All Questions
6,016 questions
15
votes
3
answers
2k
views
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I've been trying to find an asymptotic expansion of the following series
$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$
and
$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
- 153
15
votes
1
answer
1k
views
Characterizing $\mathbf{R}$ as an ordered group
A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ...
- 18.7k
15
votes
2
answers
1k
views
Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?
I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...
- 181
15
votes
1
answer
2k
views
Real polynomials that go to infinity in all directions: how fast do they grow?
Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
- 26.9k
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 536
15
votes
1
answer
1k
views
Existence of double eigenvalue
Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent.
Must there exist a nonzero real linear combination $aA + bB$ which has a repeated ...
- 42.8k
15
votes
4
answers
1k
views
Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...
- 41.1k
15
votes
2
answers
680
views
Are Fourier transforms of L^p stable under diffeomorphisms?
Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
- 2,639
15
votes
1
answer
595
views
Solving a non linear equation
I've been trying to prove that the following equation has a unique solution in interval 0 < x < 1 :
$$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$
Where K is a number (integer, if it ...
- 159
15
votes
2
answers
6k
views
Linearly constrained eigenvalue problem
Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& \...
- 723
15
votes
2
answers
1k
views
If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?
Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. We say $x \in \mathbb R^n$ is a strong Lebesgue point of $f$ if
$$\lim_{r \to 0} \frac{\int_{B_r (x)} |f(y) - f(x)| \, dy}{r^{n+1+\...
- 6,155
15
votes
2
answers
620
views
Maximum dimension of space of matrices with a real eigenvalue
Let $M_n(\mathbb{R})$ denote the space of all $n\times n$ real
matrices. What is the maximum dimension $f(n)$ of a subspace $V$ of
$M_n(\mathbb{R})$ such that every matrix in $V$ has at least one real
...
- 50.8k
15
votes
1
answer
1k
views
Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$
The following question has a 500 points bounty on MSE that soon comes to an end, and no answer
as expected was given yet. How would a professional solve the problem? Wish you succcess.
https://math....
- 253
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
15
votes
1
answer
904
views
Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?
Willie Wong asked here (MO) and here (MSE) very interesting question.
As he phrased it:
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ ...
- 225
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
15
votes
1
answer
764
views
Does there exist a nowhere smooth function, that has arbitrary many derivatives?
I'm sorry if my title sounds misleading, I don't know a better way to word my question briefly. But I have the following question about functions.
First, as long as $A$ is a dense subset of $\mathbb{R}...
- 700
15
votes
1
answer
649
views
On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
- 178
15
votes
2
answers
530
views
Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets
Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets.
Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
- 807
15
votes
2
answers
1k
views
Converse of mean value theorem
Note: This is an attempt to narrow down conditions under which the conjecture stated in this previous post is true. As stated, it is false as shown by the counterexample provided in the answers by the ...
- 6,155
15
votes
3
answers
903
views
Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $
I am trying to prove or disprove
$$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$
where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
- 5,474
15
votes
1
answer
2k
views
Interpolating between piecewise linear functions, with a family of smooth functions
Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that:
there is a number $k\in\mathbb N-\{0\}$ ...
- 4,312
15
votes
3
answers
2k
views
Can the Riemann integral be defined through a closure/completion process?
Let us consider real-valued functions on the bounded interval $[0,1]$. A "step function" means an element of the vector space spanned by indicator functions of (points and) intervals in $[0,1]$ (the ...
- 32.5k
15
votes
2
answers
2k
views
Where does the Lebesgue differentiation theorem fail?
The Lebesgue differentiation theorem says that for certain metric spaces $X$ (see below), any Borel measure $\mu$ that is finite on bounded sets and any $f: X \rightarrow \mathbb{R}$ locally $\mu$-...
- 1,368
15
votes
1
answer
602
views
Integrability property of polynomials in several variables
This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
- 151
15
votes
0
answers
244
views
Natural examples of Borel surjections without right inverse
As discussed in this question, in general a Borel surjection $f:\mathbb{R}\rightarrow\mathbb{R}$ may not have a Borel right inverse, namely a $g$ such that $f\circ g=id$, although there is always a ...
- 623
15
votes
0
answers
477
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
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 $\mathbb R^\omega$ (...
- 41.8k
15
votes
0
answers
510
views
Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
- 6,541
14
votes
3
answers
2k
views
How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...
- 1,326
14
votes
3
answers
2k
views
Every function on reals a sum of two surjective real functions?
From this question, and the answer thereof, we can see that every real valued function on reals is a sum of two injective functions. Is the same true if we replace injectivity by surjectivity.
For ...
- 2,089
14
votes
3
answers
878
views
Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$
Let $a_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a_i\neq a_j$.
I have the following theorem:
For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which $[a_k,...
user79942
14
votes
2
answers
2k
views
Generalisation of Cauchy's mean value theorem
I apologise in advance if this is an elementary question more fitted for Math Stack Exchange. The reason why I have decided to post here is that the question I am used to seeing on that site are not ...
- 413
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 ...
- 149
14
votes
6
answers
3k
views
What's a natural candidate for an analytic function that interpolates the tower function?
I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
- 4,466
14
votes
2
answers
871
views
Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?
Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has fixed distinct singular values $0<\sigma_1<\sigma_2$ and an everywhere positive determinant (which is the product $\...
- 6,741
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 43.2k
14
votes
2
answers
807
views
Integral of power of binomials equal to sum of power of binomials?
Inspired by this MO question about integrating binomial coefficients and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And ...
- 4,014
14
votes
3
answers
1k
views
Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?
Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set
$$
\operatorname{perim}(S)=\| \nabla 1_S\|_{TV}
$$
...
- 1,247
14
votes
1
answer
974
views
Positive roots of a polynomial
Let $a_i>0$, $i=1,\dots,n$, and put $\overline{a}:=\frac{1}{n}\sum_{i=1}^n a_i$. Assuming not all $a_i$'s are equal, take
$$
p(x):=\sum_{i=1}^n a_i (a_i-\overline{a})\prod_{k=1,\dots,n\;k\neq i} (x+...
- 959
14
votes
2
answers
2k
views
Is the composition of two nowhere differentiable functions still nowhere differentiable?
Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has
$$
\limsup\limits_{x\to x_0}\...
- 938
14
votes
3
answers
547
views
Recognizing Lipschitz functions up to change of target metric
Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.
Is ...
- 143
14
votes
1
answer
655
views
Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure
Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows:
$$S:=\{t\in [0,\infty):nt\...
- 632
14
votes
1
answer
401
views
Conjecture: Finitely many points where gravitational field due to N masses vanishes
Given a configuration $C$ of $N$ distinct fixed points of equal mass in the plane (eventually in space), let $f_C(N)$ denote the number of points $P$ for which the gravitational field at $P$ vanishes. ...
- 147
14
votes
2
answers
540
views
Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
$M(M(a,b),M(a,c))=...
- 5,103
14
votes
1
answer
481
views
A question on a real sequence
Let $\{a_n\}_{n\ge1}$ be a real sequence that decays faster than any algebraic speed, that is, $\lim_{n\to \infty} n^pa_n = 0$ for every positive integer $p$. Assume that $$\sum_{n\ge 1}(n+1)^kn^ka_n =...
- 903
14
votes
1
answer
900
views
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
Volterra ...
- 4,897
14
votes
2
answers
2k
views
Is this property equivalent to Lusin's property (N) for continuous functions?
A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...
- 1,601
14
votes
1
answer
918
views
Was Cantor aware of Lebesgue theory of integration?
Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
- 747