All Questions
1,732 questions
0
votes
1
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204
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Is the linear production game a convex game?
In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex ...
- 17
0
votes
1
answer
543
views
Convert general optimization problem to LP problem
I am trying to convert the following problem into a linear programming problem:
There are $M\times N$ matrix $T$ of real numbers between 0 and 1 and $N\times 1$ vector $w$ of real numbers between 0 ...
- 99
0
votes
1
answer
82
views
Introducton books for $\frak{E}_p(I)$
Are there any good books different from abstract harmonic analysis by hewitt to study $\frak{E}_p(I)$. where $\frak{E}_p(I)$ is: Let $I$ be an arbitrary index set. For each $i\in I$ let $H_i$ ...
- 209
0
votes
2
answers
1k
views
The condition number of a scaled Vandermonde matrix
Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...
- 265
0
votes
1
answer
169
views
Finding the distribution of a random variable numerically with sample data? [closed]
Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...
- 113
0
votes
1
answer
216
views
Upper bound for a ratio of modified Bessel functions
I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...
- 1
0
votes
1
answer
179
views
A kind of Discrete Fourier Transform
Given a $z\in \mathbb{C}^N$, the DFT of $z$ is given for every $k\in [0,N-1]_\mathbb{N}$ by
$$DFT_z(k)=\frac{1}{N} \sum_{j=0}^{N-1} z_j\, \omega^{-k j}$$ where I have denoted by $\omega$ the $N$-th ...
- 364
0
votes
2
answers
118
views
Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
- 2,239
0
votes
1
answer
525
views
Improving Newton's Inequalities using the Taylor Theorem
Newton's inequalities say that if $f(x) = \sum \binom{n}{k} a_k x^k$ is a polynomial with all real roots then $ a_k^2 > a_{k-1}a_{k+1}$.
The proof this result uses that if $f(x)$ has all real ...
- 22.8k
0
votes
1
answer
546
views
Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
0
votes
2
answers
708
views
Approximate solution to large mixed integer programming problem
What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...
user24451
0
votes
1
answer
452
views
Large scale least squares of non symmetric and non square problems
Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric ...
- 125
0
votes
1
answer
100
views
generalization from linear programming solution [closed]
I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...
- 109
0
votes
1
answer
85
views
About the suboptimality of linear estimators
Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$.
...
- 29
0
votes
1
answer
2k
views
eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
0
votes
1
answer
321
views
How to handle a scalar product in an integral?
I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it.
Here's a simplification of the problem:
$ \...
0
votes
1
answer
536
views
Solving system of nonlinear equations
Dear all,
Can anyone tell me all the algorithms that are available for finding all solutions of a system of nonlinear equations?
I am particularly interested in solving problems of the form:
X1=f1(...
- 1
0
votes
1
answer
488
views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
0
votes
2
answers
189
views
solving non linear equations
When solving non-linear equations via Newton's method, load increments are often used to improve convergence. In mechanics for example, if the final load in 90N, one could choose 3 load steps of 30N ...
- 3
0
votes
1
answer
193
views
Ease of calculation of norm
I have SPD matrix A and two vectors z and b.
Is there exist a norm where I can calculate $||A^{1/2}b-z||$ without having to calculate $A^{1/2}b$ explicitly ?
- 489
0
votes
2
answers
376
views
Convergence of Fourier series for $C^p$ functions
Let $f \in C^p[0,2\pi]$ and periodic. Denote $\omega_p$ as the moduli of continuous of $f^{(p)}$. Then
$
|f - S_Nf| \le K \frac{\log{N}}{N^p}\omega_p(2\pi/N),
$
where $S_N$ is the Fourier partial sum ...
- 13
0
votes
1
answer
205
views
SDP Algorithms/ maximally complementary solutions
Hello,
I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold.
...
- 1
0
votes
1
answer
504
views
$\ell_o$ Minimization (Minimizing the support of a vector)
I have been looking into the problem
$\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...
- 11
0
votes
1
answer
226
views
Continuity of Lexicographic Minimum Solution of a parametrized LP problem
Given a parametrized LP problem
find x, that minimizes F*x
such that Ax <=Bt+D
where t is a parameter.
And suppose C(t) is a set of all optimal solutions of LP with parameter t.
Let x_L(t) be ...
- 29
0
votes
1
answer
207
views
Regular Perturbation Series soln to eqn
I want to find the a 3 term perturbation soln of
(i) $(1+x)^3 = ex$ where $e\ll1$
Direct substitution of the regular perturbation series $x = x_0 + ex_1 + e^2x_2$
into (i) does not work
I ...
0
votes
1
answer
533
views
Differences between the Poisson's and elliptic Monge-Ampere equations?
I am working with a numerical application where some known techniques solve $\Delta u=\rho$ for $\rho(x,y)\geq0$ defined on a planar rectangle (given as the density of a large pointset). Since the ...
- 341
0
votes
3
answers
1k
views
Numerical solution to Fisher-Kolmogorov equation
Can you use the Crank-Nicolson method to get a numerical approximation to the fisher-kolmogorov equation?
If not what would be the easiest way to model the equation using matlab?
Thanks and sorry its ...
0
votes
1
answer
695
views
Minimum distance between two data sets
Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...
0
votes
1
answer
409
views
Need help to find an efficient algorithm for the following problem!
Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...
- 131
0
votes
1
answer
307
views
Comparing iterative methods for linear systems
For a tridiagonal matrix of the from
\begin{bmatrix}
a & -b & \newline
-b & a & -b \newline
& \ddots & \ddots & \ddots \newline
& & & & -...
0
votes
1
answer
2k
views
Solving 5 eqns with 6 unknowns in a 2x3 contingency matrix, is there a unique solution? [closed]
Background
I have the following equations:
$$a+b+c=6$$
$$d+e+f=15$$
$$a+d=5$$
$$b+e=7$$
$$c+f=9$$
This is a 2x3 matrix $[a b c, d e f]$ where the marginal totals are fixed. In addition, all of the ...
- 225
0
votes
1
answer
592
views
Delta notation used for describing numerical stencil
While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order ...
- 157
0
votes
1
answer
65
views
Newton method for polynomials with random starting points
I know that this question exists, but unfortunately it doesn't cover my issue sufficiently.
Assume that we have a polynomial $p(x)$ of degree $n$ with real coefficients, we can assume that all its ...
- 1,171
0
votes
0
answers
57
views
Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
- 67
0
votes
0
answers
21
views
Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
0
votes
0
answers
25
views
Numerical computation of spectral abscissa of operator
I would like to numerically compute the spectral abscissa of an unbounded linear operator $A$ on a Hilbert space. To give you an idea my operator has the form:
$$Af(x,y) = a(y) \partial_x f(x,y) - b(x)...
- 143
0
votes
0
answers
48
views
A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214
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54
views
How to deal with minimizing a flat objective function
Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
- 121
0
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0
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83
views
Interpolating points and orthogonal polynomials on varying intervals
In general, a Lebesgue-Stieltjes integral, $\int (\cdot) \, d \alpha(x)$, that defines an inner product on the space of polynomials establishes a notion of orthogonality.
Suppose we have a sequence of ...
- 13
0
votes
0
answers
32
views
Finding measure representation for rank 2 moment matrices
Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it.
$$
\begin{bmatrix}
1 & 1 \\
...
- 275
0
votes
0
answers
101
views
Simulation of Markov processes with exponential timestepping
Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way:
Choose an initial ...
- 167
0
votes
0
answers
115
views
Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
- 1
0
votes
0
answers
72
views
Probability of being inside a convex n-dimensional polytop
I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays.
As part of the research, each individual transmission is modelled using a probability ...
0
votes
0
answers
30
views
Peak search for NLS equation solutions
By solving the initial value problem of the NLS equation, I can get a matrix of numerical solutions. I hope to perform a peak search on the matrix of this numerical solution, that is, there are ...
- 33
0
votes
0
answers
72
views
Initial guess in shifted QR algorithm
I'm comparing timings of two implementations of algorithms for the computation of Gauss-Legendre nodes.
1 - The first is a Newton algorithm to find the roots of the Gauss-Legendre polynomials. Quite ...
- 297
0
votes
0
answers
49
views
Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)
What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
- 163
0
votes
0
answers
136
views
Antiderivatives via Taylor series and the FT of Calculus
If $f$ is a real function on an interval $[a,b]$ such that
$f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
- 827
0
votes
0
answers
98
views
Interpolation polynomials with constraints
Lets consider a collection of $n$ points $\{Z_i\}_{i=1}^n\subseteq \mathbb{D}^k(1)=\{(z_1,\ldots,z_k)\in\mathbb{C}^k:\forall j\leq k, |z_j|\leq 1\}$. Let $h: \{Z_i\}_{i=1}^n \to \mathbb{D}^1(1)$ be a ...
- 502
0
votes
0
answers
67
views
Concentration of bilinear forms
This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
- 6,962
0
votes
0
answers
39
views
Max-flow modeling with unified vehicle and commodity variables
I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...
- 101