Questions tagged [duality]
Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.
312 questions
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$L_p$ estimate in mixed boundary problem for elliptic equation
Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...
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Does coercivity/supercoercivity conjugates?
According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if,
$$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$
and it is super-coercive if
$$\...
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Self-dual hypergraph on $\omega$
Let $H=(V,E)$ be a hypergraph. For $v\in V$ we let $v^* = \{e\in E:v\in e\}$. We define the dual of $H$ by $H^*= (E, V^*)$ where $V^* = \{v^*: v\in V\}$. We say that a $H$ is self-dual if $H \cong H^*$...
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Does rigidity imply a unique dualizing functor?
Let $\mathcal{C}$ be a rigid, monoidal category. Can I talk about $\mathcal{C}$ as having a unique, well-defined, dualizing functor (i.e. one that maps objects and morphisms onto their respective ...
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Evolution PDE in dual space : Generalization of a result of Gelfand
The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand :
Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
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Category with binary biproducts but no zero morphism
Is there a category with binary biproducts but no zero morphism?
I'm wondering if the definition of biproducts as objects that are simultaneously products and coproducts that obey some identities on ...
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Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
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Effect of dualization of density
Let $D\subset X$ be a dense subset of a complete separable locally convex space $X$ over $\mathbb{R}$. Though the question seems simple enough, I can't seem to find the answer in the literature:
If $...
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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 ...
- 3,513
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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 ...
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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 ...
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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 $\mathcal{S}(\mathbb{R}^{d})$ ...
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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 ...
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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(...
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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 ...
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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)))_{...
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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. ...
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How to inference the dual form of perceptron?
The model of perceptron is a linear binary classifier, which is $f(x)=\mathbb{sign}(w^Tx+b)$. $x$ is the datapoint as $w$ as well as $b$ are the parameters.
The cost function of Primal Perceptron is $...
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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 ...
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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&...
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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
$$
\...
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Duality of finite signed measures and bounded continuous functions
Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...
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Calculating vertex weights
Vertex weights are a metaphor for a constant value $\pi_i$ that is added to the weight of every edge $e_{ij}$ that is adjacent to vertex $v_i$ in a symmetric graph $G(V,E)$ with weighted edges.
The ...
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From $f$-divergence to its dual: the transformation of convex functions on $\mathbb R_+$ by $f^*(t) = 1 f(\frac 1 t)$
I would like to understand the relationship between minimising the KL divergence $P \mapsto D_{KL}[P,Q]$ and the reverse KL divergence $P\mapsto D_{KL}^*[P,Q]=D_{KL}[Q,P]$ for probability measures $P$ ...
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Self-duality of cones associated with elementary symmetric polynomials
Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates:
$$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$
...
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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; ...
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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$
...
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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 ...
- 63.9k
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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 ...
- 2,306
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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 ...
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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 ...
- 63.9k
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Access to a classic reference of Dold-Puppe
There is an old reference that I am unable tofind. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as:
A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the ...
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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 ...
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Show that duality functor is anti-monoidal
Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{U,V}: U^*\otimes V^*\to (V\otimes U)^*$ be the canonical isomorphism for every objects $...
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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$.
...
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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}...
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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}$ ...
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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 ...
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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^∗...
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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 ...
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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 $ ...
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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 ...
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Closed embeddings of monoidal categories in *-autonomous ones
It's often very convenient for objects in a monoidal category to have duals. Hence, it's natural to wonder whether an arbitrary monoidal category can be embedded in one where all objects have duals. ...
CommunityBot
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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 ...
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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-...
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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 ...
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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 (...
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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 $...
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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 ...
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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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