All Questions
5,857 questions
2
votes
0
answers
70
views
Essentially anti-Cauchy functions
Call a function $f: \mathbb R+ \to \mathbb R$ essentially $C^\infty$ if there exists a sequence $f_n$ $(n \geq 0)$ such that each $f_n$ is differentiable a.e., $f_0 = f$ a.e., and $f_n’$ is equal to $...
- 2,079
1
vote
1
answer
112
views
Orthogonal complement vector space
Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...
- 13
1
vote
1
answer
137
views
Concave functions of different behaviour in the neighbourhood of $0$ from the Shannon function
I'm looking for an example of a concave function $g \colon [0,1] \to \mathbb{R}$, $g(0)=0$ such that:
$$\liminf_{x\to 0^+}\frac{g(x)}{-x\ln x}\neq \limsup_{x\to 0^+}\frac{g(x)}{-x\ln x}.$$
Moreover, ...
- 51
10
votes
2
answers
2k
views
A result attributed to Whitney
One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...
- 737
0
votes
2
answers
132
views
Dirichlet problem for capillary equation over convex domain
Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary.
Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function.
Let $L$ be a quasilinear elliptic ...
- 823
6
votes
1
answer
396
views
Well definition of a function
I've edited, just skip the first attempt and go to the second one.
THE FRAMEWORK: let us consider a real topological vector space $V$.
We denote with $\mathscr C_k(V)$ the set of all continous ...
- 779
2
votes
1
answer
224
views
Strongly continuous semigroup: continuous or continuous componentwise?
Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$
Then, one can consider the function
$f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space ...
- 536
1
vote
1
answer
103
views
Is $X = \{ B \in L^\infty(\mathbb R^n,\mathbb R^n): \nabla \cdot B \in L^\infty(\mathbb R^n,\mathbb R^n) \}$ a dense subspace?
The Sobolev space $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ is not dense in $L^\infty(\mathbb R^n,\mathbb R^n)$. In fact the functions in $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ are Lipshitz, and not ...
- 13
20
votes
0
answers
634
views
Is $\sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
This question has been posted in MSE for two years without an answer. A094082 seems to suggest that it is not rational. Is it still an ...
user14319
9
votes
1
answer
299
views
Sequence of nested sets in $[0, 1]$ with bound on gaps
What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...
user78817
7
votes
2
answers
477
views
Characterizing the Radon transforms of log-concave functions
$f:\mathbf{R}^d\to \mathbf{R}_{\ge 0}$ is log-concave if $\log(f)$ is concave (and the domain of $\log(f)$ is convex).
Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$,
$$
...
- 5,992
4
votes
0
answers
459
views
Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?
Is there any closed form expression for the following serie?
$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$
Or at least a proof that it is an irrational number. The ...
9
votes
3
answers
934
views
local behavior of a finite Borel measure
Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall,...
- 1,839
1
vote
0
answers
74
views
Removability of the isolated singularity of real analytic mappings with nondegenerate Jacobian
Let $B^n=\{x\in \mathbf{R}^n: |x|<1\}$$(n>2)$. Consider real analytic mappings $f_1:B^n\setminus \{0\}\to B^n$, $f_2:B^n\setminus \{0\}\to \mathbf{R}^n$ and $f_3: B^n\setminus \{0\}\to S^n$ ...
- 89
4
votes
0
answers
633
views
Problem with an integral equation taken from a paper
I am reading a paper (the 2015 paper by A. Falkowski and L. Slominski Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation, page 353, proof of the Lemma 3.1)...
- 779
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
0
votes
2
answers
503
views
A Jordan arc in the unit disk
Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
- 105
10
votes
2
answers
344
views
A moment problem
Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...
- 205
1
vote
1
answer
1k
views
Intermediate value property and continuity
We say that a function $f:\mathbb{R}\to\mathbb{R}$ has the intermediate value property (ivp) if for $a<b$ in $\mathbb{R}$ we have $$f([a,b]) \supseteq [\min\{f(a),f(b)\}, \max\{f(a), f(b)\}].$$
The ...
- 48.9k
5
votes
1
answer
914
views
Extension of a function from almost everywhere to everywhere
The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...
- 7,182
11
votes
3
answers
3k
views
Is the supremum of continuous functions integrable?
Let $f_\alpha$ be a family of continuous positive functions $\mathbb R\to \mathbb R$
where the index $\alpha$ runs in a compact metric space
and the map $\alpha\to f_\alpha$ is continuous
with ...
- 29.1k
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
0
votes
1
answer
662
views
A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
- 309
3
votes
2
answers
383
views
Looking for some function
Is there a continuous function $F: R\to R$ such that $F$ is a surjection but not an injection, $F(Q)\subset Q$ and the restriction $F: Q\to Q$ is an injection, but not a surjection. Here $Q$ denotes ...
- 1,835
3
votes
1
answer
213
views
A really simple probabilistic inequality on the unit interval
Given a probability distribution on the interval $[0,1]$, is there any relationship between the quantity $$\sup_{S}{\mathbb{E}(X|X\in S)^{2}\Pr(X\in S)}$$ over all measurable subsets $S$, and the ...
4
votes
1
answer
2k
views
A continuity/bootstrap argument
I am trying to understand how one can prove the following assertion using a continuity argument:
Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to [0,\...
- 175
4
votes
0
answers
174
views
Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user127450
4
votes
2
answers
436
views
Variation of Radon transform for probability measures on $\mathbb C$
Let $\mu$ be a probability measure on $\mathbb C$. For $z \in \mathbb C$, let $$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$ be the function $f^z(\lambda) = |\lambda - z|$. Consider now the family $(\...
- 25.5k
2
votes
0
answers
122
views
A community effort: equilibrium in quitting games [closed]
This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...
- 745
4
votes
1
answer
184
views
Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...
- 391
4
votes
0
answers
105
views
On a much weaker version of the Normal conjecture
I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...
- 515
3
votes
1
answer
142
views
PDE satisfied by projection of a function onto a subspace
Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$}...
- 261
3
votes
1
answer
670
views
A specific mollified functions in the Sobolev space H^1(R)
Let $u>0$ be in $H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of $C^{\infty}$ functions with compact support are dense in the Sobolev space $H^{1}(\mathbb{R})$. Hence, we have a ...
- 31
4
votes
1
answer
461
views
Continuous non-constant function with infinite intersections with horizontal line on a compact interval?
The title might be misleading, but whether such a function exists is what boggles me about the following problem:
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all $...
- 281
1
vote
1
answer
250
views
Question about the stationary phase method and the smooth function used
A statement of the stationary phase method I know is the following.
Suppose $\phi(x_0) = \phi'(x_0) = 0$ and $\phi''(x_0) \not = 0$. If $\psi$ is a smooth function supported in a sufficiently small ...
- 3,625
3
votes
1
answer
196
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
- 33
4
votes
2
answers
519
views
Closed-Form solution for system of simple nonlinear equations
I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...
- 155
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 ...
- 698
0
votes
0
answers
40
views
Derivative of a valuation map for a second order ODE
Let $V:\mathbb{R}^+\to\mathbb{R}$ a smooth potential. Given $\lambda\in\mathbb{R}$, let $\psi_{\lambda}$ be the solution to
$$-\psi_{\lambda}''+\lambda V\psi_{\lambda}=0$$
with initial condition $\...
- 943
2
votes
1
answer
182
views
Proof of existence and uniqueness of solution to f(c)=0
I have a function $f:R^n_+\rightarrow R^n$ for which I want to show the following:
$$\exists c\in R^n_+ \quad \forall i,j\,\,f_i(c)=f_j(c)$$
Where $f_i (c)$ are the different coordinates of $f$.
$f$ ...
- 139
2
votes
1
answer
127
views
Variation of trace of symmetric powers
Consider the space $\mathrm{SU}(2)^\natural$ of conjugacy classes in $\mathrm{SU}(2)$. It has a natural identification with the interval $[0,\pi]$ with Haar measure $\frac{2}{\pi} \sin^2\theta\, \...
- 5,759
1
vote
1
answer
116
views
To what extent do integral moments determine a function?
Suppose that $f$ is a many-times integrable function on $[-1, 1]$. We can consider integral moments of $f$, given by
$$ I_n(f) := \int_{-1}^1 \big( f(x) \big)^n dx.$$
My question is: to what extent do ...
- 1,570
4
votes
1
answer
370
views
Convergence of a series
Let $F(z)=\displaystyle \sum_{k=0}^\infty a_kz^k,\;|z|<R $ and $F(R)=\displaystyle \sum_{k=0}^\infty a_kR^k$ (the series converges).
Assume that $F(\alpha_j)=0,\;j=1,2,\dots ,m$, where all $|\...
- 783
6
votes
2
answers
378
views
Slight variation on law of the iterated logarithm
Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability one,$$\...
user80013
12
votes
1
answer
1k
views
A generalization of intermediate value theorem on R^k
Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...
- 207
1
vote
0
answers
62
views
Regularity of a shrunken domain
I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner.
Let $\Omega\subset\Bbb R^d$ be an open bounded (...
- 1,101
2
votes
0
answers
69
views
Extension of a $\delta$-subharmonic function that is subharmonic on a reduced domain
Suppose $B$ is a ball in $\mathbb{R}^{m}$ and $u$ and $s$ are subharmonic on $B$. Suppose there is a closed subset $F$ of the closure of $B$ with no interior such that $v=u-s$ is subharmonic on $B\...
- 411
4
votes
3
answers
1k
views
Introductory texts to mathematics [closed]
I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would ...
0
votes
1
answer
135
views
Example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition
I'm looking for the example of a concave function $g \colon [0,1] \mapsto \mathbb{R}$, with $g(0)=0$, for which
$\lim\limits_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$, and
$\lim\limits_{x\to 0^+}\frac{\...
- 51
3
votes
2
answers
666
views
Brownian motion, quadratic variation, existence of partitions?
Let $B_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t_{k, n} : k = 0, 1, \dots, k_n\}$ $$0 = t_{0, n} < t_{1, n} < \dots < t_{k_n, n} = ...
- 33