# Questions tagged [duality]

Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.

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### References: properties of $L_p$ spaces involving time and probability space

Questions are from the theory of PDEs\SPDEs Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of ...
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### Gross-Hopkins duality

$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
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### Lovasz extension and underlying matroid

Let $F \colon 2^V \rightarrow \mathbf{R}$ be a set function for some ground set $V = \{ 1, \dots, n\}$. The domain of $F$ is $\{ 0,1\}^n$ , under the usual identification of a set $S$ with its ...
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### Is there Sperner-type lemma where the colors are on the edges?

I am looking for Sperner-type lemmas where: The colors are on the edges (1-dimensional simplices) - rather than on the vertices; The target is a fully-colored vertex (a vertex adjacent to n edges of ...
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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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### Dual norm for weighted Sobolev space

Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm: \begin{equation} \|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{...
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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 ...
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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  authors define the following ...
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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 $U(... 1 vote 0 answers 81 views ### Dual of union of Reproducing Kernel Hilbert Spaces I have a union of Reproducing Kernel Hilbert Spaces$\mathcal{B}$. I am interested in finding the dual of$\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ... 4 votes 0 answers 299 views ### Wasserstein distance and Monge-Kantorovich-Rubinstein duality The definition of Wasserstein$p$-distance between two measures$\mu$and$\nu$on a Polish space$X$is given by $$W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\... 9 votes 1 answer 274 views ### Origin and context of adjunctions inducing equivalences between full subcategories The following is well-known. Theorem. Let F\dashv U be a pair of adjoint functors$$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 0 answers 48 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. ... 7 votes 2 answers 307 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 ... 6 votes 1 answer 278 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 1 answer 630 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 1 answer 190 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 1 answer 162 views ### Duality argument for elliptic regularity M. Dauge proved in  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 1 answer 204 views ### Is every sequentially \sigma(E',E)-continuous linear functional on a dual Banach space E' necessarily a point evaluation? \newcommand{\bf}{\mathbb #1}\newcommand{\sc}{\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 1 answer 214 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(\...
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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 ...