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Andrius Kulikauskas
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Book.Duality istorija
Paslėpti nežymius pakeitimus  Rodyti kodo pakeitimus
Pakeistos 364370 eilutės iš
 Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} y \\ x \end{pmatrix}$} {$\begin{pmatrix} ix \\ iy \end{pmatrix}$} where in each case two of three are applied: flipping, multiplying by i, multiplying by 1.
į:
 Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} y \\ x \end{pmatrix}$} {$\begin{pmatrix} ix \\ iy \end{pmatrix}$} where in each case two of three are applied: flipping, multiplying by i, multiplying by 1.
 Adjunction is a form of duality.
 Equality and equivalence, in general, are forms of duality.
 Duality is an extension of equivalence where the two sides of the equation are somehow different. For example, one side may be a variable and the other side a value that it is set to.
 Conjugates i and j are the form of duality that is the same as equality. Except that they are not identified as such.
 Duality (opposites coexist) and equality (all is the same) form a duality, as in the twosome.
Pakeistos 360364 eilutės iš
 Linear functionals. Onedimensional economic thinking is like linear functionals  the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinitedimensional, which is to say, it can't be captured by a onedimensional shadow?
į:
 Linear functionals. Onedimensional economic thinking is like linear functionals  the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinitedimensional, which is to say, it can't be captured by a onedimensional shadow?
Opposites
 Exercise: Find all matrices with eigenvalues 1 and 1. {$\lambda=\frac{a_{11}+a_{22} \pm \sqrt{(a_{11}+a_{22})^2  4A}}{2}$} so A=1. {$\begin{pmatrix} \pm \sqrt{1 + a_{12}a_{21}} & a_{12} \\ a_{21} & \mp \sqrt{1 + a_{12}a_{21}} \end{pmatrix}$} such as {$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$}{$\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}${$\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$}
 Multiplying by {$\begin{pmatrix} x \\ y \end{pmatrix}$} yields three ways of coding opposites: {$\begin{pmatrix} iy \\ ix \end{pmatrix}$} {$\begin{pmatrix} y \\ x \end{pmatrix}$} {$\begin{pmatrix} ix \\ iy \end{pmatrix}$} where in each case two of three are applied: flipping, multiplying by i, multiplying by 1.
Pakeistos 329360 eilutės iš
 complex numbers i=j iš kurio atsiveria 1<>1, i<>i. Paprastai i > j > i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai 1111 amžinai.
į:
 complex numbers i=j iš kurio atsiveria 1<>1, i<>i. Paprastai i > j > i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai 1111 amžinai.
Notes
Duality
 Duality of nonrelevance in system (kernel) and nonrelevance beyond system (cokernel).
 Compare the kinds of dualities with the kind of transformations of perspectives and then with the kinds of geometric transformations.
 In the automata hierarchy, consider how to model duality of internal structure and external network.
 Duality of vectors and covectors (linear functionals)  flipsides of composition  what is "inside" and "outside" the composer.
 Conjugates can be thought of as "twins", whereas +1 and 1 are "spouses".
 Conjugate = mystery = false. (Hidden distinction).
 i>j is asymmetric, onedirectional. i<>i* is symmetric, twodirectional, breaks antisymmetry, hides antisymmetry (which is i and which is j?)
 Algebra  geometry duality. (Pullback). Morphism <> ring homomorphism. Intrinsic and extrinsic geometry. Ambient space. Relation between two spaces. Varieties [morphisms X to Y] <> Affine Falgebras [Flinear ring homomorphisms F[Y] to F[x]]
 Study the idea behind linear functionals, fundamental representations, eigenvectors, cohomology, and other maps into one dimension.
 Duality examples (conjugates)
 complex number "i" is not one number  it is a pair of numbers that are the square roots of 1
 spinors likewise
 Dn where n=2
 the smallest crosspolytope with 2 vertices
 taking a sphere and identifying antipodal elements  this is a famous group
 polar conjugates in projective geometry (see Wildberger)
 Study how Set breaks duality (the significance of initial and terminal objects).
 Each physical force is related to a duality:
 Charge (matter and antimatter)  electromagnetism
 Weak force  time reversal
 So the types of duality should give the types of forces.
 John Baez on duality: Dyson's Threefold Way: either X is not isomorphic to its dual (the complex case), or it is isomorphic to its dual (in the real or quaternionic cases). Study his slides!
 John Baez on duality in logic and physics
 Algebra (of the observer) and analysis (of the observed) exhibit a duality of worldviews.
 The interpretation {$\mathbb{C} \leftrightarrow \mathbb{R}^2$} gives meaning to the two axes. One axis the opposites 1 and 1 and the other axis is the opposites i and j. And they become related 1 to i to 1 to j. Thus multiplication by i is rotation by 90 degrees. It returns us not to 1 but sends us to 1.
 Composition algebra. Doubling is related to duality.
 Linear functionals. Onedimensional economic thinking is like linear functionals  the dual space of the multidimensional reality. In finite dimensions, the dual dual is the same as what we started. But in infinite dimensions not necessarily. Does this suggest that our life is infinitedimensional, which is to say, it can't be captured by a onedimensional shadow?
Pridėta 71 eilutė:
 Tannaka duality: A simple case of Tannaka duality is that of Gsets of a group, i.e. representations on a set. In this case, Tannaka duality follows entirely from repeated application of the ordinary Yoneda lemma.
Pridėta 70 eilutė:
 Tannaka duality An algebra A is reconstructible from the fiber functor on the category of all its modules. (I think: an algebra A is reconstructible from all of its representations.)
Pakeistos 2021 eilutės iš
į:
 Does category theory distinguish between automorphisms and isomorphisms?
Pridėta 72 eilutė:
 An automorphism (mapping a structure to itself) is an internal isomorphism. Typically, isomorphisms are external.
Pridėta 70 eilutė:
 This is the point of Hopf algebras: multiplication (generating external structure) is dual to comultiplication (generating internal structure)
Pridėtos 293296 eilutės:
Cohomology at nLab: Cohomology is dual to homotopy (as an operation): the cohomology of X with coefficients in A is the homotopy of A with cocoefficients in X. Cohomology should be thought of as "cohomotopy". For any {$A\in H$} the set {$H(S^n,A)$} is equivalently:
 the Acohomology of {$S^n$}
 the nth homotopy group of A
Pakeistos 315320 eilutės iš
So I'm very interested where such dualities and opposites come up in math.
į:
So I'm very interested where such dualities and opposites come up in math.
Unmarked opposite
 turinys = raiška. "Those things are which show themselves to be." buvimo pagrindas
 inner 2cycle, kurio paprastai nebūna.
 complex numbers i=j iš kurio atsiveria 1<>1, i<>i. Paprastai i > j > i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai 1111 amžinai.
Pakeistos 306315 eilutės iš
 Representation theory: Langlands dual of certain nonAbelian groups
į:
 Representation theory: Langlands dual of certain nonAbelian groups
Another key concept for me is the idea of an "unmarked opposite" vs. a "marked opposite".
 We can have what is beyond a system be identified with an opposite within the system. For example, a blank sheet of paper can be noted by the empty set. The empty set is opposite to nonempty sets. And there is a sense in which the empty set is preferred, is central. Or we can have the identity element of a group which expresses no action at all. Similarly, good can be distinguished from bad by claiming that God is good, where God is what is beyond the system. So here in this sense I say that good is the marked opposite, the one identified with what is beyond the system.
 Also, in a different sense, in a system, a marked opposite is when you have two opposites (choices) that are clearly distinguished and one is the default (thus preferred) because it is unmarked, whereas the other one is marked to distinguish it and thus secondary. For example, 1 and 1. 1 is unmarked and 1 is marked (and it actually has an extra mark). And they are clearly distinct: 1 x 1 = 1 whereas 1 x 1 = 1.
 Finally, we can have "unmarked opposites" where two choices are distinct but otherwise not distinguishable. They have yet to be marked. For example, the two square roots of 1. One will imagined clockwise, the other counterclockwise, perhaps. But which is which doesn't matter. Only when we name them using + and , only when we attribute such a prejudice to them, do we lose touch with their original indistinguishability, thus ending up with i and i, forgetting that i is no less basic than +i.
So I'm very interested where such dualities and opposites come up in math.
Pakeistos 292306 eilutės iš
į:
Tannaka duality
Examples in Math Companion:
 Platonic solids
 Points and lines in the projective plane
 Sets and their complements
 Dual vector spaces
 Polar bodies
 Duals of Abelian groups
 Homology and cohomology
 Differential forms: kforms and kdimensional surfaces
 Distributions: ...
 Mirror symmetry: (conjectural) CalabiYau manifolds and mirror manifolds
 Fourier transform?
 Representation theory: Langlands dual of certain nonAbelian groups
Pakeista 124 eilutė iš:
 Duality %28projective geometry%29. 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.)
į:
 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.)
Pakeista 60 eilutė iš:
į:
Pakeista 124 eilutė iš:
 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.)
į:
 Duality %28projective geometry%29. 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.)
Pakeista 137 eilutė iš:
Duality: Complements: Plane duality
į:
Duality: Complements: Plane duality
Pakeista 142 eilutė iš:
 Dual polygon: rectification; polar reciprocation (pole and polar); projective duality; combinatorially.
į:
 Dual polygon: rectification; polar reciprocation (pole and polar); projective duality; combinatorially.
Pakeista 201 eilutė iš:
į:
Pakeista 211 eilutė iš:
 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:
į:
 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:
Pakeista 259 eilutė iš:
 Electricity and magnetism  Electrical circuits In special relativity, applying the Lorentz transformation to the electric field transforms it into a magnetic field.
į:
 Electricity and magnetism  Electrical circuits In special relativity, applying the Lorentz transformation to the electric field transforms it into a magnetic field.
Pakeista 266 eilutė iš:
į:
Pakeistos 290292 eilutės iš
This duality between compact and noncompact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. Wikipedia: List of simple Lie groups: Symmetric spaces
į:
This duality between compact and noncompact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. Wikipedia: List of simple Lie groups: Symmetric spaces
Tannaka duality
Pakeistos 288290 eilutės 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". "Turn" does not need to be marked, but "noturn" needs to be marked ("no"). Although, contentwise, notturning is unmarked and turning is marked.
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.
This duality between compact and noncompact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. Wikipedia: List of simple Lie groups: Symmetric spaces
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.
Pakeistos 286288 eilutės iš
 Electrical duality Topology and Elementary Electric Circuit Theory, II: Duality Tony Phillips
į:
 Electrical duality 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".
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 Topology and Elementary Electric Circuit Theory, II: Duality Tony Phillips
Ištrintos 89 eilutės:
Pakeista 10 eilutė iš:
į:
Pridėtos 14 eilutės:
Pridėtos 69 eilutės:
Investigation: Understand mathematics as the discrimination of a variety of dualities.
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).
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.
Pridėta 49 eilutė:
 Wikipedia Category: Duality theories
Pakeista 15 eilutė iš:
 Dual problems in linear programming. What sort of duality is this?
į:
 Dual problems in linear programming. What sort of duality is this? Is it related to adjoints?
Pridėta 164 eilutė:
 Geometric Langlands duality from six dimensions Edward Witten
Pakeista 2 eilutė iš:
į:
Pridėta 16 eilutė:
Pakeista 265 eilutė iš:
 The CayleyDickson construction is all about duality breaking. It yields commutativity, associativity, etc.
į:
 The CayleyDickson construction is all about duality breaking. It thereby yields noncommutativity, nonassociativity, etc.
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).
Pridėta 18 eilutė:
See: John Baez: Duality in Logic and Physics
Pridėta 152 eilutė:
 Internal vs. external geometry = implicit vs. embedding = vector space vs. dual space (functionals)
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.
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.
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.
Pridėta 14 eilutė:
 Study the duality between 1^N and N in symmetric functions (Young tableaux) but also Catalan numbers, etc.
Pakeistos 259261 eilutės iš
Duality breaking allows that God is good and not bad. Because we want to break the duality of good and bad, increasing and decreasing slack. Orientation is a complete, absolute, total distinction between inside and outside, their complete segregation and isolation. (In contrast to the yinyang symbol.) So it is highly tenuous  it can break at any single point  but it can eternally grow more weighty.
į:
Duality breaking
 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.
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.
Pridėtos 252257 eilutės:
More:
The 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
Pridėtos 1213 eilutės:
 Dual problems in linear programming. What sort of duality is this?
Pakeista 147 eilutė iš:
į:
 Vectors and covectors. A vector is 1dimensional and a covector is n1 dimensional hyperplane (tangent plane), see Penrose chapter 12.
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 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 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.
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.
Pakeista 33 eilutė iš:
 In ring theory, there is a subtle distinction between the ascending chain condition  Artinian rings, and the descending chain condition  Noetherian rings.
į:
 In ring theory, there is a subtle distinction between the descending chain condition  Artinian rings, and the ascending chain condition  Noetherian rings.
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.
Pridėta 78 eilutė:
 Conjugation (ab)* = b*a* is very important in the CayleyDickson construction of the numbers: real, complex, quaternion, octonion.
Pridėta 91 eilutė:
 For a normal subgroup, the left cosets match the right cosets.
Pridėta 11 eilutė:
 Make a list of the central objects in combinatorics  look at Stanley's books.
Pridėta 10 eilutė:
 Make a list of the central mathematical examples to study and relate.
Pridėta 9 eilutė:
 Relate my combinatorial proof of the CayleyHamilton theorem to Nakayama's lemma making use of Atiyah's observation.
Pakeista 32 eilutė iš:
į:
Pakeista 225 eilutė iš:
Duality in Math and Physics by Sir Michael Atiyah
į:
Duality in Mathematics and Physics by Sir Michael Atiyah
Pridėtos 224225 eilutės:
Duality in Math and Physics by Sir Michael Atiyah
Pridėta 227 eilutė:
Pakeistos 229234 eilutės iš
į:
Functionals
 Tduality
 AdS/CFT correspondence
 SYZ conjecture
 Mirror symmetry
Pridėta 236 eilutė:
Pakeista 226 eilutė iš:
 Sduality strong coupling  weak coupling.
į:
 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.
Pridėtos 222226 eilutės:
Duality in Physics
 Electricity and magnetism  Electrical circuits In special relativity, applying the Lorentz transformation to the electric field transforms it into a magnetic field.
 Sduality strong coupling  weak coupling.
Pakeista 164 eilutė iš:
 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.
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. Peter Smith. The Galois Connection between Syntax and Semantics.
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.
Pridėta 163 eilutė:
 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.
Pakeista 101 eilutė iš:
 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. Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
į:
 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. 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.
Pridėta 138 eilutė:
 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.
Pridėta 8 eilutė:
 How do 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?
Pakeista 100 eilutė iš:
 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.
į:
 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. Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
Pakeistos 165168 eilutės iš
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.
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.
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?
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:
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.
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.
 de Rham cohomology links algebraic topology and differential topology
 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.
 Isbell duality relates higher geometry with higher algebra.
 Topos links geometry and logic.
 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.
 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 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.
 Pontryagin duality of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. The reciprocal lattice is related to this.
 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.
 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. An Elementary Introduction to the Langlands Program by Stephen Gelbart.Langland Frenkel. 6D (2,0) superconformal field theory.
 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 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 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 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 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 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 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 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.
 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.
 Weil pairing is generalized by Cartier duality, which is an analogue of Pontryagin duality for noncommutative schemes.
Pakeistos 152164 eilutės iš
į:
Duality: Adjunction
 Adjoint bendrai ir 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:
Adjoints
 Adjoint bendrai ir 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 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.
 Pontryagin duality of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. The reciprocal lattice is related to this.
 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.
 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. An Elementary Introduction to the Langlands Program by Stephen Gelbart.Langland Frenkel. 6D (2,0) superconformal field theory.
 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 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 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 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 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 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 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 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.
 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.
 Weil pairing is generalized by 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.
 de Rham cohomology links algebraic topology and differential topology
 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.
 Isbell duality relates higher geometry with higher algebra.
 Topos links geometry and logic.
 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.
 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.
Ištrinta 58 eilutė:
Ištrinta 61 eilutė:
Pakeista 67 eilutė iš:
Duality: Bottomup and Topdown
į:
Duality: Bottomup and Topdown
Pakeistos 7395 eilutės iš
į:
Duality: Complements
 Center and totality for a simplex and other infinite families of complex polytopes.
 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.)
 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. Verdier duality is a generalization.
 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.
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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.
 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.
 In differential geometry, the 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.
 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. 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, 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 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 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.
 Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary.
Duality: Complements: 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.
 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.
 Dual polygon: rectification; polar reciprocation (pole and polar); projective duality; combinatorially.
 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.
 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
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.
Pakeistos 111116 eilutės iš
Duality: Reversing the maps
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
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: Complements
 Center and totality for a simplex and other infinite families of complex polytopes.
 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.)
 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. Verdier duality is a generalization.
 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.
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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.
 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.
 In differential geometry, the 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.
 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. 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, 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 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 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.
 Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary.
Duality: Complements: 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.
 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.
 Dual polygon: rectification; polar reciprocation (pole and polar); projective duality; combinatorially.
 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.
 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:
Pakeistos 4950 eilutės iš
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.)
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:
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 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
 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 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 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 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
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 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
 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 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 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
Duality: Existing and nonexisting
 Switching of "existing" and "nonexisting", for example, edges in a graph. This underlies 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:
Pridėta 45 eilutė:
 Duality between matrices expressed explicitly (in terms of entries) and implicitly (in terms of eigenvalues).
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.
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.
Pakeistos 88102 eilutės iš
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. 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.
 de Rham cohomology links algebraic topology and differential topology
 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.
 Isbell duality relates higher geometry with higher algebra.
 Topos links geometry and logic.
 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.
 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:
Pridėtos 9192 eilutės:
Duality: Halving space: Rotation: Reversing orientation
Pridėtos 97100 eilutės:
Duality: Reflection
 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 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 Langlands dual group is the complex connected reductive group whose root datum is dual to that of G.
Pakeistos 114118 eilutės iš
Duality: Reflection
 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 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 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.
 de Rham cohomology links algebraic topology and differential topology
 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.
 Isbell duality relates higher geometry with higher algebra.
 Topos links geometry and logic.
 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.
 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.
Pridėtos 46 eilutės:
 Understand the bijective proof between involutions and standard tableau. Understand
 SchurWeyl duality
 RobinsonSchenstedKnuth correspondence and relation to Schur functions, and implications for symmetric functions of eigenvalues.
Pakeistos 1213 eilutės iš
į:
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
Pridėtos 17 eilutės:
 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.
Pakeista 9 eilutė iš:
 Stone's representation theorem for Boolean algebras. Every Boolean algebra is isomorphic to a certain field of sets.
į:
 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.
Pridėtos 2734 eilutės:
The fundamental antiduality (SchurWeyl duality) between external representations and internal structure
 Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finitedimensional representations of the general linear and symmetric groups.
 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:
Duality: 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.
Ištrinta 95 eilutė:
 Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finitedimensional representations of the general linear and symmetric groups.
Pridėta 25 eilutė:
 Tableaux as the square root of a matrix.
Pridėta 24 eilutė:
 My dream of Young tableaux whose entries were short and long dashes  "this is the fundamental unit of information".
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?
Ištrintos 9699 eilutės:
Pullback?
 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.
 Weil pairing is generalized by Cartier duality, which is an analogue of Pontryagin duality for noncommutative schemes.
Pridėtos 166167 eilutės:
 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.
 Weil pairing is generalized by Cartier duality, which is an analogue of Pontryagin duality for noncommutative schemes.
Pridėta 121 eilutė:
 Center and totality for a simplex and other infinite families of complex polytopes.
Pakeista 97 eilutė iš:
į:
Pridėta 99 eilutė:
 Weil pairing is generalized by Cartier duality, which is an analogue of Pontryagin duality for noncommutative schemes.
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...
Pridėta 41 eilutė:
 Covariance and contravariance
Pakeista 54 eilutė iš:
 Adjoint bendrai ir 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:
 Adjoint bendrai ir 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.
Pridėtos 1316 eilutės:
Sources of examples of duality
 List of dualities (Wikipedia)
 nLab: Duality.
Ištrintos 187190 eilutės:
Examples of duality
 List of dualities (Wikipedia)
 nLab: Duality.
Pakeistos 6869 eilutės iš
į:
 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 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:
 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
 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.
Pakeistos 6768 eilutės iš
į:
 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. An Elementary Introduction to the Langlands Program by Stephen Gelbart.Langland Frenkel. 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
 An Elementary Introduction to the Langlands Program by Stephen Gelbart
 Langland Frenkel
 6D (2,0) superconformal field theory ?
Pridėta 140 eilutė:
 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.
Pridėta 73 eilutė:
 Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finitedimensional representations of the general linear and symmetric groups.
Pakeista 109 eilutė iš:
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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:
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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.
Pridėta 108 eilutė:
 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.
Pridėta 113 eilutė:
 In Galois cohomology, 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:
 Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary.
Ištrintos 194203 eilutės:
 Dualizing complex
 Dualizing sheaf
 Esakia duality
 Fenchel's duality theorem
 Haag duality
 Hodge dual
 Jónsson–Tarski duality
 Lagrange duality
 Langlands dual
 Lefschetz duality
Pridėta 16 eilutė:
 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.
Pridėta 113 eilutė:
 The Hodge isomorphism or 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.
Pridėta 17 eilutė:
 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.
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 dually equivalent.
Pakeistos 108110 eilutės iš
į:
 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. 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 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 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
Pakeistos 89 eilutės iš
 Generalized by Stone's duality: categorical dualities between certain categories of topological spaces and categories of partially ordered sets.
į:
 Generalized by 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 pointless topology and are exploited in theoretical computer science for the study of formal semantics.
 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:
 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.
Pridėta 135 eilutė:
 The dual space X' of a 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.
Pakeistos 2942 eilutės iš
 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:
 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.
Pridėta 29 eilutė:
 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.
Pridėtos 119120 eilutės:
 A 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 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 Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures.
Pridėta 117 eilutė:
 In functional analysis and related areas of mathematics a 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.
Pridėta 107 eilutė:
 A dual vector space (or just 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 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.
Pridėta 50 eilutė:
 Coxeter groups are built from reflections. Reflections are dualities.
Pakeista 52 eilutė iš:
 Coxeter groups are built from reflections. Reflections are dualities.
į:
 If G is a group and ρ is a linear representation of it on the vector space V, then the 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.
Pakeistos 9798 eilutės iš
į:
 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 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
Pakeista 91 eilutė iš:
Duality: Complements: Planar
į:
Duality: Complements: Plane duality
Pakeistos 9697 eilutės iš
į:
 Dual polygon: rectification; polar reciprocation (pole and polar); projective duality; combinatorially.
Ištrinta 154 eilutė:
Pridėta 109 eilutė:
 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:
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 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.
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 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.
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.
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.
 Dual object is a category theory generalization of the concept of dual space in linear algebra.
Pakeistos 152153 eilutės iš
į:
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.
Pakeistos 6768 eilutės iš
 Duality  parity  išsiaiškinimo rūšis.
į:
 Duality  parity  išsiaiškinimo rūšis
Duality: Reflection
 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 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 Langlands dual group is the complex connected reductive group whose root datum is dual to that of G.
Pakeistos 105106 eilutės iš
 Pontryagin duality of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T.
į:
 Pontryagin duality of a locally compact abelian group G is the group given by maps (characters) from it to the circle group T. The reciprocal lattice is related to this.
 Given the lattice of characters of a maximal torus, the dual lattice is given by the 1parameter subgroups.
Pakeista 96 eilutė iš:
į:
Pakeistos 100101 eilutės iš
į:
 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:
Pakeista 86 eilutė iš:
Duality: Complements: Graph
į:
Duality: Complements: Planar
Ištrintos 123125 eilutės:
 Faces of an object and corners of an object. (Why are they dual?)
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.
 Dual polyhedron
Pakeistos 138140 eilutės iš
 https://en.wikipedia.org/wiki/Dual_polyhedron
į:
Pakeista 142 eilutė iš:
 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.
į:
Pridėtos 1215 eilutės:
Intersections and Unions
 Inclusionexclusion principle
 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:
Pridėtos 6264 eilutės:
Pullback
 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:
 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. Verdier duality is a generalization.
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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.
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/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
 Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros
Complements
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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
 Inclusionexclusion principle
 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:
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.
Pullback
 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 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.
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 Eckmann–Hilton duality.
Ištrinta 93 eilutė:
 Wikipedia: Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called Eckmann–Hilton duality.
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
 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 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:
Pakeistos 2122 eilutės iš
 Adjoint bendrai ir 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:
Adjoints
 Adjoint bendrai ir 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.
 Isbell duality relates higher geometry with higher algebra.
 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:
 List of dualities (Wikipedia)
 nLab: Duality.
Pakeistos 9091 eilutės iš
Read nLab: Duality. Here are examples to consider:
į:
Pakeistos 100103 eilutės iš
 Isbell duality relates higher geometry with higher algebra.
 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
 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.
List of dualities (Wikipedia)
į:
 Duality  parity  išsiaiškinimo rūšis. Įvairios simetrijos  išsiaiškinimo būdų sandaros
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.
 de Rham cohomology links algebraic topology and differential topology
 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
 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:
 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.)
į:
Pakeistos 8384 eilutės iš
 Cotangent space and tangent space
 de Rham cohomology links algebraic topology and differential topology
į:
Pakeistos 8892 eilutės iš
 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.
į:
Pakeista 16 eilutė iš:
 Opposite category? Morphisms can be organized from left to right or from right to left. The opposite category turns all of the arrows around.
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
 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..
į:
Pakeistos 1532 eilutės iš
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.
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: 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 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
 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.
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.
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 Eckmann–Hilton duality.
Pakeistos 18 eilutės iš
Study duality as the basis of logic, and mathematics as ways of altering duality.
(internal structure mirrors external structure  duality of category theory)
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 God's Dance.
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 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.
Pakeistos 910 eilutės iš
į:
Pakeistos 1617 eilutės iš
Mathematics: Near duality  Duality breaking
į:
Logic in Mathematics: The Kinds of Duality
Mathematics: Almost Duality  Duality Breaking
Pakeistos 2628 eilutės iš
Equivalence and uniqueness
į:
Mathematical Tension: Equivalence and Uniqueness
Ištrintos 3036 eilutės:
 Langlands program
 An Elementary Introduction to the Langlands Program by Stephen Gelbart
 Langland Frenkel
 6D (2,0) superconformal field theory ?
Pridėtos 164168 eilutės:
 Langlands program
 An Elementary Introduction to the Langlands Program by Stephen Gelbart
 Langland Frenkel
 6D (2,0) superconformal field theory ?
Pakeistos 107108 eilutės iš
 Dual abelian variety. There is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety.
į:
Pullback
 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
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
į:
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.
Pridėta 101 eilutė:
Pakeistos 104108 eilutės iš
 De Groot dual
 Dual abelian variety
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.
Pridėta 91 eilutė:
Pakeistos 93101 eilutės iš
 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}}
 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
 Inclusionexclusion principle
 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:
Pakeistos 9194 eilutės iš
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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.
 Alvis–Curtis duality
į:
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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.
 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}}
Pakeistos 8993 eilutės iš
į:
 Jordan curve theorem (separating the inside and outside of a curve) generalized by the Jordan–Brouwer separation theorem, generalized by 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:
 Spanier–Whitehead duality
į:
Pakeistos 910 eilutės iš
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).
 Stone's representation theorem for Boolean algebras. Every Boolean algebra is isomorphic to a certain field of sets.
 Generalized by 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:
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:
į:
Ištrinta 136 eilutė:
Pakeistos 136138 eilutės iš
Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia.
į:
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.
Pridėtos 135136 eilutės:
Kategorijų teorijos prieštaringumas yra, kad pavyzdžiai yra "objektai" su vidinėmis sandaromis, nors tai kertasi su kategorijų teorijos dvasia.
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)
Pridėtos 1318 eilutės:
 Langlands program
 An Elementary Introduction to the Langlands Program by Stephen Gelbart
 Langland Frenkel
 6D (2,0) superconformal field theory ?
Pridėtos 121122 eilutės:
Duality of silence (topdown) and speaking (bottomup).
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.
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).
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?
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 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.
Pridėtos 1108 eilutės:
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? 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 Eckmann–Hilton duality.
 Adjoint bendrai ir 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 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 nLab: Duality. Here are examples to consider:
 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
 de Rham cohomology links algebraic topology and differential topology
 Modularity theorem.
 Langlands program
 general Stokes theorem: duality between the boundary operator on chains and the exterior derivative
 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.
 Isbell duality relates higher geometry with higher algebra.
 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.
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".

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