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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dual problem of SDP [closed]
suppose we have the following optimization problem:
\begin{array}{l}
\mathop {\min }\limits_{{\bf{X}},{\bf{x}}} \,\,Tr\left( {{\bf{XA}}} \right) + 2{{\bf{a}}^H}{\bf{x}} + b\\
s.t:\,\,\,\,\left[ {\...
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Equivalent Norms for the Dual of Sobolev / Bessel Spaces
Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb ...
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Is the closed ball of a normed space closed in any Hausdorff locally convex topology, weaker than the norm topology?
Assume that we have a normed space $X$ and a subspace $Y$ of $X^{*}$ such that $Y^{\perp}=\{0\}$. They form a non-degenerate dual pare.
Moreover, $\|y\|=\sup_{x\in B_{X}}|\langle x,y\rangle|$, where $...
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Is dual cone unique? [closed]
Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition ...
- 187
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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 (...
- 440
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90
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Poincaré Duality of a quasi-free algebra
I'm completely stumped on this one (yet I feel it is obviously true or obviously false)
If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality?
All i need to find is a protective ...
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Duality between large and small scale structures
A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
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Simultaneously extending the functionals of a subspace of a Banach space to the whole space
Let $X$ be a Banach space and $Y$ a closed subspace of $X$. If $\varphi\in Y^*$, then Hahn-Banach allows us to extend $\varphi$ to a $\tilde\varphi\in X^*$, such that $\|\tilde\varphi\|=\|\varphi\|$. ...
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Do any Stone-like dualities have some self-dualities hidden inside them?
This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
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hyperfunctions and analytic duals
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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Electromagnetic duality symmetry
This question arose while reading http://prd.aps.org/abstract/PRD/v13/i6/p1592_1 (Duality transformations of Abelian and non-Abelian gauge fields, by Stanley Deser and Claudio Teitelboim). It is well ...
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Grothendieck duality for stacks
Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...
- 8,998
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dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
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Self-dual surfaces in $\mathbb P^3$ with isolated singularities
I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties:
the smooth quadric;
Kummer surfaces;
The surface with the equation $...
- 1,939
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singularities of the dual variety of a surface
I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the ...
- 3,779
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Question on convex optimization and dual norms [closed]
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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Topological problems solved by lattice duality
It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
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Looking for concrete description of a category derived from abelian groups
The category of abelian groups $\mathsf{Ab}$ is the $\mathcal{Ind}$-completion of the full subcategory of finitely presentable abelian groups $\mathsf{Ab}_{fp}$. This is not so special, since the ...
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Confusion about the dual/predual to the tangent plane to a Teichmüller space
I apologize in advance if this question is not considered research-level.
I am reading material on Teichmüller theory and I am getting confused as to the nature of the space $Q(R)$ of all integrable, ...
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Which categories are the categories of models of a Lawvere theory?
Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
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Duality between K-theory and K-homology in the non-spin^c case.
I posted this question on Math.SE (https://math.stackexchange.com/questions/409444/), but got no answer. So I repost it here.
Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times ...
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Is Khovanov's Frobenius algebra self-dual over the integers?
Khovanov's Frobenius algebra (used in the definition of Khovanov homology) is $\mathbb{Z}[X]/X^2$ with the comultiplication. $\Delta(X)=X\otimes X, \Delta(1)=1\otimes X+X\otimes 1$ and the trace $\...
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Duality between K-theory and K-homology in the non-compact, spin$^c$ case
Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\...
- 2,998
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Existence of dominating measure for weak*-compact set of measures
I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-...
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compact embedding in duals of weighted Sobolev spaces
On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding
$$
L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset (W^{k,q}(\mathbb{R}^d,\...
- 5,380
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Matlis' dual of injective modules
Let $(R, \mathfrak{m})$ be a commutative Noetherian complete local rings ($R$ can be regular, if you need). Let $E(R/\mathfrak{p})$ be injective hull of $R/\mathfrak{p}$, if $\mathfrak{p}= \mathfrak{m}...
- 2,773
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Characterizing the Dual of $W_0^{s,p}$
I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $...
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Looking for a uniform explanation of algebras with canonical generators.
Let $\mathcal{V}$ be a finitary variety i.e. the algebras for a signature whose operations have finite arity and for some arbitrary set of equations. Then any algebra $A \in \mathcal{V}$ has a ...
- 1,291
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Base change of trace for Gorenstein or Cohen-Macaulay morphisms
This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'...
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Twisted duality in a symmetric monoidal category
I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?
Definition. Let $\mathcal{C}$ be a ...
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What is the dual of a pre-injective map?
In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math.
Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of ...
- 2,452
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Dual of a weighted projective space
I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space ...
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Why the preimage rather than image in Stone-type dualities.
I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.
Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ ...
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Why are possibility and necessity dual?
Hello,
Recently, I'm studying modal logic for my master's thesis, and my research background is category theory.
So, I naturally have a question that why it is said that necessity (box) and ...
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Characterization of schemes whose dualizing complex is perfect
I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, ...
- 5,052
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How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?
Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in $V_s$...
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What is a de Vries algebra?
I've come across a set of slides by Guram Bezhanishvili where he claims the category of compact hausdorff spaces is related by duality to de Vries algebras. What are they?
- 2,083
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A bijective correspondence induced by Fourier Transform
Let $G$ be a discrete Abelian group and denote by $\widehat G$ the (compact) Pontryagin-Van Kampen dual of $G$. I was reading in a paper of Justin Peters that Fourier Transform induces a bijection ...
- 2,452
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Dual Norm For Sum of 2-Norms
What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = ...
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connections between Grothendieck's and Serre's duality
Hi,
I would like to show that
if $f: X \rightarrow Y=Spec \, \mathbb{C}$, where $X$ is a nonsingular complex projective variety, is the projection to a point, then the complex $f^! \mathcal{O}_Y$, ...
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Does equality of Hodge star and symplectic star imply Kähler structure?
Question
The question asked is:
On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does $\ast=\...
- 1,389
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A name for a weak topology
Let $V$ be a real vector space and let $V'$ be the algebraic dual of $V$, i.e. the space of all the linear functionals $V\to\mathbb{R}$. Then there exists the weakest topology $\tau$ which makes all ...
- 3,749
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Conjugate function for matrix mixed norm
I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:
$$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\...
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In what sense is a generically submersive morphism of varieties subermersive over singular points?
Background/Motivation
I'm currently interested in the duality theorem for projective varieties and more specifically in properties of the conormal variety over the dual variety.
Let $V$ be a $k$-...
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624
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Dualizable classifying spaces
If $G$ is a finitely generated free group, then its classifying space $B G$ can be presented as a finite CW complex (a finite bouquet of circles), and therefore is Spanier-Whitehead dualizable. Are ...
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Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
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Examples for "nice" Boolean algebras that are not complete or not atomic
A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).
Boolean Algebras that are complete as ...
- 1,498
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Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?
I am attempting to solve the argument maximization problem
$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f_1$ and $f_2$ are concave but ...
- 101
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712
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Alexander duality theorem for CW-complexes and stable homotopy theory
In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem
Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$
for finite CW-complexes with a "nice ...
- 2,871
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319
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Generalized fourier transform and convolution?
Let $a(t)$ and $b(t)$ be two equal length sequences indexed by time index $t$.
We know that $a(t) * b(t)$ corresponds to $A(\omega) \odot B(\omega)$ in the frequency domain where $A(\omega)$ and $B(\...
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