Questions tagged [duality]
Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.
312 questions
2
votes
1
answer
253
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 ...
- 63.9k
1
vote
0
answers
303
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_{...
- 31
2
votes
1
answer
602
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 ...
- 40.3k
2
votes
0
answers
66
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 ...
- 984
1
vote
0
answers
49
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 ...
- 12.9k
9
votes
1
answer
909
views
Grothendieck-Verdier duality without the noetherian condition
The Grothendieck-Verdier duality:
$$
Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\...
- 2,923
0
votes
1
answer
99
views
Finding dual of a scheduling LP formulation
Suppose I have an LP formulation as such:
$\min\ \ \sum\limits_{i,j,t}\ w_{ij}x_{ijt} (\frac{t-r_j}{p_{ij}}+0.5)$
$\sum\limits_{i,t}\frac{x_{ijt}}{p_{ij}}=1\,\forall\ j$
$\sum\limits_{j}x_{ijt}\leq ...
- 5,429
1
vote
0
answers
43
views
How do I find the Rockafellar-Fenchel dual derivation of reguarlized objective function?
I'm stuck finding the dual of this optimization problem:
$$ \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\|...
- 383
3
votes
0
answers
166
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 ...
- 1,623
2
votes
0
answers
48
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 $s$ ...
- 1,766
7
votes
0
answers
262
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 ...
CommunityBot
- 1
3
votes
1
answer
853
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 ...
- 16.5k
2
votes
0
answers
122
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 ...
- 42k
3
votes
0
answers
188
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 ...
CommunityBot
- 1
9
votes
0
answers
217
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 ...
- 63.9k
0
votes
0
answers
59
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 ...
- 4,713
5
votes
0
answers
185
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 ...
- 1,623
5
votes
0
answers
346
views
Weak to weak$^*$ continuity of the duality mapping
Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
CommunityBot
- 1
4
votes
1
answer
596
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 \{+\...
- 576
1
vote
1
answer
213
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 ...
- 13k
5
votes
0
answers
383
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 $...
- 512
13
votes
3
answers
607
views
GOE/GSE duality and Bott periodicity
Many papers in random matrix theory make passing references to duality between eigenvalue statistics of the GOE and GSE, for which the most concrete reference I can find is https://arxiv.org/pdf/math-...
- 188k
5
votes
0
answers
141
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 \...
- 1,593
2
votes
0
answers
158
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 ...
- 5,407
9
votes
2
answers
513
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 ...
- 1,286
3
votes
0
answers
152
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 ...
- 1,013
4
votes
2
answers
1k
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 ...
- 623
4
votes
0
answers
347
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 ...
- 435
4
votes
1
answer
215
views
Dual cone of 'positive' Bochner integrable functions
If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...
- 283
2
votes
1
answer
194
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 (...
CommunityBot
- 1
6
votes
1
answer
123
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 ...
- 984
12
votes
1
answer
591
views
Uniqueness of dualizing objects
One definition of (symmetric) star-autonomous category is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to (...
- 64k
3
votes
0
answers
102
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) ...
- 1,127
1
vote
1
answer
172
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 ...
- 1,586
2
votes
0
answers
42
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 ...
- 6,853
5
votes
1
answer
581
views
Strong duality for a particular moment problem
Reading the paper in this Link (see pag 13) with the objective of understanding a topic related to stochastic optimization I came across a problem in demonstrating one of the theorems. The situation ...
CommunityBot
- 1
7
votes
1
answer
422
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}, \...
- 118k
8
votes
4
answers
734
views
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 ...
- 5,094
11
votes
1
answer
300
views
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 ...
- 213
10
votes
3
answers
2k
views
Dual cell structures on manifolds
Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $...
- 41
19
votes
1
answer
1k
views
What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras?
Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the ...
- 5,094
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 761
4
votes
0
answers
141
views
Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
49
votes
4
answers
4k
views
Why is there a duality between spaces and commutative algebras?
1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety ...
- 9,491
3
votes
1
answer
247
views
Support of cohomology of a dualizing complex
Let $A$ be a commutative noetherian local ring, and let $D$ be a dualizing complex over $A$. Let $i$ be the minimal integer such that $H^i(D) \ne 0$ (I am assuming cohomological grading, so the ...
- 28.8k
11
votes
1
answer
314
views
What do you call $C$ if $[D,C] = D^\vee \otimes C$ for all $D$?
This is different from $C$ being dualizable ($[C,D] = C^\vee \otimes D$). (EDIT: It turns out to be the same -- see Mike Shulman's answer!) But for example, if $C$ is a locally free sheaf of finite ...
- 64k
1
vote
0
answers
154
views
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 ...
2
votes
1
answer
702
views
Duality of Bochner $L^{\infty}$ space
Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
- 26
1
vote
1
answer
118
views
For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials
Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following ...
- 3,085
0
votes
0
answers
72
views
Discrete primal-duality in optimization
I would like to inquire about the existence of something perhaps similar to the duality theorem in the convex analysis or convex programming in the discrete setting. Here is a concrete example.
Let $...
- 2,251