Questions tagged [duality]

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

Filter by
Sorted by
Tagged with
3
votes
0answers
51 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{...
6
votes
1answer
59 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^∗...
2
votes
0answers
104 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
243 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 $ ...
14
votes
1answer
694 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 ...
0
votes
0answers
16 views

Derivation of dual for infinite linear program

I'm reading the section on Linear Programming in Barbu and Precupanu's Convexity and Optimization in Banach Spaces, and had a couple of questions concerning their derivation of the dual problem for ...
8
votes
0answers
231 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
44 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-...
6
votes
1answer
294 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 ...
1
vote
1answer
277 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
194 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
232 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
146 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
94 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
193 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_{...
2
votes
1answer
164 views

Pushforward in Compactly Supported Cohomology

Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
2
votes
0answers
45 views

Reference request: Matroid cryptomorphisms for arbitrary monomial ideals

For a matroid $M$ let $C$ be the circuit ideal of $M$, that is, the Stanley-Reisner ideal of independence complex of $M$. Then there are simple ideal-theoretic operations that take $C$ to the facet ...
1
vote
0answers
36 views

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 ...
6
votes
1answer
443 views

Grothendieck-Verdier duality without the noetherian condition

The Grothendieck-Verdier duality: $$ Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\...
8
votes
1answer
119 views

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. ...
0
votes
1answer
41 views

Finding dual of a scheduling LP formulation

Suppose I have an LP formulation as such: $\min\ \ \sum\limits_{i,j,t}\ w_{ij}x_{ijt} (\frac{t-r_j}{p_{ij}}+0.5)$ $\sum\limits_{i,t}\frac{x_{ijt}}{p_{ij}}=1\,\forall\ j$ $\sum\limits_{j}x_{ijt}\leq ...
0
votes
0answers
35 views

Convert the maximization problem with log objective function into exponential cone programming

Below is the problem definition and the detailed process I converted it into a formulation solvable by exponential cone solver. There are other constraints but for simplicity I omit them. However my ...
1
vote
0answers
28 views

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\|...
3
votes
0answers
167 views

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 $...
3
votes
0answers
111 views

Naturality of Poincaré–Lefschetz

Let $X$ be compact and Hausdorff, $A\subseteq B\subseteq X$ both closed such that $X\setminus A$ is an open orientable $d$-manifold. Then also $X\setminus B$ is an open orientable $d$-manifold. We ...
2
votes
0answers
34 views

Nondegenerate linear maps functorially associated to algebras

In the sequel, "$k$-algebra" means "associative unital finite dimensional $k$-algebra. Apologies for the very long exposition. If $A$ is a $k$-algebra and $s:A\to k$ is $k$-linear, we say that $s$ ...
3
votes
1answer
192 views

Which complexes of coherent sheaves are dual to perfect ones?

Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ of coherent sheaves. Then (if I understand these matters and the answer to the question Characterization of ...
6
votes
0answers
205 views

Dual Abelian scheme (relative Picard functor) vs Ext sheaf

Let $A$ be an abelian scheme over some base scheme $S$. Let $A^\vee$ be the dual abelian scheme, defined as $\text{Pic}^0_{A/S}$ where $\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$. (maybe some ...
2
votes
0answers
103 views

A new topology on the dual of a locally convex space?

Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...
3
votes
0answers
120 views

Is the category of abstract $\sigma$-algebras the pro-category of countable discrete spaces?

I am wondering if the category of abstract $\sigma$-algebras is contravariantly equivalent to the pro-category of the category of countable sets. Note: I have tweaked this question a little. Now it ...
7
votes
0answers
135 views

Point-free topology, but with $\sigma$-algebras instead of spaces

I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had: If abstract $\sigma$-algebras (i.e. certain boolean ...
0
votes
0answers
56 views

Nests on Banach spaces and their duals

Let $X$ be a Banach space and $\mathcal{E}$ a nest on $X$. Take $f\in X^{*}$ and suppose: $N \in\mathcal{E}$ is the largest element of the nest so that $f \in N^\bot$ $N=\bigcap_{M>N}M$ Is there ...
3
votes
0answers
114 views

Dual Steenrod squares

Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
1
vote
1answer
126 views

Cassels Pairing for Fine Selmer groups

Let $S_n$ be the Selmer group of $E/K_n$ where $K_n/K$ is the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $C_n$ be the torsion part of $S_n$. By Cassels pairing, we know that ...
4
votes
0answers
134 views

Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
4
votes
1answer
359 views

Monge-Kantorovich duality with a $\{0,1\}$ cost function

Consider the usual Monge-Kantorovich transportation problem where $X$ and $Y$ are Polish spaces, $\mu$ and $\nu$ are probability measures on $X$ and $Y$, and $c:X\times Y \to \mathbb{R}^+ \cup \{+\...
5
votes
0answers
115 views

Poincare duality in families of smooth, projective curves

Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
2
votes
0answers
70 views

The dual of the space of continuous sections in a vector bundle

If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
5
votes
1answer
232 views

Generalized Gelfand Triples

Normally, when we talk about Gelfand triple we have three Hilbert spaces $$\newcommand{\X}{\mathcal{X}} \X_+ \subset \X_0 \subset \X_- $$ such that the subsets are dense, the embedding mappings are ...
15
votes
1answer
780 views

References on dualities on quantum field theory for mathematicians

Dualities on QFT–also called Quantum Field Theory dynamics–is a huge and fundamental research area. However, despite underpinning major mathematical breakthroughs such as the work of Kapustin and ...
3
votes
0answers
141 views

I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module

I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...
4
votes
2answers
644 views

If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am ...
3
votes
0answers
139 views

How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?

$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\hom}{\mathcal{Hom}} \DeclareMathOperator{\ox}{\mathcal{O}_X}$Let $f:X \to Y$ be a proper morphism. In section 6.4. of Liu's book he introduces ...
2
votes
1answer
163 views

What is the dual of generating Boolean subalgebra by subexpressions of a modal formula?

I am supposed to be answering this question rather than asking it but I really cannot figure out. There is a variation on Stone duality linking algebraic and (descriptive) Kripke semantics for (...
9
votes
2answers
395 views

Divided power algebra is artinian as a module over the polynomial ring

I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow. In the paper Homological algebra on a complete intersection, with ...
3
votes
0answers
65 views

Legendre transform on signed measure space

Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
1
vote
1answer
97 views

Examples of isomorphic W* algebra with non-homeomorphic weak topology

Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
6
votes
1answer
88 views

Reflection-invariant monomial ideals and Alexander duality

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...
1
vote
0answers
34 views

Dual representation of problems involving $f$-divergences

Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems. Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
6
votes
1answer
236 views

Dualizable presheaves with respect to Day convolution

This question was posted on MSE and got very little attention, so I'm also posting it here. Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...