All Questions
5,856 questions
2
votes
0
answers
120
views
A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
- 69
0
votes
1
answer
416
views
Necessary conditions for convergence of convolution
In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
- 61
0
votes
1
answer
252
views
Reference request: log Sobolev inequality for uniform measure (uniform distribution over discrete set)
Suppose that $N \in \mathbb N_+$ is fixed and denote by $\mu = (\mu_0,\ldots,\mu_N)$ the uniform distribution on the set $\{0,1,\ldots,N\}$ (i.e., $\mu_n = \frac{1}{N+1}$ for each $0\leq n\leq N$). I ...
- 730
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
0
votes
0
answers
126
views
A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
0
votes
1
answer
76
views
Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$
We fix $p \in [1, \infty)$. We have for every $f \in L^p (\mathbb R^d)$ that $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$. I wonder if there is an estimate of above decay, i.e.,
Is there a ...
- 825
0
votes
0
answers
280
views
Are there "gaps" between Lipschitz functions and $C^1$ functions?
I was wondering about this fact. By Rademacher theorem, it is well known that a Lipschitz function in $\mathbb{R}^d$ is $\mathcal{L}^d$-a.e. differentiable. Of course, a $C^1$ function is everywhere ...
- 697
1
vote
1
answer
162
views
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
3
votes
1
answer
146
views
Behaviour of the solution of a second order ODE
I am currently studying the following second order ODE
\begin{cases}
\ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\
y(0) = 0\\
\dot y(T) = c
\end{...
- 452
1
vote
1
answer
187
views
Bound the distance between two vectors on the probability simplex
Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$
$$\sup_{x>0} \...
- 135
6
votes
0
answers
220
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
7
votes
1
answer
356
views
High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?
Let $d\geq 1$ be a positive integer. If $\{\vec x_n\}_{n=1}^\infty$ is a sequence of $d$-dimensional vectors satisfying $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...
- 517
15
votes
1
answer
2k
views
Real polynomials that go to infinity in all directions: how fast do they grow?
Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
- 26.9k
0
votes
0
answers
112
views
Characterization for the multipliers of Schwartz space
Is the following true?
A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following:
For every $\alpha$ ...
- 938
2
votes
0
answers
159
views
Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
- 826
2
votes
1
answer
140
views
Invertibility of message passing with invertible parametrization
Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...
1
vote
1
answer
57
views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 825
3
votes
1
answer
132
views
Existence of a density
Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite ...
- 301
3
votes
1
answer
161
views
Equivalent definition for Skorokhod metric
I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$:
$$
d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
- 177
0
votes
1
answer
47
views
Everywhere existence of marginals
Let $f\in L^1(\mathbb{R}^2)$ be a (joint) probability density function which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$.
What is a necessary and sufficient condition under which the ...
- 3,574
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
37
votes
12
answers
5k
views
Examples where existence is harder than evaluation
In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, ...
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
0
votes
0
answers
29
views
$ \sup_{\theta \in [0,2\pi)}\max_{r\leq \delta}\frac{\log\left(\frac{f(r,\theta)}{f(\delta,\theta)}\right)}{\log(r)}<\infty,$ $f$ real analytic
$\textbf{Conjecture.}$
Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $\delta <1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and suppose that $(0,0)$ is the only ...
- 171
2
votes
1
answer
87
views
Symmetric and anti-symmetric matrices and maximal eigenvalues
Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
- 1,755
3
votes
4
answers
353
views
What real distributions solve $f'=0$? [closed]
I mean specifically real-valued Schwartz distributions on the real line. That is linear functionals on $C^{\infty}_c(\mathbb{R})$ continuous in the canonical LF topology. My question is, what are ...
- 11.1k
9
votes
4
answers
2k
views
How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074
2
votes
1
answer
117
views
When is a $p$-th order stationary point of a polynomial actually a local minimum?
Definition: For integer $p\geq 1$, we say $x\in \mathbb{R}^d$ is a $p$-th order stationary point of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ if there exists a $C>0$ and an $\epsilon>0$ ...
- 637
5
votes
1
answer
224
views
A limit related to quasi-periodic function
Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that
$$
\frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2}
$$
...
- 375
2
votes
1
answer
309
views
Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
- 363
1
vote
1
answer
234
views
Zeroes of elementary polynomials without involving closed-form solutions
Consider the following two polynomials, where $n$ is an integer:
$$
p_n(x) = x^3-\frac1nx-\frac2n, \\
q_n(x) = x^2-\frac2n.
$$
For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
- 21
8
votes
1
answer
496
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
- 4,143
0
votes
1
answer
150
views
Property of $p$-norm in the $n$-simplex
Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that
$$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$
implies that
$$\lVert x\rVert_p \...
0
votes
1
answer
58
views
An expansion for 2d Euler equation
Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$:
$$
-\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
- 471
3
votes
1
answer
255
views
Is this constraint convex?
I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
- 31
0
votes
0
answers
80
views
Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
- 85
0
votes
0
answers
103
views
Who first gave a result stronger-or-equal to this one on ODEs
After some thinking I've come to the following conclusion.
Consider the initial value problem $$\text{(P)}\begin{cases}x'(t)=f(t,x(t)),\quad t\geq t_0\\x(t_0)=x_0 \end{cases}$$ where $f:D\subset\...
- 637
1
vote
0
answers
79
views
How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
- 111
2
votes
0
answers
80
views
Limit of lacunar power series at $1^-$
I've asked this question on MSE but I didn't get a convincive answer so I'm trying here. Here is the question :
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider ...
- 186
3
votes
2
answers
354
views
General version of Weyl's lemma
The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega)$ satisfies
$$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$
then $u$ is harmonic in $\Omega.$ What I want ...
- 379
25
votes
2
answers
3k
views
What is the origin/history of the following very short definition of the Lebesgue integral?
Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
- 32.5k
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
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
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
$$ \int_{\...
- 83
2
votes
2
answers
329
views
$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
- 178
3
votes
2
answers
429
views
Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 4,074
4
votes
1
answer
668
views
$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
- 1,149
1
vote
1
answer
113
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 825
2
votes
2
answers
274
views
Is a simple closed curve always a free boundary arc?
Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?
For a simple closed curve $\...
- 23
5
votes
1
answer
589
views
On the Riemannian integrability of the bounded derivative
Let $f:[a,b]\to\mathbb R$ be a differentiable function with $f'$ bounded. According to this post, $f'$ is not necessarily Riemann integrable on $[a,b]$, see also Volterra's function.
I wonder, if $f'$...
- 299