All Questions
Tagged with real-analysis or linear-algebra
11,369 questions
0
votes
0
answers
109
views
Linear independence in $\mathbb{Z}_q^n$
Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...
- 503
3
votes
0
answers
138
views
What is the probability that the absolute value of the root of a polynomial is greater than $x$?
Note: This question was unanswered in MSE for a month so posting it in MO.
Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we ...
- 5,470
4
votes
1
answer
686
views
Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 41.8k
0
votes
0
answers
21
views
Unimodality of distribution from Lévy symbol
Also posted in MSE.
Assume that one want to study a distribution $f$ on $\mathbb{R}$ for which the Lévy symboln, i.e.:
$$
\forall u\in\mathbb{R},\quad\psi(u) := \log \mathbb{E}\left[e^{iuX}\right]
$$
...
- 393
2
votes
1
answer
110
views
Equivalence among these functions
Let $\Phi$ be the CDF of a standard Gaussian distribution, i.e.
$$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-y^2/2}dy,\quad \forall~ x\in \mathbb R.$$
Denote by $\Phi^{-1}$ its inverse ...
- 1,389
3
votes
0
answers
212
views
Differentiability along hyperplanes for rational functions
This is a follow up to my previous question.
Let $f\colon \mathbb R^3\to \mathbb R$ be a continuous function that is rational and differentiable along all planes through $0$, that is, we assume:
...
- 779
9
votes
1
answer
492
views
Dispersion points of Lipschitz functions
For a function $f: \mathbb R^n \to \mathbb R^m$ with $m < n$, we say that $x \in \mathbb R^n$ is a dispersion point of $f$ if
$$\liminf_{y \to x} \frac{|f(y) - f(x)|}{|y - x|} > 0.$$
Question: ...
- 6,155
2
votes
0
answers
160
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 593
3
votes
1
answer
193
views
Differentiability along hyperplanes
Definition. Let us say that a function $f\colon \mathbb R^d\to \mathbb R$ is differentiable along hyperplanes in the point $0\in \mathbb R^d$, if $f\circ \varphi\colon \mathbb R^{d-1}\to \mathbb R$ is ...
- 779
1
vote
1
answer
161
views
An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$.
We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
- 387
4
votes
1
answer
314
views
Some but not all eigenvectors mutually orthogonal
Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
1
vote
0
answers
56
views
Extension of this maximisation problem : finite or not?
$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
- 1,389
1
vote
1
answer
114
views
Sum of squares of $k\times k$ cofactors is $1$ for an orthonormal matrix [closed]
Let $n,k\in \mathbb N$ with $k\leq n$. Let $A$ be an $n\times n$ real orthonormal matrix. Fix any $k$ rows of $A$ and from there consider every possible $k\times k$ cofactors and there will be exactly ...
- 49
6
votes
0
answers
130
views
Bent vectors and $\pm 1$ eigenvectors with respect to non-Sylvester Hadamard matrices
A Hadamard matrix is an $n\times n$-matrix $H$ where each entry in $H$ is $\pm 1$ and where $H/\sqrt{n}$ is orthogonal. It is well-known that if $H$ is an $n\times n$-Hadamard matrix, then $n<3$ or ...
7
votes
1
answer
390
views
Questions on symmetric Hadamard matrices
Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.
If $A$ is a symmetric matrix, then $A = A^T$ and if $...
- 696
7
votes
1
answer
271
views
Can a differentiable function be nowhere locally $\alpha$-Hölder for all $\alpha > 0$?
Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder ...
- 6,155
0
votes
0
answers
56
views
How explicit the optimiser of this optimisation problem can be?
Provided the given parameters as follows :
$\mu\in\mathbb R, \sigma\in\mathbb R_+$ are constant, $\kappa, r, \alpha, \beta: \mathbb R_+\to\mathbb R_+ $ are measurable functions such that $\kappa(y)\...
- 1,334
6
votes
3
answers
851
views
Almost everywhere-periodic functions with many periods
Let $f : \mathbb{R} \to \mathbb{R}$ be a Lebesgue measurable function and $D$ be a countable dense subset of $\mathbb{R}$.
Suppose that for a.e. $x \in \mathbb{R}$ we have
\begin{equation*}
f(x + d) = ...
2
votes
1
answer
360
views
Asymptotics of an oscillatory integral
For $a > 0$ and $n \in \mathbb Z_+$, consider the oscillatory integral
$$\int_{0}^1 f(x) f(ax) \dots f(a^n x) \, dx,$$
where $f$ is an integrable function on $[0, 1]$, which we extend by ...
- 6,155
7
votes
2
answers
178
views
Separating domains in $\mathbb{R}^{2n}$ by a real algebraic variety
Suppose $\Omega_1$ and $\Omega_2$ are two disjoint unbounded domains in $\mathbb{R}^{2n}$, $n \in \mathbb{N}$. Can there be conditions on $\Omega_1$ and $\Omega_2$ so that these two domains can be ...
- 171
4
votes
1
answer
128
views
Lower bound of mean curvature implies that the set is subset of a given ball
If a simply connected set $\Omega\subset\mathbb{R}^n$ has $C^2$ boundary such that the mean curvature $H$ of $\partial \Omega$ satisfies:
$$H\geq 1$$
Does this imply that $\Omega\subset B_1$ after ...
- 421
1
vote
0
answers
40
views
From a constraint satisfaction problem (CSP) to a sudoku grid [closed]
one of the existing methods of solvin a sudoku grid is via constraints satisfaction (CSP), but can we do the inverse ie convert a CSP problem into a sudoku grid and then solve it ?
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
2
votes
1
answer
215
views
Forming real positive semidefinite matrices from complex matrices
I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices.
Let $Q \in \...
- 31
14
votes
2
answers
851
views
Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
- 143
20
votes
1
answer
2k
views
Is $1/F$ Schwartz if $F$ is "reverse Schwartz"?
Let's call a positive function $F:\mathbb{R}\to\mathbb{R}$ "reverse Schwartz" if $F$ is smooth and
$$\forall n \forall k,\quad\lim_{x\to\infty}\frac{|x|^n}{|\partial_x^k F(x)|}=0\quad .$$
In ...
- 23.3k
4
votes
1
answer
205
views
Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
- 45
1
vote
1
answer
118
views
Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{...
- 835
6
votes
1
answer
817
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,155
0
votes
0
answers
143
views
A Poincaré inequality holds for $p>2$ but fails for $p\leqslant 2$
I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).
Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset ...
- 69
0
votes
1
answer
131
views
Function of eigenvalues of Laplacian matrix
Let $G$ be a simple $n$-vertex graph and let $\mu_n\geq\mu_{n-1}\geq\dots\geq\mu_1$ be the eigenvalues of its Laplacian matrix, how can I find a function $$f(\mu_1,\mu_2,\dots\mu_n) \text{ such that } ...
- 53
3
votes
0
answers
146
views
Two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number
Are there two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number?
- 356
5
votes
1
answer
245
views
Are singular functions dense in the space of Hölder continuous functions?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
For every positive $\alpha < 1$, is the set of ...
- 6,155
5
votes
0
answers
583
views
Dimension inequality for subspaces in field extensions
Let $K\subset L$ be a field extension and $A, B\subset L$ be $K$-subspaces of $L$ of finite positive dimensions. Assume further that for every $a, b \in L$ and every nontrivial proper finite ...
- 429
0
votes
0
answers
125
views
Has anyone seen such a function/quantity?
I am dealing with a problem wherein I encounter the following quantity-
$$
Q_{d, \epsilon}(t_0) = \sup_{t' \notin B(t_0, \epsilon)} \inf_{t \in B(t_0, \epsilon)} \frac{d(t') - d(t)}{t'-t}.
$$
Here,...
- 11
4
votes
1
answer
101
views
Limits along lines for the gradient of a convex function
It is easy to see that if a function $f: \mathbb{R} \to \mathbb{R}$ is strictly convex, $C^1$ and $f'$ has bounded image, then as $t\to \infty$ the limit
$$
\lim_{t\to\infty} f'(t) = \lim_{t\to\infty} ...
- 89
8
votes
2
answers
509
views
Condition to guarantee that an inhabited and bounded set of reals has a supremum
This question is about constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the symbol $\mathbb{R}$) refer to ...
- 32.5k
-1
votes
1
answer
223
views
Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball
Hi I just happened to have a small question. If we have
$$\frac{\int_\Omega x}{|\Omega|}=\frac{\int_{\partial\Omega} x}{|\partial\Omega|}$$
for a simply connected set $\Omega$ with analytic boundary. ...
- 421
2
votes
0
answers
78
views
Partitions of bent vectors
Let $H=\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}.$ Let $A^{\otimes N}$ denote the tensor product of the matrix $A$ with itself taken $N$ times. We say that a vector $v$ of ...
1
vote
1
answer
101
views
Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>2 $
Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>2 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.
...
- 1,219
0
votes
0
answers
86
views
Solve equation with three square roots
I am trying to solve a more general question and I have the following subproblem:
Find $x>0$ that satisfies for fixed $ i \geq 3$,
$$\left(1 + \frac{1}{b^2}\right) x = \frac{\sum_i a_i^2} {b^2} + \...
- 49
2
votes
1
answer
106
views
Lower bounds for the expectation of log ratio between the posterior and prior Beta densities
The quantity I'm interested in is expressed as follows:
$$
I = \mathbb{E}_{k\sim \text{Binom}(n,p)} \left[\ln \frac{\text{Beta}(p;a+k,b+n-k)}{\text{Beta}(p;a,b)}\right]
$$
The term inside the ...
- 63
2
votes
1
answer
103
views
Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
- 171
6
votes
0
answers
111
views
Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
2
votes
0
answers
43
views
Good Polynomial lower estimates for beta function
I'm looking for polynomial lower estimates for beta function, and what I've found so far is this, which can be found in proposition 2.3 in this paper
Proposition 2.3 1. If $0<𝑞<1$ and $𝑝 \geq ...
- 677
9
votes
1
answer
845
views
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
- 541
4
votes
0
answers
87
views
Non-zero element in diagonal of cofactor matrix of symmetric 0-1 matrix with non-zero determinant, zero diagonal and odd number of rows
Is the following statement correct?
Let $A$ be a symmetric 0-1 matrix with non-zero determinant, all diagonal elements equal to 0 and an odd number of rows. The diagonal of the cofactor matrix of $A$ ...
- 41
2
votes
1
answer
246
views
Inequality with Hermite polynomials
Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by
$$\sqrt{\sqrt{\pi} 2^n n!}$$
for the purpose of normalization.
These are orthogonal with respect to the weight function $e^{...
- 43.2k
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
0
votes
0
answers
109
views
A Lipschitz function induced by the infimum of the length of curves
Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of ...
- 69