All Questions
5,910 questions
13
votes
1
answer
1k
views
Is there an algebra for divergent series summation operators?
Let $D$ denote a divergent series and let $C$ denote a convergent series.
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
CommunityBot
- 1
0
votes
1
answer
693
views
the double dual of "little l one" sequence space
I remember a professor remarking a while back that the double dual of the sequence space $l_1^{\infty}(\mathbb{R})$ is a very complicated space. I understand it must contain a copy of the original ...
CommunityBot
- 1
68
votes
2
answers
2k
views
Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
CommunityBot
- 1
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
CommunityBot
- 1
4
votes
1
answer
251
views
Superadditivity of the lower density
Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) $\mu^\star(\...
CommunityBot
- 1
5
votes
1
answer
320
views
Two Concepts of Monotonicity
Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that
$F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that
$$
\langle F(y)-F(x), y-x\...
CommunityBot
- 1
3
votes
2
answers
432
views
A logarithmic cotangent inequality
I must be a terrible googling searcher but I cannot find a reference to the following inequality:
$$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$
I have just ...
- 7,500
2
votes
1
answer
101
views
Convergence of energy of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by
\begin{equation}
f_r(x):=
\begin{cases}
...
- 24.2k
1
vote
2
answers
183
views
Convergence of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$.
Question 1: Can we ...
CommunityBot
- 1
51
votes
6
answers
2k
views
Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?
Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that
$$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$
for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...
CommunityBot
- 1
7
votes
0
answers
628
views
Proving Richardson's theorem for constants
(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
CommunityBot
- 1
8
votes
1
answer
458
views
An interesting integration
For any positive integer $n$, let
$$A_n=\idotsint\limits_{\substack{x_1+\cdots+x_n+y_1+\cdots+y_n\leq1\\x_1,\cdots,x_n,y_1,\cdots,y_n\geq0}}\prod_{i,j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots dy_n.$$
It ...
CommunityBot
- 1
12
votes
1
answer
448
views
An interesting inequality
Let $\mathbb{R}$ be the real field. For any homogeneous polynomial $f(X_1,\cdots,X_n)$ in $\mathbb{R}[X_1,\cdots,X_n]$, we use $S_f(X_1,\cdots,X_n)$ to denote the following homogeneous symmetric ...
CommunityBot
- 1
11
votes
2
answers
595
views
When does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$
On a (simply connected) domain $\Omega$ for a smooth vector field $F\colon \Omega \to \mathbb{R}^3$, when does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$. I know that $n\cdot(\nabla\...
- 156k
0
votes
0
answers
322
views
Comparison of Parameter estimation using maximum likelihood and Maximum entropy
I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
- 495
5
votes
1
answer
481
views
A continuous path between two Sobolev functions
Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...
- 679
21
votes
1
answer
1k
views
If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity?
This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, ...
- 156k
2
votes
0
answers
65
views
Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$
Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below
$$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$
$$G = ...
- 123
3
votes
1
answer
68
views
Determine a sign of the limitation of a certain integral
I can't determine a sign of an integral written below and it has hit a dead end.
My setting is rather special.
Let $a\in(0,1)$ be a given constant and $(x_{\varepsilon},y_{\varepsilon})\in[0,a)\times[...
- 15.8k
12
votes
1
answer
5k
views
Points of continuity of Baire class one functions
This is an idle question motivated by two comments I made to a previous MO question (which I just searched for, unsuccessfully). That question asked if the characteristic function of the rationals is ...
- 3
5
votes
0
answers
216
views
Existence or construction of a sequence of orthogonal matrices with three properties
This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...
- 63.9k
5
votes
1
answer
181
views
Continuity of solutions to $Av=b$
Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of complex-valued, continuous functions on $X$. Let $A$ be a matrix with entries from $C(X)$ of size $m\times n$ and $b\in \...
- 109k
23
votes
0
answers
939
views
A question about small sets of reals
In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is ...
- 3,104
1
vote
0
answers
158
views
On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $
This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $...
- 1,396
2
votes
0
answers
421
views
Derivatives of Minkowski function?
Let $A\subset \mathbb R^n$ and $M$ be the convex hull of the set $A$, e.g., $M:=Conv(A)$. The Minkowski function on $M$ is defined as follows
\begin{align*}
&f: \mathbb R^n \to \mathbb R\\
&f(...
- 52.3k
0
votes
3
answers
125
views
Bounds on derivative of integrable, monotonically decreasing, differentiable functions on $\mathbb R_+$
The following three conditions have shown up as hypotheses in some recent work, and despite not having been able to find an example, we assume the third is not implied by the former two. We're hoping ...
- 54.2k
2
votes
0
answers
108
views
Distributive law
I was wondering whether there is any reference that deals with the distributive law for infinitely many elements, i.e.
$$
\prod_{i\in \mathbb N} \sum_{k\in \mathbb N} \alpha_{i,k} = \sum_{(k_i)_{i\in ...
10
votes
2
answers
2k
views
A result attributed to Whitney
One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...
- 52.3k
12
votes
3
answers
440
views
Is a certain subset of the disc a convex set?
Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that?
Draw a Cantor set $C$ on the circle ...
CommunityBot
- 1
3
votes
1
answer
355
views
convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?
Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
CommunityBot
- 1
-1
votes
1
answer
226
views
separable BV space for PDE's, Whats stopping us? [closed]
Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
- 16.2k
1
vote
2
answers
270
views
Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]
In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...
CommunityBot
- 1
1
vote
0
answers
157
views
Laplace method with "bad" zero set
It is well-known that if $\phi:\mathbb{R}^n \longrightarrow \mathbb{R}$ is a function with $\phi(0) = 0$, $\phi(x)>0$ if $x \neq 0$ and $D^2\phi(0) \geq 0$, then the integral
$$\int_{\mathbb{R}^n} ...
- 9,853
-2
votes
1
answer
80
views
Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [closed]
Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative $\lim_{\epsilon->0}\frac{h(x+\epsilon)-2h(x)+h(x-\epsilon)}{\epsilon^...
- 19.3k
0
votes
0
answers
471
views
Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
4
votes
0
answers
500
views
Properties of the solution of the heat equation
Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...
CommunityBot
- 1
6
votes
0
answers
396
views
Recurrence Formula for Zernike polynomials
I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...
- 8,597
0
votes
1
answer
482
views
Complement of a finite union of convex sets
Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...
- 31.2k
0
votes
1
answer
169
views
An increasing sequence of real numbers [closed]
This was first posted to SE, but now I think its better to be posted here.
For what positive real numbers $\alpha$, the sequence $a_n = \frac{\lfloor n\alpha\rfloor}n $ is (not necessary strictly) ...
7
votes
1
answer
226
views
Unusual isoperimetry and maximizing the measure of unions of translates of a set
Let me state a standard result first. Let a $A\subset \mathbb{R}^d$ be a set of fixed volume. Define $A_t$ to be the set of all points at distance at most $t$ from $A$. Then the volume of $A_t$ is ...
CommunityBot
- 1
1
vote
1
answer
195
views
A metric on the set of BV functions, is it mentioned/studied in literature?
I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$.
Given any $x,y \in ...
CommunityBot
- 1
0
votes
1
answer
319
views
Double Integral Equations
In my research I've come across a handful of double integral equations, and I'm nearly at a total loss for how to derive anything useful from such things.
I've been lead to believe that even single ...
- 54.2k
3
votes
0
answers
375
views
An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only
The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
- 128k
7
votes
2
answers
1k
views
Intermediate value for a vector-valued function
Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f_1(x), ..., f_n(x)\}$ with $x\in[0,1]^n$, where the $f_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the ...
4
votes
1
answer
393
views
How can I show that "almost all function" have property P?
The following is cross-posted from
https://math.stackexchange.com/questions/1391293/is-almost-all-function-a-well-defined-concept
since I didn't (yet) get an answer there.
(I hope that's okay?)
...
CommunityBot
- 1
0
votes
1
answer
96
views
Optimal covering with finite subcollection of open sets
This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for ...
- 1,407
0
votes
0
answers
344
views
Beurling density $D(X)$ of $X=\{x_j\in\mathbb R, \ |x_j-x_{i}|>\gamma>0: \ i,j\in\mathbb Z\}$
Beurling density of set $X$ is defined (see, for example "Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces" by Aldroubi and Grochenig) as:
$$D(X)=\lim_{r\rightarrow \infty} \inf_{y\in\...
- 297
2
votes
2
answers
411
views
Asymptotics of the derivatives of analytic functions
Are there sources that treat questions like the following ones?
Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x\...
- 60.6k
4
votes
1
answer
875
views
comparing norms of tensor product of two Hilbert spaces
Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...
- 25.8k
2
votes
1
answer
572
views
Extension of Dynkin's formula, conclude that process is a martingale
This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
CommunityBot
- 1