All Questions
985 questions
11
votes
0
answers
234
views
When is cohomology of a finitely presented dg-algebra computable?
Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...
- 3,772
0
votes
1
answer
126
views
An otherwise linear matrix equation with the presence of a signum function : reference request
Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$
$\pmb{c}$ is a $n\times1$ matrix.
$G$ is a $n\times n$ matrix which is also positive definite.
matrices $G$ and $c$ are real.
$L$ is a $n\...
- 698
5
votes
2
answers
200
views
Fast algorithms for calculating the distance between measures on finite ultrametric spaces
Let $X$ be a finite ultrametric space and $P(X)$ be the space of probability measures on $X$ endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula
$$\...
- 41.8k
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
16
votes
3
answers
652
views
How to plot this fractal
I'm a graphic designer and my client has asked me to use this fractal in a design that I'm working on. As you can see, it's not a very good copy, so I'm trying to see if I can generate a high-...
- 263
0
votes
0
answers
108
views
Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
1
vote
0
answers
68
views
Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 13.9k
8
votes
1
answer
231
views
Counting polygons in arrangements
For an arrangement of lines $\cal{A}$ in the plane, an
inducing polygon $P$ is a simple polygon satisfying:
(a) every edge $e$ of $P$ lies on some line $\ell$ of $\cal{A}$, and
(b) every line $\ell \...
- 151k
1
vote
0
answers
147
views
Convergence of infinite linear programming
Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
- 53
-1
votes
1
answer
103
views
How to solve MILP problem on several linear subspaces
I have a set of close mixed-integer programming problems. More exactly, all the problems share the same set of (binary and continuous) variables, the same set of linear inequality constraints, and the ...
- 39
0
votes
0
answers
42
views
Construct pairs of $n$-dimensional convex bodies with given ratios ($p$) of volumes
Given a dimension $n$ and a number $p \in (0,1)$, to what extent is it possible (in what cases) to construct a convex set $A$--not a hypersphere--and a "snugly" inscribed (InscribedFigure) ...
- 723
2
votes
0
answers
46
views
Notion of distance between linear programs
Consider the linear programming problem
\begin{align}
\max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1
\end{align}
where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
- 1,421
6
votes
1
answer
424
views
Probability of intersecting a rectangle with random straight lines
We are given a rectangle $R$ with sides lengths $r_1$ and $r_2$, contained in a square $S$, with sides lengths $s_1=s_2\ge r_1$ and $s_2=s_1\ge r_2$. $R$ and $S$ are axis-aligned in a cartesian plane $...
- 1,677
5
votes
1
answer
388
views
Calculating $n$-dimensional hypervolumes ($n \sim 50$), for example
I have a question regarding efficient and possibly simple algorithms for computing volumes of $n$-dimensional polytopes.
The polytope of concern isn't arbitrary: it is obtained by applying a linear ...
- 51
26
votes
2
answers
4k
views
Why did Robertson and Seymour call their breakthrough result a "red herring"?
One of the major results in graph theory is the graph structure theorem from Robertson and Seymour
https://en.wikipedia.org/wiki/Graph_structure_theorem. It gives a deep and fundamental connection ...
- 290
5
votes
1
answer
156
views
On folding a polygonal sheet
Consider a polygonal sheet $P$ of area $A$ with $N$ vertices (it material is not stretchable or tearable). Let $n$ be a positive integer >=2.
Question: Let $P$ lie on a flat plane. We need to fold ...
- 5,979
1
vote
0
answers
82
views
What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,826
1
vote
1
answer
138
views
Packing L's in Tans and L's in L's
I'm a young researcher and I'm pretty new in this field. I want to work on packing problem "L's in Tans" and "L's in L's" as presented on https://erich-friedman.github.io/packing ....
- 121
2
votes
0
answers
113
views
Computing whether a set of polynomials cuts out a projective variety
I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....
- 980
1
vote
1
answer
464
views
Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram
A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal ...
- 19
0
votes
1
answer
1k
views
Fast way to generate random points in 2D according to a density function
I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...
- 121
1
vote
0
answers
78
views
Reference for the algorithm to find the intersection between a subspace and positive orthant
I came across this algorithm, in this question Algorithm for the intersection of a vector subspace with a cone of non-negative vectors ;
Is there any reference for the algorithm described in the ...
0
votes
1
answer
139
views
Linear programming with exponential inequalities and rational variables
If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear ...
- 1,826
2
votes
1
answer
591
views
Intersection of a vector subspace with a cone
Given a set of vectors $S=\{v_1, v_2,...,v_d\} \subset \mathbb{R}^{N}, \, N>d$, is there any algorithm to decide if there exist a vector with all coordinates strictly positive in the generating ...
1
vote
0
answers
67
views
What are the corners of this polytope?
Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...
- 11
2
votes
1
answer
197
views
Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional ideals
This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...
- 697
2
votes
1
answer
192
views
On some optimal containers of a set of points on the 2D plane
Given a set of N points in general position on the plane, the problem is to give efficient algorithms to find
the smallest semicircular region (semidisk) that contains the points
the smallest ...
- 5,979
4
votes
1
answer
211
views
$\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$
I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
- 41
0
votes
1
answer
226
views
Fractional values in linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
0
votes
1
answer
320
views
Sub optimal algorithm for linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $c$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
4
votes
1
answer
451
views
Implementation of Koebe–Andreev–Thurston circle packing?
The circle packing theorem (Koebe–Andreev–Thurston theorem) claims for a planar graph, we can pack disjoint circles, such that: the circles correspond to vertices and the disks are tangent if the ...
- 1,465
2
votes
1
answer
222
views
Discrete curve-shortening flow – numerical implementation
I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
- 31
2
votes
0
answers
66
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-...
- 121
0
votes
0
answers
37
views
Restrictions on crossing edges in Delaunay triangulations
what can be said about crossing edges in Delaunay triangulations, i.e. about pairs of edges that constitute to the heaviest perfect matching int the $K_4$ induced by the quadruplet of adjacent ...
- 13.2k
0
votes
1
answer
144
views
Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
1
vote
0
answers
34
views
How do I find the portion of a cell/voxel lying within a defined surface?
We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$.
A cone-like surface is defined by some function, $z = f(x, y)$, which in ...
- 111
1
vote
0
answers
62
views
fast V representation update of polytope
Say that I have both the V and the H representation of a (possibly unbounded) polytope $P$. I want to append a some rows to the H representation, how can I quickly update the V representation to ...
- 155
5
votes
2
answers
371
views
Is there an upper bound on the number of points in point cloud for which we compute the persistent homology?
I am interested in the topic of persistent homology (topological data analysis). According to what I read, there is some roadblock in the analysis of "big data" using persistent homology as it is ...
- 1,229
3
votes
0
answers
163
views
A new "adversarial" Wasserstein distance?
Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of ...
- 555
-1
votes
1
answer
120
views
IntersectInP bug of Macaulay2 [closed]
I am trying to use the intersectInP command in Macaulay2, inside package ReesAlgebra. However, I tried to follow the exact code in the user-guide, but it doesn't run in my Ubuntu app (of win 10). Can ...
- 287
11
votes
2
answers
964
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
2
votes
1
answer
485
views
Odd cycle transversal
Suppose we have a graph G. Say B a fundamental basis of the cycle space of G. Say LP a linear programming problem where there is a variable for each vertex of G, each variable can take value $\geq 0$, ...
2
votes
0
answers
71
views
grobner basis of an ideal dependent on some parameter
Suppose $I = \langle f_1, ... , f_l \rangle$ is an ideal generated by polynomials $f \in k[x_1,\dots,x_n]$, where $k$ is a field of rational functions in some parameters $s_1,\dots,s_m$.
What are the ...
- 1,435
1
vote
2
answers
958
views
Integer linear programming (ILP) formulation of connectivity of induced subgraph
Can anyone assist me to find out what should be the ILP formulation of a case when I try to label vertices by say $0$, $1$ and $2$ and want the subgraph of graph $(V,E)$ made by same vertex set but ...
2
votes
1
answer
871
views
Interior point of a convex polytope
Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to ...
2
votes
1
answer
237
views
Sampling algorithms on convex polytopes
Let $f=\mathbf{c}\cdot\mathbf{x}$ be the optimization objective function whose parameter vector $\mathbf{x}\in\mathbb{R}^n$ is subject to the following constraints in the very well-known linear-...
3
votes
0
answers
106
views
finding a good term order for grobner basis
What are the tricks to pick a "good" monomial order to find a Grobner basis for a given ideal?
By good I mean one in which the final Grobner basis has a simple expression in terms of the ...
- 1,435
5
votes
0
answers
85
views
special classes of ideals (eg. toric) that admit faster Buchberger algorithm?
I have heard that toric ideals allow one to speed up the Buchberger algorithm considerably (see Grobner bases of toric ideals, Remark 2,3). My question is two-fold:
What are the precise complexity-...
- 1,221
4
votes
2
answers
335
views
Software for finding conjugates in the braid group
The conjugacy problem for the braid group was solved by Garside, and gives an algorithm for determining whether two braids are conjugate. Since this algorithm is rather tedious, I was wondering if ...
- 469
12
votes
1
answer
382
views
Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries related to octonionic multiplication
I consider $1,i,j,k,l,m,n,o$ the standard basis of the (complexified if you like) octonions ($\mathbb{O}$ for the octonions).
Let $a = x_1.1 +\ldots + x_8.o$, $b = x_9.1+ \ldots + x_{16}.o$ and $c = ...
- 7,300