All Questions
7,280 questions
0
votes
0
answers
257
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What is the integer form of a projector into the intersection of the ranges of two integer projection matrices?
Consider two square integer matrices $X$ and $Y$ of the same dimension with the following properties:
$X^2=rX$, and $Y^2=sY$ for integers $r$ and $s$. The $\gcd$ of the entries of $X$ is 1 and the $\...
17
votes
1
answer
3k
views
2x2 subdeterminants of a matrix
If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...
- 1,309
1
vote
0
answers
1k
views
Properties of a rational function of multiple variables
Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...
1
vote
0
answers
296
views
Finding lower triangular matrix of an indefinite matrix
So I have the system $M = RS = RQQ^{-1}S $ and I have $R$ and $S$ currently.
I impose some constraints on $R$ in the form of $r^T$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
- 11
2
votes
1
answer
760
views
Apollonian gasket and the degree of convergence
Let $r_1,r_2\dots$ be the radii of Apollonian gasket.
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$
I know that if three circles $A$, $B$ and $C$ ...
CommunityBot
- 1
1
vote
0
answers
133
views
Extension of a function
Hello,
Given a $\mathcal{C}^\infty$ function $\varphi$ defined on a portion of a surface $\Sigma^-$ and let $\Sigma$ be a closed surface or union of surfaces bounding a compact volume $\Omega \...
CommunityBot
- 1
2
votes
0
answers
160
views
Radius of convergence to be proved more precisely (differential equation)
There is a differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
It is possible to get a solution which is a power series (see below). However, I am looking for an ...
5
votes
4
answers
8k
views
Proving a determinant = 0
The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...
- 51
9
votes
2
answers
1k
views
Rescaling positive definite matrices to force a unit eigenvector
Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones.
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$...
- 193
1
vote
1
answer
193
views
If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative?
Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the ...
- 3,245
4
votes
0
answers
340
views
Viscosity solution of the PDE
Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and
$$|Du|-f(x,u)=0$$
where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
- 61
2
votes
1
answer
351
views
Operation on measurable sets in lines, containing an interval?
Question 1: In $\mathbb{R}^2$, let $l_1$,$l_2$ be two parallel lines and $l_3$ another line which is not parallel to $l_1$. Given two measurable sets $E_1$ and $E_2$ in $l_1$ and $l_2$ respectively, ...
- 435
13
votes
1
answer
1k
views
Is it necessary to use AC to solve this problem ?
Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, $A\...
CommunityBot
- 1
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 8,246
1
vote
2
answers
508
views
Sufficient conditions for inverse-positivity
I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the ...
CommunityBot
- 1
8
votes
4
answers
7k
views
Positive solutions of linear Diophantine equations
Let $A$ be a non-negative integer $k\times n$-matrix (i.e. each entry is non-negative and integer) with $rank(A) = k < n$. Let $b$ be a $k$-dimensional vector with positive integer entries. ...
- 293
2
votes
0
answers
259
views
Eigenvalues of the products of a fixed unitari matrix with diagonal unitari matrices
How does the spectra of $DU$ change when $D$ runs over all diagonal unitary matrices? Here $U$ is a fixed unitary matrix. Precisely, let spec$(X)$ be a set of eigenvalues of $X$.
For a unitary matrix $...
- 723
3
votes
1
answer
399
views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\...
1
vote
1
answer
206
views
What is such an equation called?
Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?
$Ax+f(\lambda)Bx=g(\lambda)x$.
- 28.6k
5
votes
1
answer
346
views
Linear maps preserving positive semidefiniteness
I know of Choi's theorem and some related problems, but not a solution to this exact problem:
Characterize the linear maps from the space $S_n$ of symmetric $n \times n $ matrices to itself that ...
- 6,962
1
vote
1
answer
383
views
Solution of a PDE and its uniqueness
Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
$\...
- 108k
6
votes
2
answers
3k
views
Approximating erf by tanh
It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...
- 41.1k
26
votes
2
answers
12k
views
About the definition of Borel and Radon measures
I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...
CommunityBot
- 1
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 435
12
votes
4
answers
2k
views
Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(...
- 7,831
6
votes
2
answers
425
views
orderings of the field R((x, y))
I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the ...
- 328
4
votes
2
answers
1k
views
Minimum eigenvalue of a Affine Combination of two Hermitian matrices
Consider two $N \times N$ hermitian indefinite matrices $A_1$ and $A_2$. Consider their affine combination
\begin{align}
M(t)=(1-t)A_1+tA_2
\end{align}
I am interested in the minimum eigenvalue of $M(...
- 52.3k
1
vote
1
answer
4k
views
How to solve this optimization with the orthogonal constraint?
Problem
Supposing that $A$ is a symmetric real matrix and $\{\mathbf{w}\_i\}_{i=1}^n$ is any orthogonal basis on $\mathbb{R}^n$ such that $W^\top W=WW^\top=\mathbf{I}_n$ where $W=\left[\mathbf{w}_1\;\...
- 607
3
votes
1
answer
6k
views
Classical Derivative, Weak Derivative and Integration by Parts
Hello,
While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.
QUESTION
Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $...
- 882
29
votes
6
answers
10k
views
how to find/define eigenvectors as a continuous function of matrix?
I asked this (with background) here
https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...
CommunityBot
- 1
1
vote
1
answer
168
views
Mollification with prescribed boundary values
Suppose you are given a $C^1$-function $f:\mathbb R^n\to\mathbb R$ which restricts to a smooth function $f|_{\partial B}:\partial B\to \mathbb R$, where $B$ is the unit ball in $\mathbb R^n$. Can one ...
- 13k
8
votes
3
answers
2k
views
Does a weaker condition than vanishing derivative imply a function being constant?
I learned this question from math.stackexchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does
$$g(x):=\lim_{\epsilon\to 0}\frac{f(x+...
CommunityBot
- 1
1
vote
1
answer
1k
views
Calculating the Lebesgue decomposition of a measure [closed]
How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
- 60.5k
2
votes
3
answers
365
views
Construct a fixed-point set operator
How to find an uncountable set $S$, and construct an function $f : 2^S
\longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in
T$?
for example, let $S =\mathbb{R}$, how can I ...
- 73.2k
1
vote
1
answer
417
views
Decomposition of Matrix to its sub-matrix with constant rank
When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ...
- 54.2k
2
votes
1
answer
486
views
Is there any connection between this matrices
Matrices I discuss are all $N\times N$ hermitian matrices. Define two positive (semi)definite matrices $H_1$ and $H_2$. Define the following matrices
\begin{align}
P_1&=H_1+(I+H_2)^{-1} \\\
P_2&...
- 108k
2
votes
1
answer
300
views
Rayleight Ritz Ratio and smallest eigenvalue for a set of given matrices
I am familiar with Rayleigh Ritz Ratio for hermitian matrices. Let $A_1$ be a given $N \times N$ hermitian matrix. Then the smallest eigenvalue of $A_1$ is given by
\begin{align}
\lambda_{min}(A_1)=\...
- 52.3k
1
vote
1
answer
620
views
Smallest Lipschitz constant on non-convex domains
It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...
- 54.2k
4
votes
1
answer
319
views
Concerning strata in $C^\infty(M)$
The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf ...
- 17.5k
8
votes
1
answer
1k
views
Norm of inverse confluent Vandermonde matrix
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as
$$V=
\begin{bmatrix}
v_{1,0}&v_{2,0}&\dots&...
CommunityBot
- 1
8
votes
2
answers
2k
views
Characterizing invertible matrices with {0,1} entries
Related to the question link text I was asking myself some time ago the following. Can one precisely describe the invertible n\times n matrices with{0, 1} entries? For example, is anything special ...
CommunityBot
- 1
9
votes
1
answer
1k
views
Determinantal formula for the nullspace of a singular matrix
In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...
- 39.3k
11
votes
1
answer
2k
views
Transcendentality of all irrationals in the Cantor set
Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic ...
- 6,548
4
votes
1
answer
336
views
What is the geometry of the intersection of some cones defined by generalized inequalities?
Hello, considering that for real numbers, the intersection of intervals defined by simple inequalities has a quite simple form as
$$
\bigcap_i\{x|x\leq a_i\}=\{x|x\leq\min_i\{a_i\}\}
$$
However, what ...
- 46
5
votes
1
answer
550
views
Weakest assumption for pointwise convergence of Fourier series
This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose $f(x)=\sum_{...
- 31.5k
8
votes
3
answers
1k
views
Relating a Polynomial equation to the characteristic equation of a Hermitian matrix
This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
21
votes
0
answers
904
views
Cauchy matrices with elementary symmetric polynomials
$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...
- 28.6k
0
votes
1
answer
272
views
A certain type of quadratic problem.
I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min_{u^{H}u=1}~(u^{H}A_1u) \\\
s.t.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ ...
- 211
0
votes
1
answer
341
views
Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
0
votes
0
answers
237
views
Geometric Mean of Positive Matrices
Hello all,
My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices $(A,B)$ is given by:
$M_0(A,B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-...
- 155