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3
votes
2answers
64 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
3
votes
0answers
25 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
4
votes
2answers
38 views

Duality in quadratically constrained quadratic program

I have been given the primal quadratic program with a single quadratic constraint as given below: $$ \text{min} ~~~~~~~~~~~~~~~~~~~~~~~~~ \frac{1}{2}x^{T}Qx $$ \begin{align*} \text{subject ...
2
votes
1answer
32 views

If $X\subset Y$ then $X^*\subset Y^*$

Is the following true, If $X$ and $Y$ are Banach spaces and $X\subset Y$, then $X^*\subset Y^*$. One argument for this is the following let $i:X\to Y$ be the identity map which implies its one to ...
0
votes
0answers
10 views

Show that B is an optimal basis matrix obtained by replacing b with b*?

I need help with the following 2 part question: Let $B = [a_{B1}, a_{B2}, \ldots , a_{Bm}]$ be an optimal basis matrix for the following linear program: Maximize $CX$ Subject to $AX \leq b$, ...
0
votes
1answer
36 views

examples on double dual spaces

I am looking for examples on double dual spaces as I know $\ell_p $ is a double dual of $\ell_p$ for $1<p,q<\infty$. $\mathcal L_p $ is a double dual for $\mathcal L_p$ for ...
1
vote
0answers
46 views

Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
1
vote
1answer
42 views

Connected Pontryagin dual

The dual group to a compact abelian group is discrete so in particular very much disconnected. I was trying to invent an example of a connected locally compact abelian group with connected dual which ...
0
votes
1answer
28 views

Show that $E^*\neq \{0\}$ iff $E\neq \{0\}$ [closed]

Let $E$ be a vectorspace and $E^*$ be a algebraic dual of $E$. Show that $E^*\neq \{0\}$ iff $E\neq \{0\}$
0
votes
1answer
38 views

Duality Principle in Boolean Algebra - Why do I alway get !F instead of F?

I have the function: F = !(a && d || b || c) Now i apply the duality principle and exchange all * with + ...
0
votes
0answers
20 views

Lefschetz Duality for Simplicial Homology

What are the duality theorems for compact, orientable simplicial complexes (possibly the triangulation of a manifold with boundary)? Is there a good way to calculate this boundary just from a list of ...
0
votes
1answer
33 views

Infinite dimensional transpose?

I know that if $L$ is a linear transformation from $V$ to $W$ where $V,W$ are finite dimensional, then we can conclude that the dimension of image (rank) of $L$ is same as that of its transpose, i.e., ...
0
votes
0answers
25 views

Linear Programming question involving a data set of consumer purchases

I am from Netherlands and preparing for an interview with Two Sigma Capital, which for the position I am applying for is notorious for asking linear programming questions. I was trying to solve this ...
2
votes
1answer
73 views

Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
3
votes
1answer
49 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
1
vote
0answers
21 views

Show that it is a element of $(H^1(\Omega))'$

Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$. I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le ...
1
vote
0answers
27 views

Legendre transform and Minimax Theorems.

Denote the class of lower-semi-continuous convex functions $f:\mathbb{R}^n\to \mathbb{R}\cup\{\pm\infty\}$ by $Lscx(\mathbb{R}^n)$ ( so that only function attaining the value $-\infty$ is the constant ...
3
votes
1answer
75 views

Question on how to get back “classical” Serre-duality from its derived functor formulation

I'm really new to derived categories, so i hope this isn't a stupid question. I'm trying to understand how the duality described as for example in Residues and Duality of R. Hartshorne, using the ...
0
votes
0answers
13 views

Second derivative of Bregman divergence

Suppose I define an exponential family distribution: $$ f(x; \theta) = \exp \left( \langle x, \theta \rangle - h(x) - \psi(\theta)\right) $$ where the log-partition function is: $$ \psi(\theta) = ...
1
vote
0answers
50 views

p-direct sum and dual spaces

I have a problem with my assignment of Linear Analysis. It should be rather easy and straight-forward, but I have problems =(. Let E and F be normed spaces. For $p \in [1,\infty]$, define the ...
1
vote
1answer
13 views

Solving the euclidian distance squared to kernelize a Lagrangian dual

Homework question, looking for a hint on the following problem: I'm trying to solve this dual lagranging form (which could potentially be wrong already, but let's assume it is right) $\boldsymbol{x}$ ...
0
votes
1answer
28 views

Dual simplex doubt (unrestricted)

I have this two problems and i only want to find the dual form: $\begin{gather} max\hspace{.1cm}z =5x_1+6x_2\\ s.t\hspace{.1cm}x_1+2x_2=5\\ -x_1+5x_2 \ge 3\\ x_2 \ge 0\\ x_1\hspace{.1cm} ...
0
votes
0answers
16 views

Euler path line graphs and their duals?

Am having trouble drawing a dual of a given graph, I have tried it but I don'tvthink am correct. How should this question be done, below is also my trial;
0
votes
0answers
14 views

Equivalence of Two Statements (Duality Theory, Optimization)

Let $a$ and $a_{1}, ... , a_{m}$ be given vectors in $\mathbb{R^{n}}$. Prove that the following two statements are equivalent. $a)$ For all $x \geq 0$ we have $a'x \leq max_{i} a_{i}'x$. $b)$ There ...
0
votes
0answers
37 views

Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
0
votes
0answers
17 views

Show piecewise linearity of value function

Define, $$v(z) := \text{min}_{y\in\mathbb{R}^p_+} \{qy:Wy=z \}$$ where $W$ is a complete recourse matrix and $q$ is sufficiently expensive, i.e. $q \geq \lambda W$. Then show that $v$ is piecewise ...
1
vote
1answer
43 views

Dual of concave function is convex

If $U(x)$ is strictly increasing and strictly concave and $lim_{x \rightarrow \infty}$ U'(x) = 0, prove that its dual: $$U^{*}(y) = max_x \{U(x) - xy\}$$ is convex. Does anyone know how to prove ...
1
vote
1answer
39 views

Do lagrangian multipliers converge to dual variables in LPs?

Can anybody clarify the following to me? Consider an LP, say a maximization problem, with solution x* and optimal value Z*. Its dual will have optimal value W*=Z* (by strong duality) and optimal ...
2
votes
1answer
52 views

Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
0
votes
0answers
49 views

How to derive dual of this L1 norm approximation problem?

I am working through a question in Convex Optimization by Boyd and Vandenberghe. I've made an image with the original question, and the part of the solution I don't understand: how the dual is ...
2
votes
1answer
51 views

Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ? I know that if $E$ is finite dimension the result is true ...
0
votes
1answer
22 views

Understanding a bilinear form problem from Greub's Multilinear Algebra

I read the following problem from exercise sets of Greub's Multilinear Algebra, Chapter I, Sec. 1 Let $E$, $E^*$ be a pair of dual spaces and assume that $\mathit{\Phi}:E^{*}\times E\to\Gamma$ is ...
0
votes
0answers
25 views

Dual problem of SDP

Suppose we have the following optimization problem: \begin{array}{l} \mathop {\min }\limits_{{\bf{X}},{\bf{x}}} \,\mathrm{Tr}\left( {{\bf{XA}}} \right) + 2{{\bf{a}}^H}{\bf{x}} + b\\ ...
2
votes
1answer
35 views

Why does duality imply, that it is enough to consider p>2.

Let $f\in L^p(\Omega, \nu).$ Let $L$ be a self-adjoint operator on $L^p.$ Suppose we want, for every $p>1,$ to prove an inequality $$||Lf||_p\leq C(p)||f||_p,$$ where $C(p)$ is some function ...
0
votes
0answers
13 views

Duality gap of nonconvex problem

I have an optimization problem (presumably) nonconvex but the objective funtion is increasing, continuous, and smooth. I also have a set of linear constraints which are fulfilled with equality, i.e., ...
0
votes
0answers
24 views

Duality set for $L~p$ spaces, $1<p<\infty$.

I need to show that, given $f \in L^p$, $1<p<\infty$, the duality set $F(f)$ is equal to the point $$\|f\|_p^{2-p}|f|^{p-2}\overline{f}.$$ I have a hint: this is a consequence of convexity of ...
1
vote
0answers
52 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
2
votes
2answers
105 views

How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$?

How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$? If the polar dual of a set $A$ is $A^*=\{x\text{ in }\Bbb R^2:ax\leqslant 1\text{ for all }a\text{ in }A\}$. I study from ...
0
votes
2answers
26 views

LP: An algorithm to decide whether a polyhedron is a subst of another polyhedron

I've encountered the following question which I am unable to solve: $$ P = \{\vec x | A\vec x \geq \vec a\} \\ Q = \{\vec x | B\vec x \geq \vec b\}\\ P, Q \subseteq R^n $$ Find an algorithm to ...
0
votes
0answers
20 views

Conic programming duality - relative interior

Consider the primal/dual conic programming problems $$ \newcommand{\ip}[1]{\left< #1 \right>} \newcommand{\myvec}[1]{\mathbf{#1}} \newcommand{\bvec}[0]{\mathbf{b}} ...
0
votes
0answers
7 views

If a voronoi vertex q is an endpoint of a voronoi edge l, then the delaunay polygon dual to q has a delaunay edge dual to l as one of its edges.

If a Voronoi vertex $q$ is an endpoint of a Voronoi edge $l$, then the Delaunay polygon dual to $q$ has a Delaunay edge dual to $l$ as one of its edges. I understand and know what the ...
0
votes
0answers
78 views

The dual simplex algorithm

Following is the dual simplex algorithm, adapted from p. 283 of Daniel Solow's "Linear Programming, An Introduction to Finite Improvement Algorithms", Elsevier Science Publishing Co., Inc., 1984. I ...
2
votes
1answer
50 views

Does the Duality Theorem of Linear Programming hold only in closed convex cones

I've just read the the Duality Theorem of Linear Programming. Here is the proof from my book (and my questions after it): Duality Theorem of Linear Programming: If the primal or dual linear ...
0
votes
1answer
15 views

how to show weakly convergence is equivalent to point convergence under a certain condition?

Let $\{f_n\}$ in $C[0,1]$ satisfy $\sup_n \sup_{x\in[0,1]}|f(x)|<\infty$. Show that $$f_n \ \ \text{converges weakly to 0} \Longleftrightarrow \lim_{n}f_n(x)=0 \ \ \text{for all }x\in[0,1].$$ ...
0
votes
0answers
38 views

dual form of an optimization

Consider the following optimization in primal form $\displaystyle\max_{x_1, \ldots, x_n}\sum_{i=1}^n d_ix_i -\sum_{i=1}^n x_i\log(x_i)$ subject to $a_i\leq x_i\leq b_i$ and $\sum_{i=1}^n ...
1
vote
1answer
36 views

Why $(1,\textbf{0}) \not\in \{(r,\textbf{w}): r=tz_0-\textbf{c}^T\textbf{x}, \textbf{w}=t\textbf{b}-\textbf{Ax}, \;\textbf{x}\geq\textbf{0}, t\geq0\}$

I'm learning about linear and nonlinear programming and on the chapter about duality I have the following statement and proof I can't understand: minimize $\textbf{c}^T\textbf{x}$ subject to ...
3
votes
1answer
35 views

Trouble seeing why this is the dual of an LP

$A$ is an $m \times n$ matrix. Using the notation $x=(x_1, \ldots, x_n)$, $z=(z_1, \ldots, z_m)$, and $y=(y_1, \ldots, y_m)$, I'm reading that if the primal LP is $$ \min 0x_1 + 0x_2 + \cdots + 0x_n ...
1
vote
1answer
35 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
0
votes
0answers
31 views

Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there any relationship between dual of an optimalization problem and dual of a polyhedra?
2
votes
1answer
47 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...