All Questions
281 questions with no upvoted or accepted answers
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
14
votes
0
answers
899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
13
votes
0
answers
800
views
Reference request for a complete and formal Duality Principle in category theory
Most textbooks on category theory only sketch the meaning of the Duality Principle. But even when they do it more formally, I have only seen a version so far which concerns the language of a (single) ...
- 63.1k
10
votes
0
answers
533
views
Isbell duality between algebras and sheaves
nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...
- 1,347
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
10
votes
0
answers
948
views
Dimensions of dual vector spaces
Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
- 18.7k
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 ...
user30211
9
votes
0
answers
322
views
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 ...
- 63.1k
8
votes
0
answers
1k
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
$$
W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
- 1,407
8
votes
0
answers
451
views
Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?
Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...
- 5,094
8
votes
0
answers
1k
views
Infinite Linear Programming
I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
- 133
7
votes
0
answers
262
views
Dual Abelian scheme (relative Picard functor) vs Ext sheaf
Let $A$ be an abelian scheme over some base scheme $S$.
Let $A^\vee$ be the dual abelian scheme, defined as $\text{Pic}^0_{A/S}$ where $\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$. (maybe some ...
- 71
7
votes
0
answers
1k
views
Closed-form solution of a linear programming question
Among all the probability matrices
\begin{equation*}
P =
\left(\begin{array}{cccc}
p_{00} & p_{01} & \ldots & p_{0,J-1} \\
p_{10} & p_{11} & \ldots & p_{1,J-1} \\
\vdots & \...
6
votes
0
answers
162
views
Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
6
votes
0
answers
163
views
Explicit computations of Serre duality for elliptic curves
I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing
$$
H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R.
$$
Clearly we have that $H^0(E, \Omega_E^1)=R \...
- 109
6
votes
0
answers
133
views
The metric gives the optimal element in a class
In geometry there is plenty of examples in which the following happens:
Some elements are considered equivalent, in some topological or algebraic sense
We take the quotient
The metric is usually not ...
- 2,129
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
6
votes
0
answers
902
views
Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s?
Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so ...
- 13k
5
votes
0
answers
127
views
What are the conditions for the dual of the exterior algebra to be isomorphic to the exterior algebra of the dual?
The exterior algebra $\Lambda^*_kM$ can be defined for a $k$-module $M$, where $k$ is a commutative ring. A number of sources mention, without condition or proof, a (canonical) isomorphism $$(\Lambda^*...
- 171
5
votes
0
answers
138
views
Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)
Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
- 32.5k
5
votes
0
answers
160
views
Faltings' Cartier duality for A-modules in terms of Hopf algebras
$\newcommand\dual{^{\text{dual}}}\newcommand\GrpSch{\mathrm{GrpSch}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Vect{Vect}$If $G$ is a finite group scheme over a field $k$, we can define its ...
5
votes
0
answers
305
views
Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality
I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari.
After some arguments, we get a exact sequence
$$
\mathbf{P}^1_S(k,M^{'})^* \...
- 151
5
votes
0
answers
488
views
Dual norm for weighted Sobolev space
Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm:
\begin{equation}
\|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{...
- 407
5
votes
0
answers
60
views
Self-duality of cones associated with elementary symmetric polynomials
Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates:
$$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$
...
- 52.3k
5
votes
0
answers
270
views
Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?
The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
- 2,306
5
votes
0
answers
185
views
Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
- 1,623
5
votes
0
answers
382
views
Serre functors for non-proper categories
One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
- 512
5
votes
0
answers
141
views
Poincare duality in families of smooth, projective curves
Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
- 1,593
5
votes
0
answers
839
views
Access to a classic reference of Dold-Puppe
There is an old reference that I am unable tofind. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as:
A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the ...
- 2,871
5
votes
0
answers
162
views
Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
- 1,150
5
votes
0
answers
345
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 ...
- 799
5
votes
0
answers
64
views
Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
- 289
5
votes
0
answers
130
views
Quantum subgroups of Locally Compact Groups and Parabolic Induction
In the classical theory , parabolic induction is used to construct the (reduced) dual of a (semi-simple) Lie Group. However, for this we need subgruops. Given that the theory of "quantum subgroups" of ...
- 561
5
votes
0
answers
167
views
A specific case of the $p$-center problem
Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
- 125
5
votes
0
answers
194
views
A linear optimization problem on a graph
Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
- 2,288
5
votes
0
answers
343
views
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
5
votes
0
answers
448
views
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 ...
- 131
5
votes
0
answers
204
views
A polytope associated with the Hadamard Transform
In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
- 4,636
5
votes
0
answers
581
views
When is polytope compatible with network flow?
A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
- 3,475
4
votes
0
answers
46
views
Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
4
votes
0
answers
128
views
Length of dual module
It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...
- 223
4
votes
0
answers
149
views
Examples of $\ast$-autonomous $(\infty,1)$-categories
A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ ...
- 764
4
votes
0
answers
176
views
Is the test function topology a Mackey topology?
I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...
- 41
4
votes
0
answers
495
views
Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175
4
votes
0
answers
112
views
Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.1k
4
votes
0
answers
180
views
Spanier-Whitehead dual of space of natural transformations
Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
...
4
votes
0
answers
123
views
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 ...
- 3,549
4
votes
0
answers
150
views
How does Gabriel–Ulmer duality extend to (limit, colimit) sketches?
$\newcommand\Sketch{\mathit{Sketch}}\newcommand\Set{\mathit{Set}}
\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\Colim{Colim}\DeclareMathOperator\Mod{Mod}\newcommand\mod{\operatorname{mod}}\...
- 63.9k
4
votes
0
answers
166
views
Is the Serre dualizing complex local in the analytic topology?
There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, ...
- 7,952