All Questions
563 questions with no upvoted or accepted answers
19
votes
0
answers
841
views
I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?
Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:
$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$
with the following ...
- 5,215
15
votes
0
answers
398
views
References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology
Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
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
13
votes
0
answers
591
views
What are the difficulties in proving almost-everywhere stability of Gaussian elimination?
It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
13
votes
0
answers
458
views
Descartes rule of signs for a noncommutative polynomial in matrix variables
Recently, while studying certain notions of "averaging" a set of input matrices, I obtained a nonlinear polynomial in matrix variables. A simple example is
\begin{equation*}
\mathcal{G}(X) := X^n - \...
- 28.6k
13
votes
0
answers
1k
views
Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
- 1,839
12
votes
0
answers
599
views
Once differentiable, piecewise degree three polynomials on triangulated planar domains
Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of ...
- 121
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
439
views
Evaluating Shintani cone zeta functions
Hi everyone
I am trying the evaluate sums of the form
$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$
...
- 265
9
votes
0
answers
561
views
Padé approximations of $e$
The following question came up in the analysis of some algorithm.
Let $R_{s,t}(z)$ be the Padé approximants of $e^z$, and define $r_{s,t} = R_{s,t}(1)$. Using the explicit expression for the error ...
- 1,906
8
votes
0
answers
481
views
Problems where Conjugate gradient works much better than GMRES
I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
- 489
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
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 & \...
7
votes
0
answers
317
views
An inequality which involves a sum of integrals
Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
- 71
7
votes
0
answers
209
views
Numerical linear algebra: how to compute $B^TC^{−1}B$ efficiently
Hi,
my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$.
The matrix $C$ is huge ($n$ up to a ...
- 619
6
votes
0
answers
233
views
Newton type method for finite fields?
I have a polynomial $p(x)$ in $\mathbb{Z}/q\mathbb{Z}$ that is easy to compute for any $x$ but has an absurdly large degree $d > 2^{256}$. I know for a fact that it has a zero and I would like to ...
- 591
6
votes
0
answers
137
views
Why wavelet methods are not popular anymore in nonparametric statistics?
Back in my master years, I took a nonparametric statistics class. In this class, a few nonparametric methods were presented, but I remember spending a lot of times on methods based on wavelet ...
- 73
6
votes
0
answers
197
views
Where to cut off a double sum?
I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form
$$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$
where $|a_{m,n}|\...
- 20.2k
6
votes
0
answers
394
views
Numerical analysis with p-adic numbers
How should one go about doing numerical analysis with $p$-adic numbers?
By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian ...
- 3,065
6
votes
0
answers
320
views
Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
- 596
6
votes
0
answers
206
views
Degree of Chebyshev polynomial necessary
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
- 13.9k
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\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$$ where $x_1,x_2\...
6
votes
0
answers
379
views
Numerical Methods for stochastic PDE, from rough paths to backward equations
this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular,
Have the numerical implications, if any, of the results in ...
- 254
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
6
votes
0
answers
565
views
What are the eigenvectors of the Lagrange interpolation matrix?
Let $F$ be a field.
Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field.
Consider the $k\times k$ matrix that in position $i$, $j$ has the element
$\frac{\prod_{l\neq i}(y_i - ...
- 179
5
votes
0
answers
75
views
What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
5
votes
0
answers
211
views
Numerical analytic continuation/asymptotics
I posted this question, quite a while ago, on math.stackexchange.com, here. I received an interesting answer but not sufficiently accurate for my purposes, so I'm trying here.
I have a class of ...
- 516
5
votes
0
answers
225
views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
- 83
5
votes
0
answers
77
views
Numerical and computational approaches to limit cycle theory
I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach.
I would like to ask ...
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
$$...
- 1,150
5
votes
0
answers
122
views
How to solve this operator equation numerically?
I would like to know how one solves Sturm-Liouville problems on $(0,\infty)$ NUMERICALLY for the eigenvalues that are of the form
$$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$
So even if there ...
- 501
5
votes
0
answers
2k
views
Find the axis of symmetry in a point cloud
I have some dataset which describes a spherical cloud of points in 4D space. Actually, the coordinates of the points are the coefficients of unit quaternions, so you get the idea on what the data is ...
- 151
5
votes
0
answers
131
views
On existence of rapid Arithmetic geometric procedure?
We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...
- 13.9k
5
votes
0
answers
161
views
Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function
Setup:
Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function.
For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} f(\...
- 616
5
votes
0
answers
167
views
A specific case of the $p$-center problem
Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
- 125
5
votes
0
answers
143
views
Error of midpoint method for differentiable functions
Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$?
...
- 19.7k
5
votes
0
answers
2k
views
Computing a 2D Fourier transform on a disk
I am looking for a reliable and fast way of evaluating an integral like
$$
F(r, \phi)= \int_0^1 \int_0^{2\pi} f(\rho, \theta) e^{2\pi i \rho r \cos(\theta - \phi)}\rho\, d\theta\,d\rho,
$$
where $f$ ...
- 702
5
votes
0
answers
194
views
A linear optimization problem on a graph
Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
- 2,288
5
votes
0
answers
392
views
Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix
I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
- 500
5
votes
0
answers
190
views
Pair of two-variable polynomial equations of high order
I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N \...
- 548
5
votes
0
answers
301
views
Inadmissibility of Simpson's rule
(An earlier version of this at stackexchange got no answers.)
Bayesianism says that all uncertainties, or at least all uncertainties about the truth or falsity of propositions, can be expressed by ...
5
votes
0
answers
321
views
Approximation by polynomials
The following is a well-known theorem (see e.g. The Chebyshev Polynomial by Rivlin):
If $p(x) = x^n + a_{n_1} x^{n-1} + \ldots + a_0$, then $\max_{-1\leq x \leq 1} |p(x)| \geq 2^{1-n}$ for $n \geq 1$ ...
- 111
5
votes
0
answers
500
views
Parabolic cylinder functions - explicit estimates?
I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker).
Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in ...
- 20.2k
5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 922
5
votes
0
answers
204
views
A polytope associated with the Hadamard Transform
In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
- 4,636
5
votes
0
answers
412
views
Approximations of negative Sobolev norms
Consider the standard Cahn-Hilliard free energy, augmented by a nonlocal interaction term which measures the $H^{-1}$ norm of a zero-mean function. Could someone point me to a reference where this ...
- 1,171
5
votes
0
answers
581
views
When is polytope compatible with network flow?
A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
- 3,475
5
votes
1
answer
124
views
Is Sun's spectral variation bound for normal matrices optimal?
In On the variation of the spectrum of a normal matrix, Sun proves the following result (Corollary 1.2):
Let $A$ be an $n$-square normal matrix and $B$ an arbitrary $n$-square matrix. Then $$ \min_{\...
- 283
4
votes
0
answers
198
views
Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
- 51
4
votes
1
answer
88
views
Simulation of SDEs using Karhunen Loeve expansion
A very common and easy way to simulate the solution of a SDE is to use the Euler-Maruyama method. At each time step the only random part comes from the realization of the increment of the Brownian.
It ...
