Questions tagged [duality]
Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.
224
questions
3
votes
0answers
102 views
Reference for duality inducing bijections between subobjects and quotients?
I'm not the most category-theoretic person but I have run into the following statement in my work. Suppose we have a category duality $\mathcal F:\mathcal C\to\mathcal D$. For my needs you can ...
9
votes
0answers
100 views
Finitely presentable grids and co-presentable topological spaces
In the 90s there has been some interest for the category $\mathsf{Top}^\circ,$ the dual of the category of spaces. Most of the relevant papers on the topic are co-authored by Pedicchio.
Barr, ...
6
votes
1answer
157 views
Top cohomology of profinite Poincaré duality group
The paper "Cohomology of p-adic analytic groups" by Symonds and Weigel is considered one of the main references for continuous cohomology of profinite groups. There is a passage I do not ...
35
votes
12answers
3k views
No canonical isomorphism [duplicate]
I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
5
votes
1answer
210 views
Generalization of Bernstein’s inequality
I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim:
Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...
1
vote
1answer
66 views
Confusing definition of homogeneous Sobolev norm of order -1
Let $\Omega \subset \mathbb{R}^{d}$ and $\|.\|$ is the standard euclidean $2$-norm. I came across a definition of $\dot{H}^{-1}(\Omega)$ which is a bit confusing. In [1] authors define the following ...
2
votes
0answers
172 views
Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry
Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...
0
votes
0answers
35 views
Dual of union of Reproducing Kernel Hilbert Spaces
I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...
4
votes
0answers
91 views
Wasserstein distance and Monge-Kantorovich-Rubinstein duality
The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
10
votes
1answer
255 views
Origin and context of adjunctions inducing equivalences between full subcategories
The following is well-known.
Theorem. Let $F\dashv U$ be a pair of adjoint functors
$$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$
with unit $(\eta_A\colon A\to U(F(A)))_{...
3
votes
0answers
46 views
Conjugate of composition in Bochner spaces
Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
0
votes
0answers
40 views
Dual cone operation is an antitone morphism of lattices
Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. The chapter on Dual cones made me thinkin of the fallowing ...
8
votes
1answer
175 views
Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\tensor}{\otimes}$
$\DeclareMathOperator{\Sp}{Sp}$
This question is about comparing the approaches for a formal Wirthmüller ...
4
votes
1answer
144 views
Can one characterize maximal antichains in terms of distributive lattices?
This is inspired by the recent question Verification of a maximal antichain
The celebrated duality between finite posets and finite distributive lattices has several nice formulations. One of them ...
13
votes
1answer
480 views
Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
3
votes
1answer
139 views
On the dimension of the dual variety of a singular hypersurface
I was primarily interested in the following question. Let $n\geq 3$, and let $X\subset \mathbb{P}^n$ be a degree $d$ hypersurface. Assume that its singularity locus $S$ (with reduced structure) is ...
3
votes
1answer
118 views
Duality argument for elliptic regularity
M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
7
votes
1answer
146 views
Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation?
$\newcommand{\bf}[1]{\mathbb #1}\newcommand{\sc}[1]{\mathscr #1}$
A duality between two vector spaces $E$ and $F$ over $\bf K$ ($= {\bf R}$ of ${\bf C}$)
is, by definition, a bilinear form
$$
\...
8
votes
1answer
185 views
Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?
A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
1
vote
1answer
54 views
Calculating vertex weights
Vertex weights are a metaphor for a constant value $\pi_i$ that is added to the weight of every edge $e_{ij}$ that is adjacent to vertex $v_i$ in a symmetric graph $G(V,E)$ with weighted edges.
The ...
1
vote
1answer
65 views
Weak duality sign
Let $\left\{\begin{matrix}
\operatorname{min}_xc^Tx\\Ax\leq b
\\ x\in \mathbb{R}^+
\end{matrix}\right.$ be a LP primal problem and $\left\{\begin{matrix}
\operatorname{max}_yb^Ty\\A^Ty\geq c
\\ y\in ...
1
vote
3answers
262 views
Duality of finite signed measures and bounded continuous functions
Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...
1
vote
0answers
55 views
From $f$-divergence to its dual: the transformation of convex functions on $\mathbb R_+$ by $f^*(t) = 1 f(\frac 1 t)$
I would like to understand the relationship between minimising the KL divergence $P \mapsto D_{KL}[P,Q]$ and the reverse KL divergence $P\mapsto D_{KL}^*[P,Q]=D_{KL}[Q,P]$ for probability measures $P$ ...
3
votes
0answers
34 views
Self-duality of cones associated with elementary symmetric polynomials
Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates:
$$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$
...
0
votes
0answers
68 views
Preservation of the Euler-Poincaré characteristics and commutativity of the multiplication in R^d
While talking about duality for polyhedra and the analogue for polygons (where the duality is essentially trivial) with a friend, I wondered if this self-duality in dimension 2 was anyhow related to ...
2
votes
0answers
62 views
A strong duality for convex functional optimization that admits Lipschitz continuity constraints?
Problem Statement
I am looking for formal proof---hopefully textbook material---of two items:
an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...
1
vote
1answer
125 views
Dual problem with integrals
I am reading a paper where the author derives the following Lagrangian dual problem :
$\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$
...
2
votes
0answers
20 views
Weak relaxation of a strongly lower semi-continuous functional
Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...
3
votes
2answers
270 views
Bi-annihilator of a subspace of the dual of an infinite-dimensional vector space
Let $V$ be an infinite-dimensional vector space and $V^*$ its dual.
For a linear subspace $W\subset V$ define $W^ \circ\subset V^*$ as the subspace of linear forms on $V$ vanishing on $W$.
Dually, for ...
5
votes
0answers
190 views
Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?
The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
0
votes
0answers
87 views
When is the natural map of Tate cohomology an isomorphism?
First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
3
votes
1answer
96 views
Duality problem of an infinite dimensional optimization problem
I am reading the paper "OPTIMAL INEQUALITIES IN PROBABILITY THEORY: A CONVEX OPTIMIZATION APPROACH" by BERTSIMAS and POPESCU. In the paper, the authors derived a duality problem for an ...
0
votes
1answer
68 views
If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?
Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$.
...
0
votes
0answers
103 views
Dual space of a periodic Sobolev space $H^m([0,2\pi]^n)$
For $m \in \mathbb{N}$ let $H^m([0,2\pi]^n)$ be the periodic Sobolev space on $\mathbb T^n=\mathbb R^n/(2\pi\mathbb Z^n)$ equipped with the norm
$$
\|u\|_{H^m} = \left(\sum_{k \in \mathbb{Z}^n} (1+|k|^...
0
votes
0answers
82 views
Dual space of mean-free Sobolev space
I am considering the space $V:=\{v \in H^1(\Omega): \int_\Omega v = 0\}$ of mean free functions. What is the dual space of this space? Is the dual space given by $D:= \{f \in (H^1(\Omega))^*: \langle ...
5
votes
1answer
283 views
A generalization of integral Poincaré duality
In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:
An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}...
6
votes
1answer
184 views
If $X$ is separable space then $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?
Let $(X,\|.\|_{X})$ be a separable Banach space and the associated dual space is denoted
by $X^*$. By $w^*$ we shall indicate the weak$-*$ topology on $X^*$.
Let $B_{X^∗}= \{x^∗ \in X^∗ : \|x^∗\|_{X^∗...
1
vote
0answers
154 views
Dual varieties and nodal sections
Let $X$ be a(n even dimensional) smooth complex projective variety in $\mathbb{P}^N$, and let $X^{\vee}$ be its dual variety; up to an higher degree Veronese embedding of $X$, I assume that $X^{\vee}$ ...
2
votes
0answers
127 views
Do Poincaré duality algebras need to be defined over a field?
I asked the below question here on MSE, but after some time and a bounty offering I have not received an answer.
A graded commutative, connected $\mathbb{k}$-algebra $A$ is called a Poincaré duality ...
1
vote
1answer
248 views
Poincare duality-differential geometry
Let $ M $ be a smooth and compact manifold with boundary $\partial M = X \times F $ on which the structure of a smooth locally trivial bundle $$ \pi: \partial M \longrightarrow X $$
where the $ X $ ...
15
votes
1answer
779 views
What is a module over a Boolean ring?
Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
8
votes
0answers
263 views
Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?
Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...
2
votes
0answers
51 views
Proving the existence of a dual for an infinite linear program
I am concerned with proving the existence of the dual of an infinite linear program. In addition to the writings of Rockafellar, Luenberger, and Boyd & Vandenberghe on: subdifferentials, Legendre-...
4
votes
1answer
331 views
Which topological spaces admit embeddings into Euclidean spaces
I'm interested in the dual question to:
continuous images of open intervals, about surjections onto open intervals.
Namely, if $X$ is a topological space, when can we guarantee that there exists a ...
2
votes
1answer
320 views
Why are Serre functors always exact?
Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...
7
votes
2answers
206 views
Completeness of coefficient functionnals
My questions is about Schauder bases and more specifically about coefficient functionals.
Let $(x_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f_n(x) x_n$. The $...
8
votes
1answer
317 views
Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
1
vote
2answers
184 views
Is unit ball in 2-Wassestein metric weakly compact?
This might be a trivial question, but I am trying to prove equipment-coerciveness of some family of functions on the space of Probability measures on some space. I could reduce the problem to showing ...
2
votes
1answer
120 views
Characteristic polynomial of the line graph (originally dual graph)
I am quite sure I have seen somewhere the connection between the characteristic polynomial of a (finite undirected) graph and its dual. I am not able to find it currently. Could you please refer me to ...
1
vote
0answers
215 views
Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
