All Questions
5,909 questions
14
votes
6
answers
6k
views
Russian Equivalent of Big Rudin
Is there any Russian-authored textbook on Analysis equivalent to Big Rudin (Real and Complex Analysis)?
I like Russian math textbooks a lot. I am looking for Russian textbooks (either in English or ...
- 91.8k
0
votes
1
answer
166
views
Can this result be proven by using only two (or maybe three) results listed below? [closed]
I wan to show that there is no continuous real-valued function of a real variable that sends rationals to irrationals and irrationals to rationals by using only $1)$ and $2)$:
$1)$ We can use the ...
- 4,280
1
vote
1
answer
258
views
How to prove or disprove a type of states form an overcomplete basis in the Hilbert space?
I am a PhD student in Physics. Let us consider a vector in an infinite dimensional Hilbert space as
\begin{equation}
|f\rangle\equiv
\begin{bmatrix}
1 \\
z \\
z^2 \\
\vdots
\end{...
- 17.2k
2
votes
1
answer
153
views
Function in $B(\mathbb{R})$
Denote by $B(\mathbb{R})$ the set of all functions on $\mathbb{R}$ which are representable in the form $f(x)=\int_{\mathbb{R}}e^{itx}d\mu(t)$, where $\mu$ is a finite complex-valued Borel measure.
...
- 778
3
votes
0
answers
115
views
First order linear ODE with some decay condition
In Kronheimer [1, p.183], a certain statement is made of which I extract the following special case.
Let $\alpha:\mathbb{R}\to \mathrm{Mat}(n\times n,\mathbb{C})$ be smooth and suppose that there ...
- 24.2k
1
vote
1
answer
121
views
A property of a nonlinear ODE under periodic boundary conditions
Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $$u_1(0) = u_1(1),u_1'(0)= u_1'(1)$$ and $$u_2(0) = u_2(1),u_2'(0)=u_2'(1)$$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \...
- 698
3
votes
2
answers
358
views
Order-preserving surjection ${\mathbb N}^{\mathbb N}\to [0,\infty)$
This is kind of a continuation of a recent (closed) question.
Is there an order-preserving surjective function $f:{\mathbb N}^{\mathbb N}\to [0,\infty)$ (where for $a,b\in {\mathbb N}^{\mathbb N}$ we ...
- 109k
0
votes
1
answer
59
views
Improved maximum principle estimates (deleting first mode)
Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write
$$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$
where $ r=|x|$ and $ \theta = \frac{...
- 17.2k
0
votes
1
answer
186
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 91.8k
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 91.8k
2
votes
0
answers
135
views
Can we get rid of this test function?
I have a real-valued function $f$ defined on a ball $B$ of $\mathbb{R}^{N}$, $N\geq2$. I have found a constant $M>0$ such that for all $x\in B$ and $B(x,R)$ (ball of center $x$ and radius $R>...
- 411
0
votes
0
answers
124
views
Reference for the Hardy maximal function on the torus
I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
- 3,425
3
votes
1
answer
186
views
packing with special sets in high dimensional Euclidean space
Let $\lambda$ be Lebesgue measure on $[0,1]$. For $\mathbf{x}=(x_1,x_2,..,x_k)\in[0,1]^k$, define $$A(\mathbf{x}):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[...
- 23.7k
2
votes
2
answers
446
views
Entrywise modulus matrix and the largest eigenvector
Disclaimer. This is a cross-post from math.SE where I asked a variant of this question two days ago which has been positively received but not has not received any answers.
Let $A$ be a complex ...
- 4,361
4
votes
2
answers
228
views
lower bound volume of a set
Let $\lambda$ be Lebesgue measure on [0,1]. For any $x_{1},\dots,x_{k}$ in $[0,1]$, define $$A(x_1,..,x_k):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[0,1]$$ ...
CommunityBot
- 1
2
votes
1
answer
289
views
Laplacian dissipative?
is it true that the Laplacian $\Delta:=\frac{d^2}{dx^2}$ on $(0,1)$ with Neumann boundary conditions is dissipative on $C[0,1]?$
For this we have to show that there is for any $x \in D(\Delta)$a $x' \...
- 4,729
4
votes
1
answer
367
views
Dissipative operator on Banach spaces
An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$
$$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$
On a Hilbert space this is ...
- 54.2k
4
votes
1
answer
445
views
Calderon-Zygmund theorem for the kernel of spherical harmonics
I don't want to write precisely the formulation of the Calderon-Zygmund theorem for singular integrals. The details are not so important here.
So I consider the operator $T$ given by the following ...
- 17.2k
7
votes
3
answers
390
views
Bounds on polynomial values
Assume $f(x)\in\Bbb{R}[x]$ is a polynomial of degree $n$.
Question. If $\int_{-1}^1f^2(x)\,dx=1$, is it true that
$$\vert f(x)\vert\leq \frac1{\sqrt2}(n+1), \qquad \text{for $\vert x\vert\leq1$}\,\,\...
- 4,034
3
votes
2
answers
129
views
ODEs whose finite-time solutions are not L^2 on their interval of definition
Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be analytic and consider the ODE
$$x'(t)=f(x(t)).$$
It is well-known that if $(t_{min},t_{max})$ is the maximal domain of a solution $x$ and $t_{max}<\infty$, ...
- 3,085
-2
votes
1
answer
209
views
Modulus of continuity an exponential type function [closed]
Fixed $0<a<1$, define $f(x):=(1-x)^{a}$ for every $x\in [0,1]$. Recalling that the modulus of continuity of $f$ of order $\varepsilon$ is given by
$\omega(f,\varepsilon):=\sup\{|f(x)-f(y)|:|x-y|\...
- 128k
2
votes
1
answer
1k
views
Proof of Agmon's inequality in $\mathbb{R}^3$
According to Wikipedia, Agmon's inequality provides a bound on the $L^\infty$ norm of a $H^2$ function on a (regular) subset of $\mathbb{R}^3$. In the book of JC Robinson et al. "The Three-...
CommunityBot
- 1
1
vote
2
answers
257
views
Pairwise disjoint subsets of $\mathbb{N}$ with positive upper density
For $A\subseteq\mathbb{N}$ we define the upper density to be $$\text{ud}(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$
Is there an infinite set ${\cal S}$ of pairwise disjoint subsets ...
- 13.4k
1
vote
1
answer
184
views
Meeting a set of spheres in $\mathbb{R}^n$
Let $n \geq 1$ be an integer. Let us call $S\subseteq \mathbb{R}^{n+1}$ an $n$-sphere if there is $x\in \mathbb{R}^{n+1}$ and $r\in \mathbb{R}$ with $r>0$ such that $$S = \{z\in \mathbb{R}^{n+1}: \|...
- 4,713
3
votes
2
answers
303
views
Basic question related to Stieltjes integral
I am reading this paper. I am stuck on something, which I think is something basic but I haven't been able to figure it out yet, and I was hoping someone could explain it to me.
Let
$$
\sigma(u) = \...
- 24.2k
15
votes
0
answers
409
views
Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?
Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
- 42k
4
votes
1
answer
315
views
What is the number of representations of a real number?
Let $f:\omega\to\mathbb N$ be a function such that $\sum_{n=0}^\infty\frac{f(n)}{2^n}<\infty$.
We identify each natural number $n\in\mathbb N$ with the set $\{0,\dots,n-1\}$.
Then the map $$\...
- 56.9k
3
votes
1
answer
334
views
At what point does exponential integral coincide with exponential?
I'm looking for the solution to the equation $E_1(x)=e^{-x}$, where $E_1(x)$ is the exponential integral function. The solution is approximately 0.434818204, but is this constant well known/studied?
...
CommunityBot
- 1
2
votes
0
answers
79
views
One-dimensional integral equation uniquely solvable?
I recently met a question similar to this one and I would like to post it here, because I basically found nothing:
We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
- 473
2
votes
0
answers
142
views
Self-adjointness on Banach spaces
Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded ...
- 501
1
vote
1
answer
248
views
Expected value of maximal displacement in permutations of $\{1,\ldots,n\}$
For any positive integer $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijections) $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For any $\pi\in S_n$ we let the maximal displacement be ...
- 4,529
11
votes
2
answers
1k
views
Concentration compactness. Can this concept be stated in a theorem?
I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.
When I approached the speaker ...
- 27.5k
0
votes
0
answers
311
views
Approximations of Polylogarithm and Lerch transcendent?
For the Gamma function $\Gamma(x+1)$, we have beautiful approximations of the function in terms of elementary function, such as the Stirling approximation and its refinements, that give sharp ...
- 1,324
3
votes
1
answer
188
views
Equivalent Definitions of the Gaussian Surface Measure for Regular Sets
I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
- 17.2k
3
votes
1
answer
148
views
Prove existence of continuous function on $(0,1)$ with special properties [closed]
Consider the interval $I=(0,1)$ and let $f,g$ be two linearly independent continuous functions on $[0,1]$.
I am asking if there is a continuous function $h$ such that
$$\int_0^1 h(s) f(s) ds=0$$
$$...
- 18.7k
1
vote
1
answer
204
views
Why study the moment problem in one dimensional case( Hamburger moment problem)
I have been reading about moment problem and I have been curious about the following question.
What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?
I ...
- 54.2k
0
votes
1
answer
165
views
Is there an example of a one to one and onto mapping between these two spaces?
Let $\Omega$ be a convex open subset of $\mathbb{R}^d$ with a smooth boundary. Is there an example of a one to one and onto mapping of the form $$L^{d+1}(\Omega) \to W^{1,d+1}(\Omega)$$
- 2,834
8
votes
8
answers
6k
views
Is Riemannian integration sufficient in physics?
Are there any applications in physics or engineering which require the Lebesgue integral and cannot be treated by Riemannian integration
- 1
8
votes
4
answers
338
views
Scaling a set of reals to be nearly integers
A version of this question was previously asked on MSE. I'll mention progress below.
A geometric construction I'm exploring
leads to a set $R$ of $n$ positive real numbers, for example:
$$
R = \{ \pi,...
- 32.5k
7
votes
0
answers
550
views
Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
7
votes
0
answers
219
views
Results that are easier in a metric space
Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces?
In particular, I'm ...
- 10.1k
10
votes
2
answers
1k
views
Harmonic oscillator discrete spectrum
Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator
$$-\frac{d^2}{dx^2}+x^2$$
explicitly.
Is there an abstract argument why the ...
- 24.2k
2
votes
1
answer
140
views
An inequality about embedding of cube into metric spaces
A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An edge of a cube is a pair of points $\{\psi(\epsilon_1),\psi(\epsilon_2)\}$ in $X$ such that $\epsilon_1$ and $\epsilon_2 $ differ in ...
9
votes
2
answers
244
views
If normal with respect to prime base then normal for all bases
I tried to find it on internet but couldn't so m asking this here. I want to ask if a number is normal with respect to all prime number base then do we know that it is normal with respect to any base. ...
- 4,713
1
vote
1
answer
335
views
Orthonormal basis and decay
Edit: I added smoothness, hoping to simplify the problem with this additional assumption.
Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
- 17.2k
4
votes
2
answers
767
views
Possible subsets of reals that equal the set of continuity of a function
This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of this past question, though it was motivated mainly by ...
- 32.1k
1
vote
0
answers
88
views
How to show this function is increasing? [closed]
Define function
$$f(\alpha)=\frac{-(4-5\alpha-4\alpha^{2})+\sqrt{32\alpha^{3}+\alpha^{2}-40\alpha+16}}{(1-2\alpha)\alpha}$$
where $\alpha \in [0,\frac{1}{2}]$
I can verify that:
(1) $f(\alpha)\geq 0$...
- 121
22
votes
1
answer
5k
views
Are functions of bounded variation a.e. differentiable?
In general, it is well known that, on the real line, say on $[0,1]$, if a function $f$ is of (pointwise) bounded variation, meaning that
$$
\sum_{i=1}^n |f(x_i)-f(x_{i-1})| <+\infty
$$
for every ...
CommunityBot
- 1
8
votes
3
answers
446
views
How to get this integral's asymptotics?
Consider the following integral
$$
\int_0^{\infty}\frac{e^{-x}-1}{x^{2+\frac{A}{\log b-5/6}}}\frac{1}{\log(b/x)-i\pi/2}\,dx
$$
where $A>0$ and $b>0$. I am interested in the small $b$ asymptotics ...
- 128k
2
votes
0
answers
70
views
The $n$-th derivative for $z$ at $z = 0$ of $F(a,b)(z)$ is no less than that of $F(\frac{a+b}{2},\frac{a+b}{2})(z)$ [closed]
The proposition the OP wants to prove is incorrect. --Aug 8, 2017
Can we find a elegant way to prove that the $n$-th derivative for $z$ at $z = 0$ of $F(a,b)(z)$ is no less than that of $F(\frac{a+...
- 1,551