Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
Book.MathNotes istorijaRodyti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2019 birželio 01 d., 18:32
atliko 
Pakeistos 321323 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 319321 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 317319 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 310317 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 308310 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 306308 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 304306 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 302304 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 300302 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 298300 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 296298 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 292296 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 290292 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 288290 eilutės iš
In {$D_n$}, think of {$x_ix_j$} and {$x_i+x_j$} as complex conjugates. į:
In {$D_n$}, think of {$x_ix_j$} 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 286288 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 {$x_ix_j$} 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 284286 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 283284 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 278280 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 275278 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
atliko 
Pakeistos 273275 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
atliko 
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
atliko 
Pakeistos 271273 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
atliko 
Pakeistos 267271 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 265267 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
atliko 
Pakeistos 263265 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 261263 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 259261 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
atliko 
Pakeistos 257259 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 255257 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
atliko 
Pakeistos 251255 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
atliko 
Pakeistos 249251 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
atliko 
Pakeistos 247249 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 Falgebras [Flinear 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 Falgebras [Flinear 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
atliko 
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 Falgebras [Flinear 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 245247 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
atliko 
Pridėtos 244246 eilutės:
https://golem.ph.utexas.edu/category/2017/01/basic_category_theory_free_onl.html 2019 kovo 11 d., 11:02
atliko 
Pakeistos 243245 eilutės iš
Vector bundles: Identity and selfidentity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber  selfidentity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of selfidentity (or nonidentity) allows the edge to be flipped upside down. į:
Vector bundles: Identity and selfidentity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber  selfidentity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of selfidentity (or nonidentity) allows the edge to be flipped upside down. http://pi.math.cornell.edu/~hatcher/AT/ATpage.html 2019 kovo 10 d., 22:11
atliko 
Pakeistos 241243 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 selfidentity (like the ends of a regular strip or a Moebius strip). Identity of a point, identity of a fiber  selfidentity under continuity. How does a fiber relate to itself? Is it flipped or not? 2 kinds of selfidentity (or nonidentity) allows the edge to be flipped upside down. 2019 kovo 07 d., 14:30
atliko 
Pakeistos 239241 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 237239 eilutės iš
į:
DanielChanMaths Vandermonde determinant shows invertible  basis for finite Fourier transform 2019 kovo 05 d., 09:01
atliko 
Pakeista 233 eilutė iš:
į:
Primena trejybę. [[https://en.wikipedia.org/wiki/Homotopy_group  Wikipedia: Homotopy groups]] Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups: 2019 kovo 05 d., 09:00
atliko 
Pakeistos 229236 eilutės iš
monad = black box? į:
monad = black box? Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. Then there is a long exact sequence of homotopy groups Wikipedia: Homotopy groups Let p: E → B be a basepointpreserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is pathconnected. 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_{n1}(F) \rightarrow \cdots \rightarrow \pi_0(F) \rightarrow 0. $} 2019 vasario 17 d., 05:55
atliko 
Pakeistos 225228 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 219225 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 217219 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
atliko 
Pakeistos 215218 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/thesierpinskispaceanditsspecialproperty  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}$}. But that special case leverages the framework to establish all the other cases į:
In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How. 2019 vasario 13 d., 13:59
atliko 
Pakeistos 217218 eilutės iš
Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information. į:
Mathematics theorems relate information that is less explicit (leveraging the presumptions inherent in the framework) with information that is more explicit (expressing those presumptions). Thus mathematics makes information more explicit. It is revealing information and, in that sense, "creating" the explicitness of the information. * For example, in this [[https://www.math3ma.com/blog/thesierpinskispaceanditsspecialproperty  proposition]] about pullbacks, the statement about the pullback is much more explicit than that for the function f because it includes f as a special case when Z = x, for then f*(idX)={$f^{1}$}. But that special case leverages the framework to establish all the other cases. 2019 vasario 13 d., 13:56
atliko 
Pakeistos 215217 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
atliko 
Pakeistos 213215 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 211213 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 209211 eilutės iš
Category theory for me: distinguishing what observations are nontrivial  intrinsic to a subject  and what are observations are contentwise 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 contentwise 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 207209 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 contentwise trivial or universal  not related to the subject, but simply an aspect of abstraction. 2019 vasario 12 d., 08:52
atliko 
Pakeistos 203207 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 ncategory 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
atliko 
Pakeistos 201203 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
atliko 
Pakeistos 199201 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
atliko 
Pakeistos 197199 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
atliko 
Pakeistos 195197 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
atliko 
Pakeistos 193195 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
atliko 
Pakeistos 191193 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
atliko 
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
atliko 
Pakeistos 189191 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
atliko 
Pakeistos 187189 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
atliko 
Pakeistos 185187 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 181185 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
atliko 
Pakeistos 174181 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 172174 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
atliko 
Pakeistos 170172 eilutės iš
Note that 2dimensional phase space (as with a spring) is the simplest as there is no 1dimensional phase space and there can't be (we need both position and momentum). į:
Note that 2dimensional phase space (as with a spring) is the simplest as there is no 1dimensional 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
atliko 
Pakeistos 168170 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 2dimensional phase space (as with a spring) is the simplest as there is no 1dimensional 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
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 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
atliko 
Pakeistos 166168 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
atliko 
Pakeistos 164166 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 162164 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
atliko 
Pakeistos 159162 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 108110 eilutės:
* [[https://golem.ph.utexas.edu/category/2008/06/classical_string_theory_and_ca.html  Five related lectures by Christopher L. Rogers]] Pakeistos 129138 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 144147 eilutės iš
Conformal į:
CayleyDickson 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 156158 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
atliko 
Pakeistos 155157 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
atliko 
Pakeistos 153155 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 151153 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$} gives meaning to the two axes. One axis the opposites 1 and 1 and the other axis is the opposites i and j. And they become related 1 to i to 1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to 1. A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. 2019 sausio 10 d., 13:57
atliko 
Pakeistos 149151 eilutės iš
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$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$} gives meaning to the two axes. One axis the opposites 1 and 1 and the other axis is the opposites i and j. And they become related 1 to i to 1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to 1. 2019 sausio 10 d., 13:54
atliko 
Pakeista 149 eilutė iš:
Conformal geometry preserves angles. Radial coordinates distinguishes distances r and angles theta, and makes use of the exponential {$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 147149 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 145147 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
atliko 
Pakeistos 144145 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 142144 eilutės:
Each physical force is related to a duality: * Charge (matter and antimatter)  electromagnetism * Weak force  time reversal 2019 sausio 04 d., 03:03
atliko 
Pakeistos 138141 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 136138 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
atliko 
Pakeistos 134136 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
atliko 
Pakeistos 132134 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
atliko 
Pakeistos 130132 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
atliko 
Pakeistos 126129 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
atliko 
Pakeistos 124126 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
atliko 
Pakeistos 122124 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
atliko 
Pakeistos 120122 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
atliko 
Pakeistos 118120 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
atliko 
Pakeistos 116118 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
atliko 
Pakeistos 114116 eilutės iš
G2 requires three lines to get between any two points (?) Relate this to the threecycle. į:
G2 requires three lines to get between any two points (?) Relate this to the threecycle. 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
atliko 
Pakeistos 112114 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 threecycle. 2018 gruodžio 19 d., 12:07
atliko 
Pakeistos 110112 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
atliko 
Pakeistos 107110 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%2Fs0022000909519.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
atliko 
Pakeistos 105107 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
atliko 
Pakeistos 103105 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
atliko 
Pakeistos 99103 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
atliko 
Pakeistos 9799 eilutės iš
What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/20023902/S027309790100934X/S027309790100934X.pdf  John Baez, The Octonions]]. į:
What is the connection between Bott periodicity and spinors? See [[http://www.ams.org/journals/bull/20023902/S027309790100934X/S027309790100934X.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
atliko 
Pakeistos 9597 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/20023902/S027309790100934X/S027309790100934X.pdf  John Baez, The Octonions]]. 2018 gruodžio 16 d., 18:27
atliko 
Pakeistos 9395 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
atliko 
Pakeistos 9193 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 8991 eilutės iš
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory  YangMills theory]]. į:
[[https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory  YangMills 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  YangMills theory]]. 2018 gruodžio 15 d., 11:40
atliko 
Pakeistos 8689 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 8486 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
atliko 
Pakeistos 8284 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 8082 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 7880 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 7178 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 crosspolytope 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 6970 eilutės:
Root systems relate two spheres  they relate two "sheets". Logic likewise relates two sheets: a sheet and a metasheet 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 6468 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 crosspolytopes we divide the "is" into two halves, preserving the "not" intact. * for Dn we have simply "this" and "that" (notthis). 2018 gruodžio 10 d., 13:14
atliko 
Pridėtos 6263 eilutės:
Use "this" and "that" as unmarked opposites  conjugates. 2018 gruodžio 09 d., 15:01
atliko 
Pakeistos 4748 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 5761 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 8fold folding and unfolding. 2018 gruodžio 09 d., 14:59
atliko 
Pakeistos 5357 eilutės iš
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0x_1)$} but the others appear both positive and negative. į:
{$x_0$} is fundamentally different from {$x_i$}. The former appears in the positive form {$\pm(x_0x_1)$} 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 {$\pm(x_0x_1)$} but the others appear both positive and negative. 2018 gruodžio 09 d., 14:42
atliko 
Pakeistos 4753 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 selffolding. 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
atliko 
Pakeistos 3947 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
atliko 
Pakeistos 3739 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
atliko 
Pakeistos 3537 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
atliko 
Pakeistos 3335 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
atliko 
Pakeistos 3133 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
atliko 
Pakeistos 2931 eilutės iš
Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? į:
Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional 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 810 eilutės:
* Shear: sideshot Ištrintos 910 eilutės:
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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
atliko 
Ištrintos 69 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 1112 eilutės:
Ištrintos 1216 eilutės:
[[https://ac.elscdn.com/S0012365X96003500/1s2.0S0012365X96003500main.pdf?_tid=7e02ab6eb80c44b39ef033327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08  Geometry of Classical Groups over Finite Fields and Its Applications]], Zhexian 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
atliko 
Ištrintos 1819 eilutės:
Collect examples of the arithmetic hierarchy such as calculus (deltaepsilon), differentiable manifolds, etc. Ištrintos 2223 eilutės:
Ištrintos 2526 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, leftright or rightleft. 2018 lapkričio 11 d., 17:29
atliko 
Ištrintos 03 eilutės:
What kind of conjugation is that? Pakeistos 910 eilutės iš
į:
* Matematika mus moko, kaip besąlygiškumą reikšti sąlygomis, o tai įmanoma sąlygiškai. Pakeistos 2629 eilutės iš
* Category of perspectives: steppingin and steppingout 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 2830 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 3033 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 3839 eilutės:
Pakeistos 5157 eilutės iš
How does the expansion (x1 + ... + xm)N relate to the matrix of nonnegative integers? and how Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? The combinatorial interpretation of nchoosek counts placements = "external arrangements" n! x...x (nk+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). į:
Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? 2018 lapkričio 11 d., 17:20
atliko 
Ištrintos 02 eilutės:
Function can be partial, whereas a permutation maps completely. Ištrintos 45 eilutės:
Ištrintos 1824 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 dvejybinistrejybinis  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 2021 eilutės:
Ištrintos 2332 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 1dimensional (and its dimension) and its covector is n1 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 6364 eilutės:
Hyperoctahedral group is shuffling cards that can also be flipped or not (front or back). 2018 lapkričio 11 d., 17:00
atliko 
Pakeistos 12 eilutės iš
į:
Ištrintos 78 eilutės:
Pakeistos 1221 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 2023 eilutės:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. Ištrintos 2833 eilutės:
http://wwwusers.math.umn.edu/~dgrinber/ http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf See: Combinatorics and the field with one element. Witt vectors  padic integers Ištrinta 36 eilutė:
Pakeistos 4850 eilutės iš
Symplectic form is skewsymmetric  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 5253 eilutės iš
į:
Ištrintos 6163 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
atliko 
Ištrintos 23 eilutės:
Ištrintos 411 eilutės:
Symmetry group relates: * Algebraic structure, "group" * Analytic (recurring activity) transformations Axiom of infinity  can be eliminated  it is unnecessary in "implicit math". Ištrintos 1213 eilutės:
Ištrintos 1620 eilutės:
Matematikos įrodymo būdai * 6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję Ištrintos 1819 eilutės:
Ištrintos 2022 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 5356 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 5556 eilutės:
Ištrintos 6566 eilutės:
Ištrintos 6772 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 7072 eilutės iš
į:
What do inside and outside mean in symplectic (Hamiltionian, Lagrangian) mechanics? Pakeistos 8588 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 8889 eilutės iš
į:
Pakeistos 9196 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. x2x1 etc. į:
Ištrintos 9394 eilutės:
Ištrintos 103104 eilutės:
Pakeistos 122140 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 manyworlds theory the flipside of leastaction ? 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_ie_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_ie_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 nchoosek counts placements = "external arrangements" n! x...x (nk+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). Entanglement  particle and antiparticle are in the same place and time  and they have the same clock and coordinates į:
The combinatorial interpretation of nchoosek counts placements = "external arrangements" n! x...x (nk+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 187189 eilutės iš
The combinatorial interpretation of nchoosek counts placements = "external arrangements" n! x...x (nk+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). į:
The combinatorial interpretation of nchoosek counts placements = "external arrangements" n! x...x (nk+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem). Entanglement  particle and antiparticle are in the same place and time  and they have the same clock and coordinates 2018 lapkričio 09 d., 21:59
atliko 
Pakeistos 185187 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 nchoosek counts placements = "external arrangements" n! x...x (nk+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 183185 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_ie_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_ie_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_ie_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_ie_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 181183 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_ie_j, e_i+e_j etc.) because everything is explicit. And the highest energy system is A_n, given by e_ie_j, because the e_i are implicitly latent. 2018 lapkričio 07 d., 22:37
atliko 
Pakeistos 179181 eilutės iš
Is manyworlds theory the flipside of leastaction ? į:
Is manyworlds theory the flipside of leastaction ? 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 177179 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 manyworlds theory the flipside of leastaction ? 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 175177 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 173175 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 171173 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 169171 eilutės iš
Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? į:
Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional 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 167169 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)? Onedimensional 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 infinitedimensional, which is to say, it can't be captured by a onedimensional shadow? 2018 spalio 06 d., 12:10
atliko 
Pakeistos 165167 eilutės iš
Four geometries in terms of choices: paths  onedirectional, 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  onedirectional, 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  onedirectional, lines  forwards and backwards, angles  left and right, oriented areas  inside and outside. į:
Four geometries in terms of choices: paths  onedirectional, 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 161165 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  onedirectional, lines  forwards and backwards, angles  left and right, oriented areas  inside and outside. 2018 spalio 04 d., 09:59
atliko 
Pakeistos 157161 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 151157 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 149151 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 147149 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 145147 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 143145 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 141143 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 139141 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 137139 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, leftright or rightleft. 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, leftright or rightleft. 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 135137 eilutės iš
Counting (by way of the simple roots) links + and  in a chain. x2x1 etc. į:
Counting (by way of the simple roots) links + and  in a chain. x2x1 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, leftright or rightleft. 2018 rugsėjo 30 d., 21:07
atliko 
Pakeistos 131135 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. x2x1 etc. 2018 rugsėjo 29 d., 18:24
atliko 
Pakeistos 129131 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 127129 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 125127 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 123125 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 121123 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 119121 eilutės iš
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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:
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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:
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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:
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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:
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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:
(xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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? 2018 rugsėjo 04 d., 14:48
atliko 
Pakeistos 117119 eilutės iš
* 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) į:
* 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) (xa) x is unknown (conscious question); a is know (unconscious answer); and the difference xa 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. 2018 rugsėjo 03 d., 13:57
atliko 
Pakeista 117 eilutė iš:
* elementary symmetric functions are analogous to prime factorizations of numbers  monomial symmetric functions are analogous to the numbers as į:
* 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) 2018 rugsėjo 03 d., 13:57
atliko 
Pridėtos 115117 eilutės:
Fundamental theorems * elementary symmetric functions are analogous to prime factorizations of numbers  monomial symmetric functions are analogous to the numbers as such 2018 liepos 30 d., 08:23
atliko 
Pakeistos 105114 eilutės iš
Study the chaos of watersheds for the divisions of everything  the twosome, threesome, foursome, etc. Note how a "hill" arises (for example, with the fivesome) and how that hill becomes a division into two (with the sevensome). į:
Study the chaos of watersheds for the divisions of everything  the twosome, threesome, foursome, etc. Note how a "hill" arises (for example, with the fivesome) and how that hill becomes a division into two (with the sevensome). * Category theory models perspectives and attention shifting. (Or thoughts as objects?) * Category of perspectives: steppingin and steppingout 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). * 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. * Trikampis  riba (jausmai)  simplektinė geometrija. * 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. 2018 liepos 22 d., 11:23
atliko 
Pakeista 105 eilutė iš:
Study the chaos of watersheds for the divisions of everything  the twosome, threesome, foursome, etc. į:
Study the chaos of watersheds for the divisions of everything  the twosome, threesome, foursome, etc. Note how a "hill" arises (for example, with the fivesome) and how that hill becomes a division into two (with the sevensome). 2018 liepos 22 d., 11:21
atliko 
Pakeistos 103105 eilutės iš
[[https://ac.elscdn.com/S0012365X96003500/1s2.0S0012365X96003500main.pdf?_tid=7e02ab6eb80c44b39ef033327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08  Geometry of Classical Groups over Finite Fields and Its Applications]], Zhexian į:
[[https://ac.elscdn.com/S0012365X96003500/1s2.0S0012365X96003500main.pdf?_tid=7e02ab6eb80c44b39ef033327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08  Geometry of Classical Groups over Finite Fields and Its Applications]], Zhexian Wan Study the chaos of watersheds for the divisions of everything  the twosome, threesome, foursome, etc. 2018 liepos 16 d., 15:04
atliko 
Pakeistos 101103 eilutės iš
What do inside and outside mean in symplectice (Hamiltionian, Lagrangian) mechanics? į:
What do inside and outside mean in symplectice (Hamiltionian, Lagrangian) mechanics? [[https://ac.elscdn.com/S0012365X96003500/1s2.0S0012365X96003500main.pdf?_tid=7e02ab6eb80c44b39ef033327ac6a6d1&acdnat=1531742605_d128ba18dbe99b9c084df275fab65e08  Geometry of Classical Groups over Finite Fields and Its Applications]], Zhexian Wan 2018 liepos 16 d., 14:58
atliko 
Pakeistos 99101 eilutės iš
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. į:
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? 2018 liepos 16 d., 14:56
atliko 
Pakeistos 9799 eilutės iš
Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. Orientation is a complete, absolute, total distinction between inside and outside, their complete segregation and isolation. (In contrast to the yinyang symbol.) SO it is highly tenuous  it can break at any single point  but it can eternally grow more weighty. į:
Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. Orientation is a complete, absolute, total distinction between inside and outside, their complete segregation and isolation. (In contrast to the yinyang symbol.) SO it is highly tenuous  it can break at any single point  but it can eternally grow more weighty. 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. 2018 liepos 16 d., 14:03
atliko 
Pakeista 97 eilutė iš:
Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. į:
Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. Orientation is a complete, absolute, total distinction between inside and outside, their complete segregation and isolation. (In contrast to the yinyang symbol.) SO it is highly tenuous  it can break at any single point  but it can eternally grow more weighty. 2018 liepos 16 d., 13:46
atliko 
Pakeistos 9597 eilutės iš
Symplectic form is skewsymmetric  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:
Symplectic form is skewsymmetric  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. Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. 2018 liepos 16 d., 13:46
atliko 
Pakeistos 9395 eilutės iš
An example of variables: a function Phi may be based on an point in the manifold whereas its coordinates may be based on a particular, explicit, specific chart. See Penrose page 186. į:
An example of variables: a function Phi may be based on an point in the manifold whereas its coordinates may be based on a particular, explicit, specific chart. See Penrose page 186. Symplectic form is skewsymmetric  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. 2018 liepos 16 d., 13:07
atliko 
Pakeistos 8993 eilutės iš
A vector is 1dimensional (and its dimension) and its covector is n1 dimensional (it is normal to the vector). In this sense they complement each other. į:
A vector is 1dimensional (and its dimension) and its covector is n1 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. An example of variables: a function Phi may be based on an point in the manifold whereas its coordinates may be based on a particular, explicit, specific chart. See Penrose page 186. 2018 liepos 16 d., 12:43
atliko 
Pakeistos 8789 eilutės iš
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. į:
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 1dimensional (and its dimension) and its covector is n1 dimensional (it is normal to the vector). In this sense they complement each other. 2018 liepos 16 d., 12:36
atliko 
Pakeista 87 eilutė iš:
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. į:
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. 2018 liepos 16 d., 12:34
atliko 
Pakeistos 8587 eilutės iš
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. į:
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. 2018 liepos 16 d., 12:27
atliko 
Pakeistos 8385 eilutės iš
Partial derivative  formal (explicit) based on change in variable, total derivative  actual (implicit) based on change in value. į:
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. 2018 liepos 13 d., 14:41
atliko 
Pakeistos 12 eilutės iš
[[https://www.youtube.com/watch?v=pXGTevGJ01o  Symplectic manifolds]] į:
[[https://www.youtube.com/watch?v=pXGTevGJ01o  Symplectic manifolds]] 2018 liepos 13 d., 14:34
atliko 
Pakeista 84 eilutė iš:
Partial derivative  formal (explicit) į:
Partial derivative  formal (explicit) based on change in variable, total derivative  actual (implicit) based on change in value. 2018 liepos 13 d., 14:33
atliko 
Pakeistos 8284 eilutės iš
Collect examples of symmetry breaking. į:
Collect examples of symmetry breaking. Partial derivative  formal (explicit), total derivative  actual (implicit). 2018 liepos 12 d., 12:21
atliko 
Pridėtos 13 eilutės:
[[https://www.youtube.com/watch?v=pXGTevGJ01o  Symplectic manifolds]] 56:48 2018 liepos 12 d., 12:07
atliko 
Pakeistos 7779 eilutės iš
Collect examples of the arithmetic hierarchy such as calculus (deltaepsilon), differentiable manifolds, etc. į:
Collect examples of the arithmetic hierarchy such as calculus (deltaepsilon), differentiable manifolds, etc. Collect examples of symmetry breaking. 2018 liepos 12 d., 12:06
atliko 
Pakeistos 7577 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:
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. Collect examples of the arithmetic hierarchy such as calculus (deltaepsilon), differentiable manifolds, etc. 2018 liepos 12 d., 11:53
atliko 
Pakeistos 7375 eilutės iš
An example of extending the domain: Lie algebra (infinitesimal) vs. Lie group (broader domain). į:
An example of extending the domain: Lie algebra (infinitesimal) vs. Lie group (broader domain). 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. 2018 sausio 31 d., 22:01
atliko 
Pakeistos 7173 eilutės iš
What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? į:
What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? An example of extending the domain: Lie algebra (infinitesimal) vs. Lie group (broader domain). 2018 sausio 22 d., 10:58
atliko 
Pakeistos 6971 eilutės iš
Intrinsic ambiguity of propositions  every proposition is a general rule, which can be questioned or applied. į:
Intrinsic ambiguity of propositions  every proposition is a general rule, which can be questioned or applied. What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix? 2018 sausio 19 d., 19:52
atliko 
Pakeistos 5769 eilutės iš
* 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. į:
* 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. Darij Grinberg http://wwwusers.math.umn.edu/~dgrinber/ http://www.cip.ifi.lmu.de/~grinberg/about/rs.pdf See: Combinatorics and the field with one element. Witt vectors  padic integers Matematikos savokų pagrindas yra dvejybinistrejybinis  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. Jiri Raclavsky  Frege, Tichy  Twodimensional conception of inference. Inference rules operate on derivations. Go from one truth to another truth, not from one assumption to another assumption. Intrinsic ambiguity of propositions  every proposition is a general rule, which can be questioned or applied. 2018 sausio 19 d., 19:47
atliko 
Pakeistos 5456 eilutės iš
Mandelbrot aibė skiria vidines ir išorines veiklas, besilaikančias erdvės ir nesilaikančios jokios erdvės. 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. į:
Mandelbort * 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. 2018 sausio 19 d., 19:36
atliko 
Pakeistos 4956 eilutės iš
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. į:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. Montažas ir geometrinės permainos: * Shear: sideshot Mandelbrot aibė skiria vidines ir išorines veiklas, besilaikančias erdvės ir nesilaikančios jokios erdvės. 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. 2018 sausio 19 d., 14:38
atliko 
Pakeistos 4970 eilutės iš
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. Analysis is based on the "looseness" by which a local property (the slope locally) may not maintain globally. And this looseness is of different kinds: * nonlooseness of path  discrete (integer, rational) affine * looseness of line  reals  projective * looseness of angle  complexes  conformal * looseness of orientation (cross product)  quaternions  symplectic So looseness is the flipside of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process  so taking the limit in all directions for the complexes relates to preserving angles. What would be the notion of differentiation for a function on the quaternions? http://math.ucr.edu/home/baez/symplectic.html https://en.m.wikipedia.org/wiki/Hyperkähler_manifold Dyson's threefold way R C H https://arxiv.org/abs/1101.5690 http://math.ucr.edu/home/baez/tenfold.html Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215. http://www.scholarpedia.org/article/Symplectic_maps į:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. 2018 sausio 18 d., 00:02
atliko 
Pakeistos 6870 eilutės iš
Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215. į:
Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215. http://www.scholarpedia.org/article/Symplectic_maps 2018 sausio 17 d., 11:57
atliko 
Pakeistos 6568 eilutės iš
Dyson's threefold way R C į:
Dyson's threefold way R C H https://arxiv.org/abs/1101.5690 http://math.ucr.edu/home/baez/tenfold.html Freeman J. Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215. 2018 sausio 17 d., 00:16
atliko 
Pridėtos 6263 eilutės:
https://en.m.wikipedia.org/wiki/Hyperkähler_manifold 2018 sausio 17 d., 00:15
atliko 
Pakeistos 5961 eilutės iš
What would be the notion of differentiation for a function on the quaternions? į:
What would be the notion of differentiation for a function on the quaternions? http://math.ucr.edu/home/baez/symplectic.html 2018 sausio 16 d., 16:50
atliko  2018 sausio 16 d., 16:50
atliko 
Pakeistos 5759 eilutės iš
So looseness is the flipside of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process  so taking the limit in all directions for the complexes relates to preserving angles. į:
So looseness is the flipside of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process  so taking the limit in all directions for the complexes relates to preserving angles. What would be the notion of differentiation for a function on the quaternions? 2018 sausio 16 d., 12:40
atliko 
Pridėta 53 eilutė:
* nonlooseness of path  discrete (integer, rational) affine Pakeista 57 eilutė iš:
So looseness is the flipside of invariance. į:
So looseness is the flipside of invariance. We see the role of equivalence as based on limits. And also we see the qualitative distinction based on the nature of the limiting process  so taking the limit in all directions for the complexes relates to preserving angles. 2018 sausio 16 d., 12:30
atliko 
Pakeistos 4956 eilutės iš
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. į:
Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. Analysis is based on the "looseness" by which a local property (the slope locally) may not maintain globally. And this looseness is of different kinds: * looseness of line  reals  projective * looseness of angle  complexes  conformal * looseness of orientation (cross product)  quaternions  symplectic So looseness is the flipside of invariance. 2017 gruodžio 20 d., 13:44
atliko 
Pakeistos 4749 eilutės iš
How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? į:
How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. 2017 spalio 25 d., 21:01
atliko 
Pakeistos 4547 eilutės iš
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. į:
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. How do special rim hook tableaux (which depend on the behavior of their endpoints) relate to Dynkin diagrams (which also depend on their endpoints)? 2017 spalio 24 d., 09:04
atliko 
Pakeistos 4345 eilutės iš
* Apibendrinimas yra "objekto" kūrimas. į:
* Apibendrinimas yra "objekto" kūrimas. 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. 2017 spalio 24 d., 07:47
atliko 
Ištrintos 2021 eilutės:
Ištrinta 22 eilutė:
Ištrinta 30 eilutė:
2017 spalio 24 d., 07:47
atliko 
Ištrintos 614 eilutės:
* Cube: all vertices have a genealogy, a combination of +s and s. * Halfcube defines + for all, thus defines marked opposites. Dual: * Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory). * Crosspolytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory). Ištrintos 1213 eilutės:
Ištrintos 1623 eilutės:
Symmetric group action on an octahedron is marked, 1 and 1, the octahedron itself is unmarked. A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other. * Constructiveness  closed sets any intersections and finite unions are open sets constructive Pakeistos 2528 eilutės iš
If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable. į:
Ištrintos 2932 eilutės:
Trikampis  išauga požiūrių skaičius apibudinant: affinetaškai0, projectivetiesės1, conformalkampai2, symplecticplotai3. Pakeistos 3235 eilutės iš
Matematika skiria vidines sandaras (semantika) ir išorinius santykius (sintaksė). Užtat labai svarbu mąstyti apie "viską", kuriam nėra išorinių santykių. Panašiai gal būtų galima mąstyti apie nieką, kur nėra vidinės sandaros. Nors viskas irgi neturi vidinės sandaros. Užtat viskam semantika ir sintaksė yra atitinkamai visiškai paprasta. į:
Matematikos įrodymo būdai * 6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję Pakeistos 4047 eilutės iš
Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę. į:
Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę. Variables * 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. Apibendrinimas * Protas apibendrina. Kaip nagrinėti apibendrinimą? Suvokti neurologiškai (arba tinklais). Jeigu keli pavyzdžiai (ar netgi vienas) turi tam tikras bendras savybes, tada tas apibendrintas savybes gali naujai priskirti naujoms jų apibudintoms sąvokoms. * Apibendrinimas yra "objekto" kūrimas. 2017 rugsėjo 06 d., 16:09
atliko 
Pakeistos 5266 eilutės iš
Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern. į:
Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern. Trikampis  išauga požiūrių skaičius apibudinant: affinetaškai0, projectivetiesės1, conformalkampai2, symplecticplotai3. The mind is augmented through the "symmetric group" which is the system that augments our imagination. 6 matematikos irodymo budai skiriaisi nuo issiaiskinimo budu taciau kaip jie susiję Matematika skiria vidines sandaras (semantika) ir išorinius santykius (sintaksė). Užtat labai svarbu mąstyti apie "viską", kuriam nėra išorinių santykių. Panašiai gal būtų galima mąstyti apie nieką, kur nėra vidinės sandaros. Nors viskas irgi neturi vidinės sandaros. Užtat viskam semantika ir sintaksė yra atitinkamai visiškai paprasta. Apibrėžti "gebėjimus" ir kaip matematinis mąstymas suveda skirtingus gebėjimus suvokti kelis, keliolika, keliasdešimts, tūkstančius ir t.t. daiktų Kodėl yra tiek daug būdų įrodyti Pitagoro teoremą? Gramatika visada turi prasmę. Matematika yra tai, kas pavaldu logikai, bet nebūtinai turi prasmę. 2017 sausio 08 d., 23:22
atliko 
Pakeistos 5052 eilutės iš
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. į:
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. Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern. 2017 sausio 05 d., 13:15
atliko 
Pakeista 50 eilutė iš:
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". į:
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. 2017 sausio 05 d., 13:15
atliko 
Pakeistos 4850 eilutės iš
[[http://wwwpersonal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf  What is Mathematical Music Theory?]] į:
[[http://wwwpersonal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf  What is Mathematical Music Theory?]] 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". 2016 gruodžio 20 d., 20:34
atliko 
Pakeistos 4648 eilutės iš
If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable. į:
If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable. [[http://wwwpersonal.umd.umich.edu/~tmfiore/1/FioreWhatIsMathMusTheoryBasicSlides.pdf  What is Mathematical Music Theory?]] 2016 gruodžio 15 d., 17:43
atliko 
Pakeistos 4446 eilutės iš
Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes? į:
Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes? If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable. 2016 gruodžio 15 d., 17:06
atliko 
Pakeistos 4244 eilutės iš
Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis. į:
Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis. Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes? 2016 gruodžio 15 d., 17:00
atliko 
Pridėtos 3942 eilutės:
Long exact sequence from short exact sequence: derived functors. Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis. 2016 gruodžio 15 d., 16:44
atliko 
Pridėtos 3738 eilutės:
Kan extension  extending the domain  every concept is a Kan extension https://en.wikipedia.org/wiki/Kan_extension 2016 gruodžio 15 d., 16:32
atliko 
Pridėtos 3536 eilutės:
Relate triangulated categories with representations of threesome 2016 gruodžio 13 d., 23:19
atliko 
Pridėtos 3334 eilutės:
* Constructiveness  closed sets any intersections and finite unions are open sets constructive 2016 lapkričio 24 d., 20:12
atliko 
Pridėtos 3132 eilutės:
A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other. 2016 liepos 04 d., 17:20
atliko 
Pakeista 1 eilutė iš:
Time: A moving point is a line, a moving line is a plane, a moving plane is a volume... Time is the addition of a scalar (from a field), thus the addition of choice. Time relates affine and projective space. Compare time with space. į:
Time: A moving point is a line, a moving line is a plane, a moving plane is a volume... Time is the addition of a scalar (from a field), thus the addition of choice. Time relates affine and projective space. Compare time with space. A moving "center" is a point: the center is what moves, thus what has time. 2016 liepos 04 d., 17:19
atliko 
Pakeista 1 eilutė iš:
į:
Time: A moving point is a line, a moving line is a plane, a moving plane is a volume... Time is the addition of a scalar (from a field), thus the addition of choice. Time relates affine and projective space. Compare time with space. 2016 birželio 23 d., 08:43
atliko 
Pakeista 1 eilutė iš:
Baez Rep 4, į:
Baez Rep 4, 1:01 min. 2016 birželio 23 d., 00:29
atliko 
Pakeista 28 eilutė iš:
Symmetric group action on an octahedron is marked, the octahedron itself is unmarked. į:
Symmetric group action on an octahedron is marked, 1 and 1, the octahedron itself is unmarked. 2016 birželio 23 d., 00:25
atliko 
Pridėtos 2728 eilutės:
Symmetric group action on an octahedron is marked, the octahedron itself is unmarked. 2016 birželio 23 d., 00:25
atliko 
Pridėtos 2526 eilutės:
Is the fusion of vertices in the demicube related to the fusion of edges of a square to create a torus, or of vertices to create a circle, etc.? 2016 birželio 23 d., 00:22
atliko 
Pridėtos 1924 eilutės:
Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the crosspolytope. The factoring (number of simplexes n choose k  dependent simplex) x (number of flags on k  independent Euclidean) x (number of flags on nk  independent Euclidean) = (number of flags on n) What kind of conjugation is that? 2016 birželio 23 d., 00:10
atliko 
Pridėtos 1718 eilutės:
Axiom of infinity  can be eliminated  it is unnecessary in "implicit math". 2016 birželio 23 d., 00:07
atliko 
Pridėtos 1316 eilutės:
Symmetry group relates: * Algebraic structure, "group" * Analytic (recurring activity) transformations 2016 birželio 23 d., 00:05
atliko 
Pridėtos 1012 eilutės:
Dual: * Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory). * Crosspolytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory). 2016 birželio 22 d., 23:59
atliko 
Pakeistos 59 eilutės iš
Unmarked opposites: crosspolytope. Each dimension independently + or  (all or nothing). Cube: all vertices have a genealogy, a combination of +s and s. Halfcube defines + for all, thus defines marked opposites. į:
* Unmarked opposites: crosspolytope. Each dimension independently + or  (all or nothing). * Cube: all vertices have a genealogy, a combination of +s and s. * Halfcube defines + for all, thus defines marked opposites. 2016 birželio 22 d., 23:58
atliko 
Pridėtos 59 eilutės:
Unmarked opposites: crosspolytope. Each dimension independently + or  (all or nothing). Cube: all vertices have a genealogy, a combination of +s and s. Halfcube defines + for all, thus defines marked opposites. 2016 birželio 22 d., 23:44
atliko 
Pridėtos 14 eilutės:
Function can be partial, whereas a permutation maps completely. 
MathNotesNaujausi pakeitimai 
Puslapis paskutinį kartą pakeistas 2019 birželio 01 d., 18:32
