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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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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"L^2_loc mod constants" as a reflexive space
In an article of Sten Kaijser ("A note on dual Banach spaces") I find the assertion that $E = L^2_{\text{loc}}({\mathbb R})$ modulo constants is a reflexive space.
Question 1: which is the '...
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From relative convexity to modulus of continuity estimates for the dual gradient mapping
Let $F: \mathbf{R}^d \to \mathbf{R}$ be a convex function, let $m > 0$, and define $Q_m: \mathbf{R}^d \to \mathbf{R}$ to be the mapping $x \mapsto \frac{m}{2} \| x \|_2^2$. One says that $F$ is $m$-...
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Legendre transformation of vector valued function
Good afternoon. Is there any generalisation of Legendre--Fenchel transformation to the vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$?
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Product-coproduct duality
Let $T$ be a set, $R$ be a ring with $1$ and $B, S_t$ be $R$-modules $\forall t \in T$
My task is to state and prove the dual to the following statement:
Given momomorphisms $j_t: S_t \rightarrow B$. ...
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Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?
Let $\Omega \subset \mathbb{R}^n$ be open and bounded.
What does weak-* convergence for a sequence of functions $\{f_k\}_{k \in \mathbb{N}}$ in $W^{1,\infty}(\Omega)$ mean? It seems to me that there ...
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Do doubly infeasible Linear Programming problems always have doubly infeasible bases?
Consider a Linear Programming problem in dictionary form,
$$\max\Big\{f^\pi+\!\!\!\sum_{j\in D(\pi)}\!\! d^\pi_jx_j~\Big|~\forall~i\!\in\!B(\pi)~~~ b^\pi_i+\!\!\!\sum_{j\in D(\pi)}\!\! G^\pi_{ij}x_j\...
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Does projective duality preserve arithmetic-Cohen-Macaulay-ness?
Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring $\...
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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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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 ...
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duality of sobolev spaces. Representation of elements in the dual
I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
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Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
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Duality in Spc with ∧ and [-,-]
I am thinking about two duality theorems for H-spaces and their actions.
By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider ...
user30211
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Knot invariants in WZW CFT via Holographic Principle
In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...
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$BMO$ is not reflexive
It is well known that $BMO$ is the dual space of the Hardy space $H^1$, which is the dual space of $VMO$. I believe that $BMO$ is not reflexive, but I am not quite sure that the above information is ...
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Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?
Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...
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Application of the $\operatorname{BMO}$, $H^1$ duality
Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
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Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
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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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$L_p$ 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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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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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 ...
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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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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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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_{...
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Weakly reflexive algebra vs proper (residually finite-dimensional) algebra
Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called ...
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How do I find the Rockafellar-Fenchel dual derivation of reguarlized objective function?
I'm stuck finding the dual of this optimization problem:
$$ \max_{\tilde{u},\tilde{v}} \int \left( \frac{1}{2}\left\|x\right\|^2 + \tilde{u}(x)\right) \, ds(x) + \int \left( \frac{1}{2}\left\|y\right\|...
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Weak$^*$ topology on Bochner $L^p$-spaces
Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
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Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function
We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by
$$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$
...
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Is the category of profunctors $Prof(A,B)$ equivalent to $Prof(B,A)^{op}$?
$\def\Prof{\mathsf{Prof}}\def\Set{\mathsf{Set}}\def\tobar{\mathrel{\mkern3mu \vcenter{\hbox{$\scriptscriptstyle+$}}\mkern-12mu{\to}}}$Let $A$ and $B$ be categories. Define a profunctor $A\tobar B$ to ...
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Characterization of the weak completion of $L^2(\mathbb{R}^d)$
The completion $\overline{L^2_w(\mathbb{R}^d)}$ of $L^2_w(\mathbb{R}^d)$ (i.e. the completion of $L^2(\mathbb{R}^d)$ endowed with the $\sigma(L^2(\mathbb{R}^d),L^2(\mathbb{R}^d))$ topology) is ...
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Strong Duality of Mixed Integer Linear Program
The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
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Proof/reference for a variant of Pontryagin duality
Let $X,X'$ be locally compact abelian groups with a non-degenerate quadratic form
$\left<\bullet ,\bullet \right>\colon X\times X' \to \mu_{l}$,
where $l$ is a prime, and $\mu_l$ the group of $...
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How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?
Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.
Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...
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Duality of plurisubharmonic functions
Let $F$ be a cone of upper bounded upper semicontinuous functions on a compact set set $X$ containing all the constants. Let $z\in X $ and define a class of positive measure by $$M_z^F=\{ \mu : u(z)...
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Finding the Lagrangian dual problem for a quadratic programm [closed]
I've problems to find the Lagrangian dual problem to
\begin{align*}
\min \limits_{x \in \mathbb{R}^n} \; \frac{1}{2} x^{ T} Q x + q^{T} x \\
\text{s.t.} \quad
Ax &=b \\
x &\geq 0
\end{...
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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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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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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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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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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 ...
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$f=0$ in $H^{-1}(\Omega)$ implies $f=0$ almost everywhere
Does $f=0$ in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ implies $f=0$ almost everywhere in $\Omega$?
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Is division considered the mathematical dual of multiplication? [closed]
I'm doing a bit of research for a tech presentation that touches on the subject of mathematical duality. (To be clear, my presentation is not on mathematics or duality, but mentions duality in passing....
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Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]
Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
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Natural Poisson brackets on $S(V^*)$
Let $V$ be a finite dimensional vector space and $S(V)$ the corresponding symmetric algebra. Suppose that we have a Poisson bracket $\lambda = \{,\}: S(V) \otimes S(V) \to S(V)$. Let $V^*$ be the dual ...
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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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Kernel of restriction in étale cohomology of curves over number fields
Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \...
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Calculating vertex potentials from optimal matchings
Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...
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