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Book.Duality istorijaPaslėpti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2019 birželio 08 d., 09:58
atliko 
Pakeista 288 eilutė iš:
Note that turns left and right are conjugates but not a division of everything because there can be no turn. Instead, the twosome is "turn" (left/right) and "noturn". į:
Note that turns left and right are conjugates but not a division of everything because there can be no turn. Instead, the twosome is "turn" (left/right) and "noturn". "Turn" does not need to be marked, but "noturn" needs to be marked ("no"). Although, contentwise, notturning is unmarked and turning is marked. 2019 birželio 08 d., 09:56
atliko 
Pakeistos 286288 eilutės iš
* Electrical duality [[https://www.ams.org/publicoutreach/featurecolumn/fc201905  Topology and Elementary Electric Circuit Theory, II: Duality]] Tony į:
* Electrical duality [[https://www.ams.org/publicoutreach/featurecolumn/fc201905  Topology and Elementary Electric Circuit Theory, II: Duality]] Tony Phillips Note that turns left and right are conjugates but not a division of everything because there can be no turn. Instead, the twosome is "turn" (left/right) and "noturn". 2019 gegužės 19 d., 22:27
atliko 
Pakeistos 284286 eilutės iš
* Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates). į:
* Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates). * Electrical duality [[https://www.ams.org/publicoutreach/featurecolumn/fc201905  Topology and Elementary Electric Circuit Theory, II: Duality]] Tony Phillips 2019 vasario 13 d., 15:17
atliko 
Pakeista 3 eilutė iš:
See: [[Math Notebook]] į:
See: [[Math Notebook]], [[Category duality]] 2019 vasario 13 d., 15:17
atliko 
Ištrintos 89 eilutės:
Pakeista 10 eilutė iš:
į:
2019 vasario 13 d., 15:16
atliko 
Pridėtos 14 eilutės:
>>bgcolor=#E9F5FC<<  See: [[Math Notebook]] Pridėtos 69 eilutės:
'''Investigation: Understand mathematics as the discrimination of a variety of dualities.'''  >><< 2019 vasario 13 d., 12:57
atliko 
Pridėta 204 eilutė:
* Every isomorphism is a duality in that it goes hand in hand with its inverse. If the domain and codomain are the same, then it is selfdual. Pakeistos 278280 eilutės iš
* Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates) * Every isomorphism is a duality in that it goes hand in hand with its inverse į:
* Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates). 2019 vasario 13 d., 12:56
atliko 
Pakeistos 277279 eilutės iš
* Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates). į:
* Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates). * Every isomorphism is a duality in that it goes hand in hand with its inverse. 2019 vasario 10 d., 09:00
atliko 
Pridėta 49 eilutė:
* [[https://en.wikipedia.org/wiki/Category:Duality_theories  Wikipedia Category: Duality theories]] 2019 vasario 05 d., 13:36
atliko 
Pakeista 15 eilutė iš:
* Dual problems in [[https://en.wikipedia.org/wiki/Linear_programming  linear programming]]. What sort of duality is this? į:
* Dual problems in [[https://en.wikipedia.org/wiki/Linear_programming#Duality  linear programming]]. What sort of duality is this? Is it related to adjoints? 2019 sausio 20 d., 20:37
atliko 
Pridėta 164 eilutė:
* [[http://www.crm.umontreal.ca/Bott08/pdf/witten.pdf  Geometric Langlands duality from six dimensions]] Edward Witten 2019 sausio 19 d., 22:54
atliko 
Pakeista 2 eilutė iš:
į:
 Pridėta 16 eilutė:
 2019 sausio 19 d., 22:54
atliko 
Pakeista 265 eilutė iš:
* The CayleyDickson construction is all about duality breaking. It yields į:
* The CayleyDickson construction is all about duality breaking. It thereby yields noncommutativity, nonassociativity, etc. 2019 sausio 19 d., 22:53
atliko 
Pakeistos 265266 eilutės iš
į:
* The CayleyDickson construction is all about duality breaking. It yields commutativity, associativity, etc. Pakeistos 270272 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. * Kai užsimirštame, kai žinojimas tampa nežinojimu (pavyzdžiui, metant monetą) tada iškyla jungtinės priešingybės (conjugates). 2019 sausio 03 d., 22:31
atliko 
Pridėta 18 eilutė:
See: [[http://math.ucr.edu/home/baez/dual/  John Baez: Duality in Logic and Physics]] 2018 gruodžio 09 d., 19:35
atliko 
Pridėta 152 eilutė:
* Internal vs. external geometry = implicit vs. embedding = vector space vs. dual space (functionals) 2018 lapkričio 11 d., 17:31
atliko 
Pridėtos 266267 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., 17:19
atliko 
Pridėtos 263265 eilutės:
* 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. 2018 lapkričio 11 d., 16:55
atliko 
Pakeista 14 eilutė 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. Study nonassociativity, as with the Lie bracket or subtraction. 2018 lapkričio 11 d., 16:54
atliko 
Pridėta 14 eilutė:
* Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc. 2018 lapkričio 11 d., 16:37
atliko 
Pakeistos 259261 eilutės iš
Duality breaking į:
Duality breaking * 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. * 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 lapkričio 11 d., 16:35
atliko 
Pakeistos 259262 eilutės iš
>><< 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 lapkričio 08 d., 13:53
atliko 
Pridėtos 252257 eilutės:
 More: The [[https://en.wikipedia.org/wiki/Transpose_of_a_linear_map  transpose of a linear map]] between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces 2018 rugsėjo 03 d., 12:11
atliko 
Pridėtos 1213 eilutės:
* Dual problems in [[https://en.wikipedia.org/wiki/Linear_programming  linear programming]]. What sort of duality is this? 2018 liepos 14 d., 13:03
atliko 
Pakeista 147 eilutė iš:
* Vectors and covectors. į:
* Vectors and covectors. A vector is 1dimensional and a covector is n1 dimensional hyperplane (tangent plane), see Penrose chapter 12. 2018 birželio 01 d., 12:47
atliko 
Pakeista 16 eilutė iš:
I am studying the various cases of duality in math. I imagine that at the heart is the duality between zero and infinity by way of one as in [[Gods Dance  God's Dance]]. Duality is the basis for logic, and mathematics gives the ways of deviating from duality. į:
I am studying the various cases of duality in math. I imagine that at the heart is the duality between zero and infinity by way of one as in [[Gods Dance  God's Dance]]. Duality is the basis for logic, and mathematics gives the ways of deviating from duality. Duality is also the structural mirror established within the foursome, fivesome, sixsome and sevensome. 2018 gegužės 11 d., 12:20
atliko 
Pridėta 24 eilutė:
* Logic deals with syntax  external relationships, as in category theory, because it is the syntactic form of the argument which is independent of the actual content. 2018 vasario 26 d., 11:35
atliko 
Pakeista 33 eilutė iš:
* In ring theory, there is a subtle distinction between the į:
* In ring theory, there is a subtle distinction between the descending chain condition  Artinian rings, and the ascending chain condition  Noetherian rings. 2018 vasario 26 d., 11:35
atliko 
Pridėtos 2733 eilutės:
'''Math is subtle deviations from pure duality''' These subtle deviations seem to leverage infinity. * Topology is based on defining open sets to require the inclusion of arbitrary unions but only finite intersections of open sets. * In ring theory, there is a subtle distinction between the ascending chain condition  Artinian rings, and the descending chain condition  Noetherian rings. 2018 sausio 07 d., 17:29
atliko 
Pridėta 78 eilutė:
* Conjugation (ab)* = b*a* is very important in the CayleyDickson construction of the numbers: real, complex, quaternion, octonion. 2017 lapkričio 28 d., 09:20
atliko 
Pridėta 91 eilutė:
* For a normal subgroup, the left cosets match the right cosets. 2017 lapkričio 04 d., 19:15
atliko 
Pridėta 11 eilutė:
* Make a list of the central objects in combinatorics  look at Stanley's books. 2017 lapkričio 04 d., 19:11
atliko 
Pridėta 10 eilutė:
* Make a list of the central mathematical examples to study and relate. 2017 lapkričio 04 d., 19:09
atliko 
Pridėta 9 eilutė:
* Relate my combinatorial proof of the CayleyHamilton theorem to [[https://en.wikipedia.org/wiki/Nakayama%27s_lemma  Nakayama's lemma]] making use of Atiyah's observation. 2017 spalio 31 d., 22:03
atliko 
Pakeista 32 eilutė iš:
* [[https://mathoverflow.net/questions/73711/theconceptofduality  Math į:
* [[https://mathoverflow.net/questions/73711/theconceptofduality  Math Overflow]] 2017 spalio 31 d., 22:01
atliko 
Pridėta 32 eilutė:
* [[https://mathoverflow.net/questions/73711/theconceptofduality  Math Exchange]] 2017 spalio 31 d., 22:00
atliko 
Pakeista 225 eilutė iš:
[[http://www.iecl.univlorraine.fr/~Wolfgang.Bertram/AtiyahDuality.pdf  Duality in į:
[[http://www.iecl.univlorraine.fr/~Wolfgang.Bertram/AtiyahDuality.pdf  Duality in Mathematics and Physics by Sir Michael Atiyah]] 2017 spalio 31 d., 22:00
atliko 
Pridėtos 224225 eilutės:
[[http://www.iecl.univlorraine.fr/~Wolfgang.Bertram/AtiyahDuality.pdf  Duality in Math and Physics by Sir Michael Atiyah]] 2017 spalio 31 d., 20:42
atliko 
Pridėta 227 eilutė:
Complements Pakeistos 229234 eilutės iš
į:
Functionals * [[https://en.wikipedia.org/wiki/Tduality  Tduality]] * [[https://en.wikipedia.org/wiki/AdS/CFT_correspondence  AdS/CFT correspondence]] * [[https://en.wikipedia.org/wiki/SYZ_conjecture  SYZ conjecture]] * [[https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory)  Mirror symmetry]] Pridėta 236 eilutė:
2017 spalio 31 d., 19:51
atliko 
Pakeista 226 eilutė iš:
* [[https://en.wikipedia.org/wiki/Sduality  Sduality]] strong coupling  weak coupling. į:
* [[https://en.wikipedia.org/wiki/Sduality  Sduality]] strong coupling  weak coupling. Realizations include Seiberg duality, Montonen–Olive duality, Generalizes Maxwell duality. Anton Kapustin and Edward Witten suggested that the geometric Langlands correspondence can be viewed as a mathematical statement of Montonen–Olive duality. 2017 spalio 31 d., 19:20
atliko 
Pridėtos 222226 eilutės:
[++Duality in Physics++] * [[https://en.wikipedia.org/wiki/Duality_(electricity_and_magnetism)  Electricity and magnetism]]  [[https://en.wikipedia.org/wiki/Duality_(electrical_circuits)  Electrical circuits]] In special relativity, applying the Lorentz transformation to the electric field transforms it into a magnetic field. * [[https://en.wikipedia.org/wiki/Sduality  Sduality]] strong coupling  weak coupling. 2017 spalio 31 d., 10:35
atliko 
Pakeista 164 eilutė iš:
* A very general comment of William Lawvere į:
* A very general comment of William Lawvere is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G. [[http://www.logicmatters.net/resources/pdfs/Galois.pdf  Peter Smith. The Galois Connection between Syntax and Semantics]]. 2017 spalio 31 d., 10:17
atliko 
Pridėta 168 eilutė:
* Consider an object Y in a category with pullbacks. Any morphism f : X → Y induces a functor f ∗ : Sub ( Y ) ⟶ Sub ( X ) on the category that is the preorder of subobjects. It maps subobjects T of Y (technically: monomorphism classes of T → Y to the pullback X × Y T ). If this functor has a left or right adjoint, they are called ∃ f and ∀ f, respectively.[3] They both map from Sub ( X ) back to Sub ( Y ) . Very roughly, given a domain S ⊂ X to quantify a relation expressed via f over, the functor/quantifier closes X in X × Y T and returns the thereby specified subset of Y. 2017 spalio 31 d., 10:04
atliko 
Pridėta 163 eilutė:
* [[https://mathoverflow.net/questions/6551/whatisanintuitiveviewofadjointsversion1categorytheory  Example of adjoint functors]] Given inclusion i:Z>R, with morphism x>y in R whenever x<=y, then the right adjoint is the floor function and the left adjoint is the ceiling function. A pair of adjoint functors is what is needed to make two categories compatible in their objects and morphisms. 2017 spalio 29 d., 07:53
atliko 
Pakeista 101 eilutė iš:
* The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. [[https://en.wikipedia.org/wiki/Grothendieck_local_duality  Grothendieck local duality]] is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. į:
* The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. [[https://en.wikipedia.org/wiki/Grothendieck_local_duality  Grothendieck local duality]] is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. The Grothendieck–Riemann–Roch theorem from about 1956 is usually cited as the key moment for the introduction of this circle of ideas. The more classical types of Riemann–Roch theorem are recovered in the case where S is a single point (i.e. the final object in the working category C). Using other S is a way to have versions of theorems 'with parameters', i.e. allowing for continuous variation, for which the 'frozen' version reduces the parameters to constants. In other applications, this way of thinking has been used in topos theory, to clarify the role of set theory in foundational matters. Assuming that we don’t have a commitment to one 'set theory' (all toposes are in some sense equally set theories for some intuitionistic logic) it is possible to state everything relative to some given set theory that acts as a base topos. 2017 spalio 29 d., 07:40
atliko 
Pridėta 138 eilutė:
* [[https://en.wikipedia.org/wiki/Grothendieck%27s_relative_point_of_view  Grothendieck's relative point of view]] studies and object X by considering instead morphisms f: X → S where S is a fixed object. This idea is made formal in the idea of the slice category of objects of C 'above' S. To move from one slice to another requires a base change; from a technical point of view base change becomes a major issue for the whole approach (see for example Beck–Chevalley conditions). A base change 'along' a given morphism g: T → S is typically given by the fiber product, producing an object over T from one over S. 2017 spalio 29 d., 07:34
atliko 
Pridėta 8 eilutė:
* How do [[https://en.wikipedia.org/wiki/Six_operations  Grothendieck's six operations]] (inverse image, direct image, proper direct image, proper inverse image, internal tensor product, internal Hom) fit in the map of dualities? 2017 spalio 29 d., 07:30
atliko 
Pakeista 100 eilutė iš:
* The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. į:
* The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. [[https://en.wikipedia.org/wiki/Grothendieck_local_duality  Grothendieck local duality]] is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. 2017 spalio 29 d., 07:08
atliko 
Pakeistos 165168 eilutės iš
* Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) * Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. į:
Pridėtos 190192 eilutės:
Duality: Reversing the ordering * Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) * Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. 2017 spalio 29 d., 07:04
atliko 
Pakeistos 78 eilutės iš
į:
* Ištirti simetrinių funkcijų dualumus: elementary ir homogeneous, Schur ir power. Pridėtos 4748 eilutės:
Dualities in the symmetric functions: Elementary and homogeneous; Schur and power; monomial and forgotten? 2017 spalio 29 d., 06:55
atliko 
Pakeistos 151152 eilutės iš
į:
* The Yoneda Lemma gives our connection to Why, and collapsing a network's node or relating it to its arrows. Relationship with Why as given by the eightfold way. Ištrinta 217 eilutė:
2017 spalio 29 d., 06:27
atliko 
Pridėtos 4763 eilutės:
AntiDuality: Symmetry and Structure * A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance". Duality: Translating structures * Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. * Galois theory: field extensions (solutions of polynomials) and groups * Lie groups: solutions to differential equations. Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology * [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz  Hilbert's Nullstellensatz]] * Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. * In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. * [[https://ncatlab.org/nlab/show/Isbell+duality  Isbell duality]] relates higher geometry with higher algebra. * [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. * [[https://en.wikipedia.org/wiki/AGT_correspondence  The AGT correspondence]] is a relationship between Liouville field theory on a punctured Riemann surface and a certain fourdimensional SU(2) gauge theory obtained by compactifying the 6D (2,0) superconformal field theory on the surface. * [[https://en.wikipedia.org/wiki/Modularity_theorem  The modularity theorem]] (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Pakeistos 128148 eilutės iš
į:
'''Duality: Functionals''' * Vectors and covectors. * A dual vector space (or just [[https://en.wikipedia.org/wiki/Dual_space  dual space]] for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. * dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V∗. If it does span V∗, then B∗ is called the dual basis or reciprocal basis for the basis B. * Dual basis in a field extension * Dual bundle of a vector bundle π : E → X is a vector bundle π∗ : E∗ → X whose fibers are the dual spaces to the fibers of E. * [[https://en.wikipedia.org/wiki/Pontryagin_duality  Pontryagin duality]] of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. The [[https://en.wikipedia.org/wiki/Reciprocal_lattice  reciprocal lattice]] is related to this. * [[https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality  Tannaka–Krein duality theory]] concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. ... In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category Π(G) with some additional structures, formed by the finitedimensional representations of G. The idea of Tannaka–Krein duality: category of representations of a group. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids. * Given the lattice of characters of a maximal torus, the dual lattice is given by the 1parameter subgroups. * [[https://en.wikipedia.org/wiki/Langlands_program  The Langlands program]] seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. * The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups. To make this theory explicit, there must be defined the concept of Lhomomorphism of an Lgroup into another. That is, Lgroups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but Lhomomorphisms must be 'over' the Weil group. * Langlands program. [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart.[[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]]. [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]]. * [[https://en.wikipedia.org/wiki/Dual_object  Dual object]] is a category theory generalization of the concept of dual space in linear algebra. * When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field F = C or R. A [[https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space  Continuous dual space]] or topological dual is a linear subspace of the algebraic dual space V and V'. For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide. * In functional analysis and related areas of mathematics a [[https://en.wikipedia.org/wiki/Dual_topology  dual topology]] is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. * A [[https://en.wikipedia.org/wiki/Dual_pair  dual pair]] or dual system is a pair of vector spaces with an associated bilinear map to the base field. A dual pair generalizes this concept of continuous dual to arbitrary vector spaces, with the duality being expressed as a bilinear map. Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces. * A [[https://en.wikipedia.org/wiki/Dual_wavelet  dual wavelet]] is the dual to a wavelet. In general, the wavelet series generated by a square integrable function will have a dual series, in the sense of the [[https://en.wikipedia.org/wiki/Riesz_representation_theorem  Riesz representation theorem]]. The Hilbert space representation theorem establishes an important connection between a Hilbert space and its (continuous) dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically antiisomorphic. The (anti) isomorphism is a particular, natural one. * The [[https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem  Riesz–Markov–Kakutani representation theorem]] relates linear functionals on spaces of continuous functions on a locally compact space to measures. * The dual space X' of a [[https://en.wikipedia.org/wiki/Stereotype_space  stereotype space]] is defined as the space of all linear continuous functionals f : X → C endowed with the topology of uniform convergence on totally bounded sets in X. * [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. * [[https://en.wikipedia.org/wiki/Weil_pairing  Weil pairing]] is generalized by [[https://en.wikipedia.org/wiki/Cartier_duality  Cartier duality]], which is an analogue of Pontryagin duality for noncommutative schemes. Pakeistos 152164 eilutės iš
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'''Duality: Adjunction''' * [[https://en.m.wikipedia.org/wiki/Adjoint  Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors  Adjoint functors]]. The minimialistic solution  the maximalist problem solved. The most efficient solution  the most difficult problem solved. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. * A functor F : C ← D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F to X. A functor G : C → D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y to G. * A counit–unit adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and two natural transformations... * A homset adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and a natural isomorphism... * A very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G. * division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. * Tensor products are adjoint to a set of homomorphisms. * The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. Duality: Reversing the ordering * Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) * Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. Ištrintos 188241 eilutės:
* [[https://en.m.wikipedia.org/wiki/Adjoint  Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors  Adjoint functors]]. The minimialistic solution  the maximalist problem solved. The most efficient solution  the most difficult problem solved. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. * A functor F : C ← D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F to X. A functor G : C → D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y to G. * A counit–unit adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and two natural transformations... * A homset adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and a natural isomorphism... * A very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G. * division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. * Tensor products are adjoint to a set of homomorphisms. * The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. Duality: Reversing the ordering * Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) * Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. Duality: Symmetry and Structure * A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance". Duality: Functionals * Vectors and covectors. * A dual vector space (or just [[https://en.wikipedia.org/wiki/Dual_space  dual space]] for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. * dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V∗. If it does span V∗, then B∗ is called the dual basis or reciprocal basis for the basis B. * Dual basis in a field extension * Dual bundle of a vector bundle π : E → X is a vector bundle π∗ : E∗ → X whose fibers are the dual spaces to the fibers of E. * [[https://en.wikipedia.org/wiki/Pontryagin_duality  Pontryagin duality]] of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. The [[https://en.wikipedia.org/wiki/Reciprocal_lattice  reciprocal lattice]] is related to this. * [[https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality  Tannaka–Krein duality theory]] concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. ... In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category Π(G) with some additional structures, formed by the finitedimensional representations of G. The idea of Tannaka–Krein duality: category of representations of a group. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids. * Given the lattice of characters of a maximal torus, the dual lattice is given by the 1parameter subgroups. * [[https://en.wikipedia.org/wiki/Langlands_program  The Langlands program]] seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. * The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups. To make this theory explicit, there must be defined the concept of Lhomomorphism of an Lgroup into another. That is, Lgroups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but Lhomomorphisms must be 'over' the Weil group. * Langlands program. [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart.[[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]]. [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]]. * [[https://en.wikipedia.org/wiki/Dual_object  Dual object]] is a category theory generalization of the concept of dual space in linear algebra. * When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field F = C or R. A [[https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space  Continuous dual space]] or topological dual is a linear subspace of the algebraic dual space V and V'. For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide. * In functional analysis and related areas of mathematics a [[https://en.wikipedia.org/wiki/Dual_topology  dual topology]] is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. * A [[https://en.wikipedia.org/wiki/Dual_pair  dual pair]] or dual system is a pair of vector spaces with an associated bilinear map to the base field. A dual pair generalizes this concept of continuous dual to arbitrary vector spaces, with the duality being expressed as a bilinear map. Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces. * A [[https://en.wikipedia.org/wiki/Dual_wavelet  dual wavelet]] is the dual to a wavelet. In general, the wavelet series generated by a square integrable function will have a dual series, in the sense of the [[https://en.wikipedia.org/wiki/Riesz_representation_theorem  Riesz representation theorem]]. The Hilbert space representation theorem establishes an important connection between a Hilbert space and its (continuous) dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically antiisomorphic. The (anti) isomorphism is a particular, natural one. * The [[https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem  Riesz–Markov–Kakutani representation theorem]] relates linear functionals on spaces of continuous functions on a locally compact space to measures. * The dual space X' of a [[https://en.wikipedia.org/wiki/Stereotype_space  stereotype space]] is defined as the space of all linear continuous functionals f : X → C endowed with the topology of uniform convergence on totally bounded sets in X. * [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. * [[https://en.wikipedia.org/wiki/Weil_pairing  Weil pairing]] is generalized by [[https://en.wikipedia.org/wiki/Cartier_duality  Cartier duality]], which is an analogue of Pontryagin duality for noncommutative schemes. Duality: Translating structures * Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. * Galois theory: field extensions (solutions of polynomials) and groups * Lie groups: solutions to differential equations. Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology * [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz  Hilbert's Nullstellensatz]] * Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. * In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. * [[https://ncatlab.org/nlab/show/Isbell+duality  Isbell duality]] relates higher geometry with higher algebra. * [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. * [[https://en.wikipedia.org/wiki/AGT_correspondence  The AGT correspondence]] is a relationship between Liouville field theory on a punctured Riemann surface and a certain fourdimensional SU(2) gauge theory obtained by compactifying the 6D (2,0) superconformal field theory on the surface. * [[https://en.wikipedia.org/wiki/Modularity_theorem  The modularity theorem]] (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. 2017 spalio 29 d., 06:21
atliko 
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Duality: Bottomup and Top į:
'''Duality: Bottomup and Topdown''' Pakeistos 7395 eilutės iš
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Duality: Complements * Center and totality for a simplex and other infinite families of complex polytopes. * [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29  Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.) * [[https://en.wikipedia.org/wiki/Poincaré_duality  Poincare duality]] states that if M is an ndimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k. [[https://en.wikipedia.org/wiki/Verdier_duality  Verdier duality]] is a generalization. * [[https://en.wikipedia.org/wiki/Twisted_Poincar%C3%A9_duality  Twisted Poincaré duality]] is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. * [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [[https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. * Betadual space is a certain linear subspace of the algebraic dual of a sequence space. * The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. * In differential geometry, the [[https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem  Atiyah–Singer index theorem]], proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Riemann–Roch theorem, as special cases * Dualizing sheaf. * [[https://en.wikipedia.org/wiki/Artin%E2%80%93Verdier_duality  ArtinVerdier duality]] is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Artin and Verdier (1964), that generalizes Tate duality. [[https://en.wikipedia.org/wiki/Tate_duality  Tate duality]] or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field. * In Galois cohomology, [[https://en.wikipedia.org/wiki/Local_Tate_duality  local Tate duality]] (or simply local duality) is a duality for Galois modules for the absolute Galois group of a nonarchimedean local field. It is named after John Tate who first proved it. It shows that the dual of such a Galois module is the Tate twist of usual linear dual. This new dual is called the (local) Tate dual. Local duality combined with Tate's local Euler characteristic formula provide a versatile set of tools for computing the Galois cohomology of local fields. Tate duality is a version for global fields. * The [[https://en.wikipedia.org/wiki/Verdier_duality  dualizing complex]] DX on X is defined to be ... where p is the map from X to a point. Part of what makes Verdier duality interesting in the singular setting is that when X is not a manifold (a graph or singular algebraic variety for example) then the dualizing complex is not quasiisomorphic to a sheaf concentrated in a single degree. * The Hodge isomorphism or [[https://en.wikipedia.org/wiki/Hodge_star_operator  Hodge star operator]] is an important linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finitedimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element is called the element's Hodge dual. * [[https://en.wikipedia.org/wiki/Lefschetz_duality  Lefschetz duality]] is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Duality: Complements: [[https://en.wikipedia.org/wiki/Duality_(projective_geometry)  Plane duality]] * Dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. * [[https://en.wikipedia.org/wiki/Dual_polyhedron  Dual polyhedron]] * Tangents * Dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. * [[https://en.wikipedia.org/wiki/Dual_polygon  Dual polygon]]: [[https://en.wikipedia.org/wiki/Rectification_(geometry)  rectification]]; [[https://en.wikipedia.org/wiki/Pole_and_polar#Reciprocation_and_projective_duality  polar reciprocation]] (pole and polar); projective duality; combinatorially. * [[https://en.wikipedia.org/wiki/Fenchel%27s_duality_theorem  Fenchel's duality theorem]] ... Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents. * [[https://en.wikipedia.org/wiki/Stokes%27_theorem  Stokes' theorem]]. Stokes' theorem is a vast generalization of the fundamental theorem of calculus, which states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f: General Stokes theorem: duality between the boundary operator on chains and the exterior derivative. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the dual of ∂ on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. Ištrinta 98 eilutė:
Pakeistos 106108 eilutės iš
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Generated by complements * De Groot dual of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X, τ). Saturated subset is an intersection of open subsets. Pakeistos 111116 eilutės iš
Duality: į:
Duality: Actions and Objects * We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome. '''Duality: Reversing the maps''' Ištrinta 148 eilutė:
Pakeistos 154183 eilutės iš
* We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome. Duality: Complements * Center and totality for a simplex and other infinite families of complex polytopes. * [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29  Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.) * [[https://en.wikipedia.org/wiki/Poincaré_duality  Poincare duality]] states that if M is an ndimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k. [[https://en.wikipedia.org/wiki/Verdier_duality  Verdier duality]] is a generalization. * [[https://en.wikipedia.org/wiki/Twisted_Poincar%C3%A9_duality  Twisted Poincaré duality]] is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. * [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [[https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. * Betadual space is a certain linear subspace of the algebraic dual of a sequence space. * The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. * In differential geometry, the [[https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem  Atiyah–Singer index theorem]], proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Riemann–Roch theorem, as special cases * Dualizing sheaf. * [[https://en.wikipedia.org/wiki/Artin%E2%80%93Verdier_duality  ArtinVerdier duality]] is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Artin and Verdier (1964), that generalizes Tate duality. [[https://en.wikipedia.org/wiki/Tate_duality  Tate duality]] or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field. * In Galois cohomology, [[https://en.wikipedia.org/wiki/Local_Tate_duality  local Tate duality]] (or simply local duality) is a duality for Galois modules for the absolute Galois group of a nonarchimedean local field. It is named after John Tate who first proved it. It shows that the dual of such a Galois module is the Tate twist of usual linear dual. This new dual is called the (local) Tate dual. Local duality combined with Tate's local Euler characteristic formula provide a versatile set of tools for computing the Galois cohomology of local fields. Tate duality is a version for global fields. * The [[https://en.wikipedia.org/wiki/Verdier_duality  dualizing complex]] DX on X is defined to be ... where p is the map from X to a point. Part of what makes Verdier duality interesting in the singular setting is that when X is not a manifold (a graph or singular algebraic variety for example) then the dualizing complex is not quasiisomorphic to a sheaf concentrated in a single degree. * The Hodge isomorphism or [[https://en.wikipedia.org/wiki/Hodge_star_operator  Hodge star operator]] is an important linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finitedimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element is called the element's Hodge dual. * [[https://en.wikipedia.org/wiki/Lefschetz_duality  Lefschetz duality]] is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Duality: Complements: [[https://en.wikipedia.org/wiki/Duality_(projective_geometry)  Plane duality]] * Dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. * [[https://en.wikipedia.org/wiki/Dual_polyhedron  Dual polyhedron]] * Tangents * Dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. * [[https://en.wikipedia.org/wiki/Dual_polygon  Dual polygon]]: [[https://en.wikipedia.org/wiki/Rectification_(geometry)  rectification]]; [[https://en.wikipedia.org/wiki/Pole_and_polar#Reciprocation_and_projective_duality  polar reciprocation]] (pole and polar); projective duality; combinatorially. * [[https://en.wikipedia.org/wiki/Fenchel%27s_duality_theorem  Fenchel's duality theorem]] ... Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents. * [[https://en.wikipedia.org/wiki/Stokes%27_theorem  Stokes' theorem]]. Stokes' theorem is a vast generalization of the fundamental theorem of calculus, which states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f: General Stokes theorem: duality between the boundary operator on chains and the exterior derivative. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the dual of ∂ on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. Generated by complements * De Groot dual of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X, τ). Saturated subset is an intersection of open subsets. į:
2017 spalio 29 d., 06:13
atliko 
Pakeistos 4950 eilutės iš
Duality: Conjugation į:
'''Duality: Conjugation''' * Square roots of i. There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.) Pakeista 53 eilutė iš:
Duality: Halving space: Rotation: Reversing orientation į:
'''Duality: Halving space''': Rotation: Reversing orientation Ištrintos 121122 eilutės:
Ištrinta 124 eilutė:
2017 spalio 29 d., 06:05
atliko 
Pridėtos 4777 eilutės:
[+Internal, Implicit Dualities+] Duality: Conjugation * '''Square roots of i.''' There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.) * Conjugation establishes the twosome by way of the Complex numbers. The Reals give the onesome. And this is followed by the Quaternions defining the threesome and so presumably the Octonions define the foursome. Duality: Halving space: Rotation: Reversing orientation * Coxeter groups are built from reflections. Reflections are dualities. * A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix. * If G is a group and ρ is a linear representation of it on the vector space V, then the [[https://en.wikipedia.org/wiki/Dual_representation  dual representation]] ρ* is defined over the dual vector space V* as follows: ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all g ∈ G. A general ring module does not admit a dual representation. Modules of Hopf algebras do, however. Duality: Reflection * [[https://en.wikipedia.org/wiki/Root_system#Dual_root_system_and_coroots  Dual root system]]  roots and coroots, given by the inner product, thus by reflection to match the shorter root with the longer root. This is generalized by the [[https://en.wikipedia.org/wiki/Root_datum  root datum]] of an algebraic group, which furthermore determines the center of a group. The dual root datum is gotten by switching the characters with the 1parameter subgroups, and the roots with the coroots. * Given a connected reductive algebraic group G, the [[https://en.wikipedia.org/wiki/Langlands_dual_group  Langlands dual group]] is the complex connected reductive group whose root datum is dual to that of G. Duality: Reversing order of operations * Normality says conjugate invariancy: gN = Ng. * Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices. * This is related to the duality between left and right multiplication. Examples include Polish notation. Duality: Bottomup and Topdown * Coordinate systems can be organized "bottom up" or "top down". This yields the duality in projective geometry. * Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R. * Cotangent space and tangent space (or is this between covariant and contravariant?) * For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact. * Duality of silence (topdown) and speaking (bottomup). Duality: Existing and nonexisting * Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem  Ramsey's theorem]]. Tao: "the Ramseytype theorem, each one of which being a different formalisation of the newly gained insight in mathematics that complete disorder is impossible." * Duality  parity  išsiaiškinimo rūšis Pridėtos 8687 eilutės:
[+External, Explicit Dualities+] Pakeistos 123140 eilutės iš
* '''Square roots of i.''' There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.) * Conjugation establishes the twosome by way of the Complex numbers. The Reals give the onesome. And this is followed by the Quaternions defining the threesome and so presumably the Octonions define the foursome. Duality: Halving space: Rotation: Reversing orientation * Coxeter groups are built from reflections. Reflections are dualities. * A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix. * If G is a group and ρ is a linear representation of it on the vector space V, then the [[https://en.wikipedia.org/wiki/Dual_representation  dual representation]] ρ* is defined over the dual vector space V* as follows: ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all g ∈ G. A general ring module does not admit a dual representation. Modules of Hopf algebras do, however. Duality: Reflection * [[https://en.wikipedia.org/wiki/Root_system#Dual_root_system_and_coroots  Dual root system]]  roots and coroots, given by the inner product, thus by reflection to match the shorter root with the longer root. This is generalized by the [[https://en.wikipedia.org/wiki/Root_datum  root datum]] of an algebraic group, which furthermore determines the center of a group. The dual root datum is gotten by switching the characters with the 1parameter subgroups, and the roots with the coroots. * Given a connected reductive algebraic group G, the [[https://en.wikipedia.org/wiki/Langlands_dual_group  Langlands dual group]] is the complex connected reductive group whose root datum is dual to that of G. Duality: Reversing order of operations * Normality says conjugate invariancy: gN = Ng. * Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices. * This is related to the duality between left and right multiplication. Examples include Polish notation. į:
Pakeistos 128138 eilutės iš
* Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem  Ramsey's theorem]]. Tao: "the Ramseytype theorem, each one of which being a different formalisation of the newly gained insight in mathematics that complete disorder is impossible." * Duality  parity  išsiaiškinimo rūšis Duality: Bottomup and Topdown * Coordinate systems can be organized "bottom up" or "top down". This yields the duality in projective geometry. * Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R. * Cotangent space and tangent space (or is this between covariant and contravariant?) * For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact. * Duality of silence (topdown) and speaking (bottomup). į:
2017 spalio 28 d., 19:22
atliko 
Pridėta 45 eilutė:
* Duality between matrices expressed explicitly (in terms of entries) and implicitly (in terms of eigenvalues). 2017 spalio 28 d., 19:04
atliko 
Pridėta 91 eilutė:
* Conjugation establishes the twosome by way of the Complex numbers. The Reals give the onesome. And this is followed by the Quaternions defining the threesome and so presumably the Octonions define the foursome. 2017 spalio 28 d., 14:24
atliko 
Pridėta 219 eilutė:
The Yoneda Lemma gives our connection to Why, and collapsing a network's node or relating it to its arrows. Relationship with Why as given by the eightfold way. 2017 spalio 27 d., 11:32
atliko 
Pakeistos 88102 eilutės iš
Duality: * Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. * Galois theory: field extensions (solutions of polynomials) and groups * Lie groups: solutions to differential equations. Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology * [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz  Hilbert's Nullstellensatz]] * Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. * In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. * [[https://ncatlab.org/nlab/show/Isbell+duality  Isbell duality]] relates higher geometry with higher algebra. * [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. * [[https://en.wikipedia.org/wiki/AGT_correspondence  The AGT correspondence]] is a relationship between Liouville field theory on a punctured Riemann surface and a certain fourdimensional SU(2) gauge theory obtained by compactifying the 6D (2,0) superconformal field theory on the surface. * [[https://en.wikipedia.org/wiki/Modularity_theorem  The modularity theorem]] (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Duality: Reversing orientation į:
Duality: Conjugation Pridėtos 9192 eilutės:
Duality: Halving space: Rotation: Reversing orientation Pridėtos 97100 eilutės:
Duality: Reflection * [[https://en.wikipedia.org/wiki/Root_system#Dual_root_system_and_coroots  Dual root system]]  roots and coroots, given by the inner product, thus by reflection to match the shorter root with the longer root. This is generalized by the [[https://en.wikipedia.org/wiki/Root_datum  root datum]] of an algebraic group, which furthermore determines the center of a group. The dual root datum is gotten by switching the characters with the 1parameter subgroups, and the roots with the coroots. * Given a connected reductive algebraic group G, the [[https://en.wikipedia.org/wiki/Langlands_dual_group  Langlands dual group]] is the complex connected reductive group whose root datum is dual to that of G. Pakeistos 114118 eilutės iš
* [[https://en.wikipedia.org/wiki/Root_system#Dual_root_system_and_coroots  Dual root system]]  roots and coroots, given by the inner product, thus by reflection to match the shorter root with the longer root. This is generalized by the [[https://en.wikipedia.org/wiki/Root_datum  root datum]] of an algebraic group, which furthermore determines the center of a group. The dual root datum is gotten by switching the characters with the 1parameter subgroups, and the roots with the coroots. * Given a connected reductive algebraic group G, the [[https://en.wikipedia.org/wiki/Langlands_dual_group  Langlands dual group]] is the complex connected reductive group whose root datum is dual to that of G. į:
Pridėtos 177190 eilutės:
Duality: Translating structures * Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. * Galois theory: field extensions (solutions of polynomials) and groups * Lie groups: solutions to differential equations. Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology * [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz  Hilbert's Nullstellensatz]] * Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. * In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. * [[https://ncatlab.org/nlab/show/Isbell+duality  Isbell duality]] relates higher geometry with higher algebra. * [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. * [[https://en.wikipedia.org/wiki/AGT_correspondence  The AGT correspondence]] is a relationship between Liouville field theory on a punctured Riemann surface and a certain fourdimensional SU(2) gauge theory obtained by compactifying the 6D (2,0) superconformal field theory on the surface. * [[https://en.wikipedia.org/wiki/Modularity_theorem  The modularity theorem]] (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. 2017 spalio 27 d., 10:47
atliko 
Pridėtos 46 eilutės:
* Understand the bijective proof between involutions and standard tableau. Understand ** SchurWeyl duality ** [[https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth_correspondence  RobinsonSchenstedKnuth correspondence]] and relation to Schur functions, and implications for symmetric functions of eigenvalues. 2017 spalio 27 d., 10:30
atliko 
Pakeistos 1213 eilutės iš
[+Logic: Duality+] į:
[++Logic: Duality++] Pakeistos 2223 eilutės iš
[+Logic in Mathematics: The Kinds of Duality+] į:
[++Logic in Mathematics: The Kinds of Duality++] Pakeistos 3334 eilutės iš
į:
* The asymptotic growth of the number of involutions grows as the square root of n. This supports the idea of the involutions as a "square root" of... matrices? Pakeistos 189190 eilutės iš
[+Mathematics: Almost Duality  Duality Breaking+] į:
[++Mathematics: Almost Duality  Duality Breaking++] Pakeista 203 eilutė iš:
[+Mathematical Tension: Equivalence and Uniqueness+] į:
[++Mathematical Tension: Equivalence and Uniqueness++] 2017 spalio 26 d., 16:18
atliko 
Pridėtos 17 eilutės:
>>bgcolor=#FFFFC0<< * Compare dualities with perspectives and dialectics. For example, "the truth is relative" switches us from an object to its arrows  but dialectic can ground an absolute truth. >><< 2017 spalio 26 d., 13:57
atliko 
Pakeista 9 eilutė iš:
* [[https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras  Stone's representation theorem for Boolean algebras]]. Every Boolean algebra is isomorphic to a certain field of sets. į:
* [[https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras  Stone's representation theorem for Boolean algebras]]. Every Boolean algebra is isomorphic to a certain field of sets. Thank you to Eugenijus Paliokas for pointing that out. 2017 spalio 26 d., 12:43
atliko 
Pridėtos 2734 eilutės:
The fundamental antiduality (SchurWeyl duality) between external representations and internal structure * [[https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality  Schur–Weyl duality]] is a mathematical theorem in representation theory that relates irreducible finitedimensional representations of the general linear and symmetric groups. * [[https://qchu.wordpress.com/2012/11/13/fourflavorsofschurweylduality/  Four flavors of SchurWeyl duality]] Antiduality: Internal structure and external relationships * This is the duality of category theory: External relationships can restate internal structure. * Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia. * 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. Ištrintos 4246 eilutės:
* This is the duality of category theory: External relationships can restate internal structure. * Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia. * 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. Ištrinta 95 eilutė:
2017 spalio 26 d., 12:28
atliko 
Pridėta 24 eilutė:
* My dream of Young tableaux whose entries were short and long dashes  "this is the fundamental unit of information". 2017 spalio 26 d., 11:19
atliko 
Pridėta 19 eilutė:
* Involutions are counted by Young tableaux (standard tableaux). So what do special rim hook tableaux count? And can we prove therefore that there is no involution for K1 K = 1 ? For if there was an involution then we would have a way to deal with all involutions? 2017 spalio 26 d., 10:53
atliko 
Ištrintos 9699 eilutės:
* [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. * [[https://en.wikipedia.org/wiki/Weil_pairing  Weil pairing]] is generalized by [[https://en.wikipedia.org/wiki/Cartier_duality  Cartier duality]], which is an analogue of Pontryagin duality for noncommutative schemes. Pridėtos 166167 eilutės:
* [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. * [[https://en.wikipedia.org/wiki/Weil_pairing  Weil pairing]] is generalized by [[https://en.wikipedia.org/wiki/Cartier_duality  Cartier duality]], which is an analogue of Pontryagin duality for noncommutative schemes. 2017 spalio 26 d., 10:50
atliko 
Pridėta 121 eilutė:
* Center and totality for a simplex and other infinite families of complex polytopes. 2017 spalio 26 d., 10:49
atliko 
Pakeista 97 eilutė iš:
į:
Pullback? Pridėta 99 eilutė:
* [[https://en.wikipedia.org/wiki/Weil_pairing  Weil pairing]] is generalized by [[https://en.wikipedia.org/wiki/Cartier_duality  Cartier duality]], which is an analogue of Pontryagin duality for noncommutative schemes. 2017 spalio 26 d., 09:53
atliko 
Pridėtos 5961 eilutės:
* A functor F : C ← D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F to X. A functor G : C → D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y to G. * A counit–unit adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and two natural transformations... * A homset adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and a natural isomorphism... 2017 spalio 26 d., 09:31
atliko 
Pridėtos 1618 eilutės:
Involutions * [[https://en.wikipedia.org/wiki/Involution_(mathematics)  Involutions]] 2017 spalio 26 d., 09:25
atliko 
Pakeista 54 eilutė iš:
* [[https://en.m.wikipedia.org/wiki/Adjoint  Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors  Adjoint functors]]. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. į:
* [[https://en.m.wikipedia.org/wiki/Adjoint  Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors  Adjoint functors]]. The minimialistic solution  the maximalist problem solved. The most efficient solution  the most difficult problem solved. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. 2017 spalio 25 d., 21:27
atliko 
Pridėtos 1316 eilutės:
Sources of examples of duality * [[https://en.wikipedia.org/wiki/List_of_dualities  List of dualities (Wikipedia)]] * [[https://ncatlab.org/nlab/show/duality  nLab: Duality]]. Ištrintos 187190 eilutės:
Examples of duality * [[https://en.wikipedia.org/wiki/List_of_dualities  List of dualities (Wikipedia)]] * [[https://ncatlab.org/nlab/show/duality  nLab: Duality]]. 2017 spalio 25 d., 21:25
atliko 
Pakeistos 6869 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Modularity_theorem  The modularity theorem]] (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Pridėta 115 eilutė:
* In differential geometry, the [[https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem  Atiyah–Singer index theorem]], proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Riemann–Roch theorem, as special cases Pakeistos 130131 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Stokes%27_theorem  Stokes' theorem]]. Stokes' theorem is a vast generalization of the fundamental theorem of calculus, which states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f: General Stokes theorem: duality between the boundary operator on chains and the exterior derivative. Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the dual of ∂ on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. Pakeistos 158160 eilutės iš
į:
Other * Araki duality for free fields: the commuting algebra of the local algebra of a region O in spacetime is equal to the local algebra of the set of points that are spacelike separated from O. Ištrintos 187200 eilutės:
* AtiyahSinger index theorem... * Covectors and vectors * [[https://en.wikipedia.org/wiki/Modular_theorem Modularity theorem]]. * general Stokes theorem: duality between the boundary operator on chains and the exterior derivative Other * Araki duality for free fields: the commuting algebra of the local algebra of a region O in spacetime is equal to the local algebra of the set of points that are spacelike separated from O. 2017 spalio 25 d., 21:09
atliko 
Pakeistos 6768 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/AGT_correspondence  The AGT correspondence]] is a relationship between Liouville field theory on a punctured Riemann surface and a certain fourdimensional SU(2) gauge theory obtained by compactifying the 6D (2,0) superconformal field theory on the surface. Pridėta 145 eilutė:
* Langlands program. [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart.[[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]]. [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]]. Ištrinta 182 eilutė:
Ištrinta 186 eilutė:
Ištrintos 195215 eilutės:
* Local Tate duality * Poincaré duality * Twisted Poincaré duality * Poitou–Tate duality * Sduality (homotopy theory) * Schur–Weyl duality * Tannaka–Krein duality * Verdier duality * AGT correspondence * Langlands program ** [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart ** [[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]] ** [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]] ? 2017 spalio 25 d., 21:04
atliko 
Pridėta 140 eilutė:
* [[https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein_duality  Tannaka–Krein duality theory]] concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. ... In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category Π(G) with some additional structures, formed by the finitedimensional representations of G. The idea of Tannaka–Krein duality: category of representations of a group. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids. 2017 spalio 25 d., 21:00
atliko 
Pridėta 73 eilutė:
* [[https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality  Schur–Weyl duality]] is a mathematical theorem in representation theory that relates irreducible finitedimensional representations of the general linear and symmetric groups. 2017 spalio 25 d., 20:55
atliko 
Pakeista 109 eilutė iš:
* [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. į:
* [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [[https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. 2017 spalio 25 d., 20:53
atliko 
Pridėta 108 eilutė:
* [[https://en.wikipedia.org/wiki/Twisted_Poincar%C3%A9_duality  Twisted Poincaré duality]] is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. 2017 spalio 25 d., 20:52
atliko 
Pridėta 113 eilutė:
* In Galois cohomology, [[https://en.wikipedia.org/wiki/Local_Tate_duality  local Tate duality]] (or simply local duality) is a duality for Galois modules for the absolute Galois group of a nonarchimedean local field. It is named after John Tate who first proved it. It shows that the dual of such a Galois module is the Tate twist of usual linear dual. This new dual is called the (local) Tate dual. Local duality combined with Tate's local Euler characteristic formula provide a versatile set of tools for computing the Galois cohomology of local fields. Tate duality is a version for global fields. Pakeistos 116117 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Lefschetz_duality  Lefschetz duality]] is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Ištrintos 194203 eilutės:
* Dualizing sheaf * Esakia duality * Fenchel's duality theorem * Haag duality * Hodge dual * Jónsson–Tarski duality * Lagrange duality * Langlands dual * Lefschetz duality 2017 spalio 25 d., 20:36
atliko 
Pridėta 16 eilutė:
* [[https://en.wikipedia.org/wiki/General_frame#J.C3.B3nsson.E2.80.93Tarski_duality  Jónsson–Tarski duality]] General frames bear close connection to modal algebras. ... In the opposite direction, it is possible to construct the dual frame ... A frame and its dual validate the same formulas, hence the general frame semantics and algebraic semantics are in a sense equivalent. 2017 spalio 25 d., 20:32
atliko 
Pridėta 113 eilutė:
* The Hodge isomorphism or [[https://en.wikipedia.org/wiki/Hodge_star_operator  Hodge star operator]] is an important linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finitedimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element is called the element's Hodge dual. 2017 spalio 25 d., 20:25
atliko 
Pridėta 17 eilutė:
* [[https://en.wikipedia.org/wiki/Esakia_duality  Esakia duality]] is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an ordertopological representation of Heyting algebras via Esakia spaces. 2017 spalio 25 d., 20:23
atliko 
Pridėta 27 eilutė:
** If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are [[https://en.wikipedia.org/wiki/Equivalence_of_categories  dually equivalent]]. 2017 spalio 25 d., 20:16
atliko 
Pakeistos 108110 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Artin%E2%80%93Verdier_duality  ArtinVerdier duality]] is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Artin and Verdier (1964), that generalizes Tate duality. [[https://en.wikipedia.org/wiki/Tate_duality  Tate duality]] or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field. * The [[https://en.wikipedia.org/wiki/Verdier_duality  dualizing complex]] DX on X is defined to be ... where p is the map from X to a point. Part of what makes Verdier duality interesting in the singular setting is that when X is not a manifold (a graph or singular algebraic variety for example) then the dualizing complex is not quasiisomorphic to a sheaf concentrated in a single degree. Ištrintos 187200 eilutės:
* Dual polyhedron * Dual problem * Dual representation * Dual qHahn polynomials * Dual qKrawtchouk polynomials * Dual space * Dual topology * Dual wavelet * Duality (optimization) * Duality (order theory) * Duality of stereotype spaces * Duality (projective geometry) * Duality theory for distributive lattices 2017 spalio 25 d., 20:10
atliko 
Pakeistos 89 eilutės iš
* Generalized by [[https://en.wikipedia.org/wiki/Stone_duality  Stone's duality]]: categorical dualities between certain categories of topological spaces and categories of partially ordered sets. į:
* Generalized by [[https://en.wikipedia.org/wiki/Stone_duality  Stone's duality]]: categorical dualities between certain categories of topological spaces and categories of partially ordered sets. They form a natural generalization of Stone's representation theorem for Boolean algebras. Stonetype dualities provide the foundation for [[https://en.wikipedia.org/wiki/Pointless_topology  pointless topology]] and are exploited in theoretical computer science for the study of formal semantics. * [[https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices  Duality theory for distributive lattices]] provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This generalizes the wellknown Stone duality between Stone spaces and Boolean algebras. There are three equivalent ways of representing bounded distributive lattices. Each one has its own motivation and advantages, but ultimately they all serve the same purpose of providing better understanding of bounded distributive lattices. Pridėtos 1415 eilutės:
* [[https://en.wikipedia.org/wiki/Stone_duality  Stone duality]] are categorical dualities between certain categories of topological spaces and categories of partially ordered sets. * The dual of the category of frames is called the category of locales and generalizes the category Top of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every continuous map between topological spaces X and Y induces a map between the lattices of open sets in the opposite direction as for every continuous function f: X → Y and every open set O in Y the inverse image f 1(O) is an open set in X. 2017 spalio 25 d., 20:00
atliko 
Pridėta 135 eilutė:
* The dual space X' of a [[https://en.wikipedia.org/wiki/Stereotype_space  stereotype space]] is defined as the space of all linear continuous functionals f : X → C endowed with the topology of uniform convergence on totally bounded sets in X. 2017 spalio 25 d., 19:53
atliko 
Pakeistos 2942 eilutės iš
* [[https://en.wikipedia.org/wiki/Duality_(order_theory)  Duality (order theory)]]. Every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order. į:
* [[https://en.wikipedia.org/wiki/Duality_(order_theory)  Duality (order theory)]]. Every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order. Dual properties: ** Greatest elements and least elements ** Maximal elements and minimal elements ** Least upper bounds (suprema, ∨) and greatest lower bounds (infima, ∧) ** Upper sets and lower sets ** Ideals and filters ** Closure operators and kernel operators. * Selfdual notions: ** Being a (complete) lattice ** Monotonicity of functions ** Distributivity of lattices, i.e. the lattices for which ∀x,y,z: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) holds are exactly those for which the dual statement ∀x,y,z: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) holds ** Being a Boolean algebra ** Being an order isomorphism. * Since partial orders are antisymmetric, the only ones that are selfdual are the equivalence relations. 2017 spalio 25 d., 19:43
atliko 
Pridėta 29 eilutė:
* [[https://en.wikipedia.org/wiki/Duality_(order_theory)  Duality (order theory)]]. Every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order. 2017 spalio 25 d., 19:41
atliko 
Pridėtos 119120 eilutės:
* A [[https://en.wikipedia.org/wiki/Dual_wavelet  dual wavelet]] is the dual to a wavelet. In general, the wavelet series generated by a square integrable function will have a dual series, in the sense of the [[https://en.wikipedia.org/wiki/Riesz_representation_theorem  Riesz representation theorem]]. The Hilbert space representation theorem establishes an important connection between a Hilbert space and its (continuous) dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically antiisomorphic. The (anti) isomorphism is a particular, natural one. * The [[https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem  Riesz–Markov–Kakutani representation theorem]] relates linear functionals on spaces of continuous functions on a locally compact space to measures. 2017 spalio 25 d., 19:37
atliko 
Pridėta 117 eilutė:
* In functional analysis and related areas of mathematics a [[https://en.wikipedia.org/wiki/Dual_topology  dual topology]] is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. 2017 spalio 25 d., 19:35
atliko 
Pridėta 107 eilutė:
* A dual vector space (or just [[https://en.wikipedia.org/wiki/Dual_space  dual space]] for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. Pakeista 116 eilutė iš:
* When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field F = C or R. A [[https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space Continuous dual space]] or topological dual is a linear subspace of the algebraic dual space V and V'. For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide. į:
* When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field F = C or R. A [[https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space  Continuous dual space]] or topological dual is a linear subspace of the algebraic dual space V and V'. For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide. 2017 spalio 25 d., 19:31
atliko 
Pridėta 50 eilutė:
* Coxeter groups are built from reflections. Reflections are dualities. Pakeista 52 eilutė iš:
* į:
* If G is a group and ρ is a linear representation of it on the vector space V, then the [[https://en.wikipedia.org/wiki/Dual_representation  dual representation]] ρ* is defined over the dual vector space V* as follows: ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all g ∈ G. A general ring module does not admit a dual representation. Modules of Hopf algebras do, however. 2017 spalio 25 d., 19:27
atliko 
Pakeistos 9798 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Fenchel%27s_duality_theorem  Fenchel's duality theorem]] ... Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents. Pridėtos 124130 eilutės:
Dual values * Usually the term "dual problem" refers to the Lagrangian dual problem but other dual problems are used, for example, the Wolfe dual problem and the Fenchel dual problem. * In [[https://en.wikipedia.org/wiki/Wolfe_duality  Wolfe duality]], the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle. * In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem.[1] However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. * The duality gap is the difference between the values of any primal solutions and any dual solutions. The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. * In computational optimization, another "duality gap" is often reported, which is the difference in value between any dual solution and the value of a feasible but suboptimal iterate for the primal problem. This alternative "duality gap" quantifies the discrepancy between the value of a current feasible but suboptimal iterate for the primal problem and the value of the dual problem; the value of the dual problem is, under regularity conditions, equal to the value of the convex relaxation of the primal problem 2017 spalio 25 d., 19:09
atliko 
Pakeista 91 eilutė iš:
Duality: Complements: į:
Duality: Complements: [[https://en.wikipedia.org/wiki/Duality_(projective_geometry)  Plane duality]] Pakeistos 9697 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Dual_polygon  Dual polygon]]: [[https://en.wikipedia.org/wiki/Rectification_(geometry)  rectification]]; [[https://en.wikipedia.org/wiki/Pole_and_polar#Reciprocation_and_projective_duality  polar reciprocation]] (pole and polar); projective duality; combinatorially. Ištrinta 154 eilutė:
2017 spalio 25 d., 18:47
atliko 
Pridėta 109 eilutė:
* [[https://en.wikipedia.org/wiki/Langlands_program  The Langlands program]] seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Pakeista 144 eilutė iš:
į:
Ištrintos 145146 eilutės:
2017 spalio 25 d., 18:38
atliko 
Pridėtos 111112 eilutės:
* When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field F = C or R. A [[https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space Continuous dual space]] or topological dual is a linear subspace of the algebraic dual space V and V'. For any finitedimensional normed vector space or topological vector space, such as Euclidean nspace, the continuous dual and the algebraic dual coincide. * A [[https://en.wikipedia.org/wiki/Dual_pair  dual pair]] or dual system is a pair of vector spaces with an associated bilinear map to the base field. A dual pair generalizes this concept of continuous dual to arbitrary vector spaces, with the duality being expressed as a bilinear map. Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces. 2017 spalio 25 d., 18:22
atliko 
Pakeistos 9496 eilutės iš
į:
* Tangents * Dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. Pakeistos 109115 eilutės iš
* The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups. To į:
* The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups. To make this theory explicit, there must be defined the concept of Lhomomorphism of an Lgroup into another. That is, Lgroups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but Lhomomorphisms must be 'over' the Weil group. * [[https://en.wikipedia.org/wiki/Dual_object  Dual object]] is a category theory generalization of the concept of dual space in linear algebra. Pakeistos 152153 eilutės iš
* Dual object į:
2017 spalio 25 d., 18:09
atliko 
Pridėtos 107109 eilutės:
* The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between "good" representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between "good" representations of the corresponding groups. To make this theory explicit, there must be defined the concept of Lhomomorphism of an Lgroup into another. That is, Lgroups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but Lhomomorphisms must be 'over' the Weil group. 2017 spalio 25 d., 18:08
atliko 
Pakeistos 6768 eilutės iš
* Duality  parity  išsiaiškinimo rūšis. į:
* Duality  parity  išsiaiškinimo rūšis Duality: Reflection * [[https://en.wikipedia.org/wiki/Root_system#Dual_root_system_and_coroots  Dual root system]]  roots and coroots, given by the inner product, thus by reflection to match the shorter root with the longer root. This is generalized by the [[https://en.wikipedia.org/wiki/Root_datum  root datum]] of an algebraic group, which furthermore determines the center of a group. The dual root datum is gotten by switching the characters with the 1parameter subgroups, and the roots with the coroots. * Given a connected reductive algebraic group G, the [[https://en.wikipedia.org/wiki/Langlands_dual_group  Langlands dual group]] is the complex connected reductive group whose root datum is dual to that of G. Pakeistos 105106 eilutės iš
* [[https://en.wikipedia.org/wiki/Pontryagin_duality  Pontryagin duality]] of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. į:
* [[https://en.wikipedia.org/wiki/Pontryagin_duality  Pontryagin duality]] of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. The [[https://en.wikipedia.org/wiki/Reciprocal_lattice  reciprocal lattice]] is related to this. * Given the lattice of characters of a maximal torus, the dual lattice is given by the 1parameter subgroups. 2017 spalio 25 d., 17:13
atliko 
Pakeista 96 eilutė iš:
į:
Duality: Functionals Pakeistos 100101 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Pontryagin_duality  Pontryagin duality]] of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. Pakeista 175 eilutė iš:
į:
2017 spalio 25 d., 16:33
atliko 
Pakeista 86 eilutė iš:
Duality: Complements: į:
Duality: Complements: Planar Ištrintos 123125 eilutės:
* Faces of an object and corners of an object. (Why are they dual?) 2017 spalio 25 d., 16:31
atliko 
Pridėtos 8689 eilutės:
Duality: Complements: Graph * Dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. * [[https://en.wikipedia.org/wiki/Dual_polyhedron  Dual polyhedron]] Pakeistos 138140 eilutės iš
į:
Pakeista 142 eilutė iš:
į:
2017 spalio 25 d., 11:38
atliko  2017 spalio 25 d., 10:40
atliko 
Pridėtos 1215 eilutės:
Intersections and Unions * [[https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle  Inclusionexclusion principle]] * [[https://en.wikipedia.org/wiki/Alvis%E2%80%93Curtis_duality  Alvis–Curtis duality]] is a duality operation on the characters of a reductive group over a finite field. Kawanaka introduced a similar duality operation for Lie algebras. The dual ζ* of a character ζ of a finite group G with a split BNpair is defined to be... Pridėta 22 eilutė:
* [[https://en.wikipedia.org/wiki/Dual_(category_theory)  Dual (category theory)]] Pridėtos 6264 eilutės:
Pullback * [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. Pakeistos 6768 eilutės iš
į:
* Duality  parity  išsiaiškinimo rūšis. Pakeistos 8188 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/Poincaré_duality  Poincare duality]] states that if M is an ndimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k. [[https://en.wikipedia.org/wiki/Verdier_duality  Verdier duality]] is a generalization. * [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. * Betadual space is a certain linear subspace of the algebraic dual of a sequence space. * The [[https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem  RiemannRoch theorem]] relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. [[https://en.wikipedia.org/wiki/Serre_duality  Serre duality]] is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. [[https://en.wikipedia.org/wiki/Coherent_duality  Coherent duality]] is a generalization applying to coherent sheaves. Generated by complements * De Groot dual of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X, τ). Saturated subset is an intersection of open subsets. Pridėtos 9299 eilutės:
Linear functionals * dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V∗. If it does span V∗, then B∗ is called the dual basis or reciprocal basis for the basis B. * Dual basis in a field extension * Dual bundle of a vector bundle π : E → X is a vector bundle π∗ : E∗ → X whose fibers are the dual spaces to the fibers of E. Tangents * Dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. Pridėtos 111114 eilutės:
Ways of figuring things out * Duality of the deep and the broad. * Įvairios simetrijos  išsiaiškinimo būdų sandaros. Pakeista 126 eilutė iš:
į:
Pakeistos 134153 eilutės iš
* https://en.m.wikipedia.org/wiki/Dual_ * Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros Complements * [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. * Betadual space is a certain linear subspace of the algebraic dual of a sequence space. * The RiemannRoch theorem relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. Serre duality is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. Coherent duality is a generalization applying to coherent sheaves. Intersections and Unions * [[https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle  Inclusionexclusion principle]] * [[https://en.wikipedia.org/wiki/Alvis%E2%80%93Curtis_duality  Alvis–Curtis duality]] is a duality operation on the characters of a reductive group over a finite field. Kawanaka introduced a similar duality operation for Lie algebras. The dual ζ* of a character ζ of a finite group G with a split BNpair is defined to be... į:
* https://en.wikipedia.org/wiki/Dual_polyhedron Pridėta 138 eilutė:
* Dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. Ištrintos 140158 eilutės:
* De Groot dual of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X, τ). Saturated subset is an intersection of open subsets. Pullback * [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. Linear functionals * dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V∗. If it does span V∗, then B∗ is called the dual basis or reciprocal basis for the basis B. * Dual basis in a field extension * Dual bundle of a vector bundle π : E → X is a vector bundle π∗ : E∗ → X whose fibers are the dual spaces to the fibers of E. Tangents * Dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. Opposite category * Dual (category theory) Dual graph * Dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. 2017 spalio 25 d., 10:20
atliko 
Pridėta 23 eilutė:
** Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality  Eckmann–Hilton duality]]. Ištrinta 93 eilutė:
2017 spalio 25 d., 08:21
atliko 
Pakeistos 1416 eilutės iš
į:
* Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia. * 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. Pakeistos 6465 eilutės iš
į:
* Duality of silence (topdown) and speaking (bottomup). Pridėtos 7274 eilutės:
Duality: Symmetry and Structure * A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance". Pakeistos 187193 eilutės iš
Duality of silence (topdown) and speaking (bottomup). Category theory * Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia. * 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:
2017 spalio 25 d., 08:18
atliko 
Pakeistos 2122 eilutės iš
į:
Adjoints * [[https://en.m.wikipedia.org/wiki/Adjoint  Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors  Adjoint functors]]. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. * A very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G. * division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. * Tensor products are adjoint to a set of homomorphisms. * The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. Pakeistos 3639 eilutės iš
į:
* In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. * [[https://ncatlab.org/nlab/show/Isbell+duality  Isbell duality]] relates higher geometry with higher algebra. * [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. Pakeistos 6162 eilutės iš
į:
* For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact. Pakeistos 8386 eilutės iš
į:
* [[https://en.wikipedia.org/wiki/List_of_dualities  List of dualities (Wikipedia)]] * [[https://ncatlab.org/nlab/show/duality  nLab: Duality]]. Pakeistos 9091 eilutės iš
į:
Pakeistos 100103 eilutės iš
* [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. * For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact. * The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. į:
Pakeistos 105110 eilutės iš
* * division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. * Tensor products are adjoint to a set of homomorphisms. * Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros. * In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. [[https://en.wikipedia.org/wiki/List_of_dualities  List of dualities (Wikipedia)]] į:
* Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros 2017 spalio 25 d., 07:51
atliko 
Pakeistos 2627 eilutės iš
* Lie groups: solutions to differential equations.. į:
* Lie groups: solutions to differential equations. Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology * [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz  Hilbert's Nullstellensatz]] * Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. Pakeistos 5253 eilutės iš
į:
* Cotangent space and tangent space (or is this between covariant and contravariant?) Pridėtos 5759 eilutės:
Duality: Complements * [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29  Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.) Pakeistos 6566 eilutės iš
į:
* Meromorphic function is the quotient of two holomorphic functions, thus compares them. Ištrintos 7778 eilutės:
Pakeista 79 eilutė iš:
į:
Pakeistos 8384 eilutės iš
* [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology į:
Pakeistos 8892 eilutės iš
* Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. * Meromorphic function is the quotient of two holomorphic functions, thus compares them. į:
2017 spalio 25 d., 07:47
atliko 
Pakeista 16 eilutė iš:
* į:
* Opposite category. Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around. Pridėtos 2327 eilutės:
Duality: Translating * Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. * Galois theory: field extensions (solutions of polynomials) and groups * Lie groups: solutions to differential equations.. Pakeistos 3132 eilutės iš
į:
* Coxeter groups are built from reflections. Reflections are dualities. Pakeistos 3637 eilutės iš
į:
* This is related to the duality between left and right multiplication. Examples include Polish notation. Pakeistos 4041 eilutės iš
į:
* Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. Pridėtos 4951 eilutės:
Duality: Actions and Objects * We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome. Ištrinta 57 eilutė:
Ištrinta 61 eilutė:
Ištrintos 6670 eilutės:
* Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. * We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome. ** This is related to the duality between left and right multiplication. Examples include Polish notation. Pakeistos 6871 eilutės iš
* Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. ** Galois theory: field extensions (solutions of polynomials) and groups ** Lie groups: solutions to differential equations.. į:
2017 spalio 25 d., 07:43
atliko 
Pakeistos 1532 eilutės iš
The duality between zero and infinity, between nothing and everything, is broken in many subtle ways. Here are some examples: * By definition, a topological space includes both an entire set X and the empty set. However, the intermediary sets are closed under arbitrary unions, but only finite intersections. What would happen if they were closed under infinite intersections? * Perhaps similarly, having in mind the Zariski topology, ideals of a ring are defined with respect to multiplication (union) but not addition (intersection). [+Mathematical Tension: Equivalence and Uniqueness+] In Math, there is an everpresent tension between the notions of equivalence class and uniqueness. If something is mathematically significant, it should in some sense be unique. But math is a model and so, as such, can never be entirely unique but represents a variety of cases. Thus it is ever natural to define equivalence classes, especially in math itself. For example, a rational number is an equivalence class that establishes a proportion. >>bgcolor=#EEEEEE<< Examples of duality * '''Square roots of i.''' There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.) * A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix. ** Normality says conjugate invariancy: gN = Ng. į:
Duality: Reversing the maps Ištrintos 2022 eilutės:
** Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) ** Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality  Eckmann–Hilton duality]]. Pridėtos 2234 eilutės:
Duality: Reversing orientation * '''Square roots of i.''' There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.) * A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix. Duality: Reversing order of operations * Normality says conjugate invariancy: gN = Ng. * Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices. Duality: Reversing the ordering * Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) Duality: Existing and nonexisting Pridėtos 3637 eilutės:
Duality: Bottomup and Topdown Pakeistos 3959 eilutės iš
* į:
* Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R. [+Mathematics: Almost Duality  Duality Breaking+] The duality between zero and infinity, between nothing and everything, is broken in many subtle ways. Here are some examples: * By definition, a topological space includes both an entire set X and the empty set. However, the intermediary sets are closed under arbitrary unions, but only finite intersections. What would happen if they were closed under infinite intersections? * Perhaps similarly, having in mind the Zariski topology, ideals of a ring are defined with respect to multiplication (union) but not addition (intersection). [+Mathematical Tension: Equivalence and Uniqueness+] In Math, there is an everpresent tension between the notions of equivalence class and uniqueness. If something is mathematically significant, it should in some sense be unique. But math is a model and so, as such, can never be entirely unique but represents a variety of cases. Thus it is ever natural to define equivalence classes, especially in math itself. For example, a rational number is an equivalence class that establishes a proportion. >>bgcolor=#EEEEEE<< Examples of duality ** Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality  Eckmann–Hilton duality]]. 2017 spalio 25 d., 07:34
atliko 
Pakeistos 18 eilutės iš
(internal structure mirrors external structure   I am studying the various case of duality į:
I am studying the various cases of duality in math. I imagine that at the heart is the duality between zero and infinity by way of one as in [[Gods Dance  God's Dance]]. Duality is the basis for logic, and mathematics gives the ways of deviating from duality. Pridėtos 1213 eilutės:
Duality: Internal structure and external relationships * This is the duality of category theory: External relationships can restate internal structure. 2017 spalio 25 d., 07:32
atliko 
Pakeistos 910 eilutės iš
į:
[+Logic: Duality+] Pakeistos 1617 eilutės iš
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[+Logic in Mathematics: The Kinds of Duality+] [+Mathematics: Almost Duality  Duality Breaking+] Pakeistos 2628 eilutės iš
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[+Mathematical Tension: Equivalence and Uniqueness+] Ištrintos 3036 eilutės:
* Langlands program ** [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart ** [[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]] ** [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]] ? Pridėtos 164168 eilutės:
* Langlands program ** [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart ** [[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]] ** [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]] ? 2017 spalio 24 d., 08:46
atliko 
Pakeistos 107108 eilutės iš
į:
Pullback * [[https://en.wikipedia.org/wiki/Dual_abelian_variety  Dual abelian variety]] can be defined from an abelian variety A, defined over a field K. To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. ... the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. Linear functionals * dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V∗. If it does span V∗, then B∗ is called the dual basis or reciprocal basis for the basis B. Pakeistos 113114 eilutės iš
* Dual bundle į:
* Dual bundle of a vector bundle π : E → X is a vector bundle π∗ : E∗ → X whose fibers are the dual spaces to the fibers of E. Tangents * Dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. Opposite category Pakeistos 120123 eilutės iš
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Dual graph * Dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. 2017 spalio 24 d., 08:30
atliko 
Pridėta 101 eilutė:
Other Pakeistos 104108 eilutės iš
* De Groot į:
Generated by complements * De Groot dual of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X, τ). Saturated subset is an intersection of open subsets. * Dual abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety. 2017 spalio 24 d., 08:26
atliko 
Pridėta 91 eilutė:
Complements Pakeistos 93101 eilutės iš
ζ ∗ = ∑ J ⊆ R ( − 1 ) J ζ P J G {\displaystyle \zeta ^{*}=\sum _{J\subseteq R}(1)^{J}\zeta _{P_{J}}^{G}} {\displaystyle \zeta ^{*}=\sum _{J\subseteq R}(1 * Araki duality * Betadual space * Coherent duality į:
* Betadual space is a certain linear subspace of the algebraic dual of a sequence space. * The RiemannRoch theorem relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. First for Riemann surfaces, then for algebraic curves. Serre duality is present on nonsingular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. Coherent duality is a generalization applying to coherent sheaves. Intersections and Unions * [[https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle  Inclusionexclusion principle]] * [[https://en.wikipedia.org/wiki/Alvis%E2%80%93Curtis_duality  Alvis–Curtis duality]] is a duality operation on the characters of a reductive group over a finite field. Kawanaka introduced a similar duality operation for Lie algebras. The dual ζ* of a character ζ of a finite group G with a split BNpair is defined to be... * Araki duality for free fields: the commuting algebra of the local algebra of a region O in spacetime is equal to the local algebra of the set of points that are spacelike separated from O. Ištrinta 146 eilutė:
2017 spalio 24 d., 08:14
atliko 
Pakeistos 9194 eilutės iš
* [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by * Alvis–Curtis duality į:
* [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by [https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead_duality  Spanier–Whitehead duality]]. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. * [[https://en.wikipedia.org/wiki/Alvis%E2%80%93Curtis_duality  Alvis–Curtis duality]] is a duality operation on the characters of a reductive group over a finite field. Kawanaka introduced a similar duality operation for Lie algebras. The dual ζ* of a character ζ of a finite group G with a split BNpair is defined to be ζ ∗ = ∑ J ⊆ R ( − 1 ) J ζ P J G {\displaystyle \zeta ^{*}=\sum _{J\subseteq R}(1)^{J}\zeta _{P_{J}}^{G}} {\displaystyle \zeta ^{*}=\sum _{J\subseteq R}(1)^{J}\zeta _{P_{J}}^{G}} 2017 spalio 24 d., 08:12
atliko 
Pakeistos 8993 eilutės iš
* į:
* [[https://en.wikipedia.org/wiki/Jordan_curve_theorem  Jordan curve theorem]] (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by [[https://en.wikipedia.org/wiki/Alexander_duality  Alexander duality]] about the Betti numbers of the simplicial complex, and in the modern statement, between the reduced homology or cohomology of a compact, locally contractible subspace X of a sphere and its complement Y, Hq(X) and Hnq1(Y). Generalized by Spanier–Whitehead duality. Sphere complements determine the homology, and the stable homotopy type, though not the homotopy type. Pakeista 140 eilutė iš:
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2017 spalio 24 d., 07:55
atliko 
Pakeistos 910 eilutės iš
'''Duality breaking''' į:
'''Logic: Duality''' Duality arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other. This is driven by the sevensome in defining logic as the balancing of the conscious (not known) and the unconscious (known), what is and what is not (but complements it). * [[https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras  Stone's representation theorem for Boolean algebras]]. Every Boolean algebra is isomorphic to a certain field of sets. * Generalized by [[https://en.wikipedia.org/wiki/Stone_duality  Stone's duality]]: categorical dualities between certain categories of topological spaces and categories of partially ordered sets. '''Mathematics: Near duality  Duality breaking''' Pridėtos 2223 eilutės:
Pakeista 38 eilutė iš:
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Examples of duality Ištrinta 136 eilutė:
2017 spalio 24 d., 07:42
atliko 
Pakeistos 136138 eilutės iš
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Category theory * Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia. * 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. 2017 spalio 24 d., 07:35
atliko 
Pridėtos 135136 eilutės:
Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia. 2017 spalio 17 d., 10:36
atliko 
Pridėtos 16 eilutės:
Study duality as the basis of logic, and mathematics as ways of altering duality. (internal structure mirrors external structure  duality of category theory)  2016 gruodžio 13 d., 22:49
atliko 
Pridėtos 1318 eilutės:
* Langlands program ** [[http://www.ams.org/journals/bull/19841002/S027309791984152376/S027309791984152376.pdf  An Elementary Introduction to the Langlands Program]] by Stephen Gelbart ** [[https://arxiv.org/pdf/hepth/0512172v1  Langland Frenkel]] ** [[https://en.wikipedia.org/wiki/6D_(2,0)_superconformal_field_theory  6D (2,0) superconformal field theory]] ? 2016 rugpjūčio 23 d., 22:15
atliko 
Pridėtos 121122 eilutės:
Duality of silence (topdown) and speaking (bottomup). 2016 rugpjūčio 16 d., 16:20
atliko 
Pridėtos 811 eilutės:
'''Equivalence and uniqueness''' In Math, there is an everpresent tension between the notions of equivalence class and uniqueness. If something is mathematically significant, it should in some sense be unique. But math is a model and so, as such, can never be entirely unique but represents a variety of cases. Thus it is ever natural to define equivalence classes, especially in math itself. For example, a rational number is an equivalence class that establishes a proportion. 2016 rugpjūčio 16 d., 13:08
atliko 
Pridėta 7 eilutė:
* Perhaps similarly, having in mind the Zariski topology, ideals of a ring are defined with respect to multiplication (union) but not addition (intersection). 2016 rugpjūčio 16 d., 13:06
atliko 
Pakeista 6 eilutė iš:
* By definition, a topological space includes both an entire set X and the empty set. However, the intermediary sets are closed under arbitrary unions, but only finite intersections. į:
* By definition, a topological space includes both an entire set X and the empty set. However, the intermediary sets are closed under arbitrary unions, but only finite intersections. What would happen if they were closed under infinite intersections? 2016 rugpjūčio 16 d., 13:04
atliko 
Pridėtos 19 eilutės:
I am studying the various case of duality in math. I imagine that at the heart is the duality between zero and infinity by way of one as in [[Gods Dance  God's Dance]]. '''Duality breaking''' The duality between zero and infinity, between nothing and everything, is broken in many subtle ways. Here are some examples: * By definition, a topological space includes both an entire set X and the empty set. However, the intermediary sets are closed under arbitrary unions, but only finite intersections. 2016 birželio 21 d., 09:00
atliko 
Pridėtos 1108 eilutės:
>>bgcolor=#EEEEEE<< Dualities. Duality arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other. For example: * '''Square roots of i.''' There are two square roots of 1. One we call +i, the other i, but neither should have priority over the other. Similarly, clockwise and counterclockwise rotations should not be favored. Complex conjugation is a way of asserting this. (Note that the integer +1 is naturally favored over 1. But there is no such natural favoring for i. It is purely conventional, a misleading artificial contrivance.) * A rectangular matrix can be written out from left to right or right to left. So we have the transpose matrix. ** Normality says conjugate invariancy: gN = Ng. * '''[[Opposite category  Opposite category]]''' Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around. ** Colimits and limits ** Monomorphisms ("onetoone") and epimorphisms (forcing "onto"). ** Coproducts and products ** Initial and terminal objects ** Wikipedia: In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). If we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices. ** Wikipedia: Reversing the direction of inequalities in a partial order. (Partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element.) ** Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called [[https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_duality  Eckmann–Hilton duality]]. ** [[https://en.m.wikipedia.org/wiki/Adjoint  Adjoint]] bendrai ir [[https://en.wikipedia.org/wiki/Adjoint_functors  Adjoint functors]]. Wikipedia: It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. A construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. * Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies [[https://en.wikipedia.org/wiki/Ramsey's_theorem  Ramsey's theorem]]. Tao: "the Ramseytype theorem, each one of which being a different formalisation of the newly gained insight in mathematics that complete disorder is impossible." * Coordinate systems can be organized "bottom up" or "top down". This yields the duality in projective geometry. ** Root systems relate reflections (hyperplanes) and root vectors. Given a root R, reflecting across its hyperplane, every root S is taken to another root S, and the difference between the two roots is an integer multiple of R. But this relates to the commutator sending the differences into the module based on R. * Analysis provides lower and upper bounds on a function or phenomenon which helps define the geometry of this space. * We can look at the operators that act or the objects they act upon. This brings to mind the two representations of the foursome. ** This is related to the duality between left and right multiplication. Examples include Polish notation. * Faces of an object and corners of an object. (Why are they dual?) * Coxeter groups are built from reflections. Reflections are dualities. * Any two structures which have a nice map from one to the other have a duality in that you can start from one and go to the other. ** Galois theory: field extensions (solutions of polynomials) and groups ** Lie groups: solutions to differential equations.. Read [[https://ncatlab.org/nlab/show/duality  nLab: Duality]]. Here are examples to consider: * [[https://en.wikipedia.org/wiki/Duality_%28projective_geometry%29  Duality (projective geometry)]]. Interchange the role of "points" and "lines" to get a dual truth: The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". (Compare with the construction of an equilateral triangle and the lattice of conditions.) * AtiyahSinger index theorem... * RiemannRoch theorem * Covectors and vectors * Cotangent space and tangent space * [[https://en.m.wikipedia.org/wiki/De_Rham_cohomology  de Rham cohomology]] links algebraic topology and differential topology * [[https://en.wikipedia.org/wiki/Modular_theorem Modularity theorem]]. * [[https://en.m.wikipedia.org/wiki/Langlands_program  Langlands program]] * general Stokes theorem: duality between the boundary operator on chains and the exterior derivative * [[https://en.m.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz  Hilbert's Nullstellensatz]] * Class field theory provides a onetoone correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. * Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. * One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[79] This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complexvalued function. * Meromorphic function is the quotient of two holomorphic functions, thus compares them. * [[https://ncatlab.org/nlab/show/Isbell+duality  Isbell duality]] relates higher geometry with higher algebra. * [[https://ncatlab.org/nlab/show/topos  Topos]] links geometry and logic. * For integers, decomposition into primes is a "bottom up" result which states that a typical number can be compactly represented as the product of its prime components. The "top down" result is that this depends on an infinite number of exceptions ("primes") for which this compact representation does not make them more compact. * The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. * https://en.m.wikipedia.org/wiki/Coherent_duality https://en.m.wikipedia.org/wiki/Serre_duality https://en.m.wikipedia.org/wiki/Verdier_duality https://en.m.wikipedia.org/wiki/Poincaré_duality * https://en.m.wikipedia.org/wiki/Dual_polyhedron * a very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G. * division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. * Tensor products are adjoint to a set of homomorphisms. * Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros. * In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P. Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a vertex operator algebra called the moonshine module or monster vertex algebra, constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, having the monster group as symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proved by Richard Borcherds for the moonshine module in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. [[https://en.wikipedia.org/wiki/List_of_dualities  List of dualities (Wikipedia)]] * Alexander duality * Alvis–Curtis duality * Araki duality * Betadual space * Coherent duality * De Groot dual * Dual abelian variety * Dual basis in a field extension * Dual bundle * Dual curve * Dual (category theory) * Dual graph * Dual group * Dual object * Dual pair * Dual polygon * Dual polyhedron * Dual problem * Dual representation * Dual qHahn polynomials * Dual qKrawtchouk polynomials * Dual space * Dual topology * Dual wavelet * Duality (optimization) * Duality (order theory) * Duality of stereotype spaces * Duality (projective geometry) * Duality theory for distributive lattices * Dualizing complex * Dualizing sheaf * Esakia duality * Fenchel's duality theorem * Haag duality * Hodge dual * Jónsson–Tarski duality * Lagrange duality * Langlands dual * Lefschetz duality * Local Tate duality * Poincaré duality * Twisted Poincaré duality * Poitou–Tate duality * Pontryagin duality * Sduality (homotopy theory) * Schur–Weyl duality * Serre duality * Spanier–Whitehead duality * Stone's duality * Tannaka–Krein duality * Verdier duality * AGT correspondence * A "transformation group" is a group acting as transformations of some set S. Every transformation group is the group of all permutations preserving some structure on S, and this structure is essentially unique. The bigger the transformation group, the less structure: symmetry and structure are dual, just like "entropy" and "information", or "relativity" and "invariance". >><< 
DualityNaujausi pakeitimai 
Puslapis paskutinį kartą pakeistas 2019 birželio 08 d., 09:58
