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1answer
35 views

The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
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2answers
59 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
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2answers
27 views

Dual Mapping Preserves Linear Independence if and only if Original Mapping is Surjective

Here is my question: Let $V$ and $W$ be finite-dimensional vectors spaces over a field $F$ and $f:V \rightarrow W$ a linear map. Show that $f$ is surjective if and only if the image under ...
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0answers
21 views

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 and ...
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0answers
15 views

Solving Linear Inequalities Using Primal and Dual LP's

I've been working through my linear programming homework, and I'm having difficulty understanding how and why you would want to use the dual to find the optimal solution to the primal. I'll write up ...
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0answers
21 views

Prove one program to be another's dual program

This is the Primal problem $\min f_0(x)$, subject to $f_i(x) \leq 0$, i = $1,2,\ldots,m$ $f(x)$ is a linear program and my target is to prove that $\max g(λ)$ subject to ...
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0answers
24 views

Mixed Integer Non Linear Problem for Relaxation Approach

I have the following problem. I have meat markets$(\mathcal{T}_1)$ and vegetable markets$(\mathcal{T}_2)$. $(\mathcal{T}_1) \cup (\mathcal{T}_2) = T$ and $(\mathcal{M}) \cap (\mathcal{V}) = ...
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0answers
9 views

principle of duality in propositional formula

Can principle of duality be applied in propositional formulas? I am trying to find ways solving problems involving propositional formulas and am wondering if I could apply principle of duality here.
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0answers
13 views

dual basis and standaard basis and matrix

let $V$ be a vector space from dimantion $n$ and $V^{\star}$ be a map from $V$ to $R$ ($V^{\star}$: $V$$\mapsto$$R$) and $A$ be a matrix from a bilineare form $T$:$V$$\times$V$^{\star}$ $\mapsto$$R$ ...
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0answers
21 views

flabbiness of hyperfunctions

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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1answer
50 views

Natural map from vector space to double dual not surjective example

I recognize the fact that the natural map from an infinite dimension vector space $V$ to it's double dual space $V^{**}$ need not necessarily to be surjective because we don't have that the $\dim V$ = ...
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1answer
20 views

Maximization with the Dual using the Simplex Method.

I have an exam in a few hours. I need to understand the solution to the following question Find the Maximal to the the following $2 x_1 + 3 x_2$ is the objective function. The constraints are ...
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1answer
27 views

How to obtain primal problem from Lagrangian?

If you're trying to optimize $\min_x f_0(x)$ subject to $f_i(x) \leq 0$ then the Lagrangian would be $$L(x, \lambda) = f_0(x) + \sum_i \lambda_i f_i(x)$$ The dual problem is $\max_\lambda g(y)$ ...
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1answer
61 views

Existence of a function in Radon Measure

I am having trouble to construct a function $f$ in the following problem. Let $X$ be Locally Compact Hausdorff topological space. Let $\mu$ is a positive Radon measure on $X$ with $\mu(X)=\infty$ ...
3
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1answer
55 views

Proof of Poincare duality in Bott and Tu's book

In Bott and Tu's book, "Differential forms in Algebraic Topology", page 45, Section 5 of Chapter one, he tried to prove the Poincare duality. But I find one step mysterious, namely when he describes ...
3
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1answer
24 views

cofree comodules and embedding

For an $R$-coalgebra C, is it possible for every C-comodule M to be embeded into a C-comodule of the form $\underset{i \in I}{\bigoplus} C$?
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1answer
35 views

Proof of properties of dual cone

Show that if C $\subseteq$ D then $D^*$ $\subseteq$ $C^*$ where * is dual cone operation. Can somebody explain it.
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1answer
22 views

Calculating the dual module

Is the dual $\mathbb{Z}$-module of $\mathbb{Z}/n\mathbb{Z}$, that is ${\rm Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\mathbb{Z})$ isomorphic to $\mathbb{Z}/n\mathbb{Z}$. Looking at it briefly I think ...
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1answer
12 views

Difference between duality, symmetry, equivalency and invaraince?

Can someone difference in detail on these four terms? Thank you.
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1answer
52 views

The dual space of normed vector space $X$ is isomorphic to the dual of its completion

Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic. Why these are isomorphic? I don't know how to prove ...
4
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1answer
81 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
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2answers
60 views

Constructing Dual notions via Duality

First of all, I do not have much mathematical background and I have minimal category theory knowledge. I am just trying to understand one or two things about category theory because the concept sounds ...
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1answer
27 views

Auto-Langlands dual gruops.

Consider a semisimple Lie group $G$. We define the Langlands dual $\hat{G}$ of $G$ as the group which has as a root system, the root system generated by the coroots of $G$. Recall that given a root ...
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1answer
21 views

Duality in a finite dimension

Let $E$ and $F$ be two finite dimension vector spaces over the same field $K$, $V$ is sub-space of $E$, $L_V(E,F)$ is the set of linear maps from $E$ to $F$ which vanish on $V$. And let $W$ such that ...
1
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1answer
66 views

The convex conjugate of a quadratic form with positive semi definite matrix

I want to find the convex conjugate (Legendre transform) of a quadratic for $1/2x^{t}Qx$ when $Q$ is positive semi-definite. If Q is non-singular, the solution is easy - the gradient is $y-Qx=0$ so ...
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0answers
26 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
3
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1answer
70 views

Lagrange dual of a sum of convex functions

Given a set of convex functions $f_i(x)$ and convex sets $X_i$ in $\mathbb R^n$ I need to find the Lagrange dual problem for the problem $\min \sum{f_i(x)} , x \in X_i \forall i$. There is of course ...
0
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1answer
44 views

Lagrangian dual for the sum of norms

I would like some help in deriving the Lagrangian dual function of a sum-of-norms minimization problem : $\sum{||A_{i}x-b_{i}||}$ when $A_{i}$ are matrices, and $b_{i},x$ vectors. I understand I can ...
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1answer
138 views

Orthogonal Complements property

I have a question about how to prove a certain property of orthogonal complements of vector subspaces. Given $\mathrm E$, a vector space over a commutative field k, define: $$\phi\text{ : } \mathrm E ...
2
votes
1answer
81 views

Recovering the solution of optimization problem from the dual problem

In the context of (most of the times convex) optimization problems - I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum ...
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0answers
19 views

how to get the dual of that optimization problem

max $1^\top x$ such that $x^\top M x = 0$ and $x_i^2 = x_i$ for all i For the above problem how can I derive the dual form. My main problem is to choose matrix notation or the element-wise notation ...
0
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1answer
55 views

how to construct the Lagrangian dual problem?

The primal optimization problem is, \begin{align*}\min_x\;&f_0(x)\\ \text{s.t.}\;&f_i(x)\le0\\ &h_j(x)=0\end{align*}, to construct the dual problem, I form the Lagrangian, ...
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1answer
47 views

Dual space of a complete vector space

Let $(X, \mathcal{F}, \mathbb{P})$ be a Probability space. Consider the space of all functions with topology induced from convergence in Probability. I am interested in knowing the dual space of it. ...
2
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1answer
41 views

Riesz Representation Theorem: isomorph

Riesz' Representation Theorem states that every linear functional can be represented by a vector. This shows that the Dual can be ANTILINEARLY and norm preserving identified with the Hilbert Space ...
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0answers
21 views

how to check slater condition for a constrained optimization problem?

Given any optimization problem that you suppose to solve with Lagrange by thrusting strong duality, you need to be sure the Slater Conditions. And I guess there is no algorithmic way to solve for all ...
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2answers
51 views

How to find dual spaces?

I would greatly appreciate it if you could kindly share how to find dual spaces? For example, let X be the vector space of n-dimensional vectors with the Euclidean norm. Prove that X*=X. I know a ...
0
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1answer
56 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
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0answers
22 views

Dual of Linear Program

I was wondering what a $symmetric $ dual is. For example, the following is supposed to be a symmetric primal and dual form of LP. Primal : $$ \max c^Tx$$ subject to $$ Ax \le b $$ $$x \ge 0 $$ Dual: ...
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1answer
89 views

What is dual representation in plain English?

Can someone please explain what is Dual representation in plain English. I read its definition on wikipedia and at many other places but could not develop an intution for it. Please explain in plain ...
2
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1answer
102 views

Serre duality as a right adjoint functor

As stated on the wikipedia page, Grothendieck generalized Serre duality by stating that there exists a right adjoint functor $f^!$ to the functor $Rf_!$ when one works within the correct category. ...
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1answer
57 views

Dual of this primal optimization problem?

How would one find the dual of the following problem? $min_x 1/2 ||y-x||_2^2 + \lambda||x||_1 $ Can someone please explain to me how to do this since there are no specific constraints?
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1answer
210 views

Dual to the dual norm is the original norm (?)

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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1answer
43 views

inf sup duality in Hilbert spaces

Let $Y$ be a Hilbert space, for all $y \in Y$ and $X$ a closed subspace of $Y$, I want to prove the following duality result: $$\inf_{g \in X} || y -g|| = \sup_{(f,X)=0} \frac{(y,f)}{||f||},$$ where ...
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0answers
31 views

Finding the dual of a linear program

I have an exam next week and I would like to make sure I am doing this problem correctly and I would also appreciate if somebody could explain to me the purpose of duality? What is the ultimate goal ...
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1answer
681 views

Is it logically valid to prove DeMorgan's laws using the duality of boolean algebra?

I'm taking an introductory course in boolean algebra, and have been assigned the task of proving DeMorgan's Laws (so, disclaimer, this is homework). One line of reasoning that I came up with would be ...
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2answers
114 views

Properties of the duals of $\ell^1$ and $\ell^{\infty}$

a) True or false: (i) $(\ell^{1})^* = \ell^{\infty}$ (ii) $\ell^1 \subset (\ell^\infty)^*$ (iii) $(\ell^\infty)^* \subset \ell^1$ (iv) $(\ell^1)^{**} \subset \ell^1$ b) Give the set of dual vectors ...
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1answer
86 views

Dual of a topological vector space. Is it nontrivial?

In the case of normed spaces we know their duals are nonempty using a quick application of the Hahn Banach Theorem. If we step back to the larger class of locally convex spaces, an enthralling ...
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1answer
314 views

Help me organize these concepts — KKT conditions and dual problem

This is a long question in which I explain my current understanding of certain ideas. If anyone is interested in reading this and would like to provide any commentary/feedback that may help me ...
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0answers
19 views

Confusion related to how the dual of a problem is derived

I have this confusion about how the dual was derived of an optimization problem Here is the primal problem It's dual is like this I didn't get how the $\lambda^{max}$ appeared there. I mean ...
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1answer
32 views

Convergence weakly to measure?

Let $ w_e (x)=\frac {\partial^2}{\partial x_1^2}\sqrt {x_1^2+e}.$ Show that $ w_e $ converges weakly as $e\to 0$ in the dual of $ C (\bar {B_1}) $ to measure $\mu $ I am that dual $ C (B) $ is borel ...