All Questions
5,632 questions
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 $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...
- 517
3
votes
1
answer
255
views
Is this constraint convex?
I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
- 31
6
votes
1
answer
427
views
When are the chirp signals orthogonal?
Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
- 287
5
votes
1
answer
235
views
Can a solution to this parameterized ODE converge to zero?
Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$?
$$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$
We are also given y(0) = 2/3, $...
- 340
0
votes
1
answer
92
views
Continuous selectors of a continuous multifunctin on a compact metric space
I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
...
- 79
3
votes
1
answer
227
views
If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?
Let $a,b\in\mathbb R$ with $a<b$ and $f:[a,b]\to\mathbb R$. Assume that there exists a Riemann integrable function $g:[a,b]\to\mathbb R$ such that $f=g$ almost everywhere.
Then we can NOT conclude ...
- 299
10
votes
1
answer
572
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 299
2
votes
1
answer
211
views
Macroscopic sets - a notion of largeness for Lebesgue null sets
Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
- 6,213
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 825
1
vote
1
answer
184
views
Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
- 6,213
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
6
votes
1
answer
392
views
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
1
vote
1
answer
157
views
On the additive property of the subdifferential of lower semicontinuous functions
Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by
$$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
- 299
25
votes
2
answers
2k
views
Writing a function on $\mathbb{R}$ as a sum of two injections
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
- 4,265
-2
votes
1
answer
283
views
Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?
Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...
4
votes
1
answer
341
views
Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?
Let $S^1$ be the circle, let us consider a function $f(x,t): S^1 \times [0,\infty) \to \mathbb{R}$ such that
\begin{equation}
\int_0^T \int_{S^1} \lvert f(x,t) \rvert dxdt <\infty
\end{equation}
...
- 3,477
1
vote
1
answer
150
views
Is the Boltzmann entropy continuous in the supremum norm?
We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\...
- 825
11
votes
3
answers
1k
views
"Simple" integral equation
Let $H(z)$ be a continuous solution of the problem
$$
H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1.
$$
Is it true that $H(0)=1-\ln2$? The question ...
- 283
1
vote
1
answer
108
views
If all mixed partials of a $C^1$ function exist and are continuous, is the function $C^2$? [closed]
For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be a $C^1$ function such that the mixed partial derivatives $\partial_i \partial_j f$ exist and are continuous for all $i \neq j$. Is it true that $f$...
- 6,213
2
votes
1
answer
192
views
Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$? [closed]
This seems pretty trivial but I can't seem to figure it out. I think it's obviously true, given an unconstrained convergent sequence we just have to add some filler elements, but I'm having trouble ...
- 291
3
votes
0
answers
83
views
Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
-2
votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 109
2
votes
0
answers
116
views
Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
- 469
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
0
votes
1
answer
270
views
Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
user516435
0
votes
0
answers
98
views
Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
1
vote
1
answer
126
views
Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?
Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that
$$
\int_0^1 |y-x| f(x) \, dx = 0
$$
for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
- 136
0
votes
1
answer
117
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
- 4,058
4
votes
2
answers
485
views
How to get this inequality in Santambrogio's book about optimal transport?
Let $\hat{\varrho}, \tilde{\varrho}$ be probability density functions on $\mathbb R^d$ where $\tilde{\varrho} \in L^{\infty} (\mathbb R^d)$. For $\varepsilon \in [0, 1]$, we define $\varrho_{\...
- 825
3
votes
1
answer
140
views
Oscillation functions and similar constructs
For given $f$ from reals to reals, the associated oscillation function is defined as follows:
$$\textstyle
osc_f(x):= \lim_{n\rightarrow \infty} [\sup_{y \in B(x, \frac{1}{2^n}) } f(y)-\inf_{z \in B(x,...
- 4,359
0
votes
1
answer
507
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 101
2
votes
0
answers
100
views
Are analytic solutions for the Navier-Stokes equations sufficient?
Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space.
However, I am wondering, whether it is possible to consider just analytic ...
- 749
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
9
votes
0
answers
287
views
The approximate mean value theorem / Rolle's theorem in pure constructive mathematics
In the replies of this very similar question, there is a fascinating answer that is beautiful in its simplicity. In particular, it seems to use perhaps the most minimal assumptions one can possibly ...
- 191
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
3
votes
3
answers
340
views
Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $n \geq 2$ be an integer, and $E \subset \mathbb R^n$ be a set of finite $\mathcal H^{n-2}$ measure.
Suppose $f: \mathbb R^...
- 6,213
1
vote
0
answers
123
views
Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
1
vote
1
answer
118
views
A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 3,477
1
vote
1
answer
139
views
Does there exist a continuous choice of maximizing balls for the Hardy Littlewood maximal function?
Note: Here $\overline B_r (x)$ denotes the closed ball of radius $r$ around $x$.
Let $f \in L^1 (\mathbb R^d)$. We define the averages $A(x, r)$ for $x \in \mathbb R^d$ and $r > 0$ by
$$A(x, r) := \...
- 6,213
2
votes
1
answer
189
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 75
2
votes
0
answers
120
views
On the integer of the form p^a q^b closest to a given integer N
If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
- 69
1
vote
1
answer
345
views
Topological degree of differentiable map using line integrals?
Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$
I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
1
vote
1
answer
65
views
Boundedness of maximisers of parametric strictly concave functions
Let $L:[0,1]\times \mathbb R^m\times \mathbb R^n\to \mathbb R$ be defined by
$$L(\lambda, x,y):=\sum_{1\le i\le m}\alpha_i x_i + \sum_{1\le j\le n}\beta_j y_j -\sum_{1\le i\le m, 1\le j\le n} p_{i,j}\...
- 1,399
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
2
votes
0
answers
174
views
Product of marginals absolutely continuous with respect to a Borel probability measure
Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
- 3,599
9
votes
2
answers
584
views
Does this integral condition characterize $L^\infty$?
Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition ...
- 6,213
5
votes
0
answers
214
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
1
vote
0
answers
98
views
Periodicity in one Fourier variable
Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument.
We can thus write $f$ as $$ f(x) = \sum_{n \in \...
4
votes
0
answers
116
views
Lipschitz extension of a flow can still be a flow?
Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well ...
- 697