# Questions tagged [duality-theorems]

For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.

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### Dual space as analogous of inner products, what does it mean?

Currently reading through Optimization by Vector Space methods, specifically chapter 5 Dual Spaces. I've read several times about this topic, but there's a specific insight here that I've never ...
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### “Dual object” as set of irreps of a finite group

In Barry Simon's Representations of finite and compact groups he refers to the set of irreps (or rather equivalence classes thereof) of a given finite group $G$ as the “dual object” $\widehat G$ ...
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### How to determine that the problem is unbounded when using Dual Simplex Method? How to prove infeasibility of problem using dual simplex method?

This is how we got the dual simplex method explained: With this method we solve a primary problem, not a dual one, in the primary simplex method (that is, in the standard method for the minimization ...
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### Interpreting theorem that dual map $T'$ injective $\iff$ $T$ surjective.

Consider the following theorem (Axler, Linear Algebra Done Right 3rd Ed., Theorem 3.108) $T$ surjective is equivalent to $T'$ injective. Suppose $V$ and $W$ are finite-dimensional and $T\in L(V,W)$. ...
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### How to prove projective duality?

I am currently in Chapter 12 of 'Lemmas in Olympiad Geometry' by Titu Andreescu et al. They have stated and given a satisfactory proof Brianchon's Theorem using poles and polars as well as Pascal's ...
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### Pontryagin duality and quotient groups

I am studying the Pontryagin duality for LCA groups, and I came across two results in which I am finding some difficulty. Here I will denote by $G^*$ the dual of the group $G$, i.e. the group of all ...
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### Dual of Model Weil group $E(K)$ over a local field $K$ is $H^1(G_K,E)$

Let $K$ be a local field. Let $G_K=Gal(\overline{K}/K)$ be an absolute Galois group of $K$. Let $E/K$ be an elliptic curve over $K$. Let $E(K)^*=Hom_{\Bbb{Z}}(E(K),\Bbb{Q}/\Bbb{Z})$ be a Pontryagin ...
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### $\#imagef=\#imagef^*$, order of image of dual and image of original homomorphism is the same

Let $M$ be an abelian group Let $M^*=Hom_{\Bbb{Z}}(M,\Bbb{Q}/\Bbb{Z})$ be pontryagin dual of $M$. Let $M$ be a finite group. Let $M'$ be an abelian group. Let $f:M\to M'$ be a homomorphism of abelian ...
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### Is $k[x, x^{-1}]$ a (graded) injective $k[x]$-module

Consider $k[x]$ with the usual grading, and the graded $k[x]$-module $k[x, x^{-1}]$. Is it injective? I suppose yes, because it is torsion free and graded divisible (i.e., divisible by homogeneous ...
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### Dual of Dual problem of a simple convex Quadratic problem

I am trying to verify the the dual of the dual is the primal? using a simple convex QP: \begin{align} \min_x& \frac{1}{2} x^\top H x + h^\top x\\ \text{s.t.} &~Ax\leq b \\ &~ A_e x = b_e \...
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### Pontlyagin dual of direct sum,$\widehat{\bigoplus_{i\in \Lambda} M_i} = \prod_{i\in \Lambda} \hat{M_i}.$

Let $M_i$ (for $i \in \Lambda$) be a family of abelian groups. Let $\bigoplus_{i\in \Lambda} M_i$ denote the infinite direct sum of the groups $M_i$. Let $\hat{M}$ denote the Pontryagin dual of the ...
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### Equivalent Formulations of Variational Problems

This post is supposed to collect some Theorems and techniques which can be used to analyse variational problems by (a) finding a related variational problem s.t. their optimal values are the same or ...
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### What do Kassel, Rosso, and Turaev mean by "duality"?

In their book "Quantum Groups and Knot Invariants", Kassel, Rosso, and Turaev prove that $U_q\mathfrak{sl}(N+1)$ has a PBW basis. I'm having trouble following the last step, though. In ...
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### how to solve for wasserstein duality easily in a special case when 2-Wasserstein inequality constraint is binding

I was going through this nice paper ” A Simple and General Duality Proof for Wasserstein Distributionally Robust Optimization”, and one quick qu on applying Theorem 1 to my poject: What if in my ...
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### Dual graph relation to star-mesh duality

I'm confused about the realtionship between dual graphs and the so called star-mesh transformation: https://en.wikipedia.org/wiki/Star-mesh_transform. Take a simple triangle, its dual graph looks like ...
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### Minimal number of edges in G that must be removed to separate two nodes.

This is the question I am trying to answer: Let $G = (V,E)$ be a directed graph and $s,t$ $\epsilon$ $V$ with $s$ $\neq$ $t$, prove that the minimal number of edges in G that must be removed in order ...
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### Weird duality between integers and p-adic integers

The integers are, up to isomorphism, the unique infinite discrete abelian group that is isomorphic to all its non-trivial subgroups. This can be seen easily by Pontryagin Duality: It's equivalent to ...
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### An Ideal Correspondence For Twisted C*-Dynamical Sytems?

Back in the 90's, Nilsen proved the following result for normal $C^{*}$-dynamical systems: Suppose that $A$ is a $C^{*}$-algebra, and $G$ is a locally compact group. Let $\delta$ be a coaction of $G$ ...
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### Factor of a dual curve

Suppose that for a homogeneous polynomial $f(x,y,z)$ we have $f(x,y,z)=g(x,y,z) \cdot h(x,y,z)$. Assume $f_d$ and $g_d$ are the dual curves of $f$ and $g$ respectively. Is it true that $g_d$ is a ...