why: f:x->y in C * How: F(f):F(x)->F(y) in F(C) * What: G(f):G(x)->G(y) in G(C) * Whether: F(x)->G(y) "Does nothing" means that it has never has any effect on other actions upon composition with them. An action does something if it takes one action into another action. (Not whether it takes an object into another object.) Thus actions are maps on actions. 2020 balandžio 27 d., 11:54
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Pridėtos 111-113 eilutės:
* Switching over from a wave (or boson) point of view to a particle (or fermion) point of view. Note that multiple morphisms can have the same location, but multiple objects cannot. * Consider how the identity morphism becomes distinguished. Note that the Yoneda lemma can be explaining how an identity morphism gets inferred. Because perhaps there aren't identity morphisms in the beginning. * Consider how a system of morphisms unfolds from a single morphism, consider how a new morphism arises and consider the conditions that it has to satisfy to be consistent with the system, especially with composition. 2020 balandžio 27 d., 10:55
atliko -
Pridėta 110 eilutė:
* Think also of the orders of magnitude in the universe. And orders of scale in Alexander's principles of life. 2020 balandžio 27 d., 10:33
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Pakeistos 107-110 eilutės iš
Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? Can the Yoneda lemma help model such a phase transition? į:
Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? Can the Yoneda lemma help model such a phase transition? * There may be a shift from relations (n2) to objects (n) when n grows large so as to reduce the overhead. This would break a system down into subsystems, which can operate independently, line by line. So there should be a pressure for a system to break down into subsystems. * The Yoneda Lemma establishes the underlying isomorphism. But how can such naturally isomorphic structures be meaningfully different? It is because the interpreting perspective is different. As with entropy, the perspective matters, the coordinate system we choose. This perspective is inherent in whether we have a system or a subsystem. * Consider how symmetric functions of eigenvalues are relevant here. 2020 balandžio 26 d., 16:54
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Pakeista 107 eilutė iš:
Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? į:
Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? Can the Yoneda lemma help model such a phase transition? 2020 balandžio 26 d., 16:53
atliko -
Pakeistos 106-107 eilutės iš
* Does the number {$\sqrt{2n}$} relate to the root systems with {$2n$} simple roots? į:
* Does the number {$\sqrt{2n}$} relate to the root systems with {$2n$} simple roots? Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? 2020 balandžio 26 d., 16:20
atliko -
Pridėta 101 eilutė:
* [[https://www.quantamagazine.org/in-mysterious-pattern-math-and-nature-converge-20130205 | Natalie Wolchover. In Mysterious Pattern, Math and Nature Converge.]] 2020 balandžio 26 d., 16:14
atliko -
Pakeista 105 eilutė iš:
* Does the number {$\sqrt{2n}$} relate to the root systems with 2n simple roots? į:
* Does the number {$\sqrt{2n}$} relate to the root systems with {$2n$} simple roots? 2020 balandžio 26 d., 16:14
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Pakeista 105 eilutė iš:
* Does the number {$\ į:
* Does the number {$\sqrt{2n}$} relate to the root systems with 2n simple roots? 2020 balandžio 26 d., 16:14
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Pakeistos 104-105 eilutės iš
* [[https://www.youtube.com/watch?v=HrtJ3SRQF4E | V: What is Universality?]] į:
* [[https://www.youtube.com/watch?v=HrtJ3SRQF4E | V: What is Universality?]] * Does the number {$\root{2n}$} relate to the root systems with 2n simple roots? 2020 balandžio 26 d., 16:11
atliko -
Pridėta 101 eilutė:
* [[https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution | W: Tracy-Widom distribution]] 2020 balandžio 26 d., 16:04
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Pakeistos 98-103 eilutės iš
What is the connection between random walks (all walks) on a graph and homotopy loops on a surface? į:
What is the connection between random walks (all walks) on a graph and homotopy loops on a surface? Universality. The tendency of the eigenvalues of random matrices to space themselves out uniformly. Similarly, in nature, physical phenomena space themselves out in different orders of magnitude. Similarly, we have orders of scale in Alexander's theory. * [[https://www.quantamagazine.org/beyond-the-bell-curve-a-new-universal-law-20141015/ | At the far ends of a new universal law.]] * [[https://www.quantamagazine.org/tag/universality/ | Quanta Magazine articles on universality]] * [[https://www.youtube.com/watch?v=HrtJ3SRQF4E | V: What is Universality?]] 2020 balandžio 26 d., 11:36
atliko -
Pakeistos 96-98 eilutės iš
Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f). į:
Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f). What is the connection between random walks (all walks) on a graph and homotopy loops on a surface? 2020 balandžio 25 d., 21:30
atliko -
Pakeista 96 eilutė iš:
Yoneda lemma: A natural transformation splits a computation į:
Yoneda lemma: A natural transformation F->G splits a morphism=computation f in C into two perspectives: the execution F(f) and the outcomes G(f). 2020 balandžio 25 d., 21:28
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Pakeistos 94-96 eilutės iš
Yoneda lemma: A set of actions becomes an object. į:
Yoneda lemma: A set of actions becomes an object. Yoneda lemma: A natural transformation splits a computation into two perspectives: the execution and the outcomes. 2020 balandžio 25 d., 18:49
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Pakeistos 92-94 eilutės iš
[[https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-16/ | Category theory notes 16: Yoneda lemma (Part 3)]] Linguist Chenchen (Julio) Song works through the details in his own way. į:
[[https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-16/ | Category theory notes 16: Yoneda lemma (Part 3)]] Linguist Chenchen (Julio) Song works through the details in his own way. Yoneda lemma: A set of actions becomes an object. 2020 balandžio 24 d., 19:36
atliko -
Pakeistos 88-92 eilutės iš
[[https://bartoszmilewski.com/2015/10/28/yoneda-embedding/ | Bartosz Milewski. Yoneda Embedding.]] į:
[[https://bartoszmilewski.com/2015/10/28/yoneda-embedding/ | Bartosz Milewski. Yoneda Embedding.]] [https://www.cs.ox.ac.uk/jeremy.gibbons/publications/proyo.pdf | What You Needa Know about Yoneda. Profunctor Optics and the Yoneda Lemma. (Functional Pearl)]] Guillaume Boisseau, Jeremy Gibbons. Mention the preorder example as proofs by indirect equality. [[https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-16/ | Category theory notes 16: Yoneda lemma (Part 3)]] Linguist Chenchen (Julio) Song works through the details in his own way. 2020 balandžio 24 d., 19:17
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Pakeistos 84-88 eilutės iš
[[https://en.wikipedia.org/wiki/Continuation-passing_style | Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. į:
[[https://en.wikipedia.org/wiki/Continuation-passing_style | Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. [[https://bartoszmilewski.com/2015/09/01/the-yoneda-lemma/ | Bartosz Milewski. The Yoneda Lemma.]] [[https://bartoszmilewski.com/2015/10/28/yoneda-embedding/ | Bartosz Milewski. Yoneda Embedding.]] 2020 balandžio 24 d., 19:06
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Pakeistos 82-84 eilutės iš
[[https://math.stackexchange.com/questions/2030418/basic-example-of-yoneda-lemma | Math Stack Exchange: Basic example of Yoneda lemma]] mentions subroutines. į:
[[https://math.stackexchange.com/questions/2030418/basic-example-of-yoneda-lemma | Math Stack Exchange: Basic example of Yoneda lemma]] mentions subroutines. [[https://en.wikipedia.org/wiki/Continuation-passing_style | Continuation passing style]]: CPS transform is conceptually a Yoneda embedding. 2020 balandžio 24 d., 19:03
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Pakeistos 80-82 eilutės iš
John Baez: "Every sufficiently good analogy is yearning to become a functor." Compare with metaphors, blends. į:
John Baez: "Every sufficiently good analogy is yearning to become a functor." Compare with metaphors, blends. [[https://math.stackexchange.com/questions/2030418/basic-example-of-yoneda-lemma | Math Stack Exchange: Basic example of Yoneda lemma]] mentions subroutines. 2020 balandžio 23 d., 15:56
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Pakeistos 78-80 eilutės iš
The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, į:
The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, we John Baez: "Every sufficiently good analogy is yearning to become a functor." Compare with metaphors, blends. 2020 balandžio 23 d., 15:20
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Pakeistos 76-78 eilutės iš
Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity. į:
Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity. The desired natural transformation has components which are indexed by the object x which they send to be evaluated upon. Whereas in the other direction, given the natural transformation, we 2020 balandžio 23 d., 15:13
atliko -
Pakeistos 74-76 eilutės iš
The left side of the Yoneda lemma relates the concepts of forgetful and free. į:
The left side of the Yoneda lemma relates the concepts of forgetful and free. Morphisms only diminish internal information (though they can enrich the context). A natural transformation is the diminishment of such diminishment. A natural transformation relates parallel worlds: thus the world of the identity map is related to an object in the parallel world. And this happens by way of the relationship, the paralellism, between an object and its identity. 2020 balandžio 23 d., 12:26
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Pakeistos 72-74 eilutės iš
given. [EM45]" į:
given. [EM45]" The left side of the Yoneda lemma relates the concepts of forgetful and free. 2020 balandžio 23 d., 12:15
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Pakeista 63 eilutė iš:
"every function should have a definite class as domain and a definite class as range". Riehl quotes Eilenberg and Maclane: ". . . the whole concept of a category is essentially an auxiliary one; į:
Why I feel strange that Set is not definite: "every function should have a definite class as domain and a definite class as range". Riehl quotes Eilenberg and Maclane: ". . . the whole concept of a category is essentially an auxiliary one; 2020 balandžio 23 d., 12:15
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Pridėtos 62-72 eilutės:
"every function should have a definite class as domain and a definite class as range". Riehl quotes Eilenberg and Maclane: ". . . the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation . . . . The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether and adopt an even more intuitive standpoint, in which a functor such as “Hom” is not defined over the category of “all” groups, but for each particular pair of groups which may be given. [EM45]" 2020 balandžio 23 d., 11:48
atliko -
Pakeistos 60-61 eilutės iš
į:
Yoneda lemma models determinism: If you know where the root of the tree goes, then you know where everything goes. 2020 balandžio 23 d., 10:46
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Pakeista 59 eilutė iš:
Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces the object d to a set of the arrows from c to d. It reduces an arrow f:a to b to a set function from (c,a) to (c,b). į:
Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces the object d to a set of the arrows from c to d. It reduces an arrow f:a to b to a set function from (c,a) to (c,b). Every functor is about forgetting, and the least forgetful functor is simply an isomorphism. But the category Set has very little explicit information, so in some sense it is the space of greatest forgetting, it is a blank slate. 2020 balandžio 23 d., 10:18
atliko -
Pakeistos 59-60 eilutės iš
Functors from C to Set include many forgetful functors, such as from Group to Set. į:
Functors from C to Set include many forgetful functors, such as from Group to Set. The functor C(c,_) from C to Set reduces the object d to a set of the arrows from c to d. It reduces an arrow f:a to b to a set function from (c,a) to (c,b). 2020 balandžio 23 d., 10:14
atliko -
Pakeistos 57-60 eilutės iš
Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. The divisions of everything. What would the foursome look like as actions taking place on doing nothing, on the nullsome? Or is it the fact that the foursome is consciousness of the onesome? į:
Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. The divisions of everything. What would the foursome look like as actions taking place on doing nothing, on the nullsome? Or is it the fact that the foursome is consciousness of the onesome? Functors from C to Set include many forgetful functors, such as from Group to Set. 2020 balandžio 22 d., 20:48
atliko -
Pakeista 57 eilutė iš:
Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. į:
Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. The divisions of everything. What would the foursome look like as actions taking place on doing nothing, on the nullsome? Or is it the fact that the foursome is consciousness of the onesome? 2020 balandžio 22 d., 15:06
atliko -
Pakeistos 55-57 eilutės iš
Does the Yoneda lemma suppose commutativity of summation? as with multiplication? į:
Does the Yoneda lemma suppose commutativity of summation? as with multiplication? Yoneda Lemma: Acting on the identity is acting on doing nothing. God as a mirror. The limits of my mind. 2020 balandžio 21 d., 17:18
atliko -
Pridėtos 50-55 eilutės:
In the Yoneda lemma, the natural transformation on the left hand side seems to coordinate vertical and horizontal composition. alpha theta Hom(f,_) takes as input the entire morphism: what value it goes to X and where that goes to in Y Does the Yoneda lemma suppose commutativity of summation? as with multiplication? 2020 balandžio 20 d., 21:46
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Pakeistos 42-49 eilutės iš
So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others. į:
So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others. Objectification of morphisms * center - whether - identity morphism * balance - what - "from and to" * set - how - *elements* in a set * list - why - *list* of items Are these all internal structures? But what about external relationships? Are they given by analysis? 2020 balandžio 20 d., 15:05
atliko -
Pakeista 42 eilutė iš:
So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. į:
So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. This gives meaning to the identity morphism, which distinguishes B from the others. 2020 balandžio 20 d., 15:05
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Pakeistos 38-42 eilutės iš
Universality is unconditionality is how God thinks. I am interested not in how the various mathematicians think but how God thinks. į:
Universality is unconditionality is how God thinks. I am interested not in how the various mathematicians think but how God thinks. The identity morphism falls out as a special morphism among the possible morphisms from B to B. It is the basis for Graziano's awareness schema. So the identity morphism, which seemed optional or arbitary, become absolutely essential. The reason it becomes essential is because of its role in composition. And composition makes sense when you have a set of morphisms from B to B and if you know them all then you know How they compose. But to really make sense of it you need to be able to think in terms of the equivalences, the possibilities for external objects with morphisms too and from them. So Why requires the notion of objects which seemed superfluous. And the idea that there are Other objects, not just B. 2020 balandžio 18 d., 17:38
atliko -
Pakeistos 36-38 eilutės iš
Kan extensions are examples of "extending the domain". So consider what they say about the three-cycle in the house of knowledge, how they relate to arguments by continuity and by self-superimposition. į:
Kan extensions are examples of "extending the domain". So consider what they say about the three-cycle in the house of knowledge, how they relate to arguments by continuity and by self-superimposition. Universality is unconditionality is how God thinks. I am interested not in how the various mathematicians think but how God thinks. 2020 balandžio 18 d., 13:28
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Pakeistos 34-36 eilutės iš
In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable? į:
In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable? Kan extensions are examples of "extending the domain". So consider what they say about the three-cycle in the house of knowledge, how they relate to arguments by continuity and by self-superimposition. 2020 balandžio 18 d., 12:52
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Pakeistos 32-34 eilutės iš
Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D. į:
Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D. In category theory, how do we distinguish cardinalities? How can we distinguish countable and second countable? 2020 balandžio 18 d., 11:35
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Pakeistos 30-32 eilutės iš
Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less. į:
Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less. Isomorphism of C and D consists of four facts: a morphism f from C to D, and a morphism g from D to C, and and the fact that fg = 1_C and gf = 1_D. 2020 balandžio 18 d., 11:27
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Pakeistos 28-30 eilutės iš
After Yoneda lemma, study the Yates Index Theorem. į:
After Yoneda lemma, study the Yates Index Theorem. Functors have an ambiguity - they are morphisms (simple, external) but they also have rich internal structure. In a sense, every morphism has this ambiguity, more or less. 2020 balandžio 17 d., 17:54
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Pakeistos 24-28 eilutės iš
The crucial point of the Yoneda lemma is that the set function is on an object A and not on a morphism. į:
The crucial point of the Yoneda lemma is that the set function is on an object A and not on a morphism. The four levels Whether, What, How, Why differ in how they focus on objects or morphisms. After Yoneda lemma, study the Yates Index Theorem. 2020 balandžio 17 d., 17:30
atliko -
Pakeistos 22-24 eilutės iš
The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? į:
The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? The crucial point of the Yoneda lemma is that the set function is on an object A and not on a morphism. 2020 balandžio 17 d., 12:56
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Pakeista 22 eilutė iš:
The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? į:
The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem | Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? 2020 balandžio 17 d., 12:56
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Pakeistos 20-22 eilutės iš
In a category, if two objects have morphisms into each other, than they are isomorphic. So if we are only interested in nonisomorphic objects, then we can consider equivalence classes. And we can see that the maps only go in one way, so they are naturally partially ordered. į:
In a category, if two objects have morphisms into each other, than they are isomorphic. So if we are only interested in nonisomorphic objects, then we can consider equivalence classes. And we can see that the maps only go in one way, so they are naturally partially ordered. The [[https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem Borsuk-Ulam theorem]] seems to divide the n-sphere into two perspectives: those antipodal points that get mapped to different points and those that get mapped to the same point. What is the source of this twosome? 2020 balandžio 15 d., 17:24
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Pakeistos 18-20 eilutės iš
What this suggests is that the richest structures are inherently contradictory. But if we pull back to simpler structures then they become safer. į:
What this suggests is that the richest structures are inherently contradictory. But if we pull back to simpler structures then they become safer. In a category, if two objects have morphisms into each other, than they are isomorphic. So if we are only interested in nonisomorphic objects, then we can consider equivalence classes. And we can see that the maps only go in one way, so they are naturally partially ordered. 2020 balandžio 14 d., 17:18
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Pakeistos 17-18 eilutės iš
* vector spaces / limits = į:
* vector spaces / limits = contradictions What this suggests is that the richest structures are inherently contradictory. But if we pull back to simpler structures then they become safer. 2020 balandžio 14 d., 17:14
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Pakeistos 11-17 eilutės iš
Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome. į:
Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome. Think of monads as describing models, that is, simplified accounts of a system. Note that monads are built on adjoint functors, which express least upper bounds and greatest lower bounds. So this suggests that the four levels of logic/geometry are related to the four levels of algebra/analysis. But how, perhaps inversely? Because the bounds are third level, but the models are second level. Thus: * center / induction = variables * balance-parity / maximum-minimum = working backwards, implication * sets / bounds = models * vector spaces / limits = contradictions 2020 balandžio 14 d., 15:00
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Pakeistos 9-11 eilutės iš
Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such. į:
Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such. Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome. 2020 balandžio 14 d., 15:00
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Pakeista 9 eilutė iš:
Conjugates i and j are the form of duality that is the same as equality. į:
Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such. 2020 balandžio 14 d., 14:59
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Pakeistos 1-9 eilutės iš
[++++数学笔记++++] į:
[++++数学笔记++++] Adjunction is a form of duality. Equality and equivalence, in general, are forms of duality. Duality is an extension of equivalence where the two sides of the equation are somehow different. For example, one side may be a variable and the other side a value that it is set to. Conjugates i and j are the form of duality that is the same as equality. 2020 balandžio 13 d., 16:03
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Pakeistos 1-12 eilutės iš
[ Does the Yoneda lemma presume the Axiom of Choice? More generally, does category theory presume the Axiom of Choice? Start work on a rewrite: Math Discovery 2020 with Discussion, collecting new examples of way things were figured out, and overviewing my further conclusions what I've found so far. And start writing a second paper, an overview for the unfolding of all of math. Also, write out my talk on variables. Math develops through ambiguity. Through parallel concepts, through equations. For example, the determinant of a matrix is the product of its eigenvalues. This is because it is the last term of the characteristic equation Det(A-lambda I)=0. Through such equations, and the ambiguity they manage, math unfolds. Variables are the source of ambiguity because they relate two levels of the foursome. Collect examples of variables and ambiguity in the mathematical concepts. į:
[++++数学笔记++++] 2020 balandžio 13 d., 14:41
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Pakeista 1 eilutė iš:
See: [[Logic į:
See: [[Logic]] 2020 balandžio 12 d., 21:49
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Pridėtos 7-12 eilutės:
Start work on a rewrite: Math Discovery 2020 with Discussion, collecting new examples of way things were figured out, and overviewing my further conclusions what I've found so far. And start writing a second paper, an overview for the unfolding of all of math. Also, write out my talk on variables. Math develops through ambiguity. Through parallel concepts, through equations. For example, the determinant of a matrix is the product of its eigenvalues. This is because it is the last term of the characteristic equation Det(A-lambda I)=0. Through such equations, and the ambiguity they manage, math unfolds. Variables are the source of ambiguity because they relate two levels of the foursome. Collect examples of variables and ambiguity in the mathematical concepts. 2020 balandžio 12 d., 19:59
atliko -
Pridėtos 5-6 eilutės:
Does the Yoneda lemma presume the Axiom of Choice? More generally, does category theory presume the Axiom of Choice? 2020 balandžio 12 d., 13:08
atliko -
Pakeistos 1-2 eilutės iš
See: [[Logic]] į:
See: [[Logic]], [[Transfer Math Notes]] Ištrintos 4-254 eilutės:
* Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown - the change. Analytic continuation * Understand analytic continuation. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}? * Learn how to extend the Gamma function to the complex numbers. Choice frameworks * Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? * Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? * Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. * Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But: * for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact. * for Cn cross-polytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (not-this). * Use "this" and "that" as unmarked opposites - conjugates. Composition algebra. * Doubling is related to duality. Counting * Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo? * A) veikla kažkada prasidėjo * B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų Finite fields * Study how turning the counting around relates to cycles - finite fields. Lie theory * Special linear group has determinant 1. In general when the determinant is +/- 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse - neither is distinguished. * Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows. * {$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. * Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? * An relates to "center of mass". How does this relate to the asymmetry of whole and center? * Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. * Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu? * Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe. * {$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative. * Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu? * How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. Linear algebra * An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. * Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. * Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Linear functionals * One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? Polytopes * edge = difference Symmetric functions * What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? Bott periodicity * Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix? * Bott periodicity is the basis for 8-fold folding and unfolding. * What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]]. * Orthogonal add a perspective (Father), symplectic subtract a perspective (Son). * My dream: Sartre wrote a book "Space as World" where he has a formula that expresses Bott periodicity / my eightfold wheel of divisions. Duality examples (conjugates) * complex number "i" is not one number - it is a pair of numbers that are the square roots of -1 * spinors likewise * Dn where n=2 * the smallest cross-polytope with 2 vertices * taking a sphere and identifying antipodal elements - this is a famous group * polar conjugates in projective geometry (see Wildberger) Rotation * Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. * If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. * This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. Physics * Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement. * Massless particles acquire mass through symmetry breaking: [[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]]. * [[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we'll understand entanglement a lot better. Octonions * Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. John Baez: 24 = 6 x 4 = An x Bn [[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24. Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]]. Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. G2 requires three lines to get between any two points (?) Relate this to the three-cycle. Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? Duality * Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. Geometry * Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? Projective geometry * Homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. Variables * Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. [[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]] Attach:GeometryFormulas.png Attach:QuadrupleFormulas.png Four geometries * Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? Affine geometry * Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. * Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. Duality * John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides! Each physical force is related to a duality: * Charge (matter and antimatter) - electromagnetism * Weak force - time reversal So the types of duality should give the types of forces. Cayley-Dickson construction * John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities. Projective geometry * Desargues theorem in geometry corresponds to the associative property in algebra. * A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. Conformal geometry * Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1. Symplectic geometry * Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. Walks on trees * Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome. Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing. How do symmetries of paths relate to symmetries of young diagrams Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum). What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]]. Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? Benet linkage - keturgrandinis - lygiagretainis, antilygiagretainis * A_n points and sets * B_n inside: perpendicular (angles) and * C_n outside: line and surface area * D_n points and position * Yoneda lemma - relates to exponentiation and logarithm Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same. Representable functors - based on arrows from the same object. Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry. Six sextactic points. [[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? Study how Set breaks duality (the significance of initial and terminal objects). Show why there is no n-category theory because it folds up into the foursome. Understand the Yoneda lemma. Relate it to the four ways of looking at a triangle. The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction. "For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries? (Conscious) Learning from (unconscious) machine learning. Topology - getting global invariants (which can be calculated) from local information. Simple examples that illustrate theory. monad = black box? Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups * Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group | Wikipedia: Homotopy groups]] Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups: {$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $} * Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down. * Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. * Study the Wolfram Axiom and Nand. * Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. * Scaling is positive flips over to negative this is discrete rotation is reflection * Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. * Study homology, cohomology and the Snake lemma to explain how to express a gap. * [[https://en.wikipedia.org/wiki/Andrei_Okounkov | Andrei Okounkov]] Bridging probability, representation theory and algebraic geometry. 2020 balandžio 12 d., 11:31
atliko -
Pakeistos 237-238 eilutės iš
Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group | Wikipedia: Homotopy groups]] Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups: į:
* Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group | Wikipedia: Homotopy groups]] Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups: Pakeistos 241-245 eilutės iš
Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example Vector bundles į:
Pakeistos 244-402 eilutės iš
[[http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf | Allen Hatcher. Vector Bundles and K-Theory. (Half-written).]] [[http://pi Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism Logic * Algebra and Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. Turing machines - inner states are "states of mind" according to Turing. How do they relate to divisions of everything? If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? Study homology, cohomology and the Snake lemma to explain how to express a gap. Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. Study the Wolfram Axiom and Nand. Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements. Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. Are Nand gates (Nor gates) related to perspectives? Study how all logical relations derive from composition of Nand gates. How is a Nor gate made from [[Nand]] gates? (And vice versa.) Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc Scaling is positive flips over to negative this is discrete rotation is reflection Equations are questions Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates. In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. Choices - polytopes, reflections - root systems. How are the Weyl groups related? Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.) Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships. Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra. For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. Lietuvių kalba: * sphere - sfera * trace - pėdsakas * semisimple - puspaprastis, puspaprastė * conjugate - sujungtinis * transpose - transponuota matrica, transponavimas How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. https://en.wikipedia.org/wiki/Andrei_Okounkov Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the hyperoctahedral group. Understand the classification of Coxeter groups. Organize for myself the Coxeter groups based on how they are built from reflections. Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope. The conjugate i is evidently the part that adds a perspective. Then R is no perspective. In what sense is SU(3) related to a rotation in octonion space? If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). {$SL(n)$} is not compact, which means that it goes off to infinity. It is like the totality. We have to restrict it, which yields {$A_n$}. Whereas the other Lie families are already restricted. The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. Real forms - Satake diagrams - are like being stepped into a perspective (from some perspective within a chain). An odd-dimensional real orthogonal case is stepped-in and even-dimensional is stepped out. Complex case combines the two, and quaternion case combines them yet again. For consciousness. Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. Arnold - "Polymathematics: complexification, symplectification and all that " 1998 video. 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe." 39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem. * Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? * Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand ad-bc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc. * Think again about the combinatorial intepretation of {$K^{-1}K=I$}. * Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. * The octonions can model the nonassociativity of perspectives. * Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? * [[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras * A_n defines a linear algebra and other root systems add additional structure * A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. * [[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]] * [[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) * {$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] * [[https://www.youtube.com/watch?v=zS-LsjrJKPA | DrPhysicsA. Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] * At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). * If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? * Random phenomena organize themselves around a critical boundary. * [[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge * Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge. * Exchange particles - gauge bosons. * Wave function Smolin says is ensemble, I say bosonic sharing of space and time * [[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]] [[https://terrytao.wordpress.com/2019/07/26/twisted-convolution-and-the-sensitivity-conjecture/ | Terrence Tao: Twisted Convolution and the Sensitivity Conjecture]] Axiom of forgetfullness. [[https://www.mathpages.com/home/ | Kevin Brown]] collection of expositions of math [[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]] * Einstein field equations - energy stress tensor - is 4+6 equations. į:
* Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. * Study the Wolfram Axiom and Nand. * Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. * Scaling is positive flips over to negative this is discrete rotation is reflection * Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. * Study homology, cohomology and the Snake lemma to explain how to express a gap. * [[https://en.wikipedia.org/wiki/Andrei_Okounkov | Andrei Okounkov]] Bridging probability, representation theory and algebraic geometry. 2020 balandžio 12 d., 09:16
atliko -
Pakeistos 362-418 eilutės iš
Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? Wave function Smolin says is ensemble, I say bosonic sharing of space and time Analyze number types in terms of fractions of differences, https://en Think again about the combinatorial intepretation of {$K^{-1}K=I$}. Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. A circle, as an abelian Lie group, is The octonions can model the nonassociativity of perspectives [[https: Conjugate = mystery = false. (Hidden distinction). Triality: C at the center, three legs i->j is asymmetric, one-directional [[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]] {$U If there is a zero in At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part Random phenomena organize themselves around a critical boundary. [[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge Spin 1/2 means there are two states separated by a quanta of energy +/- h * Spin 0 total spin: onesome * Spin 1/2: fermions: twosome * Spin 1: three states: threesome * Spin 3/2: composite particles: foursome * Spin 2: gravition: fivesome ( Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge. DrPhysicsA * [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] Exchange particles - gauge bosons. * Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). į:
* Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? * Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand ad-bc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc. * Think again about the combinatorial intepretation of {$K^{-1}K=I$}. * Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. * The octonions can model the nonassociativity of perspectives. * Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? * [[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras * A_n defines a linear algebra and other root systems add additional structure * A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. * [[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]] * [[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) * {$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] * [[https://www.youtube.com/watch?v=zS-LsjrJKPA | DrPhysicsA. Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] * At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). * If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? * Random phenomena organize themselves around a critical boundary. * [[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge * Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge. * Exchange particles - gauge bosons. * Wave function Smolin says is ensemble, I say bosonic sharing of space and time Ištrintos 396-400 eilutės:
[[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]]. How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional? Ištrintos 398-401 eilutės:
Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. Ištrintos 406-410 eilutės:
Circle (three-cycle) vs. Line (link to unconditional) - sixsome - and real forms * [[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence | Curry-Howard-Lambek correspondence]] of logic, programming and category theory 2020 balandžio 12 d., 08:46
atliko -
Pakeistos 189-190 eilutės iš
Yoneda lemma - relates to exponentiation and logarithm į:
* Yoneda lemma - relates to exponentiation and logarithm Pakeistos 417-422 eilutės iš
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). [[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]] The analysis in a Lie group is all expressed by the behavior of the epsilon. į:
* Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). * [[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]] Ištrintos 427-430 eilutės:
Induction step by step is different than the outcome, the totality, which forgets the gradation. Ištrintos 433-436 eilutės:
Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations. Pakeistos 438-449 eilutės iš
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos skaitykla Yoneda Lemma * Loss of info from How to What is equal to the Loss of info from "Why for What" to "Why for How". * How: inner logic. What: external view. * Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą? Einstein field equations - energy stress tensor - is 4+6 equations. į:
* Einstein field equations - energy stress tensor - is 4+6 equations. Pakeistos 444-582 eilutės iš
[[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence | Curry-Howard-Lambek correspondence]] of logic, programming and category Pascal's triangle - the zeros on either end of each row are like Everything at start and finish of an exact sequence. Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805.00203 Study [[https://en.wikipedia.org/wiki/Orthogonal_group | orthogonal groups]] and Bott periodicity. In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? Consider the classification of Lie groups in terms of the objects for which they are symmetries. Navier-Stokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. [[https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf | Differential Forms in Algebraic Topology]], Bott & Tu {$A^TA$} is similar to the adjoint functors - they may be inverses (in the case of a unitary matrix) or they may be similar. Terrence Tao problem solving https://books.google.lt/books/about/Solving_Mathematical_Problems_A_Personal.html?id=ZBTJWhXD05MC&redir_esc=y In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \overline{-(YX-XY)}$}. Eduardo's Yoneda Lemma diagram is the foursome. Yates Index Theorem - consider substitution. One-dimensional proteins are wound up like the chain of a multidimensional Lie group. Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? Terrence Tao: It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. [[https://www.math.ucla.edu/~tao/ | Terrence Tao courses]] [[http://msc2010.org/mediawiki/index.php?title=MSC2010 | Mathematics Subject Classification wiki]] [[https://www.youtube.com/watch?v=LaTTqgchO2o | How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries| N J Wildberger]] Compare finite field behavior (division winding around) with complex number behavior (winding around). What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. Talk with Thomas * How is the Riemann sheet, winding around, going to a different Riemann sheet, related to the winding number? and the roots of polynomials? * Organic variation, variables * Differences between even and odd for orthogonal matrices as to whether they can be paired (into complex variables) or not. * {$e^{\sum i \times generator \times parameter}$} has an inverse. * Unitary T = {$e^{iX_j\alpha_j}$} where {$X_j$} are generators and {$\alpha_j$} are angles. Volume preserving, thus preserving norms. Length is one. * Understanding of effect. Physics, why does it work? How can I describe it efficiently and correctly? * Definition of entropy depends on how you choose it. Unit of phase space determines your unit of entropy. Thus observer defines phase space. * Go from rather arbitrary set of dimensions to more natural set of dimensions. Natural because they are convenient. This leads to symmetry. Thus represent in terms of symmetry group, namely Lie groups. There are dimensions. In order to write them up, we want more efficient representations. Subgroups give us understanding of causes. Smaller representations give us understanding of effects. We want to study what we don't understand. In engineering, we leverage what we don't understand. * How many parameters do I need to describe the system? (Like an object.) Minimize constraints. It becomes complicated. Multipole is abstracting the levels of relevance. Ordering them inside the dimensions I am working with. What is the important quantity? Measures quality. How transformation leaves the object invariant. Distinguish between continuous parameters that we measure against and these quantity that we want to study. We use dimensions as a language to relate the inner structure and the outer framework. To measure momentum we need to measure two different quantities. * Complex models continuous motion. Symplectic - slack in continuous motion. * Three-cycle: same + different => different ; different + different => same ; different + same => different * {$ \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} $} * same + different + different + same + ... Does the Lie algebra bracket express slack? Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i - \Delta_j$} is {$30^{\circ}$} {$\Delta_i - \Delta_j$} {$e^{\sum k_i \Delta_i}$} Has inner product iff {$AA^?=I$}, {$A{-1}=A^?$} Killing form. What is it for exceptional Lie groups? The Cartan matrix expresses the amount of slack in the world. {$A_n$} God. {$B_n$}, {$C_n$}, {$D_n$} human. {$E_n$} n=8,7,6,5,4,3 divisions of everything. 2 independent roots, independent dimensions, yield a "square root" (?) Symplectic matrix (quaternions) describe local pairs (Position, momentum). Real matrix describes global pairs: Odd and even? Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. {$A_n$} is based on differences {$x_i-x_j$}. They are a higher grid risen above the lower grid {$x_i$}. Whereas the others are aren't based on differences and collapse into the lower grid. How to understand this? How does it relate to duality and the way it is expressed. In {A_1}, the root {$x_2-x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. If two roots are separated by more than {$90^\circ$}, then adding them together yields a new root. {$cos\theta = \frac{a\circ b}{\left \| a \right \|\left \| b \right \|}$} {$120^\circ$} yields {$\frac{-1}{\sqrt{2}\sqrt{2}}=\frac{-1}{2}$} Given a chain of composition {$\cdots f_{i-1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme. * Root systems give the ways of composing perspectives-dimensions. * {$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. * A root pair {$x$} and {$-x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. In solving for eigenvalues {$\lambda_i$} and eigenvectors {$v_i$} of {$M$}, make the matrix {$M-\lambda I$} degenerate. Thus {$\text{det}(M-\lambda I)=0$}. The matrix is degenerate when one row is a linear combination of the other rows. So the determinant is a geometrical expression for volume, for collinearity and noncollinearity. Terrence Tao: Each prime p wants to have weight ln p. Compare with [[https://en.wikipedia.org/wiki/Zipf%27s_law | Zipf's law]], [[https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot_law | Zipf-Mandelbrot law]], [[https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution | Yule-Simon distribution]], [[https://en.wikipedia.org/wiki/Preferential_attachment | Preferential attachment]], [[https://en.wikipedia.org/wiki/Matthew_effect | Matthew effect]], [[https://en.wikipedia.org/wiki/Pareto_principle | Pareto's principle]]. [[https://www.amazon.com/Category-Theory-Context-Aurora-Originals/dp/048680903X | Emily Riehl. Category Theory in Context]] Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to cate-gories. Cambridge (1997) Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation toApplied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) How is substitution, as a method of proof, related to lamba calculus, and construction? Walks from A to B in category theory are morphisms and they get mapped to the morphisms from A to B. Relate this to walks on trees. Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? What can graph theory (for example, random graphs, or random order) say about category theory? Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? [[https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe | The Elliptic Umbilic Diffraction Catastrophe]]. Optics, Bott periodicity? In the category Set, how can you distinguish between a countable and uncountable set? Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? What would the category of Lists look like? And what would the Yoneda Lemma look like if the functor mapped into the category of Lists? What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? Short exact sequence. Defining a perspective relative to a base. What is the difference between an exact and a nonexact relationship? The concept of scope: Kernel: irrelevant because goes to zero. Cokernel: irrelevant because outside of scope. How do sheaves relate to gradation of symmetric functions? [[https://paulhus.math.grinnell.edu/SMPpaper.pdf | Jennifer Paulhus. Group Actions and Riemann Surfaces]] į:
* [[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence | Curry-Howard-Lambek correspondence]] of logic, programming and category theory 2020 balandžio 11 d., 16:55
atliko -
Pakeistos 599-601 eilutės iš
How do sheaves relate to gradation of symmetric functions? į:
How do sheaves relate to gradation of symmetric functions? [[https://paulhus.math.grinnell.edu/SMPpaper.pdf | Jennifer Paulhus. Group Actions and Riemann Surfaces]] 2020 balandžio 10 d., 16:23
atliko -
Pridėta 599 eilutė:
How do sheaves relate to gradation of symmetric functions? 2020 balandžio 09 d., 22:50
atliko -
Pakeistos 591-598 eilutės iš
What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? į:
What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? Short exact sequence. Defining a perspective relative to a base. What is the difference between an exact and a nonexact relationship? The concept of scope: Kernel: irrelevant because goes to zero. Cokernel: irrelevant because outside of scope. 2020 balandžio 06 d., 18:20
atliko -
Pakeistos 587-591 eilutės iš
Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? į:
Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? What would the category of Lists look like? And what would the Yoneda Lemma look like if the functor mapped into the category of Lists? What does the Yoneda Lemma say concretely about the category of Graphs, Groups, Lists, etc.? 2020 balandžio 06 d., 17:23
atliko -
Pakeistos 585-587 eilutės iš
In the category Set, how can you distinguish between a countable and uncountable set? į:
In the category Set, how can you distinguish between a countable and uncountable set? Analyze the commutative diagram proving the Yoneda lemma by thinking it, on the one level, as a statement about four sets, but on another level, as a statement about functors and a natural transformation between them. Is there an ambiguity here? 2020 balandžio 06 d., 14:35
atliko -
Pakeistos 583-585 eilutės iš
[[https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe | The Elliptic Umbilic Diffraction Catastrophe]]. Optics, Bott periodicity? į:
[[https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe | The Elliptic Umbilic Diffraction Catastrophe]]. Optics, Bott periodicity? In the category Set, how can you distinguish between a countable and uncountable set? 2020 balandžio 06 d., 14:16
atliko -
Pakeistos 581-583 eilutės iš
In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? į:
In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? [[https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe | The Elliptic Umbilic Diffraction Catastrophe]]. Optics, Bott periodicity? 2020 balandžio 06 d., 10:48
atliko -
Pakeistos 579-581 eilutės iš
Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. į:
Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same? 2020 balandžio 05 d., 14:43
atliko -
Pakeistos 577-579 eilutės iš
What can graph theory (for example, random graphs, or random order) say about category theory? į:
What can graph theory (for example, random graphs, or random order) say about category theory? Understand classification of closed surfaces: Sphere = 0. Projective plane = 1/3. Klein bottle = 2/3. Torus = 1. 2020 balandžio 05 d., 14:10
atliko -
Pakeistos 575-577 eilutės iš
Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? į:
Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? What can graph theory (for example, random graphs, or random order) say about category theory? 2020 balandžio 05 d., 14:09
atliko -
Pakeista 575 eilutė iš:
Parentheses establish a tree structure. What are walks on these trees? į:
Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another? 2020 balandžio 05 d., 14:09
atliko -
Pakeistos 571-575 eilutės iš
How is substitution, as a method of proof, related to lamba calculus, and construction? į:
How is substitution, as a method of proof, related to lamba calculus, and construction? Walks from A to B in category theory are morphisms and they get mapped to the morphisms from A to B. Relate this to walks on trees. Parentheses establish a tree structure. What are walks on these trees? 2020 balandžio 05 d., 14:06
atliko -
Pakeistos 569-571 eilutės iš
Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) į:
Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) How is substitution, as a method of proof, related to lamba calculus, and construction? 2020 balandžio 04 d., 20:28
atliko -
Pakeistos 567-569 eilutės iš
Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation toApplied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches. į:
Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation toApplied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007) 2020 balandžio 04 d., 20:25
atliko -
Pakeistos 565-567 eilutės iš
Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to cate-gories. Cambridge (1997) į:
Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to cate-gories. Cambridge (1997) Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation toApplied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2020 balandžio 04 d., 20:25
atliko -
Pakeistos 563-565 eilutės iš
[[https://www.amazon.com/Category-Theory-Context-Aurora-Originals/dp/048680903X | Emily Riehl. Category Theory in Context]] į:
[[https://www.amazon.com/Category-Theory-Context-Aurora-Originals/dp/048680903X | Emily Riehl. Category Theory in Context]] Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to cate-gories. Cambridge (1997) 2020 balandžio 04 d., 20:20
atliko -
Pakeistos 561-563 eilutės iš
Terrence Tao: Each prime p wants to have weight ln p. Compare with [[https://en.wikipedia.org/wiki/Zipf%27s_law | Zipf's law]], [[https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot_law | Zipf-Mandelbrot law]], [[https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution | Yule-Simon distribution]], [[https://en.wikipedia.org/wiki/Preferential_attachment | Preferential attachment]], [[https://en.wikipedia.org/wiki/Matthew_effect | Matthew effect]], [[https://en.wikipedia.org/wiki/Pareto_principle | Pareto's principle]]. į:
Terrence Tao: Each prime p wants to have weight ln p. Compare with [[https://en.wikipedia.org/wiki/Zipf%27s_law | Zipf's law]], [[https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot_law | Zipf-Mandelbrot law]], [[https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution | Yule-Simon distribution]], [[https://en.wikipedia.org/wiki/Preferential_attachment | Preferential attachment]], [[https://en.wikipedia.org/wiki/Matthew_effect | Matthew effect]], [[https://en.wikipedia.org/wiki/Pareto_principle | Pareto's principle]]. [[https://www.amazon.com/Category-Theory-Context-Aurora-Originals/dp/048680903X | Emily Riehl. Category Theory in Context]] 2020 balandžio 04 d., 14:52
atliko -
Pridėta 561 eilutė:
Terrence Tao: Each prime p wants to have weight ln p. Compare with [[https://en.wikipedia.org/wiki/Zipf%27s_law | Zipf's law]], [[https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot_law | Zipf-Mandelbrot law]], [[https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution | Yule-Simon distribution]], [[https://en.wikipedia.org/wiki/Preferential_attachment | Preferential attachment]], [[https://en.wikipedia.org/wiki/Matthew_effect | Matthew effect]], [[https://en.wikipedia.org/wiki/Pareto_principle | Pareto's principle]]. 2020 balandžio 02 d., 13:39
atliko -
Pakeistos 555-560 eilutės iš
Root systems give the ways of composing perspectives-dimensions. A root pair {$x$} and {$-x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. į:
* Root systems give the ways of composing perspectives-dimensions. * {$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. * A root pair {$x$} and {$-x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. Ištrintos 559-583 eilutės:
Exercise: Get the eigenvalues for a generic matrix: 2x2, 3x3, etc. Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP? Exercise: Find all matrices with eigenvalues 1 and -1. {$\lambda=\frac{a_{11}+a_{22} \pm \sqrt{(a_{11}+a_{22})^2 - 4|A|}}{2}$} so |A|=-1. {$\begin{pmatrix} \pm \sqrt{1 + a_{12}a_{21}} & a_{12} \\ a_{21} & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} such as {$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$} {$\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$} {$\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$} Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} -y \\ -x \end{pmatrix}$} {$\begin{pmatrix} ix \\ -iy \end{pmatrix}$} where in each case two of three are applied: flipping, multiplying by i, multiplying by -1. 2020 balandžio 02 d., 13:30
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Ištrintos 587-639 eilutės:
The complex Lie algebra divine threesome H, X, Y is an abstraction. The real Lie algebra human cyclical threesome is an outcome of the representation in terms of numbers and matrices, the expression of duality in terms of -1, i, and position. Jacob Lurie, Bachelor's thesis, [[http://www.math.harvard.edu/~lurie/papers/thesis.pdf | On Simply Laced Lie Algebras and Their Minuscule Representations]] I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll. Look at effect of Lie group's subgroup on a vector. (Shear? Dilation?) and relate to the 6 transformations. Study [[https://en.wikipedia.org/wiki/SL2(R) | SL(2,R)]]. Study the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation | Möbius group]]. Understand [[https://math.stackexchange.com/questions/646183/list-of-connected-lie-subgroups-of-mathrmsl2-mathbbc | the study of subgroups]]. Relate [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group | elliptic transforms]] to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: {$1 / z$}, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), {$1 − z$} which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and {$z / ( z − 1 )$} which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). Note that this relates pairs from: 1, z, z-1. The nonexistent element of {$F_1$} may be considered to not exist, or imagined to exist, but regardless, I expect that cognitively there are three ways to interpret it as 0, 1, ∞, which thereby expand upon the duality between existence and nonexistence and make it structurally richer. Note that the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group | Mobius transformations]] classify into types which accord with my six transformations: * reflection = circular * shear = parabolic * rotation = elliptic * dilation = hyperbolic * squeeze = internal of hyperbolic (e^t e^{-t}=1) * translation = internal of parabolic Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? Analyze the Mobius group in terms of what it does to circles and lines, and analyze the transformations likewise. Reconsider what Shu-Hong's thesis has to say about fractions of differences, and how they relate to the Mobius group. Consider how A_2 is variously interpreted as a unitary, orthogonal and symplectic structure. How to relate Lie algebras and groups by way of the Taylor series of the logarithm? Understand the relations between U(1) and electromagnetism, SU(2) and the weak force, SU(3) and the strong force. * [[https://en.wikipedia.org/wiki/Standard_Model | Standard Model]] * [[https://en.wikipedia.org/wiki/Electroweak_interaction | Electroweak interaction]] Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations. [[https://smile.amazon.com/Functions-Complex-Variable-Graduate-Mathematics/dp/0387903283/ref=smi_www_rco2_go_smi_g3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Functions of One Complex Variable]] John Conway Notions of dimension d (Mathematical Companion): * locally looks like d-dimensional space * the barrier between any two points is never more than d-1 dimensional * can be covered with sets such that no more than d+1 of them ever overlap * the largest d such that there is a nontrivial map from a d-dimensional manifold to a substructure of the space * the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and infinity Real line models separation (by cutting) and connectedness (by continuity). The separating cuts become locations (points) in their own right. * {$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2020 balandžio 02 d., 13:21
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Pakeistos 639-642 eilutės iš
{$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} į:
* {$\mathrm{Aut}_{H\circ f}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$} * {$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2020 balandžio 02 d., 13:17
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Pakeistos 642-646 eilutės iš
{$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the three-cycle, so they yield the foursome: 0, 120, 240, 360. Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. į:
{$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2020 balandžio 02 d., 13:15
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Pakeistos 646-674 eilutės iš
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity Hermitian: a+bi <-> a-bi, Symmetric: a <-> a, Anti-symmetric: b <-> -b How does logic come from a quadratic form? Four ways of relating level and metalevel with "and". Consider how the "inner product", including for the symplectic form, yields geometry. The skew-symmetric bilinear form says that phi(x,y)= -phi(y,x) so one of them, say, (x,y) is + (correct) and the other is - (incorrect). This is left-right duality based on nonequality (the two must be different). And it is threefold logic, the non-excluded middle 0. SO that 1 true, -1 false, 0 middle. The inner products (like unitary) are linear. They and their quadratic product get projected onto a second quadratic space (a screen) which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global (like Mobius transformations). Three inner products - relate to chronogeometry. Three inner products are products of vectors that are external sums of basis vectors as different units. Six interpretations of scalar multiplication that are internal sums of amounts. Conformal groups. Orthogonal. Noncommutative polynomial invariants of unitary group "Some Fundamental Theorems in Mathematics" (Knill, 2018) https://arxiv.org/abs/1807.08416 http://people.math.harvard.edu/~knill/media/index.html Analytic limits need neighborhoods. Categorical limits need maps. What if there is a handle (a torus) inside a sphere? How to classify that? Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to ternary operations and triality į:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. 2020 kovo 31 d., 18:18
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Pakeista 674 eilutė iš:
Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to į:
Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to ternary operations and triality. 2020 kovo 31 d., 18:18
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Pakeista 674 eilutė iš:
Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to triality. į:
Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to trilinear forms and triality. 2020 kovo 31 d., 18:17
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Pakeistos 672-674 eilutės iš
What if there is a handle (a torus) inside a sphere? How to classify that? į:
What if there is a handle (a torus) inside a sphere? How to classify that? Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to triality. 2020 kovo 31 d., 14:46
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Pakeistos 670-672 eilutės iš
Analytic limits need neighborhoods. Categorical limits need maps. į:
Analytic limits need neighborhoods. Categorical limits need maps. What if there is a handle (a torus) inside a sphere? How to classify that? 2020 kovo 30 d., 13:46
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Pridėtos 669-670 eilutės:
Analytic limits need neighborhoods. Categorical limits need maps. 2020 kovo 29 d., 23:49
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Pakeistos 666-668 eilutės iš
"Some Fundamental Theorems in Mathematics" (Knill, 2018) https://arxiv.org/abs/1807. į:
"Some Fundamental Theorems in Mathematics" (Knill, 2018) https://arxiv.org/abs/1807.08416 http://people.math.harvard.edu/~knill/media/index.html 2020 kovo 29 d., 22:19
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Pakeistos 664-666 eilutės iš
Noncommutative polynomial invariants of unitary į:
Noncommutative polynomial invariants of unitary group "Some Fundamental Theorems in Mathematics" (Knill, 2018) https://arxiv.org/abs/1807.08416 2020 kovo 27 d., 23:34
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Pakeistos 660-664 eilutės iš
Three inner products are products of vectors that are external sums of basis vectors as different units. Six interpretations of scalar multiplication that are internal sums of amounts. į:
Three inner products are products of vectors that are external sums of basis vectors as different units. Six interpretations of scalar multiplication that are internal sums of amounts. Conformal groups. Orthogonal. Noncommutative polynomial invariants of unitary group 2020 kovo 27 d., 23:09
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Pakeista 656 eilutė iš:
The inner products (like unitary) are linear. They į:
The inner products (like unitary) are linear. They and their quadratic product get projected onto a second quadratic space (a screen) which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global (like Mobius transformations). 2020 kovo 27 d., 22:55
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Pakeista 656 eilutė iš:
The inner products į:
The inner products (like unitary) are linear. They then get projected onto another space which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global (like Mobius transformations). 2020 kovo 27 d., 22:53
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Pakeistos 656-660 eilutės iš
The inner products are linear. They then get projected onto another space which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global. į:
The inner products are linear. They then get projected onto another space which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global. Three inner products - relate to chronogeometry. Three inner products are products of vectors that are external sums of basis vectors as different units. Six interpretations of scalar multiplication that are internal sums of amounts. 2020 kovo 27 d., 22:50
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Pakeistos 650-651 eilutės iš
How does logic come from a quadratic form? į:
How does logic come from a quadratic form? Four ways of relating level and metalevel with "and". Pakeistos 654-656 eilutės iš
The skew-symmetric bilinear form says that phi(x,y)= -phi(y,x) so one of them, say, (x,y) is + (correct) and the other is - (incorrect). This is left-right duality based on nonequality (the two must be different). And it is threefold logic, the non-excluded middle 0. SO that 1 true, -1 false, 0 middle. į:
The skew-symmetric bilinear form says that phi(x,y)= -phi(y,x) so one of them, say, (x,y) is + (correct) and the other is - (incorrect). This is left-right duality based on nonequality (the two must be different). And it is threefold logic, the non-excluded middle 0. SO that 1 true, -1 false, 0 middle. The inner products are linear. They then get projected onto another space which expresses their geometric nature as a tangent space for a differentiable manifold thus relating the discrete and the continuum, the infinitesimal and the global. 2020 kovo 27 d., 05:38
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Pridėtos 650-654 eilutės:
How does logic come from a quadratic form? Consider how the "inner product", including for the symplectic form, yields geometry. The skew-symmetric bilinear form says that phi(x,y)= -phi(y,x) so one of them, say, (x,y) is + (correct) and the other is - (incorrect). This is left-right duality based on nonequality (the two must be different). And it is threefold logic, the non-excluded middle 0. SO that 1 true, -1 false, 0 middle. 2020 kovo 25 d., 22:18
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Pridėtos 647-649 eilutės:
Hermitian: a+bi <-> a-bi, Symmetric: a <-> a, Anti-symmetric: b <-> -b 2020 kovo 25 d., 13:34
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Ištrintos 628-629 eilutės:
Ištrintos 635-636 eilutės:
Pakeistos 638-652 eilutės iš
Lie algebra matrix representations code for: * Sequences - simple roots * Trees - positive roots * Networks - all roots Relate walks on trees with fundamental group. [[https://en.wikipedia.org/wiki/Riemann_surface | Riemann surface]] [[http://www.math.tifr.res.in/~pablo/download/book/book.html | Riemann Surfaces Book]], Pablo Arés Gastesi {$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} į:
* {$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} Ištrintos 643-647 eilutės:
How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? and to classification in topology? Ištrintos 645-650 eilutės:
* Homologija bandyti išsakyti persitvarkymų tarpą tarp pirminės ir antrinės tvarkos. * B_>C ..... How->What * External relations -> Internal logic .... (Not What=Why) Hom C -> Hom B (Not How=Whether) * Ar savybių visuma yra simpleksas? Ar savybės skaidomos (koordinačių sistema). Ištrintos 646-649 eilutės:
Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or cross-polytopes? When you get the definitions right, the theorems are easy to prove. When the theorems are hard to prove, then the definitions are not right. (Tobias Osborne) So this shows how definitions and theorems coevolve. 2020 kovo 24 d., 14:59
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Pakeistos 676-678 eilutės iš
Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or cross-polytopes? į:
Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or cross-polytopes? When you get the definitions right, the theorems are easy to prove. When the theorems are hard to prove, then the definitions are not right. (Tobias Osborne) So this shows how definitions and theorems coevolve. 2020 kovo 23 d., 20:10
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Pakeistos 674-676 eilutės iš
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. į:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. Consider how simplexes (in differential geometry and Stokes theorem) relate to symplectic geometry. What is the relevant polytope for symplectic geometry? Coordinate systems? Or cross-polytopes? 2020 kovo 22 d., 22:35
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Pakeistos 663-670 eilutės iš
How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? [[https://smile.amazon.com/dp/144192681X/ref=smi_www_rco2_go_smi_3905707922 [[https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham | Are there other nice math books close to the style of Tristan Needham?]] The possibilities for a complex plane (extra point for sphere, point removed for cylinder) are relevant for modeling perspectives. į:
How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? and to classification in topology? Pakeistos 668-669 eilutės iš
į:
Pakeistos 674-682 eilutės iš
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Cross-ratio | anharmonic group]] and Mobius transformations. Note that the anharmonic group is based on composition of functions. Proceed from balance - note how additive balance precedes multiplicative ratio precedes possibly negative (directional) ratio precedes anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. Look for what it would mean for a ratio to be {$i$} and the product to be -1. į:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. 2020 kovo 22 d., 21:51
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Ištrintos 685-686 eilutės:
Ištrintos 686-706 eilutės:
Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy). {$\frac{\text{d}}{\text{dt}}\Psi=-i\frac{H}{\hbar}\Psi$} In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}. Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. In this way, quadrance and spread eliminate false distinctions and the problems they cause. Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle). Challenge problems: * Determine whether {$\pi + e$} is rational or irrational. * Determine whether {$\pi^e$} is rational or irrational. A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. The center of a circle is perhaps a fourth point (with every triplet) much like the identity is related to the three-cycle? 2020 kovo 22 d., 20:44
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Ištrintos 704-767 eilutės:
Are there 6+4 branches of math? How are the branches of math related to the ways of figuring things out? [[https://en.wikipedia.org/wiki/J-invariant | J-invariant]] is related to SL(2,Z) and monstrous moonshine. Universal hyperbolic geometry * The circle maps every point to a line and vice versa. Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself. Relate the levels of the foursome with the cross-ratio (for example, how-what are the two points within a circle, and why-whether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the cross-ratio upon permuting its elements. Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases? Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?) Think of harmonic pencil types as the basis for the root systems * {$A_n$} {$\pm(x-y)$} dual * {$B_n$} {$\pm x,\pm y$} yields {$x\pm y$} * {$C_n$} {$x\pm y$} yields {$\pm2x,\pm2y$} * {$D_n$} {$\pm(x\pm y)$} dual dual In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math? Explain: (B + iC)(B - iC) = (C + iB)(C - iB) Intuit SL(2,C) as three-dimensional in C (because ad-bc=1 so we lose one complex dimension - intuit that). And in what sense is that different from ad-bc=0 (a line? a one-dimensional subspace?) SL(2,C) is the spin relativistic group. Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality? Normal bundles involve embedding in an extrinsic space. {$K_0$} and {$K_1$} perhaps express perspectives, like God's trinity and the three-cycle, the two foursomes in the eight-cycle. Linear operators - something you can have more of or less of (proportionately) - transformational action (like rotation or translation). When are two vectors, lines, etc. perpendicular? 4 kinds of i: the Pauli matrices and i itself. Generating functions relate: symmetry of analytic functions, algorithms, finite combinatorial symmetry. (Think of as a vector bundle - the infinite sequence ({$x_i$}) is the base space, and the coefficient is the fiber, and the fibers are related.) Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions. Relate bundle concepts to amounts and units. Relate motion to bundles. Symplectic geometry, looseness, etc. All 4 geometries. A geometry (like hyperbolic geometry) allows for a presentation of a bundle, thus a perspective on a perspective (atsitokėjimas - atvaizdas). Compare with: įsijautimas-aplinkybė. {$2\pi$} additive factors, e multiplicative factors. [[https://www.amazon.com/Differential-Algebraic-Topology-Graduate-Mathematics/dp/0387906134 | Bott & Tu. Differential Forms in Algebraic Topology.]] Odd cohomology works like fermions, even cohomology works like bosons. Video: Ben Mares: introduction to cohomology. SL(2,C) lines (plus infinity) become circles. Do linear equations become circular equations? What does that mean? Are SL(2,C) circular equations related to the continuum? (polynomial coefficient) Ordinal/List/Analysis vs. Cardinal/Set/Algebra (polynomial root) - the coefficients and roots are related by the binomial theorem, the factors are choices. 2020 kovo 22 d., 11:25
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Ištrintos 767-779 eilutės:
Analysis is the infinity of sheets, the recurring sequence of not going beyond oneself. But algebra is the single sheet which is the self that it all goes into, where all of the actions, all of the sheets coincide as one sheet, one going beyond. Thus the cardinal (of algebra) arises from the ordinal (of analysis). The ordinal (list) is deterministic whereas the cardinal (set) is nondeterministic. SL(2) - H,X,Y is 3-dimensional (?) but SU(2) - three-cycle + 1 (God) is 4 dimensional. Is there a discrepancy, and why? Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the Cauchy-Riemann equations, etc. Bundle = restructuring (base = continuum -> fiber = discrete). A number that is "large enough" can essentially model the continuum. Origami: [[http://alum.mit.edu/www/tchow/multifolds.pdf | The power of multifolds: Folding the algebraic closure of the rational numbers]] 2020 kovo 22 d., 09:42
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Ištrintos 777-786 eilutės:
Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. So what is a pattern? And what are the limitations on patterns? Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. [[http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf | A Cohomological Viewpoint on Elementary School Arithmetic]] About "carrying". Access restricted. Consider how math variously expresses perspectives. Pakeistos 782-813 eilutės iš
* [[https://math.stackexchange.com/questions/2371364/whats-the-canonical-embedding-of-the-globe-category-into-top | What's the canonical embedding of the globe category into Top?]] * ''They are the maps of the closed n-ball to the "northern" and "southern" hemispheres of the surface of the (n+1)-ball.'' * ''This defines a functor G→Top that extends along the Yoneda embedding, yielding a geometric representation of any globular set Gˆ→Top.'' https://oeis.org/A013609 triangle for hypercubes. (1 + 2x)^n unsigned coefficients of chebyshev polynomials of the second kind e to the matrix consisting of the natural numbers on its first off-diagonal gives a triangular matrix with pascal's triangle. and how is it in the case of the cube? Pascal's triangle tilted gives Fibonacci numbers * [[https://www.math.upenn.edu/~wilf/gfology2.pdf | Generatingfunctionology]] by Herbert Wilf ------------------------------------- Mathematics is the study of structure. It is the study of systems, what it means to live in them, and where and how and why they fail or not. homology - holes - what is not there - thus a topic for explicit vs. implicit math svarbūs pavyzdžiai * https://en.m.wikipedia.org/wiki/Möbius_transformation Symmetry * Representations of the symmetric group. Symmetric - homogeneous - bosons - vectors. Antisymmetric - elementary - fermion - covectors. Euclidean space allows reflection to define inside and outside nonproblematically, thus antisymmetricity. Free vector space. Schur functions combine symmetric and antisymmetric in rows and columns. * E8 is the symmetry group of itself. What is the symmetry group of? * Meilė (simetrija) įsteigia nemirtingumą (invariant). * Totally independent dimensions: Cartesian * Totally dependent dimensions: simplex We "understand" the simplex in Cartesian dimensions but think it in simplex dimensions. We can imagine a simplex in higher dimensions by simply adding externally instead of internally. Consider (implicit + explicit)to infinity; and also (unlabelled + labelled) to infinity. Also consider (unlabelable + labelable). And (definitively labeled + definitively unlabeled). * Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/proving-q-binomial-identities | interpretation related to Young tableaux]] į:
2020 kovo 22 d., 08:14
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Ištrintos 819-831 eilutės:
Neural networks * Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems. Number theory * Prime numbers: "Cost function". The "cost" of a number may be thought of as the sum of all of its prime factors. What might this reveal about the primes? * [[http://oeis.org/A000607 | Number of ways to partition a number into primes]]. Field with one element * https://en.m.wikipedia.org/wiki/Field_with_one_element [[Catalan]], Mandelbrot, Julia sets Pakeistos 823-851 eilutės iš
* Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/proving-q-binomial-identities | interpretation related to Young tableaux]] '''Pagrindiniai matematikos dėsniai''' Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę. Prisiminti savo matematikos mokymo dėsnius: * every answer is an amount and a unit ir tt. * combine like units * list different units * a right triangle is half of a rectangle * a triangle is the sum of two right triangles * four times a right triangle is the difference of two squares * extending the domain * purposes of families of functions Basic division rings: [[http://math.ucr.edu/home/baez/week59.html | John Baez 59]] * The real numbers are not of characteristic 2, * so the complex numbers don't equal their own conjugates, * so the quaternions aren't commutative, * so the octonions aren't associative, * so the hexadecanions aren't a division algebra. Clifford algebras * Real forms are similar to Clifford algebras in that in n-dimensions you have n versions, n signatures. Each models the takng up of one of the n perspectives in a division of everything. The squeeze function defines area. Symplectic geometry defines slack. It defines motion as oriented area. į:
* Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/proving-q-binomial-identities | interpretation related to Young tableaux]] 2020 kovo 22 d., 06:02
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Pakeistos 860-864 eilutės iš
* Real forms are similar to Clifford algebras in that in n-dimensions you have n versions, n signatures. Each models the takng up of one of the n perspectives in a division of everything. į:
* Real forms are similar to Clifford algebras in that in n-dimensions you have n versions, n signatures. Each models the takng up of one of the n perspectives in a division of everything. The squeeze function defines area. Symplectic geometry defines slack. It defines motion as oriented area. 2020 kovo 21 d., 22:46
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Pakeistos 857-860 eilutės iš
* so the hexadecanions aren't a division algebra. į:
* so the hexadecanions aren't a division algebra. Clifford algebras * Real forms are similar to Clifford algebras in that in n-dimensions you have n versions, n signatures. Each models the takng up of one of the n perspectives in a division of everything. 2020 kovo 21 d., 21:20
atliko - 2020 kovo 21 d., 21:20
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Pakeistos 857-923 eilutės iš
* so the hexadecanions aren't a division algebra. '''Įdomūs, prasmingi reiškiniai matematikoje''' Polynomial powers are "twists" of a string. One end of the string is held up and then down. Each twist of the string allows for a new maximum or minimum. This is an interpretation of multiplication. Logika * Logikos prielaidos bene slypi daugybė matematikos sąvokų. Pavyzdžiui, begalybė slypi galimybėje tą patį logikos veiksnį taikyti neribotą skaičių kartų. O tai slypi tame kad logika veiksnio taikymas nekeičia loginės aplinkos. * Ar gali būti, kad aibių teorijos problemos (pavyzdžiui, sau nepriklausanti aibė) yra iš tikrųjų logikos problemos? Kaip atskirti aibių teoriją ir logiką (pavyzdžiui, axiom of choice)? [[https://en.wikipedia.org/wiki/Arithmetic_derivative | Arithmetic derivative]] ---------------------------- Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics. From dream: Space is made up of all possible curves. Physics is about the geodesics, the curves with no slack. They really are all one curve that goes through every curve and every single point. From dream: vectors A-B, B-A, consider the difference between them, the equivalence of A and B. Conjugates can be thought of as "twins", whereas +1 and -1 are "spouses". Bundle. Geometry relates analysis (continuum) and algebra (discrete) as a restructuring. When the discrete grows large does it become a continuum? Homology and cohomology are like the relation between 0->1->2->3 and 4->5->6->7. Substantiate: Affine geometry defines no perspective, projective geometry defines one perspective, conformal geometry defines a perspective on a perspective, symplectic geometry defines a perspective on a perspective on a perspective. An grows at both ends, either grows independently, no center, no perspective, affine. Bn, Cn have both ends grow dependently, pairwise, so it is half the freedom. In what sense does Dn grow, does it double the possibilities for growth? And does Dn do that internally by relating xi+xj and xi-xj variously somehow? Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness. A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series. Does the inside of a sphere have negative curvature, and the outside of sphere have positive curvature? And likewise the inside and outside of a torus? Why is negative curvature - the curvature inside - more prominent than positive curvature - the outside of a space? Integral of 1/x is ln x what does that say about ex? Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5. In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons. The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid. Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a three-cycle, what it is relating, a self to itself. In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding. The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence. Perspective relates intrinsic and extrinsic geometry by way of ambient space. Ambient space relates base and fiber (perspective) as a bundle. Lagrangian L=T-V expresses slack (or anti-slack), makes the conversion between potential and kinetic energy as smooth as possible. Hamiltonian H=T+V expresses the totality, the love. In this sense, they are dual, as per the sevensome - the Lagrangian expresses the internal slack, and the Hamiltonian expresses the wholeness of the external frame. Potential energy is bounded from below. Kinetic energy is always positive, absolutely. Its relation to momentum is absolute, unconditional. Whereas potential is defined relatively and its relation to position is variable. The wave equation is defined on phase space but in such a way that it is understood in terms of a superposition of waves for position, yielding a "blob" - a wave packet, and an analogous superposition of waves for momentum, and the two are related by the Fourier transform, by the Heisenberg uncertainty principle. Physics abstracts from the personality of other researchers. Have a common ground, communicate not on any consensus, but based on what we find, as independent witnesses. SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will. Coordinate systems are observers and they shouldn't affect what they observe. (Relate this to the kinds of polytopes.) į:
* so the hexadecanions aren't a division algebra. 2020 kovo 21 d., 21:06
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Ištrintos 922-929 eilutės:
Tensors are invariant under linear transformations but their components do change. So a tensor is a bringing together of components, which can be either covariant or contravariant. Is this stepping out and stepping in? Is a tensor a division of everything and each component a perspective? There is a duality between a tensor and its expression under a particular basis. They are interchangeable. Ištrintos 923-977 eilutės:
Electrons are particles when we look at them and waves when we don't. Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways. Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time? The Axiom of Choice is based on the notions of a perspective (a bundle) in that we can assign to each set a choice. The set is fiber and the set of sets is the base. In an exact sequence, is a perspective the group or the homomorphism? It is the group - it is a division of zero - where zero is everything. How are coordinate rings in algebraic geometry related to coordinate systems? Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that [[https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras | propagates equalities]]. The latter are chains of quadratic equations. 6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 diagonals In what sense does affine geometry not have a coordinate system? (And thus not have a notion of infinity?) Affine = local (no infinity). How do the 4 geometries (in terms of coordinate systems) relate to the 4 classical root systems? Why is projective geometry related to lines, sphere, projection, point at infinity? All you can do * with 0 coordinate system (affine) * with 1 coordinate system (projective) is reflection, * with 2 coordinates (conformal) is rotation and shear, (the origins match) * with 3 coordinates (symplectic) is dilation (scales change), squeeze (scales change), translation (origins move). The coordinate systems (0,1,2,3) separate the level and metalevel. Study the 6 transformations between these sets of coordinate systems. Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules. Why and how is Universal Hyperbolic Geometry related to conformal geometry? Three perspectives lets you define a coordinate system as a choice framework. How is exactness (the image of f1 matching the kernel of f2) related to perspectives, for example, the notion of the complement? * 1 coordinate system = 1 side of a triangle = Length. Shrinking the side can lead to a point - the two points become equal. This is like homotopy? * 2 coordinate systems = 2 sides of a triangle = Angle. Note that turning (rotating) one side around by 2 pi gets it back to where it was, and this is true for each 2 pi forwards and backwards. So by this equivalence we generate the integers Z as the winding numbers. * 3 coordinate systems = 3 sides of a triangle = Oriented area (the systems are ordered). What equivalence does this support and what does it yield? Is it related to e? Mobius transformations * Give a geometrical interpretation of e. AutomataTheory * There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices. Duality of vectors and covectors (linear functionals) - flipsides of composition - what is "inside" and "outside" the composer. Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole. Tensors stay free of a coordinate system and work with all of them. Geometry: Options for introducing a coordinate system (none, one, two, three). No coordinate system is the case of tensors. 2020 kovo 21 d., 18:02
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Pakeistos 982-986 eilutės iš
Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole. į:
Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole. Tensors stay free of a coordinate system and work with all of them. Geometry: Options for introducing a coordinate system (none, one, two, three). No coordinate system is the case of tensors. 2020 kovo 21 d., 17:57
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Pakeistos 980-982 eilutės iš
Duality of vectors and covectors (linear functionals) - flipsides of composition - what is "inside" and "outside" the composer. į:
Duality of vectors and covectors (linear functionals) - flipsides of composition - what is "inside" and "outside" the composer. Klein bottle (twisted torus) exhibits two kinds of duality: twist (inside/outside or not) and hole. 2020 kovo 21 d., 15:34
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Ištrintos 976-985 eilutės:
Pridėtos 979-980 eilutės:
Duality of vectors and covectors (linear functionals) - flipsides of composition - what is "inside" and "outside" the composer. 2020 kovo 21 d., 15:00
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Pakeistos 977-1020 eilutės iš
* Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. Exact sequences * [[https://en.wikipedia.org/wiki/Zig-zag_lemma | Zig-zag lemma]] relates to infinite revolutions along the three-cycle. Homology * Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? Map of Math * Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. * Vector spaces are basic. What is basic about scalars? They make possible proportionality. * Basic concepts are the ways of figuring things out. * Basic concept - orientation = parity. Perspective * Ker/Image - the kernel are those that can relate, that can take up the perspective * Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. * In studying perspective: How is homology used to prove the Brouwer fixed point theorem? Math Discovery - House of Knowledge * Consider how the four levels of geometry-logic bring together the four levels of analysis and the four levels of algebra, yielding the 12 topologies. And why don't the two representations of the foursome yield a third representation? * Extension of a domain - [[https://en.wikipedia.org/wiki/Analytic_continuation | Analytic continuation]] - complex numbers - dealing with divergent series. * Root systems relate two spheres - they relate two "sheets". Logic likewise relates two sheets: a sheet and a meta-sheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out. Duality * In the automata hierarchy, consider how to model duality of internal structure and external network. Music * [[http://www-personal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf | What is Mathematical Music Theory?]] Lie theory * Signal propagation - expansions. Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. * How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? * Relate the ways of breaking the duality of counting with the ways of fusing together the sides of a square to get a manifold. Lie groups Attach:ClassicalLieGroups.png Field * Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1. į:
Ištrintos 988-1021 eilutės:
Mathematical mind/brain unites logical language (left brain - logical reasoning) with spatial visual thinking (right brain - geometric intuition). Consider what this means as two parallel ways of thinking for the 4 logic-geometry levels in the ways of discovery. 2020 kovo 21 d., 14:44
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Pakeistos 1024-1048 eilutės iš
* The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. Algebra * studies particular structures and substructures Divisions * Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean? Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself. Complex numbers * 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations -i. * Given z1, z2, z3 in projective complex plane, there exists a unique Mobius transformation such that f(z1)=0, f(z2)=1, f(z3)=infinity. Note that there is a fourth symbol z, and they get paired: 0 and infinity, 1 and z. {$$f(z)=\frac{z-z_1}{z-z_3} \cdot \frac{z_2-z_3}{z_2-z_1} $$} * The Mobius transformation f(z) which sends f(0)=p, f(1)=r, f(infinity)=s is given by: {$$f(z)=\frac{zs(p-r) + p(r-s)}{z(p-r) + (r-s)} $$} The Pairing axiom can be defined from other Zermelo Fraenkel axioms. (How?) And what does that mean for my identification of the axioms with the ways of figuring things out? Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries - the rich symmetries tend solutions towards Ockham's razor. į:
Pakeistos 1028-1103 eilutės iš
Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited. If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle. Triality of the octonions * [[http://math.ucr.edu/home/baez/octonions/conway_smith/ | Baez's review of Conway and Smith]]: The octonions can be described not only as the vector representation of {$\text{Spin}(8)$}, but also the left-handed spinor representation and the right-handed spinor representation. This fact is called 'triality'. It has many amazing spinoffs, including structures like the exceptional Lie groups and the exceptional Jordan algebra, and the fact that supersymmetric string theory works best in 10-dimensional spacetime — fundamentally because {$8 + 2 = 10$}. * The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness? Logic * Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). Bott Periodicity * John Baez's comment about 4 dimensions being most interesting because half-way between 0 and 8. The Son turns around and reverses the Father. * How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition. Elliptic integrals * Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] Perspective * Think of the nullsome as the point at infinity (the eye that sees the perspective). And think of it going beyond itself and then returning back to itself, making a big circle around. And think of that circle as a discrete line with four places. Coordinate system * Coordinate system is that which has an origin, a zero. Symmetry * [[https://theoreticalatlas.wordpress.com/2014/01/10/categorifying-global-vs-local-symmetry/ | Global vs. Local symmetry]] Duality * Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). * Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. Eightsome * 0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? Restructuring * The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. Quadratic form * Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form - as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. Division of everything * Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra Bott periodicity * R nullsome * H twosome 1+i (different) j+ij (the same) * H+H threesome splits the twosome * (H2) foursome - internally doubles * (C4) fivesome * (R8) sixsome * R+R sevensome (dividing the nullsome into two perspectives) (S16) what would S mean? half of R? positive reals? Four geometries * Projective map - preserves a single line (what does that mean?) * Conformal map - preserves pairs of lines (the angle between them) * Symplectomorphism - preserves triplets of lines (their oriented volume), the noncollinearity of a triangle * In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. Lie theory * When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? Fourier series * Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? Symplectic geometry * Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. Number theory * [[http://pi.math.cornell.edu/~hatcher/TN/TNpage.html | Allen Hatcher. Topology of numbers]] į:
2020 kovo 21 d., 14:25
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Pakeistos 117-123 eilutės iš
Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? Projective geometry Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. į:
Duality * Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. Geometry * Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? Projective geometry * Homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. Variables * Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. 2020 kovo 21 d., 14:23
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Pakeistos 1058-1068 eilutės iš
Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) How can Think of the nullsome as the point at infinity (the eye that sees the perspective). And think of it going beyond itself and then returning back to itself, making a big circle around. And think of that circle as a discrete line with four places. į:
Logic * Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). Bott Periodicity * John Baez's comment about 4 dimensions being most interesting because half-way between 0 and 8. The Son turns around and reverses the Father. * How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition. Elliptic integrals * Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] Perspective * Think of the nullsome as the point at infinity (the eye that sees the perspective). And think of it going beyond itself and then returning back to itself, making a big circle around. And think of that circle as a discrete line with four places. Pakeistos 1115-1117 eilutės iš
Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? į:
Fourier series * Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? 2020 kovo 21 d., 14:05
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Pakeistos 1118-1121 eilutės iš
* Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. į:
* Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. Number theory * [[http://pi.math.cornell.edu/~hatcher/TN/TNpage.html | Allen Hatcher. Topology of numbers]] 2020 kovo 21 d., 13:37
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Pakeistos 241-246 eilutės iš
Vector bundles http://pi į:
Vector bundles * Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down. [[http://pi.math.cornell.edu/~hatcher/AT/AT.pdf | Allen Hatcher. Algebraic topology]]. [[http://pi.math.cornell.edu/~hatcher/AT/ATpage.html | Explanation]] Homotopy, homology, cohomology. We will show in Theorem 3.21 that a finite-dimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. Example 4.55: Bott Periodicity. [[http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf | Allen Hatcher. Vector Bundles and K-Theory. (Half-written).]] [[http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html | Explanation]] [[http://pi.math.cornell.edu/~hatcher/VBKT/VBKT-tc.html | Table of Contents]] Bott-periodicity. Pakeistos 250-251 eilutės iš
Algebra and geometry are linked by logic - intersections and unions make sense in both. į:
Logic * Algebra and geometry are linked by logic - intersections and unions make sense in both. 2020 kovo 21 d., 13:21
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Ištrinta 4 eilutė:
Pakeistos 7-18 eilutės iš
Symmetric functions * An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. * Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Finite fields * Study how turning į:
Analytic continuation * Understand analytic continuation. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}? * Learn how to extend the Gamma function to the complex numbers. Choice frameworks * Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? * Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? * Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. * Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But: * for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact. * for Cn cross-polytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (not-this). * Use "this" and "that" as unmarked opposites - conjugates. Composition algebra. * Doubling is related to duality. Pakeistos 32-34 eilutės iš
* edge = difference į:
Finite fields * Study how turning the counting around relates to cycles - finite fields. Pridėtos 48-52 eilutės:
Linear algebra * An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. * Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. * Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Pakeistos 56-69 eilutės iš
Choice frameworks * Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? * Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas * Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But: * for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact. * for Cn cross-polytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (not-this). * Use "this" and "that" as unmarked opposites - conjugates. į:
Polytopes * edge = difference Symmetric functions * What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? Pakeistos 1064-1065 eilutės iš
į:
Pakeistos 1067-1086 eilutės iš
Coordinate system Duality of nonrelevance in system 0=1 yields Compare The six ways of restructuring relate the Understand analytic extension. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}? į:
Coordinate system * Coordinate system is that which has an origin, a zero. Symmetry * [[https://theoreticalatlas.wordpress.com/2014/01/10/categorifying-global-vs-local-symmetry/ | Global vs. Local symmetry]] Duality * Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). * Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. Eightsome * 0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? Restructuring * The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. Quadratic form * Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form - as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. Division of everything * Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra Bott periodicity Pridėta 1101 eilutė:
Four geometries Pakeistos 1106-1109 eilutės iš
In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. į:
* In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. Lie theory * When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? Pakeistos 1115-1116 eilutės iš
Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. į:
Symplectic geometry * Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. 2020 kovo 19 d., 19:15
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Pakeistos 1094-1096 eilutės iš
Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? į:
Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? Notion of derivative is related to tangency, to tangent vector, is related to the normal of the normal, is related to duality, is related to the slack modeled by symplectic geometry. 2020 kovo 19 d., 12:43
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Pakeistos 1092-1094 eilutės iš
When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? į:
When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? Compare a Fourier series and a generating function. How is a single basis element isolated, calculated, in each case? 2020 kovo 19 d., 12:42
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Pakeistos 1090-1092 eilutės iš
In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. į:
In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. When do we have all three: orthogonal, unitary, symplectic forms? And in what sense do they relate the preservation of lines, angles, triangular areas? And how do they diverge? 2020 kovo 19 d., 11:57
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Pakeistos 1088-1090 eilutės iš
* Symplectomorphism - preserves triplets of lines (their oriented volume), the noncollinearity of a į:
* Symplectomorphism - preserves triplets of lines (their oriented volume), the noncollinearity of a triangle In physics, the constraints once-differentiable and twice-differentiable are interesting and not trivial, whereas thrice-differentiable is basically the same as infinitely differentiable. Compare this with perspective, perspective on perspective, and perspective on perspective on perspective. 2020 kovo 18 d., 11:12
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Pakeistos 1084-1088 eilutės iš
(S16) what would S mean? half of R? positive reals? į:
(S16) what would S mean? half of R? positive reals? * Projective map - preserves a single line (what does that mean?) * Conformal map - preserves pairs of lines (the angle between them) * Symplectomorphism - preserves triplets of lines (their oriented volume), the noncollinearity of a triangle 2020 kovo 16 d., 21:10
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Pakeistos 1074-1084 eilutės iš
Understand analytic extension. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}? į:
Understand analytic extension. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}? * R nullsome * H twosome 1+i (different) j+ij (the same) * H+H threesome splits the twosome * (H2) foursome - internally doubles * (C4) fivesome * (R8) sixsome * R+R sevensome (dividing the nullsome into two perspectives) (S16) what would S mean? half of R? positive reals? 2020 kovo 16 d., 21:06
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Pakeistos 1072-1074 eilutės iš
Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford į:
Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra Understand analytic extension. Can we think of it as cutting the plane into a spiral of width {$e^n-e^{n-1}$}? 2020 kovo 15 d., 22:27
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Pakeistos 1070-1072 eilutės iš
Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form - as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. į:
Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form - as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. Perspective:Division = real form:complex Lie algebra/group = real Clifford algebra / complex Clifford algebra 2020 kovo 15 d., 22:26
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Pakeistos 1068-1070 eilutės iš
The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. į:
The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form - as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens. 2020 kovo 14 d., 20:38
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Pakeistos 1066-1068 eilutės iš
Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. į:
Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. The six ways of restructuring relate the unconscious's continuum (large number) with the conscious's discrete (small number) and express the relations between the two. Consider these relations in the house of knowledge for mathematics, and how they relate to the six geometrical transformations. 2020 kovo 13 d., 14:51
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Pakeistos 1064-1066 eilutės iš
0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? į:
0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations. 2020 kovo 10 d., 20:24
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Pakeistos 1062-1064 eilutės iš
Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). į:
Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). 0=1 yields the zero ring. In what sense is this the collapse of a system? And how is it related to the finite field {$F_1$} ? 2020 kovo 08 d., 15:59
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Pakeistos 1060-1062 eilutės iš
[[https://math.stackexchange.com/questions/25455/why-should-i-care-about-adjoint-functors?rq=1 | Why should I care about adjoint functors]] į:
[[https://math.stackexchange.com/questions/25455/why-should-i-care-about-adjoint-functors?rq=1 | Why should I care about adjoint functors]] Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel). 2020 kovo 03 d., 10:07
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Pakeistos 1058-1060 eilutės iš
[[https://theoreticalatlas.wordpress.com/2014/01/10/categorifying-global-vs-local-symmetry/ | Global vs. Local symmetry]] į:
[[https://theoreticalatlas.wordpress.com/2014/01/10/categorifying-global-vs-local-symmetry/ | Global vs. Local symmetry]] [[https://math.stackexchange.com/questions/25455/why-should-i-care-about-adjoint-functors?rq=1 | Why should I care about adjoint functors]] 2020 kovo 03 d., 09:35
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Pakeistos 1056-1058 eilutės iš
Coordinate system is that which has an origin, a zero. į:
Coordinate system is that which has an origin, a zero. [[https://theoreticalatlas.wordpress.com/2014/01/10/categorifying-global-vs-local-symmetry/ | Global vs. Local symmetry]] 2020 vasario 29 d., 22:20
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Pakeistos 1052-1056 eilutės iš
Learn how to extend the Gamma function to the complex numbers. į:
Learn how to extend the Gamma function to the complex numbers. Think of the nullsome as the point at infinity (the eye that sees the perspective). And think of it going beyond itself and then returning back to itself, making a big circle around. And think of that circle as a discrete line with four places. Coordinate system is that which has an origin, a zero. 2020 vasario 29 d., 22:13
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Pakeistos 1050-1052 eilutės iš
Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] į:
Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] Learn how to extend the Gamma function to the complex numbers. 2020 vasario 29 d., 22:02
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Pakeista 1050 eilutė iš:
Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24}$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] į:
Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24})$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] 2020 vasario 29 d., 22:01
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Pakeistos 1048-1050 eilutės iš
How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. į:
How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition. Note in the Wikipedia article on [[https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function | Particular values of the gamma function]] the connection between {$\Gamma(\frac{n}{24}$} and [[https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind | Complete elliptic integrals of the first kind]] 2020 vasario 29 d., 21:56
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Pakeista 1048 eilutė iš:
How can the fact that {$\Gamma(\frac{1}{2})=\ į:
How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. 2020 vasario 29 d., 21:51
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Pakeistos 1046-1048 eilutės iš
Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). į:
Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). How can the fact that {$\Gamma(\frac{1}{2})=\frac{\sqrt{\pi}}{2}$} relate {$\pi$} and {$e$}? 2020 vasario 29 d., 16:09
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Pakeistos 1042-1046 eilutės iš
* The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness? į:
* The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness? John Baez's comment about 4 dimensions being most interesting because half-way between 0 and 8. The Son turns around and reverses the Father. Jordan algebra projections are "propositions". This makes sense with regard to the sevensome (logic) and the eightsome (the semiotic square). 2020 vasario 29 d., 14:55
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Pakeistos 1037-1042 eilutės iš
If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle. į:
If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle. Triality of the octonions * [[http://math.ucr.edu/home/baez/octonions/conway_smith/ | Baez's review of Conway and Smith]]: The octonions can be described not only as the vector representation of {$\text{Spin}(8)$}, but also the left-handed spinor representation and the right-handed spinor representation. This fact is called 'triality'. It has many amazing spinoffs, including structures like the exceptional Lie groups and the exceptional Jordan algebra, and the fact that supersymmetric string theory works best in 10-dimensional spacetime — fundamentally because {$8 + 2 = 10$}. * The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness? 2020 vasario 29 d., 13:14
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Pakeistos 1035-1037 eilutės iš
Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited. į:
Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited. If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle. 2020 vasario 29 d., 12:31
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Pakeista 1035 eilutė iš:
Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited. į:
Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited. 2020 vasario 29 d., 12:30
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Pakeistos 1033-1035 eilutės iš
Variables are where the direction is reversed as regards the four relationships between level and metalevel. Variables are, on the one hand, the conclusion of math's "brain", but on the other hand, they are the start of math's "mind". į:
Variables are where the direction is reversed as regards the four relationships between level and metalevel. Variables are, on the one hand, the conclusion of math's "brain", but on the other hand, they are the start of math's "mind". Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited. 2020 vasario 29 d., 11:27
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Pakeistos 1031-1033 eilutės iš
Mathematical mind/brain unites logical language (left brain - logical reasoning) with spatial visual thinking (right brain - geometric intuition). Consider what this means as two parallel ways of thinking for the 4 logic-geometry levels in the ways of discovery. į:
Mathematical mind/brain unites logical language (left brain - logical reasoning) with spatial visual thinking (right brain - geometric intuition). Consider what this means as two parallel ways of thinking for the 4 logic-geometry levels in the ways of discovery. Variables are where the direction is reversed as regards the four relationships between level and metalevel. Variables are, on the one hand, the conclusion of math's "brain", but on the other hand, they are the start of math's "mind". 2020 vasario 27 d., 15:39
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Pakeista 1031 eilutė iš:
Mathematical mind/brain unites logical language (left brain į:
Mathematical mind/brain unites logical language (left brain - logical reasoning) with spatial visual thinking (right brain - geometric intuition). Consider what this means as two parallel ways of thinking for the 4 logic-geometry levels in the ways of discovery. 2020 vasario 27 d., 15:00
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Pakeistos 1029-1031 eilutės iš
Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries - the rich symmetries tend solutions towards Ockham's razor. į:
Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries - the rich symmetries tend solutions towards Ockham's razor. Mathematical mind/brain unites logical language (left brain) with spatial visual thinking (right brain). Consider what this means as two parallel ways of thinking for the 4 logic-geometry levels in the ways of discovery. 2020 vasario 27 d., 14:02
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Pakeistos 1027-1029 eilutės iš
The Pairing axiom can be defined from other Zermelo Fraenkel axioms. (How?) And what does that mean for my identification of the axioms with the ways of figuring things out? į:
The Pairing axiom can be defined from other Zermelo Fraenkel axioms. (How?) And what does that mean for my identification of the axioms with the ways of figuring things out? Ockham's razor gets us to focus on the structures which are most basic in that they generate the richest symmetries - the rich symmetries tend solutions towards Ockham's razor. 2020 vasario 26 d., 18:48
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Pakeistos 1025-1027 eilutės iš
{$$f(z)=\frac{zs(p-r) + p(r-s)}{z(p-r) + (r-s)} $$} į:
{$$f(z)=\frac{zs(p-r) + p(r-s)}{z(p-r) + (r-s)} $$} The Pairing axiom can be defined from other Zermelo Fraenkel axioms. (How?) And what does that mean for my identification of the axioms with the ways of figuring things out? 2020 vasario 24 d., 12:04
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Pakeista 1023 eilutė iš:
* The Mobius transformation f(z) which į:
* The Mobius transformation f(z) which sends f(0)=p, f(1)=r, f(infinity)=s is given by: 2020 vasario 24 d., 12:02
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Pakeistos 1021-1025 eilutės iš
{$f(z)=\frac{z-z_1}{z-z_3} \cdot \frac{z_2-z_3}{z_2-z_1} $} į:
{$$f(z)=\frac{z-z_1}{z-z_3} \cdot \frac{z_2-z_3}{z_2-z_1} $$} * The Mobius transformation f(z) which send f(0)=p, f(1)=r, f(infinity)=s is given by: {$$f(z)=\frac{zs(p-r) + p(r-s)}{z(p-r) + (r-s)} $$} 2020 vasario 24 d., 11:38
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Pakeistos 1018-1021 eilutės iš
* 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations -i. į:
* 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations -i. * Given z1, z2, z3 in projective complex plane, there exists a unique Mobius transformation such that f(z1)=0, f(z2)=1, f(z3)=infinity. Note that there is a fourth symbol z, and they get paired: 0 and infinity, 1 and z. {$f(z)=\frac{z-z_1}{z-z_3} \cdot \frac{z_2-z_3}{z_2-z_1} $} 2020 vasario 24 d., 11:16
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Pakeistos 1015-1018 eilutės iš
Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself. į:
Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself. Complex numbers * 1/z swaps inside and outside as conjugates, just as it swaps counterclockwise rotations i with counterclockwise rotations -i. 2020 vasario 23 d., 22:05
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Pakeistos 1013-1015 eilutės iš
* Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean? į:
* Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean? Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself. 2020 vasario 21 d., 22:27
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Pridėtos 971-972 eilutės:
* Basic concepts are the ways of figuring things out. * Basic concept - orientation = parity. 2020 vasario 21 d., 22:23
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Pridėtos 62-63 eilutės:
* Orthogonal add a perspective (Father), symplectic subtract a perspective (Son). * My dream: Sartre wrote a book "Space as World" where he has a formula that expresses Bott periodicity / my eightfold wheel of divisions. 2020 vasario 21 d., 22:21
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Pakeistos 1006-1009 eilutės iš
* studies particular structures and į:
* studies particular structures and substructures Divisions * Function relates many dimensions (the perspectives) to one dimension (the whole), just like a division. The whole is given by the operation +1. And what do +2 and +3 mean? 2020 vasario 21 d., 21:58
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Pakeistos 986-987 eilutės iš
Lie į:
Lie theory * Signal propagation - expansions. Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. Pridėta 989 eilutė:
* Relate the ways of breaking the duality of counting with the ways of fusing together the sides of a square to get a manifold. 2020 vasario 21 d., 21:51
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Pakeistos 61-62 eilutės iš
į:
* What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]]. Pakeistos 71-79 eilutės iš
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement. Massless particles acquire mass through symmetry breaking: į:
Rotation * Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. * If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. * This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. Physics * Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement. * Massless particles acquire mass through symmetry breaking: Pakeistos 80-87 eilutės iš
Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]]. į:
* [[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we'll understand entanglement a lot better. Octonions * Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. 2020 vasario 21 d., 21:47
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Pakeistos 37-40 eilutės iš
į:
* {$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative. * Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu? * How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. Pakeistos 51-66 eilutės iš
Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix? {$x_0$} is fundamentally different from {$x_i$} Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu? How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. Bott periodicity is the basis for 8-fold folding and unfolding. Use "this" and "that" as unmarked opposites - conjugates. į:
* Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. Pakeistos 56-58 eilutės iš
į:
* Use "this" and "that" as unmarked opposites - conjugates. Bott periodicity * Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix? * Bott periodicity is the basis for 8-fold folding and unfolding. Pridėta 980 eilutė:
* Root systems relate two spheres - they relate two "sheets". Logic likewise relates two sheets: a sheet and a meta-sheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out. 2020 vasario 21 d., 21:42
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Pakeistos 5-17 eilutės iš
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo? į:
Complex analysis * Differentiability of a complex function means that it can be written as an infinite power series. So differentiation reduces complex functions to infinite power series. This is analogous to evolution abstracting the "real world" to a representation of it. And differentiation is relevant as a shift from the known to focus on the unknown - the change. Symmetric functions * What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? Linear algebra * An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. * Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. * Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. Finite fields * Study how turning the counting around relates to cycles - finite fields. Counting * Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo? Pakeistos 25-51 eilutės iš
Special linear group has determinant 1. In general when the determinant is +/- 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse - neither is Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows. {$A_n$} tracefree condition is similar Composition algebra. Doubling is related to duality. Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? An relates to "center of mass". How does this relate to the asymmetry of whole and center? Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu? Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe į:
Polytopes * edge = difference Lie theory * Special linear group has determinant 1. In general when the determinant is +/- 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse - neither is distinguished. * Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows. * {$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. * Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? * An relates to "center of mass". How does this relate to the asymmetry of whole and center? * Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. * Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu? * Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe. Linear functionals * One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? Composition algebra. * Doubling is related to duality. Choice frameworks * Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? * Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? * Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. 2020 vasario 21 d., 21:38
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Ištrintos 4-11 eilutės:
[[http://www-personal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf | What is Mathematical Music Theory?]] Signal propagation - expansions * Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. * How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? Pakeistos 810-838 eilutės iš
* Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1. Extension of a domain * [[https://en.wikipedia.org/wiki/Analytic_continuation | Analytic continuation]] - complex numbers - dealing with divergent series. Skaičius 5 * Golden mean is the "most" irrational of numbers (based on its continued fraction). Consider series of continued fractions... as sequence patterns... * Susieti most "irrationality" su "randomness". Nes ką sužinai nieko daugiau nepasako apie kas liko. Skaičius 24 * John Baez kalba. 24 = 6 (trikampių laukas) x 4 (kvadrato laukas). * 24 + 2 = 26. Dievo šokis (žmogaus trejybės naryje) veikia ant žmogaus (už šokio) tad žmogus papildo šokį dviem matais. Ir gaunasi "group action". Susiję su Monster group. * Monster group dydis susijęs su visatos dalelyčių skaičiumi? Ypatingi skaičiai * http://math.ucr.edu/home/baez/42.html * http://math.ucr.edu/home/baez/numbers/ Vector spaces are basic. What is basic about scalars? They make possible proportionality. AutomataTheory - There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices. Combinatorics * The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. Algebra * studies particular structures and substructures į:
Pakeistos 814-815 eilutės iš
Prime numbers: "Cost function". The "cost" of a number may be thought of as the sum of all of its prime factors. What might this reveal about the primes? į:
Number theory * Prime numbers: "Cost function". The "cost" of a number may be thought of as the sum of all of its prime factors. What might this reveal about the primes? Pakeistos 818-819 eilutės iš
https://en.m.wikipedia.org/wiki/Field_with_one_element į:
Field with one element * https://en.m.wikipedia.org/wiki/Field_with_one_element Pakeistos 978-979 eilutės iš
į:
* Vector spaces are basic. What is basic about scalars? They make possible proportionality. Pakeistos 987-988 eilutės iš
į:
* Extension of a domain - [[https://en.wikipedia.org/wiki/Analytic_continuation | Analytic continuation]] - complex numbers - dealing with divergent series. Pakeistos 990-1014 eilutės iš
* In the automata hierarchy, consider how to model duality of internal structure and external network. į:
* In the automata hierarchy, consider how to model duality of internal structure and external network. Music * [[http://www-personal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf | What is Mathematical Music Theory?]] Lie algebras - Signal propagation - expansions * Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. * How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? Lie groups Attach:ClassicalLieGroups.png Field * Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1. AutomataTheory * There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices. Combinatorics * The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. Algebra * studies particular structures and substructures 2020 vasario 21 d., 21:10
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Pakeistos 998-1015 eilutės iš
Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. [[https://en.wikipedia.org/wiki/Zig-zag_lemma | Zig-zag lemma]] relates to infinite revolutions along the three-cycle. In studying perspective: Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does Basic concept Ker/Image - the kernel are those that can relate, that can take up the perspective * Pamąstyti kaip geometrijos-logikos ketverybė suveda analizės ketverybę ir algebros ketverybę, gaunasi 12 aplinkybių. Ir kodėl du ketverybių atvaizdai nesukuria trečio atvaizdo į:
Mobius transformations * Give a geometrical interpretation of e. Geometry * Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. Exact sequences * [[https://en.wikipedia.org/wiki/Zig-zag_lemma | Zig-zag lemma]] relates to infinite revolutions along the three-cycle. Homology * Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? Map of Math * Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. Perspective * Ker/Image - the kernel are those that can relate, that can take up the perspective * Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. * In studying perspective: How is homology used to prove the Brouwer fixed point theorem? Math Discovery - House of Knowledge * Consider how the four levels of geometry-logic bring together the four levels of analysis and the four levels of algebra, yielding the 12 topologies. And why don't the two representations of the foursome yield a third representation? Duality 2020 vasario 21 d., 20:11
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Pakeistos 1012-1016 eilutės iš
Ker/Image - the kernel are those that can relate, that can take up the į:
Ker/Image - the kernel are those that can relate, that can take up the perspective * Pamąstyti kaip geometrijos-logikos ketverybė suveda analizės ketverybę ir algebros ketverybę, gaunasi 12 aplinkybių. Ir kodėl du ketverybių atvaizdai nesukuria trečio atvaizdo? * In the automata hierarchy, consider how to model duality of internal structure and external network. 2020 vasario 21 d., 14:36
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Pakeistos 1010-1012 eilutės iš
Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. į:
Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. Ker/Image - the kernel are those that can relate, that can take up the perspective 2020 vasario 21 d., 13:53
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Pakeistos 1008-1010 eilutės iš
Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? į:
Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? Basic concept: Orientation (of a simplex). Relates to determinant, homology, etc. 2020 vasario 21 d., 13:32
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Pakeista 1008 eilutė iš:
Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. į:
Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. How does the boundary express orientation? 2020 vasario 21 d., 13:19
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Pakeistos 1006-1008 eilutės iš
In studying perspective: How is homology used to prove the Brouwer fixed point theorem? į:
In studying perspective: How is homology used to prove the Brouwer fixed point theorem? Interpret the boundary of a simplicial complex. Explain how the boundary of a boundary of a simplicial complex is zero. 2020 vasario 21 d., 12:42
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Pakeistos 1004-1006 eilutės iš
[[https://en.wikipedia.org/wiki/Zig-zag_lemma | Zig-zag lemma]] relates to infinite revolutions along the three-cycle. į:
[[https://en.wikipedia.org/wiki/Zig-zag_lemma | Zig-zag lemma]] relates to infinite revolutions along the three-cycle. In studying perspective: How is homology used to prove the Brouwer fixed point theorem? 2020 vasario 21 d., 12:39
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Pakeistos 1002-1004 eilutės iš
Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. į:
Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. [[https://en.wikipedia.org/wiki/Zig-zag_lemma | Zig-zag lemma]] relates to infinite revolutions along the three-cycle. 2020 vasario 21 d., 12:01
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Pakeista 1002 eilutė iš:
Homology groups measure how far a chain complex is from being an exact sequence. į:
Relate to perspectives: Homology groups measure how far a chain complex is from being an exact sequence. 2020 vasario 21 d., 12:00
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Pakeistos 1000-1002 eilutės iš
Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. į:
Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. Homology groups measure how far a chain complex is from being an exact sequence. 2020 vasario 20 d., 18:44
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Pakeistos 998-1000 eilutės iš
Give a geometrical interpretation of e. į:
Give a geometrical interpretation of e. Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}. 2020 vasario 14 d., 21:37
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Pakeistos 992-998 eilutės iš
How is exactness (the image of f1 matching the kernel of f2) related to perspectives, for example, the notion of the complement? į:
How is exactness (the image of f1 matching the kernel of f2) related to perspectives, for example, the notion of the complement? * 1 coordinate system = 1 side of a triangle = Length. Shrinking the side can lead to a point - the two points become equal. This is like homotopy? * 2 coordinate systems = 2 sides of a triangle = Angle. Note that turning (rotating) one side around by 2 pi gets it back to where it was, and this is true for each 2 pi forwards and backwards. So by this equivalence we generate the integers Z as the winding numbers. * 3 coordinate systems = 3 sides of a triangle = Oriented area (the systems are ordered). What equivalence does this support and what does it yield? Is it related to e? Give a geometrical interpretation of e. 2020 vasario 14 d., 20:18
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Pakeistos 990-992 eilutės iš
Three perspectives lets you define a coordinate system as a choice framework. į:
Three perspectives lets you define a coordinate system as a choice framework. How is exactness (the image of f1 matching the kernel of f2) related to perspectives, for example, the notion of the complement? 2020 vasario 14 d., 14:41
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Pakeista 984 eilutė iš:
į:
The coordinate systems (0,1,2,3) separate the level and metalevel. Study the 6 transformations between these sets of coordinate systems. 2020 vasario 14 d., 13:22
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Pakeistos 986-990 eilutės iš
Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules. į:
Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules. Why and how is Universal Hyperbolic Geometry related to conformal geometry? Three perspectives lets you define a coordinate system as a choice framework. 2020 vasario 14 d., 13:21
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Pridėtos 977-986 eilutės:
Why is projective geometry related to lines, sphere, projection, point at infinity? All you can do * with 0 coordinate system (affine) * with 1 coordinate system (projective) is reflection, * with 2 coordinates (conformal) is rotation and shear, (the origins match) * with 3 coordinates (symplectic) is dilation (scales change), squeeze (scales change), translation (origins move). Study the 6 transformations between these sets of coordinate systems. Tensor: function in all possible coordinate spaces such that it (its values) obey certain transformation rules. 2020 vasario 14 d., 12:57
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Pridėtos 974-975 eilutės:
How do the 4 geometries (in terms of coordinate systems) relate to the 4 classical root systems? 2020 vasario 13 d., 23:35
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Pakeistos 971-974 eilutės iš
6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 į:
6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 diagonals In what sense does affine geometry not have a coordinate system? (And thus not have a notion of infinity?) Affine = local (no infinity). 2020 vasario 13 d., 13:45
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Pakeistos 969-971 eilutės iš
Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that [[https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras | propagates equalities]]. The latter are chains of quadratic equations. į:
Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that [[https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras | propagates equalities]]. The latter are chains of quadratic equations. 6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 diagonals 2020 vasario 13 d., 13:30
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Pakeista 969 eilutė iš:
Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that į:
Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that [[https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras | propagates equalities]]. The latter are chains of quadratic equations. 2020 vasario 13 d., 11:48
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Pakeistos 967-969 eilutės iš
How are coordinate rings in algebraic geometry related to coordinate systems? į:
How are coordinate rings in algebraic geometry related to coordinate systems? Homology calculations involve systems of equations. They propagate equalities. Compare this with the determinant of the Cartan matrix of the classical Lie groups, how that propagates equalities. 2020 vasario 12 d., 22:37
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Pakeistos 965-967 eilutės iš
In an exact sequence, is a perspective the group or the homomorphism? It is the group - it is a division of zero - where zero is everything. į:
In an exact sequence, is a perspective the group or the homomorphism? It is the group - it is a division of zero - where zero is everything. How are coordinate rings in algebraic geometry related to coordinate systems? 2020 vasario 12 d., 21:04
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Pakeista 965 eilutė iš:
In an exact sequence, is a perspective the group or the homomorphism? į:
In an exact sequence, is a perspective the group or the homomorphism? It is the group - it is a division of zero - where zero is everything. 2020 vasario 12 d., 21:03
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Pakeistos 963-965 eilutės iš
The Axiom of Choice is based on the notions of a perspective (a bundle) in that we can assign to each set a choice. The set is fiber and the set of sets is the base. į:
The Axiom of Choice is based on the notions of a perspective (a bundle) in that we can assign to each set a choice. The set is fiber and the set of sets is the base. In an exact sequence, is a perspective the group or the homomorphism? 2020 vasario 11 d., 23:37
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Pakeistos 961-963 eilutės iš
Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time? į:
Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time? The Axiom of Choice is based on the notions of a perspective (a bundle) in that we can assign to each set a choice. The set is fiber and the set of sets is the base. 2020 vasario 11 d., 21:50
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Pakeistos 959-961 eilutės iš
Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways. į:
Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways. Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time? 2020 vasario 09 d., 19:03
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Pakeistos 957-959 eilutės iš
Electrons are particles when we look at them and waves when we don't. į:
Electrons are particles when we look at them and waves when we don't. Quantum mechanics superposition of states may be based on borrowing of energy. When the energy debt is too high, as when it gets entangled with an observer, then the debt has to be paid. Otherwise, the debt can be restructured in complicated ways. 2020 vasario 09 d., 18:55
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Pakeistos 955-957 eilutės iš
Coordinate systems are observers and they shouldn't affect what they observe. (Relate this to the kinds of polytopes.) į:
Coordinate systems are observers and they shouldn't affect what they observe. (Relate this to the kinds of polytopes.) Electrons are particles when we look at them and waves when we don't. 2020 vasario 09 d., 09:51
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Pakeistos 953-955 eilutės iš
There is a duality between a tensor and its expression under a particular basis. They are interchangeable. į:
There is a duality between a tensor and its expression under a particular basis. They are interchangeable. Coordinate systems are observers and they shouldn't affect what they observe. (Relate this to the kinds of polytopes.) 2020 vasario 09 d., 09:46
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Pridėtos 950-951 eilutės:
So a tensor is a bringing together of components, which can be either covariant or contravariant. Is this stepping out and stepping in? Is a tensor a division of everything and each component a perspective? 2020 vasario 09 d., 09:45
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Pakeistos 947-951 eilutės iš
How do observables relate to perspective? į:
How do observables relate to perspective? Tensors are invariant under linear transformations but their components do change. There is a duality between a tensor and its expression under a particular basis. They are interchangeable. 2020 vasario 08 d., 20:04
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Pakeistos 945-947 eilutės iš
SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will. į:
SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will. How do observables relate to perspective? 2020 vasario 08 d., 09:18
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Pakeistos 933-945 eilutės iš
Perspective relates intrinsic and extrinsic geometry by way of ambient space. Ambient space relates base and fiber (perspective) as a bundle. į:
Perspective relates intrinsic and extrinsic geometry by way of ambient space. Ambient space relates base and fiber (perspective) as a bundle. Lagrangian L=T-V expresses slack (or anti-slack), makes the conversion between potential and kinetic energy as smooth as possible. Hamiltonian H=T+V expresses the totality, the love. In this sense, they are dual, as per the sevensome - the Lagrangian expresses the internal slack, and the Hamiltonian expresses the wholeness of the external frame. Potential energy is bounded from below. Kinetic energy is always positive, absolutely. Its relation to momentum is absolute, unconditional. Whereas potential is defined relatively and its relation to position is variable. The wave equation is defined on phase space but in such a way that it is understood in terms of a superposition of waves for position, yielding a "blob" - a wave packet, and an analogous superposition of waves for momentum, and the two are related by the Fourier transform, by the Heisenberg uncertainty principle. Physics abstracts from the personality of other researchers. Have a common ground, communicate not on any consensus, but based on what we find, as independent witnesses. SL(2,C) models the transformation in the relationship between two wills: human's and God's. The complex variable describes the will. 2020 vasario 08 d., 09:05
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Pakeistos 931-933 eilutės iš
The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence. į:
The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence. Perspective relates intrinsic and extrinsic geometry by way of ambient space. Ambient space relates base and fiber (perspective) as a bundle. 2020 vasario 06 d., 23:19
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Pakeistos 927-931 eilutės iš
Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a three-cycle, what it is relating, a self to itself. į:
Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a three-cycle, what it is relating, a self to itself. In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding. The derivative of an infinite power sequence shows that it is related to counting because we get coefficients 1, 2, 3, 4 etc. for the generating sequence. 2020 vasario 06 d., 21:39
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Pakeistos 925-927 eilutės iš
The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid. į:
The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid. Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a three-cycle, what it is relating, a self to itself. 2020 vasario 06 d., 15:48
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Pakeista 925 eilutė iš:
The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. į:
The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid. 2020 vasario 06 d., 15:43
atliko -
Pakeista 925 eilutė iš:
The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. į:
The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. 2020 vasario 06 d., 15:43
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Pakeistos 923-925 eilutės iš
In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons. į:
In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons. The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. 2020 vasario 06 d., 12:24
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Pakeistos 921-923 eilutės iš
Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5. į:
Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5. In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons. 2020 vasario 04 d., 09:56
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Pakeistos 919-921 eilutės iš
Integral of 1/x is ln x what does that say about ex? į:
Integral of 1/x is ln x what does that say about ex? Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5. 2020 vasario 04 d., 00:23
atliko - 2020 vasario 03 d., 19:38
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Pakeistos 917-919 eilutės iš
Why is negative curvature - the curvature inside - more prominent than positive curvature - the outside of a space? į:
Why is negative curvature - the curvature inside - more prominent than positive curvature - the outside of a space? Integral of 1/x is ln x what does that say about ex? 2020 vasario 03 d., 18:07
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Pakeistos 913-917 eilutės iš
A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series. į:
A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series. Does the inside of a sphere have negative curvature, and the outside of sphere have positive curvature? And likewise the inside and outside of a torus? Why is negative curvature - the curvature inside - more prominent than positive curvature - the outside of a space? 2020 vasario 03 d., 17:43
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Pakeista 913 eilutė iš:
A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. į:
A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. Preservation of angles implies existence of Taylor series. 2020 vasario 03 d., 17:41
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Pakeistos 911-913 eilutės iš
Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness. į:
Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness. A function from the complex plane to the complex plane which preserves angles (of intersecting curves) necessarily is analytic. This seems related to the fact that exponentiation {$e^{i\theta}$} makes multiplication additive. And that brings to mind trigonometric functions. 2020 sausio 29 d., 18:36
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Pridėtos 909-911 eilutės:
An grows at both ends, either grows independently, no center, no perspective, affine. Bn, Cn have both ends grow dependently, pairwise, so it is half the freedom. In what sense does Dn grow, does it double the possibilities for growth? And does Dn do that internally by relating xi+xj and xi-xj variously somehow? Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness. 2020 sausio 29 d., 18:19
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Pakeistos 905-908 eilutės iš
Homology and cohomology are like the relation between 0->1->2->3 and 4->5->6->7. į:
Homology and cohomology are like the relation between 0->1->2->3 and 4->5->6->7. Substantiate: Affine geometry defines no perspective, projective geometry defines one perspective, conformal geometry defines a perspective on a perspective, symplectic geometry defines a perspective on a perspective on a perspective. 2020 sausio 28 d., 13:35
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Pridėtos 903-905 eilutės:
Bundle. Geometry relates analysis (continuum) and algebra (discrete) as a restructuring. When the discrete grows large does it become a continuum? Homology and cohomology are like the relation between 0->1->2->3 and 4->5->6->7. 2020 sausio 28 d., 13:30
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Pakeistos 899-902 eilutės iš
From dream: vectors A-B, B-A, consider the difference between them, the equivalence of A and B. į:
From dream: vectors A-B, B-A, consider the difference between them, the equivalence of A and B. Conjugates can be thought of as "twins", whereas +1 and -1 are "spouses". 2020 sausio 28 d., 13:29
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Pakeistos 895-899 eilutės iš
Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics. į:
Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics. From dream: Space is made up of all possible curves. Physics is about the geodesics, the curves with no slack. They really are all one curve that goes through every curve and every single point. From dream: vectors A-B, B-A, consider the difference between them, the equivalence of A and B. 2020 sausio 27 d., 18:33
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Pakeistos 891-895 eilutės iš
[[https://en.wikipedia.org/wiki/Arithmetic_derivative | Arithmetic derivative]] į:
[[https://en.wikipedia.org/wiki/Arithmetic_derivative | Arithmetic derivative]] ---------------------------- Much of my work may be considered pre-mathematics, the grounding of the structures that ultimately allow for mathematics. 2020 sausio 27 d., 17:29
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Pakeistos 801-891 eilutės iš
* [[https://www.math.upenn.edu/~wilf/gfology2.pdf | Generatingfunctionology]] by Herbert į:
* [[https://www.math.upenn.edu/~wilf/gfology2.pdf | Generatingfunctionology]] by Herbert Wilf ------------------------------------- Mathematics is the study of structure. It is the study of systems, what it means to live in them, and where and how and why they fail or not. homology - holes - what is not there - thus a topic for explicit vs. implicit math svarbūs pavyzdžiai * https://en.m.wikipedia.org/wiki/Möbius_transformation Symmetry * Representations of the symmetric group. Symmetric - homogeneous - bosons - vectors. Antisymmetric - elementary - fermion - covectors. Euclidean space allows reflection to define inside and outside nonproblematically, thus antisymmetricity. Free vector space. Schur functions combine symmetric and antisymmetric in rows and columns. * E8 is the symmetry group of itself. What is the symmetry group of? * Meilė (simetrija) įsteigia nemirtingumą (invariant). Field * Sieja nepažymėtą priešingybę (+) -1, 0, 1 ir pažymėtą priešingybę (x) -1, 1. Extension of a domain * [[https://en.wikipedia.org/wiki/Analytic_continuation | Analytic continuation]] - complex numbers - dealing with divergent series. Skaičius 5 * Golden mean is the "most" irrational of numbers (based on its continued fraction). Consider series of continued fractions... as sequence patterns... * Susieti most "irrationality" su "randomness". Nes ką sužinai nieko daugiau nepasako apie kas liko. Skaičius 24 * John Baez kalba. 24 = 6 (trikampių laukas) x 4 (kvadrato laukas). * 24 + 2 = 26. Dievo šokis (žmogaus trejybės naryje) veikia ant žmogaus (už šokio) tad žmogus papildo šokį dviem matais. Ir gaunasi "group action". Susiję su Monster group. * Monster group dydis susijęs su visatos dalelyčių skaičiumi? Ypatingi skaičiai * http://math.ucr.edu/home/baez/42.html * http://math.ucr.edu/home/baez/numbers/ Vector spaces are basic. What is basic about scalars? They make possible proportionality. AutomataTheory - There is a qualitative hierarchy of computing devices. And they can be described in terms of matrices. Combinatorics * The mathematics of counting (and generating) objects. I think of this as the basement of mathematics, which is why I studied it. Algebra * studies particular structures and substructures Neural networks * Very powerful and simple computational systems for which Sarunas Raudys showed a hierarchy of sophistication as learning systems. Prime numbers: "Cost function". The "cost" of a number may be thought of as the sum of all of its prime factors. What might this reveal about the primes? * [[http://oeis.org/A000607 | Number of ways to partition a number into primes]]. https://en.m.wikipedia.org/wiki/Field_with_one_element [[Catalan]], Mandelbrot, Julia sets * Totally independent dimensions: Cartesian * Totally dependent dimensions: simplex We "understand" the simplex in Cartesian dimensions but think it in simplex dimensions. We can imagine a simplex in higher dimensions by simply adding externally instead of internally. Consider (implicit + explicit)to infinity; and also (unlabelled + labelled) to infinity. Also consider (unlabelable + labelable). And (definitively labeled + definitively unlabeled). * Gaussian binomial coefficients [[http://math.stackexchange.com/questions/214065/proving-q-binomial-identities | interpretation related to Young tableaux]] '''Pagrindiniai matematikos dėsniai''' Kaip [[http://www.selflearners.net/Math/DeepIdeas | matematikos pagrindus]] pristatyti svarbiausiais dėsniais, pavyzdžiais ir žaidimais? Kuo žaidimai yra vertingi, kaip jie suveikia? Kuriu atitinkamas mokymosi priemones, tapau drobę. Prisiminti savo matematikos mokymo dėsnius: * every answer is an amount and a unit ir tt. * combine like units * list different units * a right triangle is half of a rectangle * a triangle is the sum of two right triangles * four times a right triangle is the difference of two squares * extending the domain * purposes of families of functions Basic division rings: [[http://math.ucr.edu/home/baez/week59.html | John Baez 59]] * The real numbers are not of characteristic 2, * so the complex numbers don't equal their own conjugates, * so the quaternions aren't commutative, * so the octonions aren't associative, * so the hexadecanions aren't a division algebra. '''Įdomūs, prasmingi reiškiniai matematikoje''' Polynomial powers are "twists" of a string. One end of the string is held up and then down. Each twist of the string allows for a new maximum or minimum. This is an interpretation of multiplication. Logika * Logikos prielaidos bene slypi daugybė matematikos sąvokų. Pavyzdžiui, begalybė slypi galimybėje tą patį logikos veiksnį taikyti neribotą skaičių kartų. O tai slypi tame kad logika veiksnio taikymas nekeičia loginės aplinkos. * Ar gali būti, kad aibių teorijos problemos (pavyzdžiui, sau nepriklausanti aibė) yra iš tikrųjų logikos problemos? Kaip atskirti aibių teoriją ir logiką (pavyzdžiui, axiom of choice)? [[https://en.wikipedia.org/wiki/Arithmetic_derivative | Arithmetic derivative]] 2020 sausio 27 d., 13:27
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Pakeistos 799-801 eilutės iš
Pascal's triangle tilted gives Fibonacci į:
Pascal's triangle tilted gives Fibonacci numbers * [[https://www.math.upenn.edu/~wilf/gfology2.pdf | Generatingfunctionology]] by Herbert Wilf 2020 sausio 27 d., 12:59
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Pakeistos 793-799 eilutės iš
* ''This defines a functor G→Top that extends along the Yoneda embedding, yielding a geometric representation of any globular set Gˆ→Top.'' į:
* ''This defines a functor G→Top that extends along the Yoneda embedding, yielding a geometric representation of any globular set Gˆ→Top.'' https://oeis.org/A013609 triangle for hypercubes. (1 + 2x)^n unsigned coefficients of chebyshev polynomials of the second kind e to the matrix consisting of the natural numbers on its first off-diagonal gives a triangular matrix with pascal's triangle. and how is it in the case of the cube? Pascal's triangle tilted gives Fibonacci numbers 2020 sausio 27 d., 12:47
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Pakeistos 786-793 eilutės iš
Bundle = restructuring (base = continuum -> fiber = discrete). A number that is "large enough" can essentially model the continuum. į:
Bundle = restructuring (base = continuum -> fiber = discrete). A number that is "large enough" can essentially model the continuum. Origami: [[http://alum.mit.edu/www/tchow/multifolds.pdf | The power of multifolds: Folding the algebraic closure of the rational numbers]] What do "globe", "globe category", "globular set" mean? * [[https://math.stackexchange.com/questions/2371364/whats-the-canonical-embedding-of-the-globe-category-into-top | What's the canonical embedding of the globe category into Top?]] * ''They are the maps of the closed n-ball to the "northern" and "southern" hemispheres of the surface of the (n+1)-ball.'' * ''This defines a functor G→Top that extends along the Yoneda embedding, yielding a geometric representation of any globular set Gˆ→Top.'' 2020 sausio 27 d., 12:35
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Pakeista 721 eilutė iš:
* {$B_n$} {$\ į:
* {$B_n$} {$\pm x,\pm y$} yields {$x\pm y$} Pakeista 723 eilutė iš:
* {$D_n$} {$\pm(x\ į:
* {$D_n$} {$\pm(x\pm y)$} dual dual Pakeistos 782-786 eilutės iš
[[http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf | A Cohomological Viewpoint on Elementary School Arithmetic]] About "carrying". Access restricted. į:
[[http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf | A Cohomological Viewpoint on Elementary School Arithmetic]] About "carrying". Access restricted. Consider how math variously expresses perspectives. Bundle = restructuring (base = continuum -> fiber = discrete). A number that is "large enough" can essentially model the continuum. 2020 sausio 26 d., 21:49
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Pakeistos 780-782 eilutės iš
Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. į:
Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. [[http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf | A Cohomological Viewpoint on Elementary School Arithmetic]] About "carrying". Access restricted. 2020 sausio 26 d., 21:22
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Pakeistos 780-782 eilutės iš
Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors The calculus world is the "exponential" of the discrete world. One of the reasons that Lie groups and Lie algebras are important is because they link together the "calculus world" (Lie groups are "differentiable manifolds") and the "discrete world" (Lie algebras are based on "root systems" that are geometric reflections) į:
Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. 2020 sausio 26 d., 19:21
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Pakeistos 780-782 eilutės iš
Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. į:
Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. The calculus world is the "exponential" of the discrete world. One of the reasons that Lie groups and Lie algebras are important is because they link together the "calculus world" (Lie groups are "differentiable manifolds") and the "discrete world" (Lie algebras are based on "root systems" that are geometric reflections). 2020 sausio 26 d., 18:13
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Pakeistos 778-780 eilutės iš
Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. So what is a pattern? And what are the limitations on patterns? į:
Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. So what is a pattern? And what are the limitations on patterns? Prime numbers introduce determinism in that when a prime divides a product, then it must divide one of the factors. 2020 sausio 24 d., 15:08
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Pakeista 778 eilutė iš:
Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. į:
Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. So what is a pattern? And what are the limitations on patterns? 2020 sausio 24 d., 15:08
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Pakeistos 776-778 eilutės iš
Primes are both atoms and gaps. And they elucidate the gap of all numbers with respect to infinity. į:
Primes are both atoms and gaps. And they elucidate the gap of all numbers with respect to infinity. Riemann hypothesis: is the unfolding of the primes "pattern free"? Or are there one or more hidden patterns there? A zero of the Riemann zeta function indicates a pattern. 2020 sausio 24 d., 14:51
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Pakeista 776 eilutė iš:
Primes are both atoms and gaps. į:
Primes are both atoms and gaps. And they elucidate the gap of all numbers with respect to infinity. 2020 sausio 24 d., 14:50
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Pakeistos 774-776 eilutės iš
Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the Cauchy-Riemann equations, etc. į:
Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the Cauchy-Riemann equations, etc. Primes are both atoms and gaps. 2020 sausio 24 d., 13:40
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Pakeistos 772-774 eilutės iš
SL(2) - H,X,Y is 3-dimensional (?) but SU(2) - three-cycle + 1 (God) is 4 dimensional. Is there a discrepancy, and why? į:
SL(2) - H,X,Y is 3-dimensional (?) but SU(2) - three-cycle + 1 (God) is 4 dimensional. Is there a discrepancy, and why? Note that x and y axes are separated by 90 degrees. This is the grounds for the degree four of i, the trigonometric functions, the Cauchy-Riemann equations, etc. 2020 sausio 23 d., 22:21
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Pakeistos 770-772 eilutės iš
The ordinal (list) is deterministic whereas the cardinal (set) is nondeterministic. į:
The ordinal (list) is deterministic whereas the cardinal (set) is nondeterministic. SL(2) - H,X,Y is 3-dimensional (?) but SU(2) - three-cycle + 1 (God) is 4 dimensional. Is there a discrepancy, and why? 2020 sausio 20 d., 21:28
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Pridėtos 764-770 eilutės:
(polynomial coefficient) Ordinal/List/Analysis vs. Cardinal/Set/Algebra (polynomial root) - the coefficients and roots are related by the binomial theorem, the factors are choices. geometry (bundle) links algebra (fiber) with analysis (base) and the latter manifold is also understood (ambiguously) algebraically as a Lie group. Algebra is a (finite) cognitive pattern that restructures the (infinite) Analysis, the model of the world. Together they are a restructuring. In the house of knowledge for mathematics, the three-cycle relates analysis and algebra as structuring and restructuring. Thus this restructuring (the six restructurings) is the output of the house of knowledge and the content of mathematics, its branches, concepts, statements, problems, etc. Analysis is the infinity of sheets, the recurring sequence of not going beyond oneself. But algebra is the single sheet which is the self that it all goes into, where all of the actions, all of the sheets coincide as one sheet, one going beyond. Thus the cardinal (of algebra) arises from the ordinal (of analysis). The ordinal (list) is deterministic whereas the cardinal (set) is nondeterministic. 2020 sausio 20 d., 07:42
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Pakeistos 760-763 eilutės iš
Video: Ben Mares: introduction to cohomology. į:
Video: Ben Mares: introduction to cohomology. SL(2,C) lines (plus infinity) become circles. Do linear equations become circular equations? What does that mean? Are SL(2,C) circular equations related to the continuum? 2020 sausio 16 d., 21:22
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Pakeistos 758-760 eilutės iš
Odd cohomology works like fermions, even cohomology works like bosons. į:
Odd cohomology works like fermions, even cohomology works like bosons. Video: Ben Mares: introduction to cohomology. 2020 sausio 16 d., 21:19
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Pakeistos 756-758 eilutės iš
[[https://www.amazon.com/Differential-Algebraic-Topology-Graduate-Mathematics/dp/0387906134 | Bott & Tu. Differential Forms in Algebraic Topology.]] į:
[[https://www.amazon.com/Differential-Algebraic-Topology-Graduate-Mathematics/dp/0387906134 | Bott & Tu. Differential Forms in Algebraic Topology.]] Odd cohomology works like fermions, even cohomology works like bosons. 2020 sausio 16 d., 20:25
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Pakeistos 754-756 eilutės iš
{$2\pi$} additive factors, e multiplicative factors. į:
{$2\pi$} additive factors, e multiplicative factors. [[https://www.amazon.com/Differential-Algebraic-Topology-Graduate-Mathematics/dp/0387906134 | Bott & Tu. Differential Forms in Algebraic Topology.]] 2020 sausio 16 d., 17:18
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Pakeistos 748-754 eilutės iš
Relate bundle concepts to amounts and units. į:
Relate bundle concepts to amounts and units. Relate motion to bundles. Symplectic geometry, looseness, etc. All 4 geometries. A geometry (like hyperbolic geometry) allows for a presentation of a bundle, thus a perspective on a perspective (atsitokėjimas - atvaizdas). Compare with: įsijautimas-aplinkybė. {$2\pi$} additive factors, e multiplicative factors. 2020 sausio 16 d., 13:45
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Pakeistos 746-748 eilutės iš
Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions. į:
Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions. Relate bundle concepts to amounts and units. 2020 sausio 16 d., 12:30
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Pakeistos 744-746 eilutės iš
Generating functions relate: symmetry of analytic functions, algorithms, finite combinatorial symmetry. (Think of as a vector bundle - the infinite sequence ({$x_i$}) is the base space, and the coefficient is the fiber, and the fibers are related.) į:
Generating functions relate: symmetry of analytic functions, algorithms, finite combinatorial symmetry. (Think of as a vector bundle - the infinite sequence ({$x_i$}) is the base space, and the coefficient is the fiber, and the fibers are related.) Try to express the symmetries of an object, like a polyhedron, in terms of bundle conceptions. 2020 sausio 16 d., 12:16
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Pakeistos 742-744 eilutės iš
4 kinds of i: the Pauli matrices and i itself. į:
4 kinds of i: the Pauli matrices and i itself. Generating functions relate: symmetry of analytic functions, algorithms, finite combinatorial symmetry. (Think of as a vector bundle - the infinite sequence ({$x_i$}) is the base space, and the coefficient is the fiber, and the fibers are related.) 2020 sausio 16 d., 11:55
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Pakeistos 738-742 eilutės iš
Linear operators - something you can have more of or less of (proportionately) - transformational action (like rotation or translation). į:
Linear operators - something you can have more of or less of (proportionately) - transformational action (like rotation or translation). When are two vectors, lines, etc. perpendicular? 4 kinds of i: the Pauli matrices and i itself. 2020 sausio 15 d., 19:22
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Pakeistos 736-738 eilutės iš
{$K_0$} and {$K_1$} perhaps express perspectives, like God's trinity and the three-cycle, the two foursomes in the eight-cycle. į:
{$K_0$} and {$K_1$} perhaps express perspectives, like God's trinity and the three-cycle, the two foursomes in the eight-cycle. Linear operators - something you can have more of or less of (proportionately) - transformational action (like rotation or translation). 2020 sausio 13 d., 14:44
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Pakeistos 734-736 eilutės iš
Normal bundles involve embedding in an extrinsic space. į:
Normal bundles involve embedding in an extrinsic space. {$K_0$} and {$K_1$} perhaps express perspectives, like God's trinity and the three-cycle, the two foursomes in the eight-cycle. 2020 sausio 13 d., 12:22
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Pakeistos 732-734 eilutės iš
Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality? į:
Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality? Normal bundles involve embedding in an extrinsic space. 2020 sausio 06 d., 20:40
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Pakeista 732 eilutė iš:
Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? į:
Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? What imposition is made on duality? 2020 sausio 06 d., 20:40
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Pakeistos 726-732 eilutės iš
Explain: (B + iC)(B - iC) = (C + iB)(C - iB) į:
Explain: (B + iC)(B - iC) = (C + iB)(C - iB) Intuit SL(2,C) as three-dimensional in C (because ad-bc=1 so we lose one complex dimension - intuit that). And in what sense is that different from ad-bc=0 (a line? a one-dimensional subspace?) SL(2,C) is the spin relativistic group. Investigate: In what sense do the properties of being an inverse (Cramer's rule) dictate the symmetry of SL(2,C)? Because the inverse has to maintain the same form. So why do the b, c switch sign and the a, d switch places? 2020 sausio 05 d., 22:11
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Pridėtos 3-4 eilutės:
[++++数学笔记++++] Ištrintos 5-6 eilutės:
[++++数学笔记++++] 2020 sausio 01 d., 14:29
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Pakeistos 722-724 eilutės iš
In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math? į:
In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math? Explain: (B + iC)(B - iC) = (C + iB)(C - iB) 2019 gruodžio 31 d., 22:48
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Pakeistos 715-722 eilutės iš
Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?) į:
Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?) Think of harmonic pencil types as the basis for the root systems * {$A_n$} {$\pm(x-y)$} dual * {$B_n$} {$\pmx,\pmy$} yields {$x\pm y$} * {$C_n$} {$x\pm y$} yields {$\pm2x,\pm2y$} * {$D_n$} {$\pm(x\pmy)$} dual dual In what sense do these ground four geometries? And how do 6 pairs relate to ways of figuring things out in math? 2019 gruodžio 30 d., 21:56
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Pakeistos 713-715 eilutės iš
Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases? į:
Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases? Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?) 2019 gruodžio 30 d., 12:43
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Pakeistos 711-713 eilutės iš
Relate the levels of the foursome with the cross-ratio (for example, how-what are the two points within a circle, and why-whether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the cross-ratio upon permuting its elements. į:
Relate the levels of the foursome with the cross-ratio (for example, how-what are the two points within a circle, and why-whether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the cross-ratio upon permuting its elements. Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases? 2019 gruodžio 30 d., 12:35
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Pakeistos 709-711 eilutės iš
Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself. į:
Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself. Relate the levels of the foursome with the cross-ratio (for example, how-what are the two points within a circle, and why-whether are the two points outside.) And likewise relate the six pairs of four levels with geometric concepts, the six lines that relate the four points of an inscribed quadrilateral, or the six possible values of the cross-ratio upon permuting its elements. 2019 gruodžio 30 d., 12:25
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Pakeistos 707-709 eilutės iš
* The circle maps every point to a line and vice versa. į:
* The circle maps every point to a line and vice versa. Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself. 2019 gruodžio 30 d., 10:39
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Pakeistos 704-707 eilutės iš
[[https://en.wikipedia.org/wiki/J-invariant | J-invariant]] is related to SL(2,Z) and monstrous moonshine. į:
[[https://en.wikipedia.org/wiki/J-invariant | J-invariant]] is related to SL(2,Z) and monstrous moonshine. Universal hyperbolic geometry * The circle maps every point to a line and vice versa. 2019 gruodžio 28 d., 21:33
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Pakeistos 702-704 eilutės iš
Are there 6+4 branches of math? How are the branches of math related to the ways of figuring things out? į:
Are there 6+4 branches of math? How are the branches of math related to the ways of figuring things out? [[https://en.wikipedia.org/wiki/J-invariant | J-invariant]] is related to SL(2,Z) and monstrous moonshine. 2019 gruodžio 28 d., 17:51
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Ištrintos 698-702 eilutės:
Classify mathematical constants * [[https://en.wikipedia.org/wiki/Category:Mathematical_constants | Wikipedia: Category: Mathematical constants]] * [[https://en.wikipedia.org/wiki/Mathematical_constant | Wikipedia: Mathematical constant]] * [[https://en.wikipedia.org/wiki/List_of_mathematical_constants | List of mathematical constants]] 2019 gruodžio 28 d., 16:31
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Pakeistos 705-707 eilutės iš
A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. į:
A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. The center of a circle is perhaps a fourth point (with every triplet) much like the identity is related to the three-cycle? Are there 6+4 branches of math? How are the branches of math related to the ways of figuring things out? 2019 gruodžio 28 d., 15:27
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Pakeistos 703-705 eilutės iš
* [[https://en.wikipedia.org/wiki/List_of_mathematical_constants | List of mathematical constants]] į:
* [[https://en.wikipedia.org/wiki/List_of_mathematical_constants | List of mathematical constants]] A circle (through polarity) defines triplets of points, and triplets of lines, thus sixsomes. 2019 gruodžio 28 d., 14:06
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Pakeistos 698-703 eilutės iš
* Determine whether {$\pi^e$} is rational or irrational. į:
* Determine whether {$\pi^e$} is rational or irrational. Classify mathematical constants * [[https://en.wikipedia.org/wiki/Category:Mathematical_constants | Wikipedia: Category: Mathematical constants]] * [[https://en.wikipedia.org/wiki/Mathematical_constant | Wikipedia: Mathematical constant]] * [[https://en.wikipedia.org/wiki/List_of_mathematical_constants | List of mathematical constants]] 2019 gruodžio 28 d., 13:41
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Pakeistos 694-698 eilutės iš
Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle). į:
Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle). Challenge problems: * Determine whether {$\pi + e$} is rational or irrational. * Determine whether {$\pi^e$} is rational or irrational. 2019 gruodžio 28 d., 13:04
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Pakeistos 692-694 eilutės iš
Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. In this way, quadrance and spread eliminate false distinctions and the problems they cause. į:
Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. In this way, quadrance and spread eliminate false distinctions and the problems they cause. Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle). 2019 gruodžio 27 d., 22:54
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Pakeista 692 eilutė iš:
Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. į:
Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. In this way, quadrance and spread eliminate false distinctions and the problems they cause. 2019 gruodžio 27 d., 22:53
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Pakeistos 690-692 eilutės iš
In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}. į:
In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}. Quadrance (distance squared) is more correct than distance because quadrance makes positive distance and negative distance equivalent. Spread (absolute value of the sine of angle) is more correct than angle because the value of the spread is the same for all angles at an intersection, which is to say, for both theta and pi minus theta. 2019 gruodžio 25 d., 20:04
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Pakeista 690 eilutė iš:
In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{frac{H}{kT}}$}, so here we likewise have the quantumization factor {${frac{H}{kT}}$}. į:
In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{\frac{H}{kT}}$}, so here we likewise have the quantumization factor {${\frac{H}{kT}}$}. 2019 gruodžio 25 d., 20:04
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Pakeistos 688-690 eilutės iš
{$\frac{\text{d}}{\text{dt}}\Psi=-i\frac{H}{\hbar}\Psi$} į:
{$\frac{\text{d}}{\text{dt}}\Psi=-i\frac{H}{\hbar}\Psi$} In statistics, the probability that a system at a given temperature T is in a given microstate is proportional to {$e^{frac{H}{kT}}$}, so here we likewise have the quantumization factor {${frac{H}{kT}}$}. 2019 gruodžio 25 d., 18:22
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Pakeista 688 eilutė iš:
{$\frac{\text{d}}{\text{dt}}\Psi=i\frac{H}{\hbar}\Psi$} į:
{$\frac{\text{d}}{\text{dt}}\Psi=-i\frac{H}{\hbar}\Psi$} 2019 gruodžio 25 d., 18:21
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Pakeista 688 eilutė iš:
{$frac{\text{d}}{\text{dt}}\Psi= į:
{$\frac{\text{d}}{\text{dt}}\Psi=i\frac{H}{\hbar}\Psi$} 2019 gruodžio 25 d., 18:21
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Pakeista 688 eilutė iš:
{$frac{text{d}}{text{dt}}\Psi=ifrac{H}{\hbar}\Psi$} į:
{$frac{\text{d}}{\text{dt}}\Psi=ifrac{H}{\hbar}\Psi$} 2019 gruodžio 25 d., 18:20
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Pakeistos 686-688 eilutės iš
Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy). į:
Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy). {$frac{text{d}}{text{dt}}\Psi=ifrac{H}{\hbar}\Psi$} 2019 gruodžio 25 d., 18:18
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Pakeistos 684-686 eilutės iš
Look for what it would mean for a ratio to be {$i$} and the product to be -1. į:
Look for what it would mean for a ratio to be {$i$} and the product to be -1. Schroedinger's equation relates position and momentum through complex number i. Slack in one (as given by derivative) is given by the value of the other times the ratio of the Hamiltonian over Planck's constant (the available quantum slack as given by the energy). 2019 gruodžio 23 d., 12:52
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Pakeistos 682-684 eilutės iš
Relate harmonic ranges and harmonic pencils to Lie algebras and to the restatement of {$x_i-x_j$} in terms of {$x_i$}. į:
Relate harmonic ranges and harmonic pencils to Lie algebras and to the restatement of {$x_i-x_j$} in terms of {$x_i$}. Look for what it would mean for a ratio to be {$i$} and the product to be -1. 2019 gruodžio 23 d., 12:50
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Pakeista 680 eilutė iš:
Proceed from balance - note how ratio precedes anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. į:
Proceed from balance - note how additive balance precedes multiplicative ratio precedes possibly negative (directional) ratio precedes anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. 2019 gruodžio 23 d., 12:49
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Pakeista 680 eilutė iš:
Proceed from balance - note how anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. į:
Proceed from balance - note how ratio precedes anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. 2019 gruodžio 23 d., 12:41
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Pakeistos 680-682 eilutės iš
Proceed from balance - note how anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. į:
Proceed from balance - note how anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. Relate harmonic ranges and harmonic pencils to Lie algebras and to the restatement of {$x_i-x_j$} in terms of {$x_i$}. 2019 gruodžio 23 d., 12:26
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Pakeistos 678-680 eilutės iš
Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Cross-ratio | anharmonic group]] and Mobius transformations. Note that the anharmonic group is based on composition of functions. į:
Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Cross-ratio | anharmonic group]] and Mobius transformations. Note that the anharmonic group is based on composition of functions. Proceed from balance - note how anharmonic ratio precedes cross-ratio - and see how balance gets variously extended, as with the anharmonic group and the Moebius transformations. And relate that to the Balance as a way of figuring things out. 2019 gruodžio 23 d., 10:53
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Pakeista 678 eilutė iš:
Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Cross-ratio | anharmonic group]] and Mobius transformations. į:
Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Cross-ratio | anharmonic group]] and Mobius transformations. Note that the anharmonic group is based on composition of functions. 2019 gruodžio 23 d., 10:46
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Pakeistos 676-678 eilutės iš
į:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of {$\frac{\pi}{n}$}, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. Relate God's dance to {0, 1, ∞} and the [[https://en.wikipedia.org/wiki/Cross-ratio | anharmonic group]] and Mobius transformations. 2019 gruodžio 15 d., 23:33
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Pakeista 676 eilutė iš:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of pi į:
2019 gruodžio 15 d., 23:33
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Pakeista 676 eilutė iš:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x į:
Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x}{n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of pi/n, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. 2019 gruodžio 15 d., 23:32
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Pakeistos 674-676 eilutės iš
* Ar savybių visuma yra simpleksas? Ar savybės skaidomos (koordinačių sistema). į:
* Ar savybių visuma yra simpleksas? Ar savybės skaidomos (koordinačių sistema). Math Companion section on Exponentiation. The natural way to think of {$e^x$} is the limit of {$(1 + \frac{x,n})^n$}. When x is complex, {$\pi i$}, then multiplication of y has the effect of a linear combination of n tiny rotations of pi/n, so it is linear around a circle, bringing us to -1 or all the way around to 1. When x is real, then multiplication of y has the effect of nonlinearly resizing it by the limit, and the resulting e is halfway to infinity. 2019 gruodžio 12 d., 22:34
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Pakeistos 669-674 eilutės iš
Determinant expresses oriented volume, oriented area. Real numbers: distances. Complex numbers: angles. What do quaternions express? į:
Determinant expresses oriented volume, oriented area. Real numbers: distances. Complex numbers: angles. What do quaternions express? * Homologija bandyti išsakyti persitvarkymų tarpą tarp pirminės ir antrinės tvarkos. * B_>C ..... How->What * External relations -> Internal logic .... (Not What=Why) Hom C -> Hom B (Not How=Whether) * Ar savybių visuma yra simpleksas? Ar savybės skaidomos (koordinačių sistema). 2019 lapkričio 23 d., 20:56
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Pakeistos 667-669 eilutės iš
For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the three-cycle, so they yield the foursome: 0, 120, 240, 360. į:
For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the three-cycle, so they yield the foursome: 0, 120, 240, 360. Determinant expresses oriented volume, oriented area. Real numbers: distances. Complex numbers: angles. What do quaternions express? 2019 lapkričio 23 d., 20:55
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Pakeista 667 eilutė iš:
For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the three-cycle, so they yield the foursome. į:
For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the three-cycle, so they yield the foursome: 0, 120, 240, 360. 2019 lapkričio 23 d., 20:54
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Pakeistos 665-667 eilutės iš
The possibilities for a complex plane (extra point for sphere, point removed for cylinder) are relevant for modeling perspectives. į:
The possibilities for a complex plane (extra point for sphere, point removed for cylinder) are relevant for modeling perspectives. For complex numbers, {$0 \neq 2\pi$} and so they are different when we go around the three-cycle, so they yield the foursome. 2019 lapkričio 21 d., 11:35
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Pakeistos 663-665 eilutės iš
[[https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham | Are there other nice math books close to the style of Tristan Needham?]] į:
[[https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham | Are there other nice math books close to the style of Tristan Needham?]] The possibilities for a complex plane (extra point for sphere, point removed for cylinder) are relevant for modeling perspectives. 2019 lapkričio 20 d., 14:32
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Pakeistos 661-663 eilutės iš
[[https://smile.amazon.com/dp/144192681X/ref=smi_www_rco2_go_smi_3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Naive Lie Theory]] by John Stillwell į:
[[https://smile.amazon.com/dp/144192681X/ref=smi_www_rco2_go_smi_3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Naive Lie Theory]] by John Stillwell įsigyti [[https://mathoverflow.net/questions/31879/are-there-other-nice-math-books-close-to-the-style-of-tristan-needham | Are there other nice math books close to the style of Tristan Needham?]] 2019 lapkričio 20 d., 14:29
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Pakeistos 659-661 eilutės iš
How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? į:
How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? [[https://smile.amazon.com/dp/144192681X/ref=smi_www_rco2_go_smi_3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Naive Lie Theory]] by John Stillwell įsigyti 2019 lapkričio 19 d., 14:42
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Pakeistos 657-659 eilutės iš
Why these three structures? How do they relate to the Moebius transformations? And how do these structures relate to the classical Lie families? į:
Why these three structures? How do they relate to the Moebius transformations? And how do these structures relate to the classical Lie families? How is homotopy and its [0,1]x[0,1] square related to the complex plane? and to category theory square for composition of functors? 2019 lapkričio 19 d., 10:11
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Pakeistos 651-655 eilutės iš
{$\mathrm{Aut}_{ {$\mathrm{Aut}_{ {$\mathrm{Aut}_{ į:
{$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} {$\mathrm{Aut}_{H\circ f}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$} {$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2019 lapkričio 19 d., 10:10
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Pakeistos 655-657 eilutės iš
{$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} į:
{$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} Why these three structures? How do they relate to the Moebius transformations? And how do these structures relate to the classical Lie families? 2019 lapkričio 19 d., 10:09
atliko - 2019 lapkričio 19 d., 09:48
atliko -
Pakeista 655 eilutė iš:
{$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R})$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} į:
{$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2019 lapkričio 19 d., 09:48
atliko -
Pakeista 655 eilutė iš:
{$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\ į:
{$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R})$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2019 lapkričio 19 d., 09:47
atliko -
Pakeistos 651-655 eilutės iš
{$\mathrm{Aut}_{Hof}(\mathbb į:
{$\mathrm{Aut}_{Hof}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} {$\mathrm{Aut}_{Hof}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$} {$\mathrm{Aut}_{Hof}(\mathbb{C})\simeq \mathrm{P}\delta(2,\mathbb{R})$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$} 2019 lapkričio 19 d., 09:45
atliko -
Pakeistos 649-651 eilutės iš
[[http://www.math.tifr.res.in/~pablo/download/book/book.html | Riemann Surfaces Book]], Pablo Arés į:
[[http://www.math.tifr.res.in/~pablo/download/book/book.html | Riemann Surfaces Book]], Pablo Arés Gastesi {$\mathrm{Aut}_{Hof}(\mathbb(P)^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$} 2019 lapkričio 19 d., 09:16
atliko -
Pridėtos 647-649 eilutės:
[[https://en.wikipedia.org/wiki/Riemann_surface | Riemann surface]] [[http://www.math.tifr.res.in/~pablo/download/book/book.html | Riemann Surfaces Book]], Pablo Arés Gastesi 2019 lapkričio 19 d., 09:13
atliko -
Pridėtos 644-645 eilutės:
Relate walks on trees with fundamental group. 2019 lapkričio 10 d., 18:58
atliko -
Pakeistos 638-644 eilutės iš
In the house of knowledge for mathematics, the three-cycle establishes the substructures for the symmetry group. Similarly, in physics, it establishes the scales for isolating a system (?) į:
In the house of knowledge for mathematics, the three-cycle establishes the substructures for the symmetry group. Similarly, in physics, it establishes the scales for isolating a system (?) Lie algebra matrix representations code for: * Sequences - simple roots * Trees - positive roots * Networks - all roots 2019 lapkričio 10 d., 09:12
atliko -
Pakeistos 636-638 eilutės iš
Real line models separation (by cutting) and connectedness (by continuity). The separating cuts become locations (points) in their own right. į:
Real line models separation (by cutting) and connectedness (by continuity). The separating cuts become locations (points) in their own right. In the house of knowledge for mathematics, the three-cycle establishes the substructures for the symmetry group. Similarly, in physics, it establishes the scales for isolating a system (?) 2019 lapkričio 07 d., 17:19
atliko -
Pakeistos 634-636 eilutės iš
In Lie groups, real paramater subgroups (copies of R) are important because they define arcwise connectedness, that we can move from one point (group element) to another continuously. See how this relates to the slack defined by the root system. į:
In Lie groups, real paramater subgroups (copies of R) are important because they define arcwise connectedness, that we can move from one point (group element) to another continuously. See how this relates to the slack defined by the root system. Real line models separation (by cutting) and connectedness (by continuity). The separating cuts become locations (points) in their own right. 2019 lapkričio 05 d., 17:27
atliko -
Pakeistos 632-634 eilutės iš
* the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and į:
* the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and infinity In Lie groups, real paramater subgroups (copies of R) are important because they define arcwise connectedness, that we can move from one point (group element) to another continuously. See how this relates to the slack defined by the root system. 2019 lapkričio 04 d., 19:33
atliko -
Pakeistos 625-632 eilutės iš
The empty set, or the center, has dimension -1. į:
The empty set, or the center, has dimension -1. Notions of dimension d (Mathematical Companion): * locally looks like d-dimensional space * the barrier between any two points is never more than d-1 dimensional * can be covered with sets such that no more than d+1 of them ever overlap * the largest d such that there is a nontrivial map from a d-dimensional manifold to a substructure of the space * the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and infinity 2019 lapkričio 04 d., 17:54
atliko -
Pakeistos 623-625 eilutės iš
[[https://smile.amazon.com/Functions-Complex-Variable-Graduate-Mathematics/dp/0387903283/ref=smi_www_rco2_go_smi_g3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Functions of One Complex Variable]] John į:
[[https://smile.amazon.com/Functions-Complex-Variable-Graduate-Mathematics/dp/0387903283/ref=smi_www_rco2_go_smi_g3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Functions of One Complex Variable]] John Conway The empty set, or the center, has dimension -1. 2019 spalio 31 d., 18:36
atliko -
Pakeistos 621-623 eilutės iš
Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations. į:
Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations. [[https://smile.amazon.com/Functions-Complex-Variable-Graduate-Mathematics/dp/0387903283/ref=smi_www_rco2_go_smi_g3905707922?_encoding=UTF8&%2AVersion%2A=1&%2Aentries%2A=0&ie=UTF8 | Functions of One Complex Variable]] John Conway 2019 spalio 31 d., 17:31
atliko -
Pakeistos 619-621 eilutės iš
* [[https://en.wikipedia.org/wiki/Electroweak_interaction | Electroweak interaction]] į:
* [[https://en.wikipedia.org/wiki/Electroweak_interaction | Electroweak interaction]] Mobius transformations can be composed from translations, dilations, inversions. But dilations (by complex numbers) could be understood as dilations (in positive reals), reflections, and rotations. 2019 spalio 30 d., 09:38
atliko -
Pakeistos 615-619 eilutės iš
How to relate Lie algebras and groups by way of the Taylor series of the logarithm? į:
How to relate Lie algebras and groups by way of the Taylor series of the logarithm? Understand the relations between U(1) and electromagnetism, SU(2) and the weak force, SU(3) and the strong force. * [[https://en.wikipedia.org/wiki/Standard_Model | Standard Model]] * [[https://en.wikipedia.org/wiki/Electroweak_interaction | Electroweak interaction]] 2019 spalio 25 d., 10:18
atliko -
Pakeistos 613-615 eilutės iš
Consider how A_2 is variously interpreted as a unitary, orthogonal and symplectic structure. į:
Consider how A_2 is variously interpreted as a unitary, orthogonal and symplectic structure. How to relate Lie algebras and groups by way of the Taylor series of the logarithm? 2019 spalio 24 d., 23:03
atliko -
Pridėtos 612-613 eilutės:
Consider how A_2 is variously interpreted as a unitary, orthogonal and symplectic structure. 2019 spalio 24 d., 15:48
atliko -
Pakeistos 609-611 eilutės iš
Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? į:
Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? Analyze the Mobius group in terms of what it does to circles and lines, and analyze the transformations likewise. Reconsider what Shu-Hong's thesis has to say about fractions of differences, and how they relate to the Mobius group. 2019 spalio 24 d., 14:56
atliko -
Pridėtos 599-600 eilutės:
The nonexistent element of {$F_1$} may be considered to not exist, or imagined to exist, but regardless, I expect that cognitively there are three ways to interpret it as 0, 1, ∞, which thereby expand upon the duality between existence and nonexistence and make it structurally richer. 2019 spalio 24 d., 14:51
atliko -
Pakeista 607 eilutė iš:
Need to define "internal". į:
Need to define "internal". Also, note that reflection is like rotation but more specific. Similarly, is translation like shear, but more specific? 2019 spalio 24 d., 14:41
atliko -
Pakeistos 598-607 eilutės iš
Note that this relates pairs from: 1, z, z-1. į:
Note that this relates pairs from: 1, z, z-1. Note that the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group | Mobius transformations]] classify into types which accord with my six transformations: * reflection = circular * shear = parabolic * rotation = elliptic * dilation = hyperbolic * squeeze = internal of hyperbolic (e^t e^{-t}=1) * translation = internal of parabolic Need to define "internal". 2019 spalio 24 d., 14:25
atliko -
Pakeistos 596-598 eilutės iš
Relate [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group | elliptic transforms]] to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: 1 / z į:
Relate [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group | elliptic transforms]] to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: {$1 / z$}, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), {$1 − z$} which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and {$z / ( z − 1 )$} which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). Note that this relates pairs from: 1, z, z-1. 2019 spalio 24 d., 14:22
atliko -
Pakeistos 594-596 eilutės iš
Study [[https://en.wikipedia.org/wiki/SL2(R) | SL(2,R)]]. Study the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation | Möbius group]]. Understand [[https://math.stackexchange.com/questions/646183/list-of-connected-lie-subgroups-of-mathrmsl2-mathbbc | the study of subgroups]]. į:
Study [[https://en.wikipedia.org/wiki/SL2(R) | SL(2,R)]]. Study the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation | Möbius group]]. Understand [[https://math.stackexchange.com/questions/646183/list-of-connected-lie-subgroups-of-mathrmsl2-mathbbc | the study of subgroups]]. Relate [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Subgroups_of_the_M%C3%B6bius_group | elliptic transforms]] to God's dance {0, 1, ∞} and {$F_1$}: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: 1 / z , {\displaystyle 1/z,} 1/z, which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), 1 − z {\displaystyle 1-z} 1-z, which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and z / ( z − 1 ) {\displaystyle z/(z-1)} z/(z-1) which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). 2019 spalio 24 d., 12:00
atliko -
Pakeistos 592-594 eilutės iš
Look at effect of Lie group's subgroup on a vector. (Shear? Dilation?) and relate to the 6 transformations. į:
Look at effect of Lie group's subgroup on a vector. (Shear? Dilation?) and relate to the 6 transformations. Study [[https://en.wikipedia.org/wiki/SL2(R) | SL(2,R)]]. Study the [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation | Möbius group]]. Understand [[https://math.stackexchange.com/questions/646183/list-of-connected-lie-subgroups-of-mathrmsl2-mathbbc | the study of subgroups]]. 2019 spalio 23 d., 22:57
atliko -
Pakeistos 590-592 eilutės iš
I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll. į:
I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll. Look at effect of Lie group's subgroup on a vector. (Shear? Dilation?) and relate to the 6 transformations. 2019 spalio 22 d., 12:14
atliko -
Pakeistos 588-590 eilutės iš
Jacob Lurie, Bachelor's thesis, [[http://www.math.harvard.edu/~lurie/papers/thesis.pdf | On Simply Laced Lie Algebras and Their Minuscule Representations]] į:
Jacob Lurie, Bachelor's thesis, [[http://www.math.harvard.edu/~lurie/papers/thesis.pdf | On Simply Laced Lie Algebras and Their Minuscule Representations]] I dreamed of the complex numbers as a line that curls, winds, rolls up in one way on one end, and in the mirror opposite way on the other end, like rolling up a carpet from both ends. Like a scroll. 2019 spalio 16 d., 20:27
atliko -
Pakeistos 586-588 eilutės iš
The complex Lie algebra divine threesome H, X, Y is an abstraction. The real Lie algebra human cyclical threesome is an outcome of the representation in terms of numbers and matrices, the expression of duality in terms of -1, i, and position. į:
The complex Lie algebra divine threesome H, X, Y is an abstraction. The real Lie algebra human cyclical threesome is an outcome of the representation in terms of numbers and matrices, the expression of duality in terms of -1, i, and position. Jacob Lurie, Bachelor's thesis, [[http://www.math.harvard.edu/~lurie/papers/thesis.pdf | On Simply Laced Lie Algebras and Their Minuscule Representations]] 2019 spalio 14 d., 12:54
atliko -
Pakeistos 584-586 eilutės iš
Relate the monomial, forgotten, Schur symmetric functions of eigenvalues with the matrices {$(I-A)^{-1}$} and {$e^A$}. į:
Relate the monomial, forgotten, Schur symmetric functions of eigenvalues with the matrices {$(I-A)^{-1}$} and {$e^A$}. The complex Lie algebra divine threesome H, X, Y is an abstraction. The real Lie algebra human cyclical threesome is an outcome of the representation in terms of numbers and matrices, the expression of duality in terms of -1, i, and position. 2019 spalio 13 d., 14:31
atliko -
Pridėtos 581-582 eilutės:
where in each case two of three are applied: flipping, multiplying by i, multiplying by -1. 2019 spalio 13 d., 14:30
atliko -
Pridėtos 577-580 eilutės:
Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} -y \\ -x \end{pmatrix}$} {$\begin{pmatrix} ix \\ -iy \end{pmatrix}$} 2019 spalio 13 d., 14:25
atliko -
Pakeistos 569-576 eilutės iš
{$\begin{pmatrix} \pm \sqrt{1 + a_{12}a{21}} & a_12 \\ a_21 & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} į:
{$\begin{pmatrix} \pm \sqrt{1 + a_{12}a_{21}} & a_{12} \\ a_{21} & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} such as {$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$} {$\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$} {$\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$} 2019 spalio 13 d., 14:23
atliko -
Pakeista 569 eilutė iš:
{$\begin{pmatrix} \pm sqrt{1 + a_{12}a{21}} & a_12 \\ a_21 & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} į:
{$\begin{pmatrix} \pm \sqrt{1 + a_{12}a{21}} & a_12 \\ a_21 & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} 2019 spalio 13 d., 14:23
atliko -
Pakeista 569 eilutė iš:
{$\begin{pmatrix} \pm sqrt{1 + a_{12}a{21}} & a_12 \\ a_21 & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$ į:
{$\begin{pmatrix} \pm sqrt{1 + a_{12}a{21}} & a_12 \\ a_21 & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} 2019 spalio 13 d., 14:22
atliko -
Pridėtos 566-569 eilutės:
so |A|=-1. {$\begin{pmatrix} \pm sqrt{1 + a_{12}a{21}} & a_12 \\ a_21 & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$) 2019 spalio 13 d., 14:11
atliko -
Pakeista 565 eilutė iš:
{$\lambda=\frac{a_{11}+a_{22} \pm sqrt{(a_{11}+a_{22})^2 - 4|A|}}{2}$} į:
{$\lambda=\frac{a_{11}+a_{22} \pm \sqrt{(a_{11}+a_{22})^2 - 4|A|}}{2}$} 2019 spalio 13 d., 14:10
atliko -
Pridėtos 559-560 eilutės:
Exercise: Get the eigenvalues for a generic matrix: 2x2, 3x3, etc. Pridėtos 562-565 eilutės:
Exercise: Find all matrices with eigenvalues 1 and -1. {$\lambda=\frac{a_{11}+a_{22} \pm sqrt{(a_{11}+a_{22})^2 - 4|A|}}{2}$} 2019 spalio 13 d., 14:05
atliko -
Pakeistos 559-561 eilutės iš
Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP? į:
Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP? Relate the monomial, forgotten, Schur symmetric functions of eigenvalues with the matrices {$(I-A)^{-1}$} and {$e^A$}. 2019 spalio 13 d., 13:58
atliko -
Pakeista 559 eilutė iš:
Exercise: į:
Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP? 2019 spalio 13 d., 13:55
atliko -
Pakeistos 555-559 eilutės iš
A root pair {$x$} and {$-x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. į:
A root pair {$x$} and {$-x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. In solving for eigenvalues {$\lambda_i$} and eigenvectors {$v_i$} of {$M$}, make the matrix {$M-\lambda I$} degenerate. Thus {$\text{det}(M-\lambda I)=0$}. The matrix is degenerate when one row is a linear combination of the other rows. So the determinant is a geometrical expression for volume, for collinearity and noncollinearity. Exercise: find the eigenvalues for a generic matrix. 2019 spalio 13 d., 13:17
atliko -
Pakeistos 553-555 eilutės iš
{$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. į:
{$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. A root pair {$x$} and {$-x$} yields directions such as "up" {$x––x$} and "down" {$–x–+x$}. 2019 spalio 13 d., 13:15
atliko -
Pakeista 553 eilutė iš:
{$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$frac{n(n+1)}{2}$}. į:
{$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$\frac{n(n+1)}{2}$}. 2019 spalio 13 d., 13:15
atliko -
Pakeistos 549-553 eilutės iš
Given a chain of composition {$\cdots f_{i-1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme. į:
Given a chain of composition {$\cdots f_{i-1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme. Root systems give the ways of composing perspectives-dimensions. {$A_n$} root system grows like 2,6,12,20, so the positive roots grow like 1,3,6,10 which is {$frac{n(n+1)}{2}$}. 2019 spalio 13 d., 11:17
atliko -
Pakeista 549 eilutė iš:
Given a chain of composition {$\cdots f_ į:
Given a chain of composition {$\cdots f_{i-1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards, stepping in or climbing out. There is the possiblity of switching adjacent functions at each dot. So each dot corresponds to a node in the Dynkin diagram. And the duality is affected by what happens at an extreme. 2019 spalio 13 d., 11:14
atliko -
Pakeistos 547-549 eilutės iš
{$120^\circ$} yields {$\frac{-1}{\sqrt{2}\sqrt{2}}=\frac{-1}{2}$} į:
{$120^\circ$} yields {$\frac{-1}{\sqrt{2}\sqrt{2}}=\frac{-1}{2}$} Given a chain of composition {$\cdots f_1{i-1}\circ f_i \circ f_{i+1} \cdots$} there is a duality as regards reading it forwards or backwards. 2019 spalio 13 d., 11:08
atliko -
Pakeista 547 eilutė iš:
{$120^\circ$} yields {$\frac{-1}{\sqrt{2}\sqrt{2}$} į:
{$120^\circ$} yields {$\frac{-1}{\sqrt{2}\sqrt{2}}=\frac{-1}{2}$} 2019 spalio 13 d., 11:07
atliko -
Pakeistos 545-547 eilutės iš
{$cos\theta = \frac{a\circ b}{\left \| a \right \|\left \| b \right \|}$} į:
{$cos\theta = \frac{a\circ b}{\left \| a \right \|\left \| b \right \|}$} {$120^\circ$} yields {$\frac{-1}{\sqrt{2}\sqrt{2}$} 2019 spalio 13 d., 11:04
atliko -
Pakeista 545 eilutė iš:
{$cos\theta = frac{a\circ b}{\left \| a \right \|\left \| b \right \|}$} į:
{$cos\theta = \frac{a\circ b}{\left \| a \right \|\left \| b \right \|}$} 2019 spalio 13 d., 11:04
atliko -
Pakeistos 541-545 eilutės iš
In {A_1}, the root {$x_2-x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. į:
In {A_1}, the root {$x_2-x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. If two roots are separated by more than {$90^\circ$}, then adding them together yields a new root. {$cos\theta = frac{a\circ b}{\left \| a \right \|\left \| b \right \|}$} 2019 spalio 13 d., 10:56
atliko -
Pakeistos 537-541 eilutės iš
Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. į:
Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. {$A_n$} is based on differences {$x_i-x_j$}. They are a higher grid risen above the lower grid {$x_i$}. Whereas the others are aren't based on differences and collapse into the lower grid. How to understand this? How does it relate to duality and the way it is expressed. In {A_1}, the root {$x_2-x_1$} is normal to {$x_1+x_2$}. In {A_2}, the roots are normal to {$x_1+x_2+x_3$}. 2019 spalio 13 d., 10:43
atliko -
Pakeistos 525-537 eilutės iš
Killing form. What is it for exceptional Lie groups? į:
Killing form. What is it for exceptional Lie groups? The Cartan matrix expresses the amount of slack in the world. {$A_n$} God. {$B_n$}, {$C_n$}, {$D_n$} human. {$E_n$} n=8,7,6,5,4,3 divisions of everything. 2 independent roots, independent dimensions, yield a "square root" (?) Symplectic matrix (quaternions) describe local pairs (Position, momentum). Real matrix describes global pairs: Odd and even? Root system is a navigation system. It shows that we can navigate the space in a logical manner in each direction. There can't be two points in the same direction. Therefore a cube is not acceptable. The determinant works to maintain the navigational system. If a cube is inherently impossible, then there can't be a trifold branching. 2019 spalio 13 d., 00:16
atliko -
Pakeistos 521-525 eilutės iš
{$e^{\sum k_i \Delta_i}$} į:
{$e^{\sum k_i \Delta_i}$} Has inner product iff {$AA^?=I$}, {$A{-1}=A^?$} Killing form. What is it for exceptional Lie groups? 2019 spalio 13 d., 00:14
atliko -
Pakeistos 519-521 eilutės iš
Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i - \Delta_j$} is {$30^{\circ}$} {$\Delta_i - \Delta_j$} į:
Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i - \Delta_j$} is {$30^{\circ}$} {$\Delta_i - \Delta_j$} {$e^{\sum k_i \Delta_i}$} 2019 spalio 13 d., 00:13
atliko -
Pakeista 519 eilutė iš:
Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\ į:
Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\Delta_i - \Delta_j$} is {$30^{\circ}$} {$\Delta_i - \Delta_j$} 2019 spalio 13 d., 00:12
atliko -
Pakeista 519 eilutė iš:
Every root can be a simple root. The angle between them can be made {$60^{\ į:
Every root can be a simple root. The angle between them can be made {$60^{\circ}$} by switching sign. {$\delta_i - \delta_j$} is {$30^{\circ}$} {$\delta_i - \delta_j$} 2019 spalio 13 d., 00:11
atliko -
Pakeistos 515-519 eilutės iš
* same + different + different + same + ... į:
* same + different + different + same + ... Does the Lie algebra bracket express slack? Every root can be a simple root. The angle between them can be made {$60^{\circle}$} by switching sign. {$\delta_i - \delta_j$} is {$30^{\circle}$} {$\delta_i - \delta_j$} 2019 spalio 12 d., 23:57
atliko -
Pakeistos 512-515 eilutės iš
* Complex models continuous motion. Symplectic - slack in continuous motion. į:
* Complex models continuous motion. Symplectic - slack in continuous motion. * Three-cycle: same + different => different ; different + different => same ; different + same => different * {$ \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} $} * same + different + different + same + ... 2019 spalio 12 d., 23:51
atliko -
Pakeista 507 eilutė iš:
* Unitary T = {$e^{iX_j į:
* Unitary T = {$e^{iX_j\alpha_j}$} where {$X_j$} are generators and {$\alpha_j$} are angles. Volume preserving, thus preserving norms. Length is one. 2019 spalio 12 d., 23:51
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Pakeista 502 eilutė iš:
Talk with į:
Talk with Thomas Pridėta 506 eilutė:
* {$e^{\sum i \times generator \times parameter}$} has an inverse. Pridėtos 508-511 eilutės:
* Understanding of effect. Physics, why does it work? How can I describe it efficiently and correctly? * Definition of entropy depends on how you choose it. Unit of phase space determines your unit of entropy. Thus observer defines phase space. * Go from rather arbitrary set of dimensions to more natural set of dimensions. Natural because they are convenient. This leads to symmetry. Thus represent in terms of symmetry group, namely Lie groups. There are dimensions. In order to write them up, we want more efficient representations. Subgroups give us understanding of causes. Smaller representations give us understanding of effects. We want to study what we don't understand. In engineering, we leverage what we don't understand. * How many parameters do I need to describe the system? (Like an object.) Minimize constraints. It becomes complicated. Multipole is abstracting the levels of relevance. Ordering them inside the dimensions I am working with. What is the important quantity? Measures quality. How transformation leaves the object invariant. Distinguish between continuous parameters that we measure against and these quantity that we want to study. We use dimensions as a language to relate the inner structure and the outer framework. To measure momentum we need to measure two different quantities. 2019 spalio 12 d., 23:33
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Pakeistos 505-507 eilutės iš
* Differences between even and odd for orthogonal matrices as to whether they can be paired (into complex variables) or not. į:
* Differences between even and odd for orthogonal matrices as to whether they can be paired (into complex variables) or not. * Unitary T = {$e^{iX_j/alpha_j}$} where {$X_j$} are generators and {$\alpha_j$} are angles. Volume preserving, thus preserving norms. Length is one. * Complex models continuous motion. Symplectic - slack in continuous motion. 2019 spalio 12 d., 23:09
atliko -
Pakeistos 500-505 eilutės iš
What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. į:
What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. Talk with Tomas * How is the Riemann sheet, winding around, going to a different Riemann sheet, related to the winding number? and the roots of polynomials? * Organic variation, variables * Differences between even and odd for orthogonal matrices as to whether they can be paired (into complex variables) or not. 2019 spalio 12 d., 23:05
atliko -
Pakeistos 498-500 eilutės iš
Compare finite field behavior (division winding around) with complex number behavior (winding around). į:
Compare finite field behavior (division winding around) with complex number behavior (winding around). What is the relation between the the chain of Weyl group reflections, paths in the root system, the Dynkin diagram chain, and the Lie group chain. 2019 rugsėjo 27 d., 18:29
atliko -
Pakeistos 496-498 eilutės iš
[[https://www.youtube.com/watch?v=LaTTqgchO2o | How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries| N J Wildberger]] į:
[[https://www.youtube.com/watch?v=LaTTqgchO2o | How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries| N J Wildberger]] Compare finite field behavior (division winding around) with complex number behavior (winding around). 2019 rugsėjo 23 d., 05:16
atliko -
Pakeistos 494-496 eilutės iš
[[http://msc2010.org/mediawiki/index.php?title=MSC2010 | Mathematics Subject Classification wiki]] į:
[[http://msc2010.org/mediawiki/index.php?title=MSC2010 | Mathematics Subject Classification wiki]] [[https://www.youtube.com/watch?v=LaTTqgchO2o | How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries| N J Wildberger]] 2019 rugsėjo 21 d., 13:29
atliko -
Pakeistos 492-494 eilutės iš
[[https://www.math.ucla.edu/~tao/ | Terrence Tao courses]] į:
[[https://www.math.ucla.edu/~tao/ | Terrence Tao courses]] [[http://msc2010.org/mediawiki/index.php?title=MSC2010 | Mathematics Subject Classification wiki]] 2019 rugsėjo 20 d., 22:55
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Pakeistos 488-492 eilutės iš
Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? į:
Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? Terrence Tao: It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. [[https://www.math.ucla.edu/~tao/ | Terrence Tao courses]] 2019 rugsėjo 20 d., 21:54
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Pridėtos 486-488 eilutės:
One-dimensional proteins are wound up like the chain of a multidimensional Lie group. Physics is measurement. A single measurement is analysis. Algebra gives the relationships between disparate measurements. But why is the reverse as in the ways of figuring things out in mathematics? 2019 rugsėjo 20 d., 21:50
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Pakeistos 480-485 eilutės iš
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \overline{-(YX-XY)}$}. į:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \overline{-(YX-XY)}$}. Eduardo's Yoneda Lemma diagram is the foursome. Yates Index Theorem - consider substitution. 2019 rugsėjo 20 d., 21:12
atliko -
Pakeista 480 eilutė iš:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \ į:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \overline{-(YX-XY)}$}. 2019 rugsėjo 20 d., 21:11
atliko -
Pakeista 480 eilutė iš:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \ į:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \overbar{-(YX-XY)}$}. 2019 rugsėjo 20 d., 21:11
atliko -
Pakeista 480 eilutė iš:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = bar{-(YX-XY)}$}. į:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = \bar{-(YX-XY)}$}. 2019 rugsėjo 20 d., 21:10
atliko -
Pridėtos 479-480 eilutės:
In the context {$e^iX$} the positive or negative sign of {$X$} becomes irrelevant. In that sense, we can say {$i^2=1$}. In other words, {$i$} and {$-1$} become conjugates. Similarly, {$XY-YX = bar{-(YX-XY)}$}. 2019 rugsėjo 18 d., 22:26
atliko -
Pakeistos 475-478 eilutės iš
{$A^TA$} is similar to the adjoint functors - they may be inverses (in the case of a unitary matrix) or they may be similar. į:
{$A^TA$} is similar to the adjoint functors - they may be inverses (in the case of a unitary matrix) or they may be similar. Terrence Tao problem solving https://books.google.lt/books/about/Solving_Mathematical_Problems_A_Personal.html?id=ZBTJWhXD05MC&redir_esc=y 2019 rugsėjo 17 d., 23:23
atliko -
Pakeistos 473-475 eilutės iš
[[https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf | Differential Forms in Algebraic Topology]], Bott & į:
[[https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf | Differential Forms in Algebraic Topology]], Bott & Tu {$A^TA$} is similar to the adjoint functors - they may be inverses (in the case of a unitary matrix) or they may be similar. 2019 rugsėjo 13 d., 23:01
atliko -
Pakeistos 471-473 eilutės iš
Navier-Stokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. į:
Navier-Stokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. [[https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf | Differential Forms in Algebraic Topology]], Bott & Tu 2019 rugsėjo 06 d., 14:06
atliko -
Pakeistos 469-471 eilutės iš
Consider the classification of Lie groups in terms of the objects for which they are symmetries. į:
Consider the classification of Lie groups in terms of the objects for which they are symmetries. Navier-Stokes equations: Reynolds number relates time symmetric (high Reynolds number) and time asymmetric (low Reynolds number) situations. 2019 rugpjūčio 31 d., 08:32
atliko -
Pakeistos 467-469 eilutės iš
In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? į:
In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? Consider the classification of Lie groups in terms of the objects for which they are symmetries. 2019 rugpjūčio 30 d., 19:05
atliko -
Pridėtos 466-467 eilutės:
In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships? 2019 rugpjūčio 29 d., 13:32
atliko -
Ištrintos 465-466 eilutės:
[[https://www.amazon.com/Modern-Geometry-Methods-Applications-Transformation-dp-1461287561/dp/1461287561/ref=mt_paperback?_encoding=UTF8&me=&qid= | Modern Geometry ― Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields]] Dubrovin, Fomenko, Novikov [[https://www.amazon.com/gp/product/750620133X/ref=dbs_a_def_rwt_bibl_vppi_i2 | Part II]], [[https://www.amazon.com/gp/product/146128791X/ref=dbs_a_def_rwt_bibl_vppi_i3 | Part III]] 2019 rugpjūčio 28 d., 18:16
atliko -
Pakeista 467 eilutė iš:
[[https://www.amazon.com/Modern-Geometry-Methods-Applications-Transformation-dp-1461287561/dp/1461287561/ref=mt_paperback?_encoding=UTF8&me=&qid= | Modern Geometry ― Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields]] Dubrovin, Fomenko, į:
[[https://www.amazon.com/Modern-Geometry-Methods-Applications-Transformation-dp-1461287561/dp/1461287561/ref=mt_paperback?_encoding=UTF8&me=&qid= | Modern Geometry ― Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields]] Dubrovin, Fomenko, Novikov [[https://www.amazon.com/gp/product/750620133X/ref=dbs_a_def_rwt_bibl_vppi_i2 | Part II]], [[https://www.amazon.com/gp/product/146128791X/ref=dbs_a_def_rwt_bibl_vppi_i3 | Part III]] 2019 rugpjūčio 28 d., 18:14
atliko -
Pakeistos 465-467 eilutės iš
Study [[https://en.wikipedia.org/wiki/Orthogonal_group | orthogonal groups]] and Bott periodicity. į:
Study [[https://en.wikipedia.org/wiki/Orthogonal_group | orthogonal groups]] and Bott periodicity. [[https://www.amazon.com/Modern-Geometry-Methods-Applications-Transformation-dp-1461287561/dp/1461287561/ref=mt_paperback?_encoding=UTF8&me=&qid= | Modern Geometry ― Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields]] Dubrovin, Fomenko, Novikov 2019 rugpjūčio 27 d., 12:36
atliko -
Pakeistos 461-463 eilutės iš
Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805. į:
Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805.00203 Study [[https://en.wikipedia.org/wiki/Orthogonal_group | orthogonal groups]] and Bott periodicity. 2019 rugpjūčio 22 d., 17:08
atliko -
Pakeistos 459-461 eilutės iš
Pascal's triangle - the zeros on either end of each row are like Everything at start and finish of an exact sequence. į:
Pascal's triangle - the zeros on either end of each row are like Everything at start and finish of an exact sequence. Edward Frenkel. Langlands program, vertex algebras related to simple Lie groups, detailed analysis of SU(2) and U(1) gauge theories. https://arxiv.org/abs/1805.00203 2019 rugpjūčio 22 d., 16:42
atliko -
Pakeistos 457-459 eilutės iš
[[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence | Curry-Howard-Lambek correspondence]] of logic, programming and category į:
[[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence | Curry-Howard-Lambek correspondence]] of logic, programming and category theory Pascal's triangle - the zeros on either end of each row are like Everything at start and finish of an exact sequence. 2019 rugpjūčio 19 d., 23:52
atliko -
Pakeistos 455-457 eilutės iš
Circle (three-cycle) vs. Line (link to unconditional) - sixsome - and real į:
Circle (three-cycle) vs. Line (link to unconditional) - sixsome - and real forms [[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence | Curry-Howard-Lambek correspondence]] of logic, programming and category theory 2019 rugpjūčio 19 d., 22:48
atliko -
Pakeistos 395-397 eilutės iš
[[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge į:
[[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge Ištrinta 447 eilutė:
2019 rugpjūčio 19 d., 21:13
atliko -
Pakeistos 455-457 eilutės iš
I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view. į:
I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view. Circle (three-cycle) vs. Line (link to unconditional) - sixsome - and real forms 2019 rugpjūčio 14 d., 11:39
atliko -
Pakeistos 453-455 eilutės iš
Einstein field equations - energy stress tensor - is 4+6 equations. į:
Einstein field equations - energy stress tensor - is 4+6 equations. I dreamed of the binomial theorem as having an "internal view", imagined from the inside, which accorded with the "coordinate systems". And which interweaved with the external views to yield various "moments", given by curves on the plane, variously adjusted and transformed by the internal view. 2019 rugpjūčio 10 d., 21:35
atliko -
Pakeistos 451-453 eilutės iš
* Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą? į:
* Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą? Einstein field equations - energy stress tensor - is 4+6 equations. 2019 rugpjūčio 10 d., 13:51
atliko -
Pakeistos 448-451 eilutės iš
* How: inner logic. What: external view. į:
* How: inner logic. What: external view. * Išsakyti grupės {$G_2$} santykį su jos atvirkštine. Ar ši grupė tausoja kokią nors normą? 2019 rugpjūčio 10 d., 12:28
atliko -
Pakeistos 444-448 eilutės iš
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos į:
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos skaitykla Yoneda Lemma * Loss of info from How to What is equal to the Loss of info from "Why for What" to "Why for How". * How: inner logic. What: external view. 2019 rugpjūčio 06 d., 22:26
atliko -
Pakeista 444 eilutė iš:
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] į:
[[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] VU Matematikos ir informatikos skaitykla 2019 rugpjūčio 06 d., 22:23
atliko -
Pakeistos 442-444 eilutės iš
Ways of discovery in math: [[http://www.tricki.org | Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/ | Overview by Timothy Gowers]]. į:
Ways of discovery in math: [[http://www.tricki.org | Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/ | Overview by Timothy Gowers]]. [[https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809 | The Princeton Companion to Mathematics]] 2019 rugpjūčio 06 d., 22:14
atliko -
Pakeistos 440-442 eilutės iš
[[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]] į:
[[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]] Ways of discovery in math: [[http://www.tricki.org | Tricki.org]]. [[https://gowers.wordpress.com/2010/09/24/is-the-tricki-dead/ | Overview by Timothy Gowers]]. 2019 rugpjūčio 06 d., 21:25
atliko -
Pakeistos 436-440 eilutės iš
Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations. į:
Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations. [[https://www.mathpages.com/home/ | Kevin Brown]] collection of expositions of math [[https://www.dpmms.cam.ac.uk/~wtg10/ | Timothy Gowers' webpage]] 2019 rugpjūčio 06 d., 16:58
atliko -
Pridėtos 421-422 eilutės:
[[https://terrytao.wordpress.com/2019/07/26/twisted-convolution-and-the-sensitivity-conjecture/ | Terrence Tao: Twisted Convolution and the Sensitivity Conjecture]] 2019 rugpjūčio 05 d., 10:02
atliko -
Pakeistos 432-434 eilutės iš
Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out. į:
Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out. Mathematical induction - is it possible to treat infinitely many equations as a single equation with infinitely many instantiations? Consider Navier-Stokes equations. 2019 rugpjūčio 04 d., 13:24
atliko -
Pakeistos 430-432 eilutės iš
Axiom of forgetfullness. į:
Axiom of forgetfullness. Does induction prove an infinite number of statements or their reassembly into one statement with infinitely many realizations? It proves the parallelness of intuitive meaningful stepped in and formal stepped out. 2019 rugpjūčio 04 d., 08:57
atliko -
Pakeistos 428-430 eilutės iš
Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. į:
Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. Axiom of forgetfullness. 2019 rugpjūčio 04 d., 08:53
atliko -
Pakeistos 426-428 eilutės iš
Equality holds for both value and type, amount and unit. Peano axiom. į:
Equality holds for both value and type, amount and unit. Peano axiom. Peano why can't have natural numbers have two subsets, a halfline from 0 and a full line. 2019 rugpjūčio 04 d., 08:42
atliko -
Pakeistos 424-426 eilutės iš
Induction step by step is different than the outcome, the totality, which forgets the gradation. į:
Induction step by step is different than the outcome, the totality, which forgets the gradation. Equality holds for both value and type, amount and unit. Peano axiom. 2019 rugpjūčio 04 d., 08:32
atliko -
Pakeistos 422-424 eilutės iš
Relate methods of proof and discovery, 3 systemic and 3 not. į:
Relate methods of proof and discovery, 3 systemic and 3 not. Induction step by step is different than the outcome, the totality, which forgets the gradation. 2019 rugpjūčio 04 d., 07:33
atliko -
Pakeistos 420-422 eilutės iš
How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional? į:
How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional? Relate methods of proof and discovery, 3 systemic and 3 not. 2019 rugpjūčio 01 d., 23:16
atliko -
Pakeistos 418-420 eilutės iš
[[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]]. į:
[[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]]. How is a [[https://en.wikipedia.org/wiki/Boolean_function | Boolean function]] similar to a linear functional? 2019 rugpjūčio 01 d., 22:59
atliko -
Pakeistos 416-418 eilutės iš
The analysis in a Lie group is all expressed by the behavior of the epsilon. į:
The analysis in a Lie group is all expressed by the behavior of the epsilon. [[https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/ | Complexity measures for Boolean functions]]. 2019 liepos 27 d., 17:07
atliko -
Pakeistos 414-416 eilutės iš
[[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]] į:
[[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]] The analysis in a Lie group is all expressed by the behavior of the epsilon. 2019 liepos 25 d., 19:46
atliko -
Pakeistos 412-414 eilutės iš
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). į:
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). [[https://www.youtube.com/watch?v=bFZWarP2Ef4 | Galois, Grothendieck and Voevodsky - George Shabat]] 2019 liepos 20 d., 17:45
atliko -
Pakeista 412 eilutė iš:
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief į:
Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales. Compare with coupling (of position and momentum, for example). 2019 liepos 20 d., 17:44
atliko -
Pakeistos 410-412 eilutės iš
Exchange particles - gauge bosons. į:
Exchange particles - gauge bosons. Heisenberg uncertainty principle - the slack in the vacuum - that allows the borrowing of energy on brief scales 2019 liepos 20 d., 17:26
atliko -
Pakeistos 408-410 eilutės iš
* [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] į:
* [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] Exchange particles - gauge bosons. 2019 liepos 20 d., 17:24
atliko -
Pakeistos 407-408 eilutės iš
[[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] į:
DrPhysicsA * [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] 2019 liepos 20 d., 17:18
atliko -
Pakeistos 405-407 eilutės iš
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge. į:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge. [[https://www.youtube.com/watch?v=zS-LsjrJKPA | Particle Physics 4: Rotation Operators, SU(3)xSU(2)xU(1)]] 2019 liepos 20 d., 17:18
atliko -
Pakeista 405 eilutė iš:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. į:
Weak nuclear force changes quark types. Strong nuclear force changes quark positions. Electromagnetic force distinguishes between quark properties - charge. 2019 liepos 20 d., 17:14
atliko -
Pakeistos 403-405 eilutės iš
* Spin 2: gravition: fivesome (time/space) į:
* Spin 2: gravition: fivesome (time/space) Weak nuclear force changes quark types. Strong nuclear force changes quark positions. 2019 liepos 20 d., 16:59
atliko -
Pakeistos 396-403 eilutės iš
in the light of metaphors of mathematics]] Anthony į:
in the light of metaphors of mathematics]] Anthony Judge Spin 1/2 means there are two states separated by a quanta of energy +/- h. So this is like divisions of everything: * Spin 0 total spin: onesome * Spin 1/2: fermions: twosome * Spin 1: three states: threesome * Spin 3/2: composite particles: foursome * Spin 2: gravition: fivesome (time/space) 2019 liepos 19 d., 16:53
atliko -
Pakeistos 393-396 eilutės iš
Random phenomena organize themselves around a critical boundary. į:
Random phenomena organize themselves around a critical boundary. [[https://www.laetusinpraesens.org/musings/periodt.php | Towards a Periodic Table of Ways of Knowing in the light of metaphors of mathematics]] Anthony Judge 2019 liepos 16 d., 23:25
atliko -
Pakeistos 391-393 eilutės iš
At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). į:
At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). Random phenomena organize themselves around a critical boundary. 2019 liepos 12 d., 14:42
atliko -
Pakeistos 389-391 eilutės iš
If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? į:
If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). 2019 liepos 10 d., 16:52
atliko -
Pakeistos 387-389 eilutės iš
{$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] į:
{$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] If there is a zero in the Riemann function's zone, then there is a function that it can't mimic? 2019 liepos 05 d., 16:53
atliko -
Pakeista 387 eilutė iš:
{$U(n)$} is a real form of {$GL(n,\mathbb{ į:
{$U(n)$} is a real form of {$GL(n,\mathbb{C})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] 2019 liepos 05 d., 16:52
atliko -
Pakeista 387 eilutė iš:
{$U(n)$} is a real form of {$GL(n,\mathbb į:
{$U(n)$} is a real form of {$GL(n,\mathbb{c})$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] 2019 liepos 05 d., 16:51
atliko -
Pakeistos 385-387 eilutės iš
[[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) į:
[[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) {$U(n)$} is a real form of {$GL(n,\mathbb(C))$}. [[https://www.encyclopediaofmath.org/index.php/Complexification_of_a_Lie_group | Encyclopedia: Complexification of a Lie group]] 2019 liepos 04 d., 18:35
atliko -
Pakeistos 382-385 eilutės iš
[[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]] į:
[[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]] [[https://www.youtube.com/watch?v=8KPzuPi-zKk&list=PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F | Lorenzo Sadun. Videos: Linear Algebra]] Nr.88 is SO(3) and so(3) 2019 liepos 04 d., 18:32
atliko -
Pakeistos 380-382 eilutės iš
i->j is asymmetric, one-directional. i<->i* is symmetric, two-directional, breaks anti-symmetry, hides anti-symmetry (which is i and which is j?) į:
i->j is asymmetric, one-directional. i<->i* is symmetric, two-directional, breaks anti-symmetry, hides anti-symmetry (which is i and which is j?) [[https://www.youtube.com/watch?v=wIn_dlmD8sk | Video: The rotation group and all that]] 2019 liepos 04 d., 10:38
atliko -
Pridėta 380 eilutė:
i->j is asymmetric, one-directional. i<->i* is symmetric, two-directional, breaks anti-symmetry, hides anti-symmetry (which is i and which is j?) 2019 liepos 04 d., 10:37
atliko -
Pakeista 378 eilutė iš:
į:
Triality: C at the center, three legs: quaternions, even-dimensional reals, odd-dimensional reals. Fold, fuse, link. 2019 liepos 04 d., 10:35
atliko -
Pakeistos 376-379 eilutės iš
Conjugate = mystery = false. (Hidden distinction). į:
Conjugate = mystery = false. (Hidden distinction). 2019 liepos 04 d., 10:33
atliko -
Pakeistos 374-376 eilutės iš
Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? į:
Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? Conjugate = mystery = false. (Hidden distinction). 2019 liepos 04 d., 09:48
atliko -
Pakeistos 372-374 eilutės iš
[[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition į:
[[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? 2019 liepos 04 d., 09:47
atliko -
Pakeistos 368-372 eilutės iš
A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. į:
A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. The octonions can model the nonassociativity of perspectives. [[https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) | Hurwitz's theorem]] for composition algebras 2019 liepos 04 d., 09:35
atliko -
Pakeistos 366-368 eilutės iš
Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. į:
Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. A circle, as an abelian Lie group, is a "zero", which is a link in a Dynkin diagram, linking two simple roots, two dimensions. 2019 liepos 04 d., 09:31
atliko -
Pakeistos 364-366 eilutės iš
Think again about the combinatorial intepretation of {$K^{-1}K=I$}. į:
Think again about the combinatorial intepretation of {$K^{-1}K=I$}. Symmetry: indistinguishable change, thus a lie, a nontruth, what is hidden. Hidden change, the revealing of hidden change. 2019 liepos 03 d., 10:22
atliko -
Pridėtos 363-364 eilutės:
Think again about the combinatorial intepretation of {$K^{-1}K=I$}. 2019 liepos 02 d., 19:10
atliko -
Pridėtos 361-362 eilutės:
Analyze number types in terms of fractions of differences, https://en.wikipedia.org/wiki/M%C3%B6bius_transformation , in terms of something like that try to understand ad-bc, the different kinds of numbers, the quantities that come up in universal hyperbolic geometry, etc. 2019 liepos 01 d., 10:03
atliko -
Pridėtos 359-360 eilutės:
Wave function Smolin says is ensemble, I say bosonic sharing of space and time 2019 birželio 30 d., 09:59
atliko -
Pakeista 354 eilutė iš:
39:00 Came up with the idea in 1970, while working on į:
39:00 Came up with the idea in 1970, while working on the 16th Hilbert problem. 2019 birželio 30 d., 09:58
atliko -
Pridėta 354 eilutė:
39:00 Came up with the idea in 1970, while working on a Hilbert problem. 2019 birželio 30 d., 09:55
atliko -
Pridėta 357 eilutė:
Arnold: Six geometries (based on Cartan's study of infinite dimensional Lie groups?) his list? 2019 birželio 30 d., 09:36
atliko -
Pakeistos 350-356 eilutės iš
Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. į:
Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. Arnold - "Polymathematics: complexification, symplectification and all that " 1998 video. 18:50 About his trinity, his idea: "This idea, how to apply it, and the examples that I shall discuss even, are not formalized. The theory that I will describe today is not a conjecture, not a theorem, not a definition, it is some kind of religion. I shall show you examples and in these examples, it works. So I was able, using this religion, to find correct guesses, and to find correct conjectures. And then I was able to work years or months trying to prove them. And in some cases, I was able to prove them. In other cases, other people were finally able to prove them. In other cases other people were able to prove them. But to guess these conjectures without this religion would, I think, be impossible. So what I would like to explain to you is just this nonformalized part of it. I am perhaps too old to formalize it but maybe someone who one day finds the axioms and makes a definition from the general construction from the examples that I shall describe." A_n defines a linear algebra and other root systems add additional structure 2019 birželio 26 d., 22:32
atliko -
Pakeistos 346-350 eilutės iš
The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. į:
The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. Real forms - Satake diagrams - are like being stepped into a perspective (from some perspective within a chain). An odd-dimensional real orthogonal case is stepped-in and even-dimensional is stepped out. Complex case combines the two, and quaternion case combines them yet again. For consciousness. Note that given a chain of perspectives, the possibilities for branching are highly limited, as they are with Dynkin diagrams. 2019 birželio 26 d., 20:30
atliko -
Pakeistos 342-346 eilutės iš
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). į:
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). {$SL(n)$} is not compact, which means that it goes off to infinity. It is like the totality. We have to restrict it, which yields {$A_n$}. Whereas the other Lie families are already restricted. The root systems are ways of linking perspectives. They may represent the operations. {$A_n$} is +0, and the others are +1, +2, +3. There can only be one operation at a time. And the exceptional root systems operate on these four operations. 2019 birželio 25 d., 21:16
atliko -
Pakeista 342 eilutė iš:
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. į:
If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. So the geometry is determined by the definition of M(0). 2019 birželio 25 d., 21:15
atliko -
Pakeistos 340-342 eilutės iš
In what sense is SU(3) related to a rotation in octonion space? į:
In what sense is SU(3) related to a rotation in octonion space? If SU(0) is R, then the real line is zero, and we have projective geometry for the simplexes. 2019 birželio 25 d., 18:38
atliko -
Pridėtos 337-339 eilutės:
SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope. The conjugate i is evidently the part that adds a perspective. Then R is no perspective. 2019 birželio 25 d., 18:36
atliko -
Pakeistos 335-337 eilutės iš
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? į:
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? In what sense is SU(3) related to a rotation in octonion space? 2019 birželio 25 d., 18:35
atliko -
Pakeista 335 eilutė iš:
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? į:
Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? U(1) = SU(1) x R where R gives the length. So this suggests SU(0) = R. In what sense does that make sense? 2019 birželio 25 d., 12:25
atliko -
Pakeistos 333-335 eilutės iš
Organize for myself the Coxeter groups based on how they are built from reflections. į:
Organize for myself the Coxeter groups based on how they are built from reflections. Particle physics is based on SU(3)xSU(2)xU(1). Can U(1) be understood as SU(1)xSU(0)? 2019 birželio 24 d., 14:07
atliko -
Pakeistos 331-333 eilutės iš
Understand the classification of Coxeter groups. į:
Understand the classification of Coxeter groups. Organize for myself the Coxeter groups based on how they are built from reflections. 2019 birželio 24 d., 14:04
atliko -
Pakeista 329 eilutė iš:
Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the į:
Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the hyperoctahedral group. 2019 birželio 24 d., 13:55
atliko -
Pakeistos 327-331 eilutės iš
https://en.wikipedia.org/wiki/Andrei_ į:
https://en.wikipedia.org/wiki/Andrei_Okounkov Symmetry of axes - Bn, Cn - leads, in the case of symmetry, to the equivalence of the total symmetry with the individual symmetries, so that for Dn we must divide by two the Understand the classification of Coxeter groups. 2019 birželio 19 d., 10:25
atliko -
Pakeistos 325-327 eilutės iš
the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. į:
the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. https://en.wikipedia.org/wiki/Andrei_Okounkov 2019 birželio 18 d., 19:07
atliko -
Pakeistos 323-325 eilutės iš
What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? į:
What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined. 2019 birželio 01 d., 18:32
atliko -
Pakeistos 321-323 eilutės iš
What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. į:
What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. What is the relationship between spin (and alignment to a particular axis or coordinate system) and the alignment of magnets? 2019 birželio 01 d., 12:25
atliko -
Pakeistos 319-321 eilutės iš
How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? į:
How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues. 2019 gegužės 31 d., 19:55
atliko -
Pakeistos 317-319 eilutės iš
* transpose - transponuota matrica, į:
* transpose - transponuota matrica, transponavimas How special is the Mandelbrot set? What other comparable fractals are there? Can the Mandelbrot set be understood to encompass all of mathematics? What is a combinatorial interpretation of the Mandelbrot set? How is the Mandelbrot set related to the complex numbers and numbers (normed vector spaces) more broadly? 2019 gegužės 30 d., 23:48
atliko -
Pakeistos 310-317 eilutės iš
For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. į:
For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. Lietuvių kalba: * sphere - sfera * trace - pėdsakas * semisimple - puspaprastis, puspaprastė * conjugate - sujungtinis * transpose - transponuota matrica, transponavimas 2019 gegužės 26 d., 09:35
atliko -
Pakeistos 308-310 eilutės iš
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra. į:
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra. For any {$A$} and {$B$} in Lie algebra {$\mathfrak{g}$}, {$exp(A+B) = exp(A) + exp(B)$} if and only if {$[A,B]=0$}. 2019 gegužės 26 d., 09:30
atliko -
Pakeista 308 eilutė iš:
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A$} of a Lie algebra. į:
Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A'$} of a Lie algebra. 2019 gegužės 26 d., 09:29
atliko -
Pakeistos 306-308 eilutės iš
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) į:
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) Differentiating {$AA^{-1}=I$} at {$A(0)=I$} we get {$A(A^{-1})'+A'A^{-1}=0$} and so {$A'=-(A^{-1})'$} for any element {$A$} of a Lie algebra. 2019 gegužės 26 d., 09:22
atliko -
Pakeista 306 eilutė iš:
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." į:
Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." (Ruben Arenas) 2019 gegužės 26 d., 09:21
atliko -
Pakeistos 304-306 eilutės iš
Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. į:
Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. Find the proof and understand it: "An important property of connected Lie groups is that every open set of the identity generates the entire Lie group.Thus all there is to know about a connected Lie group is encoded near the identity." 2019 gegužės 25 d., 09:09
atliko -
Pakeistos 302-304 eilutės iš
Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? į:
Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension. 2019 gegužės 24 d., 23:26
atliko -
Pakeistos 300-302 eilutės iš
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. į:
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. Study how orthogonal and symplectic matrices are subsets of special linear matrices. In what sense are R and H subsets of C? 2019 gegužės 24 d., 23:22
atliko -
Pakeista 300 eilutė iš:
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. į:
Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. In this sense, abelian Lie algebras are poorly behaved. 2019 gegužės 24 d., 23:20
atliko -
Pakeistos 298-300 eilutės iš
Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships. į:
Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships. Solvable Lie algebras are like degenerate matrices, they are poorly behaved. If we eliminate them, then the remaining semisimple Lie algebras are beautifully behaved. 2019 gegužės 18 d., 20:25
atliko -
Pakeistos 296-298 eilutės iš
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. į:
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. Internal discussion with oneself vs. external discussion with others is the distinction that category theory makes between internal structure and external relationships. 2019 gegužės 18 d., 20:24
atliko -
Pakeista 296 eilutė iš:
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective. į:
Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective, in a conditional world. 2019 gegužės 18 d., 20:24
atliko -
Pakeistos 292-296 eilutės iš
Choices - polytopes, reflections - root systems. How are the Weyl groups related? į:
Choices - polytopes, reflections - root systems. How are the Weyl groups related? Affine and projective geometries. Adding or subtracting a perspective. Such as adding or deleting a node to a Dynkin diagram. (The chain of perspectives.) Perhaps the projective geometry is the most basic, and it is based on rotation and the complexes. But perhaps an affine geometry arises, with the reals, so that we can imagine a movement of the origin (the fixed point) and thus we can have translations, a shift in origin, a shift in perspective, a relative perspective. 2019 gegužės 18 d., 20:13
atliko -
Pakeistos 290-292 eilutės iš
In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. į:
In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. Choices - polytopes, reflections - root systems. How are the Weyl groups related? 2019 gegužės 18 d., 19:48
atliko -
Pakeistos 288-290 eilutės iš
In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates. į:
In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates. In Lie root systems, reflections yield a geometry. They also yield an algebra of what addition of root is allowable. 2019 gegužės 18 d., 19:46
atliko - _
Pakeistos 286-288 eilutės iš
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. į:
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. In {$D_n$}, think of {$x_i-x_j$} and {$x_i+x_j$} as complex conjugates. 2019 gegužės 15 d., 17:18
atliko -
Pakeista 286 eilutė iš:
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. į:
Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. Local constraints can yet lead to different global solutions. 2019 gegužės 15 d., 17:17
atliko -
Pakeistos 284-286 eilutės iš
Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to į:
Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism Develop looseness - slack - freedom - ambiguity as concepts that give meaning to isomorphism, identity, structure, symmetry. 2019 gegužės 14 d., 18:27
atliko -
Pridėtos 283-284 eilutės:
Isomorphism is based on assignment but that depends on equality up to identity whereas properties define establish an object up to isomorphism 2019 gegužės 14 d., 10:12
atliko -
Pakeista 280 eilutė iš:
Scaling is positive flips over to negative this is discrete rotation is reflection į:
Scaling is positive flips over to negative this is discrete rotation is reflection 2019 gegužės 14 d., 10:00
atliko -
Pakeistos 278-280 eilutės iš
Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions į:
Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc Scaling is positive flips over to negative this is discrete rotation is reflection 2019 gegužės 14 d., 09:51
atliko -
Pakeistos 275-278 eilutės iš
Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? į:
Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? Quaternions include 3 dimensijos formos rotating (twistor) and 1 for scaling (time) and likewise for octonions etc 2019 gegužės 13 d., 14:55
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Pakeistos 273-275 eilutės iš
How is a Nor gate made from [[Nand]] gates? (And vice versa.) į:
How is a Nor gate made from [[Nand]] gates? (And vice versa.) Note how complex numbers express rotations in R2. How are quaternions related to rotations in R3? What about R4? And the real numbers? And in what sense do the complex numbers and quaternions do the same as the reals but more richly? 2019 balandžio 29 d., 13:49
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Pakeista 273 eilutė iš:
How is a Nor gate made from Nand gates? (And vice versa.) į:
How is a Nor gate made from [[Nand]] gates? (And vice versa.) 2019 balandžio 29 d., 13:41
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Pakeistos 271-273 eilutės iš
Study how all logical relations derive from composition of Nand gates. į:
Study how all logical relations derive from composition of Nand gates. How is a Nor gate made from Nand gates? (And vice versa.) 2019 balandžio 29 d., 13:39
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Pakeistos 267-271 eilutės iš
Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. į:
Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. Are Nand gates (Nor gates) related to perspectives? Study how all logical relations derive from composition of Nand gates. 2019 balandžio 29 d., 13:38
atliko -
Pakeistos 265-267 eilutės iš
Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. į:
Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. Nand gates and Nor gates (and And gates and Or gates) relate one and all. The Nand and Nor mark this relation with a negation sign. 2019 balandžio 28 d., 20:11
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Pakeistos 263-265 eilutės iš
Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements. į:
Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements. Study existential and universal quantifiers as adjunctions and as the basis for the arithmetical hierarchy. 2019 balandžio 28 d., 20:11
atliko -
Pakeistos 261-263 eilutės iš
Study the Wolfram Axiom and Nand. į:
Study the Wolfram Axiom and Nand. Mathematical induction - is infinitely many statements that are true - relate to natural transformation, which also relates possibly infinitely many statements. 2019 balandžio 25 d., 00:09
atliko -
Pakeistos 259-261 eilutės iš
Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. į:
Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. Study the Wolfram Axiom and Nand. 2019 balandžio 07 d., 22:41
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Pakeistos 257-259 eilutės iš
Study homology, cohomology and the Snake lemma to explain how to express a gap. į:
Study homology, cohomology and the Snake lemma to explain how to express a gap. Associative composition yields a list (and defines a list). Consider the identity morphism composed with itself. 2019 kovo 22 d., 10:52
atliko -
Pakeistos 255-257 eilutės iš
If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? į:
If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? Study homology, cohomology and the Snake lemma to explain how to express a gap. 2019 kovo 19 d., 14:11
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Pakeistos 251-255 eilutės iš
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. į:
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. Turing machines - inner states are "states of mind" according to Turing. How do they relate to divisions of everything? If we think of a Turing machine's possible paths as a category, what is the functor that takes us to the actual path of execution? 2019 kovo 19 d., 11:11
atliko -
Pakeista 251 eilutė iš:
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). į:
Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete. 2019 kovo 19 d., 11:10
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Pakeistos 249-251 eilutės iš
Algebra and geometry are linked by logic - intersections and unions make sense in both. į:
Algebra and geometry are linked by logic - intersections and unions make sense in both. Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). 2019 kovo 15 d., 11:13
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Pakeistos 247-249 eilutės iš
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]] į:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]] Algebra and geometry are linked by logic - intersections and unions make sense in both. 2019 kovo 14 d., 12:07
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Pakeista 247 eilutė iš:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. į:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <-> Affine F-algebras [F-linear ring homomorphisms F[Y] to F[x]] 2019 kovo 14 d., 12:00
atliko -
Pakeista 247 eilutė iš:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. į:
Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. 2019 kovo 14 d., 11:52
atliko -
Pakeistos 245-247 eilutės iš
http://pi.math.cornell.edu/~hatcher/AT/ATpage. į:
http://pi.math.cornell.edu/~hatcher/AT/ATpage.html Algebra - geometry duality. (Pullback). Morphism <-> ring homomorphism. Intrinsic and extrinsic geometry. 2019 kovo 11 d., 11:04
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Pridėtos 244-246 eilutės:
https://golem.ph.utexas.edu/category/2017/01/basic_category_theory_free_onl.html 2019 kovo 11 d., 11:02
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Pakeistos 243-245 eilutės iš
Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down. į:
Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down. http://pi.math.cornell.edu/~hatcher/AT/ATpage.html 2019 kovo 10 d., 22:11
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Pakeistos 241-243 eilutės iš
Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for į:
Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example Vector bundles: Identity and self-identity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber - self-identity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of self-identity (or non-identity) allows the edge to be flipped upside down. 2019 kovo 07 d., 14:30
atliko -
Pakeistos 239-241 eilutės iš
Vandermonde determinant shows invertible - basis for finite Fourier į:
Vandermonde determinant shows invertible - basis for finite Fourier transform Euclidean space - (algebraic) coordinate systems - define left, right, front, backwards - and this often makes sense locally - but this does not make sense globally on a sphere, for example 2019 kovo 07 d., 13:31
atliko -
Pakeistos 237-239 eilutės iš
į:
DanielChanMaths Vandermonde determinant shows invertible - basis for finite Fourier transform 2019 kovo 05 d., 09:01
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Pakeista 233 eilutė iš:
į:
Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group | Wikipedia: Homotopy groups]] Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups: 2019 kovo 05 d., 09:00
atliko -
Pakeistos 229-236 eilutės iš
monad = black box? į:
monad = black box? Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups Wikipedia: Homotopy groups Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups: {$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $} 2019 vasario 17 d., 05:55
atliko -
Pakeistos 225-228 eilutės iš
Topology - getting global invariants (which can be calculated) from local information. į:
Topology - getting global invariants (which can be calculated) from local information. Simple examples that illustrate theory. 2019 vasario 17 d., 05:55
atliko -
Pakeistos 219-225 eilutės iš
Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. į:
Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. Is Cayley's theorem (Yoneda lemma) a contentless theorem? What makes a theorem useful as a tool for discoveries? (Conscious) Learning from (unconscious) machine learning. Topology - getting global invariants (which can be calculated) from local information. 2019 vasario 15 d., 09:46
atliko -
Pakeistos 217-219 eilutės iš
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. Symbols (a and b) are equal if they refer to the same referents. But equality has different meaning for symbols, indexes, icons and things. Consider the four relations between level and metalevel. 2019 vasario 13 d., 13:59
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Pakeistos 215-218 eilutės iš
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information. * For example, in this [[https://www.math3ma.com/blog/the-sierpinski-space-and-its-special-property | proposition]] about pullbacks, the statement about the pullback is much more explicit than that for the function f because it includes f as a special case when Z = x, for then f*(idX)={$f^{-1}$}. But that special case leverages the framework to establish all the other cases į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. 2019 vasario 13 d., 13:59
atliko -
Pakeistos 217-218 eilutės iš
Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information. į:
Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information. * For example, in this [[https://www.math3ma.com/blog/the-sierpinski-space-and-its-special-property | proposition]] about pullbacks, the statement about the pullback is much more explicit than that for the function f because it includes f as a special case when Z = x, for then f*(idX)={$f^{-1}$}. But that special case leverages the framework to establish all the other cases. 2019 vasario 13 d., 13:56
atliko -
Pakeistos 215-217 eilutės iš
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information. 2019 vasario 12 d., 11:37
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Pakeistos 213-215 eilutės iš
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. į:
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. 2019 vasario 12 d., 11:24
atliko -
Pakeista 213 eilutė iš:
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). į:
In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal. 2019 vasario 12 d., 11:19
atliko -
Pakeistos 211-213 eilutės iš
"For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. į:
"For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). 2019 vasario 12 d., 10:56
atliko -
Pakeistos 209-211 eilutės iš
Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction. į:
Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction. "For all" and "there exists" are adjoints presumably because they are on opposite sides of a negation wall that distinguishes the internal structure and external relationships. (That wall also distinguishes external context and internal structure.) (And algorithms?) So study that wall, for example, with regard to recursion theory. 2019 vasario 12 d., 10:37
atliko -
Pakeistos 207-209 eilutės iš
The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. į:
The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction. 2019 vasario 12 d., 08:52
atliko -
Pakeistos 203-207 eilutės iš
Study how Set breaks duality (the significance of initial and terminal objects). į:
Study how Set breaks duality (the significance of initial and terminal objects). Show why there is no n-category theory because it folds up into the foursome. Understand the Yoneda lemma. Relate it to the four ways of looking at a triangle. The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements. 2019 vasario 11 d., 15:24
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Pakeistos 201-203 eilutės iš
Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? į:
Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? Study how Set breaks duality (the significance of initial and terminal objects). 2019 vasario 08 d., 13:58
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Pakeistos 199-201 eilutės iš
In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? į:
In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? Force (and acceleration) is a second derivative - this is because of the duality, the coupling, needed between, say, momentum and position. Is this coupling similar to adjoint functors? 2019 vasario 08 d., 13:36
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Pakeistos 197-199 eilutės iš
In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. į:
In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer? 2019 vasario 08 d., 13:34
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Pakeistos 195-197 eilutės iš
[[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. į:
[[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one. 2019 vasario 05 d., 10:02
atliko -
Pakeista 195 eilutė iš:
[[http://math.ucr.edu/home/baez/week257.html John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. į:
[[http://math.ucr.edu/home/baez/week257.html | John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. 2019 vasario 05 d., 10:02
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Pakeistos 193-195 eilutės iš
Six sextactic points. į:
Six sextactic points. [[http://math.ucr.edu/home/baez/week257.html John Baez about observables]] (see Nr.15) and the paper [[http://arxiv.org/abs/0709.4364 | A topos for algebraic quantum theory]] about C* algebras within a topos and outside of it. 2019 vasario 04 d., 15:14
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Pakeistos 191-193 eilutės iš
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry. į:
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry. Six sextactic points. 2019 vasario 04 d., 15:11
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Pakeista 191 eilutė iš:
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. į:
Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. Caustic - a phenomenon in symplectic geometry. 2019 vasario 04 d., 15:10
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Pakeistos 189-191 eilutės iš
Representable functors - based on arrows from the same object. į:
Representable functors - based on arrows from the same object. Closed curve in plane must have at least four critical points of curvature. This reflects the fourfold aspect of turning around, rotating. 2019 vasario 02 d., 13:35
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Pakeistos 187-189 eilutės iš
Whether (objects), what (morphisms), how (functors), why (natural transformations). į:
Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same. Representable functors - based on arrows from the same object. 2019 vasario 02 d., 13:10
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Pakeistos 185-187 eilutės iš
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. į:
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. Whether (objects), what (morphisms), how (functors), why (natural transformations). 2019 vasario 01 d., 11:51
atliko -
Pakeista 185 eilutė iš:
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. į:
Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. And it's essential for the meaning of the Yoneda Lemma, for example. Because two different examples may have the same structure but one example may draw richer meaning from one domain and the other may not have that richer meaning or may have other meaning from another domain. 2019 vasario 01 d., 11:48
atliko -
Pakeistos 181-185 eilutės iš
* D_n points and į:
* D_n points and position Yoneda lemma - relates to exponentiation and logarithm Category is a collection of examples that satisfy certain conditions. That is why you can have many examples that are essentially identical. 2019 sausio 27 d., 09:49
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Pakeistos 174-181 eilutės iš
Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? į:
Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? Benet linkage - keturgrandinis - lygiagretainis, antilygiagretainis * A_n points and sets * B_n inside: perpendicular (angles) and * C_n outside: line and surface area * D_n points and position 2019 sausio 27 d., 09:23
atliko -
Pakeistos 172-174 eilutės iš
What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]]. į:
What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]]. Symplectic geometry is the geometry of the "outside" (using quaternions) whereas conformal geometry is the geometry of the "inside" (using complex numbers). Then what do real numbers capture the geometry of? And is there a geometry of the line vs. a geometry of the circle? And is one of them "spinorial"? 2019 sausio 27 d., 09:07
atliko -
Pakeista 172 eilutė iš:
What is the connection between symplectic geometry and homology? į:
What is the connection between symplectic geometry and homology? See [[https://en.wikipedia.org/wiki/Morse_theory | Morse theory]]. See [[https://people.ucsc.edu/~alee150/sympl.html | Floer theory]]. 2019 sausio 27 d., 09:04
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Pakeistos 170-172 eilutės iš
Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum). į:
Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum). What is the connection between symplectic geometry and homology? 2019 sausio 27 d., 08:11
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Pakeistos 168-170 eilutės iš
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? į:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? Note that 2-dimensional phase space (as with a spring) is the simplest as there is no 1-dimensional phase space and there can't be (we need both position and momentum). 2019 sausio 27 d., 08:10
atliko -
Pakeista 168 eilutė iš:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? į:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope? and what is the significance of extremal points vs. extremal edges vs. extremal faces etc.? and homology/cohomology?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? 2019 sausio 27 d., 08:09
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Pakeista 168 eilutė iš:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points į:
Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points (convex polytope?), infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? 2019 sausio 27 d., 08:09
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Pakeistos 166-168 eilutės iš
How do symmetries of paths relate to symmetries of young į:
How do symmetries of paths relate to symmetries of young diagrams Note that symplectic geometry, in preserving areas, seems to only care about the extremal points. In what sense is that true in phase space? When and how could it not be true? What is the difference between having finitely many extremal points, infinitely many (as with a fractal boundary) or all extremal points? Can symplectic geometry be considered a study of the outside and conformal geometry a study of the inside? 2019 sausio 25 d., 11:58
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Pakeistos 164-166 eilutės iš
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing. į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing. How do symmetries of paths relate to symmetries of young diagrams 2019 sausio 22 d., 14:25
atliko -
Pakeista 164 eilutė iš:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. This brings to mind the vacillation between knowing and not knowing. 2019 sausio 22 d., 12:33
atliko -
Pakeista 164 eilutė iš:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe. The coupling models looseness - slack. 2019 sausio 22 d., 12:25
atliko -
Pakeista 164 eilutė iš:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the į:
Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling - the coupling is across the entire universe 2019 sausio 22 d., 12:25
atliko -
Pakeistos 162-164 eilutės iš
* Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome. į:
* Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome. Symplectic - basis for coupling - coupling of electric and magnetic fields - is what is responsible for the periodic nature of waves - the higher the frequency, the higher the energy, the tighter the coupling 2019 sausio 21 d., 13:17
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Pakeistos 159-162 eilutės iš
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. į:
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. Walks on trees * Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome. 2019 sausio 19 d., 12:36
atliko -
Ištrintos 108-110 eilutės:
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]] Pakeistos 129-138 eilutės iš
Affine geometry Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides! John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities. į:
Four geometries * Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? Affine geometry * Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. * Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. Duality * John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides! Pakeistos 144-147 eilutės iš
Conformal į:
Cayley-Dickson construction * John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities. Projective geometry * Desargues theorem in geometry corresponds to the associative property in algebra. * A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. Conformal geometry * Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. Pakeistos 156-158 eilutės iš
Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. į:
Symplectic geometry * Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. 2019 sausio 16 d., 11:21
atliko -
Pakeista 157 eilutė iš:
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic į:
Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy. Potential energy is based on position and not time. Kinetic energy is based on time and not position. 2019 sausio 16 d., 11:14
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Pakeistos 155-157 eilutės iš
Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. į:
Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. Lagrangian: (Force) Change in potential energy = (mass x acceleration) Change in kinetic energy 2019 sausio 16 d., 10:38
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Pakeistos 153-155 eilutės iš
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. į:
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. Kinectic energy is possibly zero and builds up from there. It can be expressed in an absolute sense. Potential is possibly infinite and subtracts from that. So it must be expressed in a relative sense. So they are coming at energy from opposite directions. 2019 sausio 15 d., 21:43
atliko -
Pakeista 153 eilutė iš:
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. į:
A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle. 2019 sausio 15 d., 21:43
atliko -
Pakeistos 151-153 eilutės iš
The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1. į:
The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1. A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. 2019 sausio 10 d., 13:57
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Pakeistos 149-151 eilutės iš
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. į:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and -1 and the other axis is the opposites i and j. And they become related 1 to i to -1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to -1. 2019 sausio 10 d., 13:54
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Pakeista 149 eilutė iš:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta į:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta}$}. 2019 sausio 10 d., 13:54
atliko -
Pakeistos 147-149 eilutės iš
Desargues theorem in geometry corresponds to the associative property in algebra. į:
Desargues theorem in geometry corresponds to the associative property in algebra. Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$e^{\pi i \theta|$}. 2019 sausio 08 d., 23:12
atliko -
Pakeistos 145-147 eilutės iš
So the types of duality should give the types of forces. į:
So the types of duality should give the types of forces. Desargues theorem in geometry corresponds to the associative property in algebra. 2019 sausio 07 d., 12:58
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Pakeistos 144-145 eilutės iš
* Weak force - time į:
* Weak force - time reversal So the types of duality should give the types of forces. 2019 sausio 07 d., 12:58
atliko -
Pridėtos 142-144 eilutės:
Each physical force is related to a duality: * Charge (matter and antimatter) - electromagnetism * Weak force - time reversal 2019 sausio 04 d., 03:03
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Pakeistos 138-141 eilutės iš
John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). į:
John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides! John Baez periodic table and stablization theorem - relate to Cayley Dickson construction and its dualities. 2019 sausio 03 d., 23:03
atliko -
Pakeistos 136-138 eilutės iš
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. į:
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). 2019 sausio 03 d., 19:32
atliko -
Pakeista 136 eilutė iš:
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra extending An. į:
Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An. 2019 sausio 03 d., 19:32
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Pakeistos 134-136 eilutės iš
Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. į:
Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra extending An. 2019 sausio 03 d., 18:51
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Pakeistos 132-134 eilutės iš
Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? į:
Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices. 2019 sausio 02 d., 14:46
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Pakeistos 130-132 eilutės iš
Attach:QuadrupleFormulas. į:
Attach:QuadrupleFormulas.png Affine geometry preserves lines. Projective geometry also preserves zeros. Conformal geometry preserves angles. Symplectic geometry preservers oriented area. What are all these objects? 2018 gruodžio 31 d., 20:13
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Pakeistos 126-129 eilutės iš
[[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]] į:
[[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]] Attach:GeometryFormulas.png 2018 gruodžio 30 d., 01:58
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Pakeistos 124-126 eilutės iš
Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. į:
Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. [[https://www.youtube.com/watch?v=7d5jhPmVQ1w | John Baez on duality in logic and physics]] 2018 gruodžio 29 d., 16:22
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Pakeistos 122-124 eilutės iš
Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. į:
Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. Homogeneous coordinates are what let us go between a simplex (when Z=1) and the coordinate system (when Z is free). Compare with the kinds of variables. 2018 gruodžio 29 d., 14:49
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Pakeistos 120-122 eilutės iš
Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? į:
Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? Projective geometry: homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z. 2018 gruodžio 22 d., 17:22
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Pakeistos 118-120 eilutės iš
Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. į:
Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. Try to express projective geometry (or universal hyperbolic geometry) in terms of matrices and thus symmetric functions. What then is algebraic geometry and how do polynomials get involved? What is analytic geometry and in what sense does it go beyond matrices? How doe all of these hit up against the limits of matrices and the amount of symmetry in its internal folding? 2018 gruodžio 21 d., 14:35
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Pakeistos 116-118 eilutės iš
Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? į:
Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews. 2018 gruodžio 21 d., 14:32
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Pakeistos 114-116 eilutės iš
G2 requires three lines to get between any two points (?) Relate this to the three-cycle. į:
G2 requires three lines to get between any two points (?) Relate this to the three-cycle. Rotation relates one dimension to two others. How does this rotation work in higher dimensions? To what extent does multiplication of rotation (through three dimensional half turns) work in higher dimensions and how does it break down? 2018 gruodžio 19 d., 12:38
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Pakeistos 112-114 eilutės iš
Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. į:
Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. G2 requires three lines to get between any two points (?) Relate this to the three-cycle. 2018 gruodžio 19 d., 12:07
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Pakeistos 110-112 eilutės iš
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]] į:
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]] Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were. 2018 gruodžio 19 d., 12:03
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Pakeistos 107-110 eilutės iš
Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. į:
Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. [[https://link.springer.com/content/pdf/10.1007%2Fs00220-009-0951-9.pdf | Categorified Symplectic Geometry and the Classical String]] * [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html | Five related lectures by Christopher L. Rogers]] 2018 gruodžio 18 d., 14:08
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Pakeistos 105-107 eilutės iš
Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]]. į:
Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]]. Dots in Dynkin diagrams are figures in the geometry, and edges are invariant relationships. A point can lie on a line, a line can lie on a plane. Those relationships are invariant under the actions of the symmetries. Dots in a Dynkin diagram correspond to maximal parabolic subgroups. They are the stablizer groups of these types of figures. 2018 gruodžio 17 d., 11:19
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Pakeistos 103-105 eilutės iš
[[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24. į:
[[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24. Discriminant of [[https://en.wikipedia.org/wiki/Elliptic_curve | elliptic curve]]. 2018 gruodžio 16 d., 23:48
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Pakeistos 99-103 eilutės iš
Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. į:
Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. John Baez: 24 = 6 x 4 = An x Bn [[https://en.wikipedia.org/wiki/Dedekind_eta_function | Dedekind eta function]] is based on 24. 2018 gruodžio 16 d., 23:20
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Pakeistos 97-99 eilutės iš
What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]]. į:
What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]]. Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization, take a look at that. 2018 gruodžio 16 d., 22:28
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Pakeistos 95-97 eilutės iš
Nobody know what E8 is the symmetry group of. (Going beyond oneself?) į:
Nobody know what E8 is the symmetry group of. (Going beyond oneself?) What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/2002-39-02/S0273-0979-01-00934-X/S0273-0979-01-00934-X.pdf | John Baez, The Octonions]]. 2018 gruodžio 16 d., 18:27
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Pakeistos 93-95 eilutės iš
Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. į:
Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. Nobody know what E8 is the symmetry group of. (Going beyond oneself?) 2018 gruodžio 16 d., 12:55
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Pakeistos 91-93 eilutės iš
[[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better. į:
[[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better. Compare trejybės ratas (for the quaternions) and Fano's plane (aštuonerybė) for the octonions. 2018 gruodžio 15 d., 13:00
atliko - 2018 gruodžio 15 d., 12:37
atliko -
Pakeistos 89-91 eilutės iš
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]]. į:
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]]. [[https://www.math.columbia.edu/~woit/wordpress/?p=5927 | Geometric unity]] I tend to agree with Roger Penrose that spin has been one of the great mysteries in quantum mechanics. As best as I can recall, he said it was one of two primary mysteries in a talk at NYU back in the late 1990’s. ... understand spin and I think we’ll understand entanglement a lot better. 2018 gruodžio 15 d., 11:41
atliko -
Pakeista 89 eilutė iš:
https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_ į:
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory | Yang-Mills theory]]. 2018 gruodžio 15 d., 11:40
atliko -
Pakeistos 86-89 eilutės iš
Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement. į:
Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement. Massless particles acquire mass through symmetry breaking: https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory 2018 gruodžio 13 d., 11:27
atliko -
Pakeistos 84-86 eilutės iš
This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. į:
This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. Which state is which amongst "one" and "another" is maintained until it is unnecessary - this is quantum entanglement. 2018 gruodžio 13 d., 11:25
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Pakeistos 82-84 eilutės iš
If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. į:
If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. This is the difference between thinking of the negative dimension as "explicitly" written out, or thinking of it as simply as one of two "implicit" states that we switch between. 2018 gruodžio 13 d., 11:19
atliko -
Pakeistos 80-82 eilutės iš
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. į:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. If we think of rotation by i as relating two dimensions, then {$i^2$} takes a dimension to its negative. So that is helpful when we are thinking of the "extra" distinguished dimensions (1). And if that dimension is attributed to a line, then this interpretation reflects along that line. But when we compare rotations as such, then we are not comparing lines, but rather rotations. In this case if we perform an entire rotation, then we flip the rotations for that other dimension that we have rotated about. So this means that the relation between rotations as such is very different than the relation with the isolated distinguised dimension. The relations between rotations is such is, I think, given by {$A_n$} whereas the distinguished dimension is an extra dimension which gets represented, I think, in terms of {$B_n$}, as the short root. 2018 gruodžio 13 d., 11:01
atliko -
Pakeista 80 eilutė iš:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. į:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. Rotation gets you on the other side of the page. 2018 gruodžio 13 d., 08:41
atliko -
Pakeista 80 eilutė iš:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. į:
Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. The usual geometric interpretation of complex numbers is then a particular reinterpretation that is possible but not canonical. 2018 gruodžio 13 d., 08:40
atliko -
Pakeistos 78-80 eilutės iš
* polar conjugates in projective geometry (see Wildberger) į:
* polar conjugates in projective geometry (see Wildberger) Usually multiplication by i is identified with a rotation of 90 degrees. However, we can instead identify it with a rotation of 180 degrees if we consider, as in the case of spinors, that the first time around it adds a sign of -1, and it needs to go around twice in order to establish a sign of +1. This is the definition that makes spin composition work in three dimensions, for the quaternions. 2018 gruodžio 11 d., 18:38
atliko -
Pridėtos 71-78 eilutės:
Duality examples (conjugates) * complex number "i" is not one number - it is a pair of numbers that are the square roots of -1 * spinors likewise * Dn where n=2 * the smallest cross-polytope with 2 vertices * taking a sphere and identifying antipodal elements - this is a famous group * polar conjugates in projective geometry (see Wildberger) 2018 gruodžio 10 d., 14:23
atliko -
Pridėtos 69-70 eilutės:
Root systems relate two spheres - they relate two "sheets". Logic likewise relates two sheets: a sheet and a meta-sheet for working on a problem. Similarly, we model our attention by awareness, as Graziano pointed out. This is stepping in and stepping out. 2018 gruodžio 10 d., 13:38
atliko -
Pridėtos 64-68 eilutės:
An simplexes allow gaps because they have choice between "is" and "not". But all the other frameworks lack an explicit gap and so we get the explicit second counting. But: * for Bn hypercubes we divide the "not" into two halves, preserving the "is" intact. * for Cn cross-polytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (not-this). 2018 gruodžio 10 d., 13:14
atliko -
Pridėtos 62-63 eilutės:
Use "this" and "that" as unmarked opposites - conjugates. 2018 gruodžio 09 d., 15:01
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Pakeistos 47-48 eilutės iš
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. į:
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Folding is the basis for substitution. Pakeistos 57-61 eilutės iš
How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. į:
How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. Keturios pasirinkimo sampratos (apimtys) visos reikalingos norint išskirti vieną paskirą pasirinkimą. Bott periodicity is the basis for 8-fold folding and unfolding. 2018 gruodžio 09 d., 14:59
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Pakeistos 53-57 eilutės iš
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative. į:
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative. Kaip dvi skaičiavimo kryptis (conjugate) sujungti apsisukimu? How to interpret possible expansions? For example, composition of function has a distinctive direction. Whereas a commutative product, or a set, does not. 2018 gruodžio 09 d., 14:57
atliko -
Pakeista 39 eilutė iš:
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. į:
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Šie pirmieji skaičiai yra išskirtiniai. Toliau gaunasi (didėjančio ir mažėjančio laisvumo palaikomas) bendras skaičiavimas, yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. 2018 gruodžio 09 d., 14:44
atliko -
Pakeista 53 eilutė iš:
{$x_0$} is fundamentally different from {$x_i$}. į:
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0-x_1)$} but the others appear both positive and negative. 2018 gruodžio 09 d., 14:42
atliko -
Pakeistos 47-53 eilutės iš
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. į:
Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. Fizikoje, posūkis yra viskas. Palyginti su ortogonaline grupe. Bott periodicity exhibits self-folding. Note the duality with the pseudoscalar. Consider the formula n(n+1)/2 does that relate to the entries of a matrix? {$x_0$} is fundamentally different from {$x_i$}. 2018 gruodžio 09 d., 14:37
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Pakeistos 39-47 eilutės iš
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. į:
Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. Esminis pasirinkimas yra: kurią pasirinkimo sampratą rinksimės? Kaip suvokiame {$x_i$}? Koks tai per pasirinkimas? Kaip sekos lankstymą susieti su baltymų lankstymu ir pasukimu? Kada pasirinkimo samprata keičiasi, visgi už visų sampratų slypi bendresnis, pirmesnis suvokimų suvokimas, taip kad renkamės pačią sampratą. 2018 gruodžio 09 d., 14:15
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Pakeistos 37-39 eilutės iš
An relates to "center of mass". How does this relate to the asymmetry of whole and center? į:
An relates to "center of mass". How does this relate to the asymmetry of whole and center? Išėjimas už savęs reiškiasi kaip susilankstymas, išsivertimas, užtat tėra keturi skaičiai: +0, +1, +2, +3. Toliau yra dešimts tūkstantys daiktų, kaip sako Dao De Jing. Trečias yra begalybė. 2018 gruodžio 05 d., 12:04
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Pakeistos 35-37 eilutės iš
Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? į:
Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? An relates to "center of mass". How does this relate to the asymmetry of whole and center? 2018 lapkričio 25 d., 09:38
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Pakeistos 33-35 eilutės iš
Composition algebra. Doubling is related to duality. į:
Composition algebra. Doubling is related to duality. Symplectic algebras are always even dimensional whereas orthogonal algebras can be odd or even. What do odd dimensional orthogonal algebras mean? How are we to understand them? 2018 lapkričio 25 d., 09:35
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Pakeistos 31-33 eilutės iš
{$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. į:
{$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. Composition algebra. Doubling is related to duality. 2018 lapkričio 17 d., 12:42
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Pakeistos 29-31 eilutės iš
One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? į:
One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? {$A_n$} tracefree condition is similar to working with independent variables in the the center of mass frame of a multiparticle system. (Sunil, Mukunda). In other words, the system has a center! And every subsystem has a center. 2018 lapkričio 13 d., 13:12
atliko - 2018 lapkričio 11 d., 17:37
atliko -
Ištrintos 8-10 eilutės:
* Shear: sideshot Ištrintos 9-10 eilutės:
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. So it's a type of completeness. What makes that true? 2018 lapkričio 11 d., 17:35
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Ištrintos 6-9 eilutės:
* Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę. * Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai. Ištrintos 11-12 eilutės:
Ištrintos 12-16 eilutės:
[[https://ac.els-cdn.com/S0012365X96003500/1-s2.0-S0012365X96003500-main.pdf?_tid=7e02ab6e-b80c-44b3-9ef0-33327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08 | Geometry of Classical Groups over Finite Fields and Its Applications]], Zhe-xian Wan * Difference between set and vector space (or list?) Set has empty set, but vector space has zero instead of the empty set. So there are no functions into the empty set, but there are functions into zero. Vector space is not distributive. Can't just take the union, need to take the span. Thus R and (S or T) is not equal to (R and S) or (R and T). A line and a plane is not a line and line or a line and line. 2018 lapkričio 11 d., 17:32
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Ištrintos 18-19 eilutės:
Collect examples of the arithmetic hierarchy such as calculus (delta-epsilon), differentiable manifolds, etc. Ištrintos 22-23 eilutės:
Ištrintos 25-26 eilutės:
Inner products are sesquilinear - they have conjugate symmetry - so as not to yield lopsided answers. If they yield a complex root as an answer, then one version should yield one root and the other version should yield the other root. In other words, in a complex field, the inner product should be thought of as yielding two answers - both answers - distinguished by the notation, left-right or right-left. 2018 lapkričio 11 d., 17:29
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Ištrintos 0-3 eilutės:
What kind of conjugation is that? Pakeistos 9-10 eilutės iš
į:
* Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai. Pakeistos 26-29 eilutės iš
* Category of perspectives: stepping-in and stepping-out as adjoints? there exists vs. for all? * Eugenia Cheng: Mathematics is the logical study of how logical things work. Abstraction is what we need for logical study. Category theory is the the math of math, thus the logical study of the logical study of how logical things work. * Category theory shines light on the big picture. Perspectives shine light on the big picture (God's) or the local picture (human's). į:
Pakeistos 28-30 eilutės iš
* Lygmuo Kodėl viską išsako ryšiais. O tas ryšys yra tarpas, kuriuo išsakomas Kitas. Kategorijų teorijoje panašiai, tikslas yra pereiti iš narių (objektų) nagrinėjimą į ryšių (morfizmų) nagrinėjimą. į:
Ištrintos 30-33 eilutės:
Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23). Ištrintos 38-39 eilutės:
Pakeistos 51-57 eilutės iš
How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). į:
One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? 2018 lapkričio 11 d., 17:20
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Ištrintos 0-2 eilutės:
Function can be partial, whereas a permutation maps completely. Ištrintos 4-5 eilutės:
Ištrintos 18-24 eilutės:
* Mandelbrot aibė skiria vidines ir išorines veiklas, besilaikančias erdvės ir nesilaikančios jokios erdvės. * Įsivaizduoti, kaip Mandelbrot aibės transformacija veikia visą plokštumą arba vieną jos kampelį. * Catalan numbers foster a duality between horizontal concatenation ()()() and vertical embedding ((())). Šis dualizmas taip pat sieja visko savybes Visaką priima (Kaip) ir Be aplinkos (Kodėl). Palyginti ir su žaidimų gramatika, su medžiai ir sekomis. Matematikos savokų pagrindas yra dvejybinis-trejybinis - operacijos jungia du narius trečiu nariu. Matricų elementai sieja du narius ir išgauna trečią. Kategorijų teorija panašiai. O geometrija lygiaverčiai sieja tris narius trikampiais, įvairiai suprastais. Tad tai paaiškintų geometrijos svarbą. Ištrintos 20-21 eilutės:
Ištrintos 23-32 eilutės:
Partial derivative - formal (explicit) based on change in variable, total derivative - actual (implicit) based on change in value. Measurement (crucial in physics) can be defined in terms of covectors, as being dual to vectors. Covectors can be thought to point in the same direction as vectors - they are complements of each other with regard to the inner product. This duality is thus fundamental to the concept of measurement. The vector and the covector divide a scalar field into its local variation (given by the vector) and its global scaling (given by the covector) which together give the value of the scalar. Thus vector and covector define the duality of local and global extremes which come together as the scalar field. A vector is 1-dimensional (and its dimension) and its covector is n-1 dimensional (it is normal to the vector). In this sense they complement each other. Vectors are described in terms of partial derivatives (based on the local coordinate systems) whereas covectors are described in terms of (total) forms dx. Ištrinta 28 eilutė:
Ištrintos 63-64 eilutės:
Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back). 2018 lapkričio 11 d., 17:00
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Pakeistos 1-2 eilutės iš
į:
Ištrintos 7-8 eilutės:
Pakeistos 12-21 eilutės iš
Apibrėžti "gebėjimus" ir kaip matematinis mąstymas suveda skirtingus gebėjimus suvokti kelis, keliolika, keliasdešimts, tūkstančius ir t.t. daiktų Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę. Apibendrinimas * Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) * Apibendrinimas yra "objekto" kūrimas į:
Signal propagation - expansions * Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives. * How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? The nature of math * Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę. Ištrintos 20-23 eilutės:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. Ištrintos 28-33 eilutės:
http://www-users.math.umn.edu/~dgrinber/ http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf See: Combinatorics and the field with one element. Witt vectors - p-adic integers Ištrinta 36 eilutė:
Pakeistos 48-50 eilutės iš
Symplectic form is skew-symmetric - swapping u and v changes sign - so it establishes orientation of surfaces - distinction of inside and outside - duality breaking. And inner product duality no longer holds. į:
Pakeistos 52-53 eilutės iš
į:
Ištrintos 61-63 eilutės:
* elementary symmetric functions are analogous to prime factorizations of numbers - monomial symmetric functions are analogous to the numbers as such ("natural" basis - "natural" numbers +1) Ištrinta 63 eilutė:
Ištrinta 65 eilutė:
Ištrinta 66 eilutė:
2018 lapkričio 11 d., 16:44
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Ištrintos 2-3 eilutės:
Ištrintos 4-11 eilutės:
Symmetry group relates: * Algebraic structure, "group" * Analytic (recurring activity) transformations Axiom of infinity - can be eliminated - it is unnecessary in "implicit math". Ištrintos 12-13 eilutės:
Ištrintos 16-20 eilutės:
Matematikos įrodymo būdai * 6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję Ištrintos 18-19 eilutės:
Ištrintos 20-22 eilutės:
* A variable is an "atom" of meaning as in my paper, The Algebra of Copyright, which can be parsed on three different levels, yielding four levels and six pairs of levels. Pakeistos 53-56 eilutės iš
Duality breaking (for slack) - disconnecting the local and the global - for example, defining locally Euclidean spaces - in lattice terms, as a consequence of limiting processes, disconnecting the inf from the sup, breaking their duality. į:
Ištrintos 55-56 eilutės:
Ištrintos 65-66 eilutės:
Ištrintos 67-72 eilutės:
Momentum can be attributed to an individual particle (as its change) but it can also be attributed to the entire system (as its change). And also, changing the momentum of a particular particle can change when (and whether) we will come to a particular state of the system. In particular, the particles are interconnected and so that makes for a complicated relation between the time evolution of each particle (in terms of its position) and the time evolution of the system. This can be compared to a computer program which may change the order of its instructions. What do inside and outside mean in symplectice (Hamiltionian, Lagrangian) mechanics? Pakeistos 70-72 eilutės iš
į:
What do inside and outside mean in symplectic (Hamiltionian, Lagrangian) mechanics? Pakeistos 85-88 eilutės iš
Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. But nature has forces (like gravity) that tend to create uncommon states. So we need to distinguish between the different forces at play and how they relate to what is common and uncommon. į:
Pakeistos 88-89 eilutės iš
į:
Pakeistos 91-96 eilutės iš
Counting (in Lie root system) can change to B, C, D only once! That puts a cap on the one end. Then the counting must continue on the other end, extending it. A second cap may not be put on that end. There cannot be a cycle. Counting (by way of the simple roots) links + and - in a chain. x2-x1 etc. į:
Ištrintos 93-94 eilutės:
Ištrintos 103-104 eilutės:
Pakeistos 122-140 eilutės iš
Symmetry group consists of distinguishable actions which accomplish nothing Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis. When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time. Is many-worlds theory the flip-side of least-action ? Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i. Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent. The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways. The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). Entanglement - particle and anti-particle are in the same place and time - and they have the same clock and coordinates į:
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). 2018 lapkričio 11 d., 12:43
atliko -
Pakeistos 187-189 eilutės iš
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). į:
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). Entanglement - particle and anti-particle are in the same place and time - and they have the same clock and coordinates 2018 lapkričio 09 d., 21:59
atliko -
Pakeistos 185-187 eilutės iš
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways. į:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways. The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). 2018 lapkričio 09 d., 15:30
atliko -
Pakeista 185 eilutė iš:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be į:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be related in one of the various ways. 2018 lapkričio 09 d., 15:30
atliko -
Pakeista 185 eilutė iš:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And į:
The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And the two must then be variously related. 2018 lapkričio 09 d., 15:29
atliko -
Pakeistos 183-185 eilutės iš
Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent. į:
Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent. The root systems, as a minimum, have to contain the e_i - e_j because they encode the Lie bracket of the e_i. The question is, what are the ways that they can be expanded? First, a dual encoding can be given e_i + e_j. And then the two can be related. 2018 lapkričio 08 d., 19:47
atliko -
Pakeistos 181-183 eilutės iš
Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i. į:
Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i. Lie algebras (root systems) are the cracks or gaps between coordinate system layers. The "higher energy" a system has, the bigger the crack, the more possibility for variation and freedom, how the system will develop or degenerate. The lowest energy system is the one that has the roots e_i (and its byproducts, e_i-e_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_i-e_j, because the e_i are implicitly latent. 2018 lapkričio 07 d., 22:37
atliko -
Pakeistos 179-181 eilutės iš
Is many-worlds theory the flip-side of least-action ? į:
Is many-worlds theory the flip-side of least-action ? Chess pieces (rooks, bishops, knights) move in very basic ways that bring to mind root systems and especially the geometries that they open up in the gap they create by being more complex e_i - e_j than the generic basis e_i. 2018 lapkričio 06 d., 15:47
atliko - 2018 lapkričio 06 d., 15:47
atliko -
Pakeistos 177-179 eilutės iš
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time. į:
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time. Is many-worlds theory the flip-side of least-action ? 2018 lapkričio 06 d., 15:34
atliko -
Pakeista 177 eilutė iš:
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? į:
When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? Comparing measurements yields "information" and it may be information which is limited by the speed of time. 2018 lapkričio 06 d., 15:27
atliko -
Pakeistos 175-177 eilutės iš
Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis. į:
Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis. When we measure spin - we impose the spin axis we are expecting - but that is an imposition of expectations related to the "waiting" that I am modeling. Also, how is that waiting related to emotional life and expectations? 2018 lapkričio 03 d., 14:11
atliko -
Pakeista 23 eilutė iš:
Express the link between algebra į:
Express the link between algebra and analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis. 2018 lapkričio 03 d., 10:51
atliko -
Pakeistos 173-175 eilutės iš
Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state. į:
Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state. Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis. 2018 lapkričio 03 d., 10:03
atliko -
Pakeistos 171-173 eilutės iš
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens. į:
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens. Symmetry group consists of distinguishable actions which accomplish nothing (leave an object invariant). So they separate the object/environment and its state. 2018 spalio 31 d., 22:54
atliko -
Pakeista 171 eilutė iš:
Path of least action (the basis for physics, namely, for Feynman diagrams) is į:
Path of least action (the basis for physics, namely, for Feynman diagrams) is violated by measurements, where we can wait and nothing happens. 2018 spalio 31 d., 22:54
atliko -
Pakeistos 169-171 eilutės iš
One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? į:
One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? Path of least action (the basis for physics, namely, for Feynman diagrams) is contradicted by measurements, where we can wait and nothing happens. 2018 spalio 08 d., 19:15
atliko -
Pakeistos 167-169 eilutės iš
How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how it yields pairs of Kostka matrices? (form and content)? į:
How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how it yields pairs of Kostka matrices? (form and content)? One-dimensional economic thinking is like linear functionals - the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinite-dimensional, which is to say, it can't be captured by a one-dimensional shadow? 2018 spalio 06 d., 12:10
atliko -
Pakeistos 165-167 eilutės iš
Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside. What are the choices with regard to? į:
Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside. What are the choices with regard to? How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how it yields pairs of Kostka matrices? (form and content)? 2018 spalio 04 d., 10:07
atliko -
Pakeista 165 eilutė iš:
Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside. į:
Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside. What are the choices with regard to? 2018 spalio 04 d., 10:07
atliko -
Pakeistos 161-165 eilutės iš
Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows. į:
Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows. Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back). Four geometries in terms of choices: paths - one-directional, lines - forwards and backwards, angles - left and right, oriented areas - inside and outside. 2018 spalio 04 d., 09:59
atliko -
Pakeistos 157-161 eilutės iš
briauna = į:
briauna = skirtingumas Special linear group has determinant 1. In general when the determinant is +/- 1 then by Cramer's rule this means that the inverse is an integer and so can have a combinatorial interpretation as such. It means that we can have combinatorial symmetry between a matrix and its inverse - neither is distinguished. Determinant 1 iff trace is 0. And trace is 0 makes for the links +1 with -1. It grows by adding such rows. 2018 spalio 04 d., 09:26
atliko -
Pakeistos 151-157 eilutės iš
Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? Study using Taylor series expansions. į:
Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? Study using Taylor series expansions. Kuom skaičius skiriasi nuo pasikartojančios veiklos - būgno mušimo? * A) veikla kažkada prasidėjo * B) kiekvienas skaičius laikomas nauju, skirtingu nuo visų kitų briauna = skirtingumas 2018 spalio 04 d., 09:24
atliko -
Pakeista 151 eilutė iš:
Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? į:
Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? Study using Taylor series expansions. 2018 spalio 04 d., 09:23
atliko -
Pakeistos 149-151 eilutės iš
Study how turning the counting around relates to cycles - finite fields. į:
Study how turning the counting around relates to cycles - finite fields. Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? 2018 spalio 04 d., 09:22
atliko -
Pakeistos 147-149 eilutės iš
What is the significance of a cube having four diagonals that can be permuted by S4? į:
What is the significance of a cube having four diagonals that can be permuted by S4? Study how turning the counting around relates to cycles - finite fields. 2018 spalio 03 d., 10:39
atliko -
Pakeistos 145-147 eilutės iš
Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. į:
Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. What is the significance of a cube having four diagonals that can be permuted by S4? 2018 spalio 03 d., 09:34
atliko -
Pakeistos 143-145 eilutės iš
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. į:
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed. 2018 spalio 03 d., 09:33
atliko -
Pakeista 143 eilutė iš:
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. į:
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ. 2018 spalio 03 d., 09:31
atliko -
Pakeistos 141-143 eilutės iš
An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. į:
An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. 2018 spalio 03 d., 09:18
atliko -
Pakeista 141 eilutė iš:
An inner product on a vector space allows it to be broken up into irreducible vector spaces. į:
An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces. 2018 spalio 03 d., 09:17
atliko -
Pakeistos 139-141 eilutės iš
Mathematical induction can be thought of as a proof by contradiction where we have organized the conditions in an infinite sequence and we are looking at the first one to fail and proving that it doesn't fail. į:
Mathematical induction can be thought of as a proof by contradiction where we have organized the conditions in an infinite sequence and we are looking at the first one to fail and proving that it doesn't fail. An inner product on a vector space allows it to be broken up into irreducible vector spaces. 2018 spalio 03 d., 09:07
atliko -
Pakeistos 137-139 eilutės iš
Inner products are sesquilinear - they have conjugate symmetry - so as not to yield lopsided answers. If they yield a complex root as an answer, then one version should yield one root and the other version should yield the other root. In other words, in a complex field, the inner product should be thought of as yielding two answers - both answers - distinguished by the notation, left-right or right-left. į:
Inner products are sesquilinear - they have conjugate symmetry - so as not to yield lopsided answers. If they yield a complex root as an answer, then one version should yield one root and the other version should yield the other root. In other words, in a complex field, the inner product should be thought of as yielding two answers - both answers - distinguished by the notation, left-right or right-left. Mathematical induction can be thought of as a proof by contradiction where we have organized the conditions in an infinite sequence and we are looking at the first one to fail and proving that it doesn't fail. 2018 spalio 03 d., 08:44
atliko -
Pakeistos 135-137 eilutės iš
Counting (by way of the simple roots) links + and - in a chain. x2-x1 etc. į:
Counting (by way of the simple roots) links + and - in a chain. x2-x1 etc. Inner products are sesquilinear - they have conjugate symmetry - so as not to yield lopsided answers. If they yield a complex root as an answer, then one version should yield one root and the other version should yield the other root. In other words, in a complex field, the inner product should be thought of as yielding two answers - both answers - distinguished by the notation, left-right or right-left. 2018 rugsėjo 30 d., 21:07
atliko -
Pakeistos 131-135 eilutės iš
Raimundas Vidūnas, deleguotas į:
Raimundas Vidūnas, deleguotas priežastingumas Counting (in Lie root system) can change to B, C, D only once! That puts a cap on the one end. Then the counting must continue on the other end, extending it. A second cap may not be put on that end. There cannot be a cycle. Counting (by way of the simple roots) links + and - in a chain. x2-x1 etc. 2018 rugsėjo 29 d., 18:24
atliko -
Pakeistos 129-131 eilutės iš
Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23). į:
Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23). Raimundas Vidūnas, deleguotas priežastingumas 2018 rugsėjo 14 d., 14:40
atliko -
Pakeistos 127-129 eilutės iš
Exponential function e^x is a symmetry from the point of view of differentiation - it is the unit element. And cos and sin are fourth roots. And look for more such roots. į:
Exponential function e^x is a symmetry from the point of view of differentiation - it is the unit element. And cos and sin are fourth roots. And look for more such roots. Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23). 2018 rugsėjo 08 d., 19:31
atliko -
Pakeistos 125-127 eilutės iš
Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai. į:
Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai. Exponential function e^x is a symmetry from the point of view of differentiation - it is the unit element. And cos and sin are fourth roots. And look for more such roots. 2018 rugsėjo 07 d., 12:31
atliko -
Pakeistos 123-125 eilutės iš
Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. But nature has forces (like gravity) that tend to create uncommon states. So we need to distinguish between the different forces at play and how they relate to what is common and uncommon. į:
Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. But nature has forces (like gravity) that tend to create uncommon states. So we need to distinguish between the different forces at play and how they relate to what is common and uncommon. Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai. 2018 rugsėjo 04 d., 20:01
atliko -
Pakeista 123 eilutė iš:
Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. į:
Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. But nature has forces (like gravity) that tend to create uncommon states. So we need to distinguish between the different forces at play and how they relate to what is common and uncommon. 2018 rugsėjo 04 d., 19:59
atliko -
Pakeistos 121-123 eilutės iš
Physics - moving backwards and forwards in time - is a (wasteful) duality. Why is it dual? į:
Physics - moving backwards and forwards in time - is a (wasteful) duality. Why is it dual? Entropy - think in terms of "common" and "uncommon" states. Uncommon states tend to common states simply because they are more frequent. 2018 rugsėjo 04 d., 19:48
atliko -
Pakeistos 119-121 eilutės iš
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. So it's a type of completeness. What makes that true? į:
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. So it's a type of completeness. What makes that true? Physics - moving backwards and forwards in time - is a (wasteful) duality. Why is it dual? 2018 rugsėjo 04 d., 14:49
atliko -
Pakeista 119 eilutė iš:
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. What makes that true? į:
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. So it's a type of completeness. What makes that true? 2018 rugsėjo 04 d., 14:49
atliko -
Pakeista 119 eilutė iš:
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. į:
(x-a) x is unknown (conscious question); a is know (unconscious answer); and the difference x-a gives the discrepancy. So a polynomial can be thought of as constructed as the product of the discrepancies - or it can be constructed as coefficients for powers. Note that EVERY polynomial with complex coefficients can be constructed as a product of such discrepancies. What makes that true? |