All Questions
5,909 questions
0
votes
1
answer
60
views
Empty interior lack of minima
Suppose that $U \subseteq \mathbb{R}^d$, and satsifies
$U$ is dense in $\mathbb{R}^d$,
U has empty interior,
Then is it possible that
$$
\inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x),
$$
...
- 63.9k
4
votes
1
answer
490
views
What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function?
Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It ...
- 188k
0
votes
1
answer
91
views
Existence of a certain type of function
Trying to find functions with the given property:
Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
- 17.2k
2
votes
2
answers
375
views
Ergodic theorem and products
If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that
$$ \lim_{n \rightarrow \infty} \frac{f_n}{...
- 1,769
2
votes
1
answer
564
views
Density in fractional Sobolev space
Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define
$$
H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right),
$$
$$
H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}.
$$
Q: Is $C^\...
- 28k
8
votes
1
answer
172
views
Distance between subalgebras and positive elements in matrices
I repost here from stackexchange, as I was not given an answer there. (https://math.stackexchange.com/questions/2956530/distance-between-subalgebras-and-commutants-in-matrix-algebras)
This is a ...
- 10.1k
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
CommunityBot
- 1
4
votes
1
answer
370
views
Proving two inequalities involving the gamma and digamma functions
I'm having trouble proving the following inequality:
$$\forall p>1 \quad \forall m\geq 0 \quad \dfrac{m^2\Gamma(\dfrac{2m}{p})\Gamma(\dfrac{2m}{q})}{\Gamma(\dfrac{2m+2}{p})\Gamma(\dfrac{2m+2}{q})}\...
- 128k
2
votes
0
answers
149
views
Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?
Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
1
vote
0
answers
99
views
How is the dominated convergence theorem applied in the proof of Lyapunov’s criterion?
Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...
- 167
7
votes
3
answers
709
views
On the inequality $\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\right)^2$
I'm have some difficulties in bounding the following inequality:
I want to find a c as small as possible s.t.
$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\...
- 63.9k
5
votes
0
answers
273
views
Is there any geometrical/homological intuition behind symmetrized gradient?
The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
- 980
29
votes
3
answers
2k
views
Wanted: Positivity certificate for the AM-GM inequality in low dimension
I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
Can one write the LHS as a sum
$\,\...
- 1,054
1
vote
0
answers
254
views
Short question on functions of bounded variation
For a function $f: \mathbb R \to \mathbb R$ of locally bounded variation, when is $$\liminf_{e \to 0} V(f)[x, x+e]/e $$finite everywhere? Here $V(f)[a, b]$ denotes the total variation of the function ...
- 59
4
votes
2
answers
159
views
$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality
$$\left\|\displaystyle\...
- 685
8
votes
2
answers
560
views
How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?
Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
- 128k
0
votes
0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 163
10
votes
4
answers
1k
views
Adventure with infinite series, a curiosity
It is easily verifiable that
$$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
...
- 977
6
votes
1
answer
351
views
Looking for infinite series resembling an exponential
I'm looking for some $f(x)$ that has the following property:
$\sum_{x=1}^\infty f(kx) = r^k$
for some real $0 < r < 1$, and at least for strictly positive integer $k$.
Does such an $f(x)$ ...
- 85.6k
11
votes
2
answers
813
views
Textbook recommendation request: Exercises to supplement Evans and Gariepy
While a great book about measure theory and real analysis in $\mathbb R^n$, the only downside is the lack of exercises. Can anyone provide a good book to supplement it with exercises? I plan to use it ...
- 28k
8
votes
1
answer
314
views
A strictly decreasing function between uncountable subsets of the reals
By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following ...
- 42k
6
votes
1
answer
1k
views
Uniformly differentiable functions
Note: Here all functions are $\mathbb R \to \mathbb R$. $Id$ denotes the identity function.
Let $g_i$ be a family of functions indexed by some (potentially uncountable) index set $I$. Given a ...
- 109k
2
votes
1
answer
300
views
Necessary and Sufficient conditions for integrable function [closed]
Suppose that $a, b$ and $c$ are constant.
Is there the necessary and sufficient conditions of $a ,b, c$ for the following integration is integrable? i.e.
$$\int_0^\infty \int_0^\infty \int_0^\infty ...
1
vote
1
answer
1k
views
Inequality for the operator norm of a product of matrices
I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that
$$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...
1
vote
0
answers
220
views
Quantitative Lusin’s theorem
We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $...
- 12.9k
12
votes
2
answers
286
views
Show that $f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c$ has at most $2n$ zeros
Let
\begin{align}
f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c
\end{align}
where $x_1<x_2<...< x_n$ and $a_i>0$. For some positive constant $c$.
Can we show that $f(t)$ has at most $2n$ ...
- 17.2k
0
votes
1
answer
116
views
Integrable function [closed]
Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...
- 128k
0
votes
1
answer
350
views
Uniformly Bounded (updating)
Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~
(1+t+z)^{a_3}}\exp\big\{-\frac{...
- 119
5
votes
0
answers
262
views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
- 1,127
1
vote
0
answers
71
views
Continuous injection of metric ball into Euclidean ball
This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...
- 5,405
2
votes
1
answer
113
views
Continuous inclusion of metric spaces of smaller capacity
If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is
$$
\kappa_X(\epsilon)\leq \...
- 28k
0
votes
0
answers
72
views
Looking for example of integral transformations that preserve number of zeros
Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros.
I am looking for non-trivial examples of integral transformation
\begin{align}
g(x)= \int f(t) h(t,x) dt
\end{align}
such that $f$ and $g$...
- 671
2
votes
2
answers
190
views
One-Sided Analyticity Condition Guarantees Analytic Function?
Let $f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying:
$f$ is differentiable infinitely many times in $(0,\infty)$, and has a right-derivative of any order at $0$.
$f$ satifsfies the ...
- 34.7k
5
votes
3
answers
638
views
Uniqueness of minimizers in a problem in the Calculus of Variations - Part II
Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which ...
- 13k
1
vote
2
answers
279
views
Reference request: Functions of bounded variation in one real variable
Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!
- 659
3
votes
0
answers
235
views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
- 31
1
vote
2
answers
2k
views
Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$
In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
It is not hard to see that for every $k < \infty$, and every $\epsilon >...
- 128k
0
votes
1
answer
419
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
- 42.8k
2
votes
1
answer
536
views
Approximation of a two-variable function by tensor products
Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function.
We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \...
- 1,848
3
votes
2
answers
226
views
Example of convex functions fulfilling a (strange) lower bound
I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
- 128k
3
votes
1
answer
217
views
Uniqueness of minimizers in the Calculus of Variations
Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...
- 17.2k
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 28k
8
votes
2
answers
891
views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
- 128k
12
votes
1
answer
520
views
Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?
Asked once on SE-mathematics.
Let $U$ be an open subset in $\mathbb{R}^n$, $m\in\mathbb{N}$, $1\leq m<n$ and let
$$\mathcal{C}^k_{\leq m}(U,\mathbb{R}^n):=\lbrace g\in\mathcal{C}^k(U,\mathbb{R}^n)\...
- 28k
2
votes
2
answers
190
views
Quantitative bound on irrational rotation recurrence time
Given an irrational $a$, the sequence $b_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified.
Given a point $p$ in $\mathbb ...
- 30.1k
1
vote
1
answer
1k
views
Smallest Lipschitz Constant of a Differentiable Function [closed]
Let $X \subset \mathbb{R}^{n}$ be compact and convex. Moreover, let $f:X \rightarrow \mathbb{R}$ be a differentiable map with $\sup_{x \in X} \|\nabla f(x)\| = K < \infty$, where $\|\cdot\|$ ...
- 42.8k
7
votes
1
answer
311
views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
- 9,486
3
votes
1
answer
240
views
A certain generalisation of the golden ratio
Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
- 128k
8
votes
0
answers
210
views
Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
- 389
7
votes
1
answer
505
views
Anisotropic perimeter and regularity of anisotropic minimal surfaces
1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...
- 1,179