All Questions
793 questions
96
votes
36
answers
17k
views
The concept of duality
I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come....
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 ...
- 9,481
38
votes
2
answers
1k
views
What are all the natural maps between iterated duals of vector spaces, and equations between these?
Fix a field. Given a vector space $V$ it has a dual $V^\ast$, which has its own dual $V^{\ast \ast}$, which has its own dual $V^{\ast \ast \ast}$, and so on ad infinitum.
There are lots of natural ...
- 22.3k
35
votes
12
answers
3k
views
No canonical isomorphism [duplicate]
I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
35
votes
4
answers
5k
views
Why are optimization problems often called "programs"?
Why are optimization problems often called programs?
linear programming
geometric programming
convex programming
Integer programming
...
- 461
35
votes
1
answer
2k
views
Are there topological versions of the idea of divisor?
I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...
- 18.4k
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 391
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 363
25
votes
3
answers
2k
views
Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
- 33.8k
20
votes
5
answers
964
views
If a $\otimes$-idempotent object has a dual, must it be self-dual?
Let $C$ be a symmetric monoidal category.
Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the ...
- 63.9k
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 ...
- 997
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
3
answers
2k
views
Classification of rings satisfying $a^4=a$
We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
- 63.1k
19
votes
2
answers
2k
views
Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
- 33.8k
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667
18
votes
2
answers
730
views
What categorical property of monoidal categories picks out the ones with duals?
Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...
- 54.6k
17
votes
3
answers
6k
views
The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
- 1,513
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
- 2,288
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 163
16
votes
1
answer
1k
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 ...
- 63.9k
16
votes
1
answer
974
views
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 ...
- 118k
16
votes
1
answer
2k
views
Questions about spectra of rings of continuous functions
I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
- 65.4k
14
votes
1
answer
1k
views
Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
- 18.4k
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
4
answers
1k
views
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 ...
- 2,998
13
votes
5
answers
4k
views
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
13
votes
2
answers
1k
views
Is there any relationship between the topologies of the clique complex and the independence complex?
Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...
- 7,857
13
votes
3
answers
605
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-...
- 568
13
votes
2
answers
664
views
Complexity of a weirdo two-dimensional sorting problem
Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
- 19.2k
13
votes
1
answer
624
views
Gabriel-Ulmer duality for $\infty$-categories
Gabriel-Ulmer duality states that 2-categories $\mathrm{Lex}$ (of small finitely complete categories and functors preserving finite limits) and $\mathrm{LFP}$ (of locally finitely presentable ...
- 4,459
13
votes
3
answers
834
views
Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
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
12
votes
3
answers
562
views
Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters
Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...
- 2,933
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
12
votes
2
answers
1k
views
Does Poincaré duality preserve algebraic cycles?
Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of ...
- 2,323
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 (...
- 66.7k
12
votes
1
answer
311
views
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 ...
- 1,271
12
votes
1
answer
266
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. ...
- 66.7k
12
votes
2
answers
409
views
Reference for free symmetric monoidal categories with duals on symmetric monoidal categories
The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.
In ...
- 37.8k
11
votes
2
answers
963
views
Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?
Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
- 1,826
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
11
votes
2
answers
3k
views
Why the circle for Pontryagin duality? [duplicate]
For a locally compact group $G$, we define the Pontryagin dual as $\hat G = Hom(G,\mathbb T)$ where $\mathbb T$ is the circle group and the homomorphisms are continuous group maps. This duality has a ...
- 7,746
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 ...
- 63.9k
11
votes
1
answer
524
views
Elementary proof of a triangular grid lemma
I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714).
Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...
- 1,174
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 ...
- 251
10
votes
2
answers
1k
views
Self-dual plane curves
Suppose that $C\subset \mathbb P^2$ is a plane projective curve (base field is $\mathbb C$) and $C^*\subset (\mathbb P^2)^*$ is its dual. What are the known examples in which $C$ is projectively (i.e.,...
- 1,839
10
votes
2
answers
3k
views
How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
- 693
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 $...
- 7,857