All Questions
5,632 questions
2
votes
1
answer
152
views
Growth rate of elementary sequences
We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
3
votes
1
answer
132
views
Is it possible to determine whether the critical values are nowhere dense in the case of a bounded set of stationary points?
Let $g:\Bbb R^{d}\rightarrow \Bbb R$ be a non-negative, continuously differentiable function satisfying the following two conditions:
The set $\{\theta\in\Bbb R^n\mid\|\nabla g(\theta)\|<\eta\}$ ...
- 51
1
vote
1
answer
249
views
Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points
Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds.
My question: Does a 'reasonable' sufficient ...
- 671
8
votes
1
answer
1k
views
Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?
Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...
- 157
0
votes
0
answers
180
views
Proof that the zeroes of certain polynomials are increasing with respect to degree
Choose $k+1$ positive integers $d_j\in\{0,1,2,3,\ldots\}$ and let $d=(d_1,\ldots,d_k)$.
Consider the following polynomial equation over the positive reals:
$$
\sum_{j=1}^{k}\frac1{x^{d_j}} = x^{d_{k+1}...
- 21
0
votes
0
answers
52
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
1
vote
2
answers
271
views
An integral inequality?
Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...
- 83
2
votes
0
answers
103
views
Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
3
votes
2
answers
218
views
Extremum placement for two-variable function
While teaching Calculus 2, one of my students asked me the following
Given 3 points $x_1$, $x_2$, $x_3$. Whether there exists one function $z = f(x,y)$ which has exactly 2 extremum and 1 saddle point:...
- 73
1
vote
1
answer
150
views
Unable to understand an application of Minkowski's inequality
Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:
with the norm $\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \...
- 90
-1
votes
1
answer
80
views
Regions when a concave function is smaller than another concave function
Let $f_1,f_2:[0,1]\mapsto\mathbb{R}$ be two bounded and concave functions. Assume $f_1(0)<f_2(0)$ and $f_1(1)<f_2(1)$. I want to investigate the set $\mathcal{X}\triangleq\{x\in[0,1]: f_1(x)>...
- 53
5
votes
1
answer
224
views
A limit related to quasi-periodic function
Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that
$$
\frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2}
$$
...
- 375
3
votes
0
answers
137
views
On the continuity with respect to the increasing convex order
For $p\ge 1$, let $\mathcal P_p(\mathbb R)$ be the set of probability measures on $\mathbb R$ of finite $p^{\rm th}$ moment. Denote by $W_p$ the Wasserstein metric of order $p$ and by $\preceq$ the ...
- 1,399
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
3
votes
1
answer
296
views
Does this condition characterise intervals, among subsets of the real line?
For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$:
$\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
- 60.5k
0
votes
1
answer
68
views
Box dimension and graph of Hölder function
In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that,
if a function $f:I^{d}\...
- 73
14
votes
3
answers
2k
views
Every function on reals a sum of two surjective real functions?
From this question, and the answer thereof, we can see that every real valued function on reals is a sum of two injective functions. Is the same true if we replace injectivity by surjectivity.
For ...
- 2,089
8
votes
1
answer
381
views
Special Schwartz function on the positive interval
Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following:
$\int \zeta(t)\: dt=1$,
$\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$,
$\operatorname{supp}(\zeta)\subset (0,...
- 263
2
votes
1
answer
289
views
Erdős–Sierpiński duality in locally compact Polish groups (e.g. $\mathbb{R}^n$)
Erdős–Sierpiński mapping for a locally compact Polish group $G$ is a bijection $f$ from $G$ to $G$ such that $A$ is a null set in $G$ with respect to the Haar measure if and only if $f(A)$ is a meager ...
- 131
4
votes
3
answers
869
views
Can these identities for the Euler-Mascheroni constant be proven?
I stumbled upon these 4 limit/integral identities involving Euler's constant aka gamma (~0.5772). They appear to be valid based on inspection but I have no idea how to prove them. In addition, I have ...
- 194
7
votes
2
answers
606
views
Countably representing all closed sets of positive measure
This may be a naive question, but I don't see an immediate argument.
Question: Does there exist a sequence $\{C_m\}_{m=1}^\infty$ of Borel subsets of $[0,1]$ with positive Lebesgue measure $|C_m|>0$...
- 1,959
2
votes
1
answer
133
views
How to calculate this integral of squared Tricomi hypergeometric function
How to solve this integral
$$
\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r
$$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...
2
votes
1
answer
138
views
Boundedness of an exit time from a campact set
Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of
\begin{align*}
& x(0)=x_0 \\
& \dot{x}=v(x).
\end{...
- 449
8
votes
1
answer
594
views
What is the minimum of this functional？
Recently I encountered an inequality from mathematical analysis.
Let $f(x)$ be twice continuously differentiable in $[0,1]$ with
$f(0)=f(1)=0$, then for all $x\in(0,1),f(x)\neq 0$, show that:$$\int_{0}...
- 505
3
votes
2
answers
615
views
A problem about how dominated convergence is used in the analysis of variation
I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
- 809
2
votes
2
answers
268
views
If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?
Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
- 605
2
votes
2
answers
152
views
Upper bound estimation for second-order variable-coefficient ODE
I'm tackling a second-order linear ordinary differential equation with variable coefficients $a(t)$ and seek advice on estimating the upper bound of
$y(t)$ s.t $|y(t)|\le M$. The equation in question ...
- 45
3
votes
2
answers
281
views
Can every $L^p$ function be written as the weak derivative of a Sobolev function?
Let $\mathbb B^n$ be the open unit ball in $\mathbb R^n$, and $g: \mathbb B^n \to \mathbb R^n$ a measurable function with $|g| \in L^p (\mathbb B^n)$. Does there exist some function $f$ in the Sobolev ...
- 6,213
0
votes
1
answer
106
views
The sequence has a stationary accumulation point
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that
(a) $\{x^k\}$ has an ...
- 21
4
votes
1
answer
194
views
What are the possible blow up limits of an $L^1$ function?
Let $f: [0, 1] \to \mathbb R$ be an $L^1$ function. Define for each $r > 0$, the blow up $f_r:[0, 1] \to \mathbb R$ by
$$f_r (x) := \frac{f(rx)}{r}.$$
Suppose $f_r$ converges in $L^1$ to some ...
- 6,213
27
votes
1
answer
2k
views
Is every real number in [0,1] a product of three (or more) Cantor set's numbers?
It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
- 60.5k
0
votes
0
answers
76
views
Implicit function theorem for non $C^1$ mappings
I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
9
votes
4
answers
742
views
Distributional derivatives are locally integrable implies the distribution is also locally integrable?
Let $T$ be a distribution on $\mathbb{R}^n$ such that there are functions $f_1,\ldots,f_n \in L^1_\text{loc}(\mathbb{R}^n)$ so that $\dfrac{\partial T}{\partial x_j} = f_j, \forall j=1,\ldots,n. $
My ...
- 93
0
votes
0
answers
272
views
How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
2
votes
1
answer
286
views
Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?
Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
21
votes
1
answer
1k
views
Does summing divergent series using cutoff functions give consistent results?
One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...
- 313
0
votes
1
answer
130
views
Explanation for Tauberian theorems for Laplace transform
I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow :
Let $F : [0,\infty) \to \mathbb{R}$ of ...
- 393
1
vote
1
answer
179
views
The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$
In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...
- 677
-1
votes
1
answer
110
views
Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 437
2
votes
2
answers
329
views
$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
- 178
3
votes
2
answers
620
views
Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?
Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
- 149
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
1
vote
1
answer
183
views
A self-consistent equation that turns into a differential equation
Suppose the function $f(x,y)$ is defined on a small neighbourhood of $(0,0)$ in $\mathbb{R} \times [0,\infty)$ and satisfies the self consistent equation
\begin{align*}
& f(x,y) = \frac{1}{1-y} + ...
- 1,295
4
votes
2
answers
2k
views
Does a function exist which is not Riemann integrable and satisfies the given condition:
I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that
$$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...
- 151
5
votes
2
answers
423
views
$C^1$ harmonic functions on a dense open set are globally harmonic
In a paper I am studying, at a certain point the authors introduce a function $u\in C^1(B_1,\mathbb{R})$ which is harmonic in a dense open subset $U$ of $B_1$. From this, they seem to conclude that $u$...
- 1,149
4
votes
2
answers
361
views
Implicit function theorem without uniqueness?
Imagine you are given $f(x,y) := y^2-\sin(x)^2$
and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$.
One idea that comes to mind is the ...
- 49
10
votes
5
answers
2k
views
Extracting a common convergent indexing from an uncountable family of sequences
Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space.
For each $\alpha \in \mathcal{A}$, let
\begin{equation}
\{ x_n^{\alpha} \}_{n=1}^\infty
\end{equation}
...
- 3,477
4
votes
1
answer
178
views
Compact-open Topology for Partial Maps?
I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow.
Compact open topology is one of the most common ways of ...
- 1,093
10
votes
1
answer
817
views
Can a nowhere locally Hölder function be differentiable almost everywhere?
Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$-Hölder continuous - that is, it is not $\alpha$-Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...
- 6,213
2
votes
2
answers
290
views
Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$
I already posted a similar question on MO and looked into the references therein.
However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form.
Let $T \in \...
- 3,477