The duality tag has no usage guidance.

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### Duality and Euler paths in graphs

I'm computer scientist and in one of my researches I'm facing this question:
if I have a planar graph that admits an Euler path (i.e. has 0 or 2 odd degree vertices, as Euler's theorem says), then his ...

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### Duality for Generalization of standard Convex

in accordance to the previous question about KKT condition for generalization to standard convex, here I look for the dual problem to the generalized convex problem. the clear questions are :
is it ...

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129 views

### Creating Duals in A Category

Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: ...

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85 views

### Natural Poisson brackets on $S(V^*)$

Let $V$ be a finite dimensional vector space and $S(V)$ the corresponding symmetric algebra. Suppose that we have a Poisson bracket $\lambda = \{,\}: S(V) \otimes S(V) \to S(V)$. Let $V^*$ be the dual ...

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### Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated algebras over $\mathbb{C}$. The equivalence associates to an affine variety its ...

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### Relative dualizing sheaf (reference, behavior)

Let $\mathcal{C}\rightarrow S$ a flat projective family of locally complete intersection projective curves over a integral noetherian scheme (say a spectrum of a local ring). I was wondering whether ...

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### Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...

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### Duality results for quadratic equality constrained optimization problem

Consider the optimization problem
$$
\begin{align}
\min_{x\in\mathbb{R}^n}&\quad f(x) = x^TA_0x+b_0^Tx+c_0\tag{P1}\\
\nonumber \text{subject to } \quad & g_i(x) = x^TA_ix+b_i^Tx+c_i = ...

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186 views

### Grothendieck-Verdier duality for affine morphisms

Suppose $X,Y$ are varieties over $\mathbb{C}$, $Y$ is smooth and $X$ is Gorenstein ($X$ is not smooth in my case). Let $f: X \to Y$ be an affine morphism, and each fibre of $f$ has the same dimension ...

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120 views

### Elementary proof of a triangular grid lemma

I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714).
Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...

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### Dual cone of 'positive' Bochner integrable functions

If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...

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### Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...

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131 views

### Predual of a subspace

Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of
$(E^*)^d$ with finite codimension.
I would like know if the space $\mathcal G$ is a dual space ...

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### Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...

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### Dual of Banach-valued $L^p$ [duplicate]

Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb ...

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### What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras?

Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the ...

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183 views

### Dual cell structures on manifolds

Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by ...

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### Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.
A poset $(P,\leq)$ is called ...

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### Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?

Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in ...

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### Is there any relationship between the topologies of the clique complex and the independence complex?

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...

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### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

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### Does projective duality preserve arithmetic-Cohen-Macaulay-ness?

Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring ...

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### Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...

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207 views

### Why is the Tate local duality pairing compatible with the Cartier duality pairing?

This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...

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### Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...

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### Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...

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### opposite category

In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$.
Is ${op}$ the instance in Cat of a more ...

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### Smooth Affine algebras are Calabi-Yau

Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/

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### dual problem of SDP [closed]

suppose we have the following optimization problem:
\begin{array}{l}
\mathop {\min }\limits_{{\bf{X}},{\bf{x}}} \,\,Tr\left( {{\bf{XA}}} \right) + 2{{\bf{a}}^H}{\bf{x}} + b\\
s.t:\,\,\,\,\left[ ...

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### Equivalent Norms for the Dual of Sobolev / Bessel Spaces

Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb ...

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### Is the closed ball of a normed space closed in any Hausdorff locally convex topology, weaker than the norm topology?

Assume that we have a normed space $X$ and a subspace $Y$ of $X^{*}$ such that $Y^{\perp}=\{0\}$. They form a non-degenerate dual pare.
Moreover, $\|y\|=\sup_{x\in B_{X}}|\langle x,y\rangle|$, where ...

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### Is dual cone unique? [closed]

Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition ...

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### Reference Request: Algebraic Serre's Duality Theorem for Curves

Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne ...

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### Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false)
If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality?
All i need to find is a protective ...

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### Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...

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### Simultaneously extending the functionals of a subspace of a Banach space to the whole space

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. If $\varphi\in Y^*$, then Hahn-Banach allows us to extend $\varphi$ to a $\tilde\varphi\in X^*$, such that $\|\tilde\varphi\|=\|\varphi\|$. ...

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### Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...

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### hyperfunctions and analytic duals

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...

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### Electromagnetic duality symmetry

This question arose while reading http://prd.aps.org/abstract/PRD/v13/i6/p1592_1 (Duality transformations of Abelian and non-Abelian gauge fields, by Stanley Deser and Claudio Teitelboim). It is well ...

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### Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...

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### dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...

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### Self-dual surfaces in $\mathbb P^3$ with isolated singularities

I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties:
the smooth quadric;
Kummer surfaces;
The surface with the equation ...

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### singularities of the dual variety of a surface

I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...

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### Question on convex optimization and dual norms [closed]

I have the following questions about dual norms :
How do you prove that the dual of the dual norm is in fact the original norm?
This is what I have so far:
If I have $||y||_* $ as the norm dual of ...

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### Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...

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### Looking for concrete description of a category derived from abelian groups

The category of abelian groups $\mathsf{Ab}$ is the $\mathcal{Ind}$-completion of the full subcategory of finitely presentable abelian groups $\mathsf{Ab}_{fp}$. This is not so special, since the ...

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### Confusion about the dual/predual to the tangent plane to a Teichmüller space

I apologize in advance if this question is not considered research-level.
I am reading material on Teichmüller theory and I am getting confused as to the nature of the space $Q(R)$ of all integrable, ...

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### Which categories are the categories of models of a Lawvere theory?

Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...

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### Duality between K-theory and K-homology in the non-spin^c case.

I posted this question on Math.SE (http://math.stackexchange.com/questions/409444/), but got no answer. So I repost it here.
Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times ...

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### Is Khovanov's Frobenius algebra self-dual over the integers?

Khovanov's Frobenius algebra (used in the definition of Khovanov homology) is $\mathbb{Z}[X]/X^2$ with the comultiplication. $\Delta(X)=X\otimes X, \Delta(1)=1\otimes X+X\otimes 1$ and the trace ...