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43 votes
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819 views

A kaleidoscopic coloring of the plane

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
Taras Banakh's user avatar
  • 41.8k
30 votes
0 answers
899 views

Three real polynomials

Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
Alexandre Eremenko's user avatar
28 votes
0 answers
1k views

Number of real roots of a polynomial

Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
Aleksei Kulikov's user avatar
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 ...
Ashutosh's user avatar
  • 9,631
21 votes
0 answers
658 views

A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}...
Zurab Silagadze's user avatar
21 votes
0 answers
1k views

Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$

A while ago, I came across the following problem, which I was not able to resolve one way or the other. Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
George Lowther's user avatar
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 ...
user avatar
19 votes
0 answers
775 views

A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions $$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$ Suppose we say that $...
Dmitry V's user avatar
  • 433
18 votes
0 answers
1k views

Does there exist a continuous open map from the closed annulus to the closed disk?

(Originally from MSE, but crossposted here upon suggestion from the comments) In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
D.R.'s user avatar
  • 831
18 votes
1 answer
2k views

Function of two sets intersection

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
pi66's user avatar
  • 1,209
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 ...
183orbco3's user avatar
  • 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 ...
Dor's user avatar
  • 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 $...
Tanya Vladi's user avatar
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 $\mathbb R^\omega$ (...
Taras Banakh's user avatar
  • 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 ...
Aryeh Kontorovich's user avatar
14 votes
0 answers
718 views

Lower bounds on analytic functions connected to Fox H

The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
Tanya Vladi's user avatar
14 votes
0 answers
633 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
Gil Kalai's user avatar
  • 24.7k
14 votes
0 answers
4k views

Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
didest's user avatar
  • 1,015
13 votes
0 answers
710 views

Minimizing total variation under constraint

For $p\in[0,1]$, we write $\mathrm{Ber}(p)$ to denote the Bernoulli measure on $\{0,1\}$; that is, $\mathrm{Ber}(p)(0)=1-p$, $\mathrm{Ber}(p)(1)=p$. For $n\in\mathbb{N}$ and $p=(p_1,\ldots,p_n)\in[0,1]...
Aryeh Kontorovich's user avatar
13 votes
0 answers
545 views

Is there a logical relationship between constructions of the reals and proof methods in real analysis?

In my elementary real analysis course three years ago, I remember noting that there seemed to be 3 main ways of proving the main theorems about continuity. There was Bolzano-Weierstrass, continuous ...
Oddly Asymmetric's user avatar
13 votes
0 answers
395 views

Converse to Riesz-Thorin Theorem

Let $T$ be an operator on simple functions on (say) $\mathbb{R}$. The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
Yonah Borns-Weil's user avatar
12 votes
0 answers
435 views

Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
Fred Dashiell's user avatar
11 votes
0 answers
374 views

A game of harmonic series(s)

Given a set $A\subseteq\mathbb{R}_{>0}$, consider the following (two-player, perfect-information, length-$\omega$) game $H_A$: Players $1$ and $2$ alternately play strictly increasing natural ...
Noah Schweber's user avatar
11 votes
0 answers
615 views

Is every Baire metric space a complete metric space in disguise?

I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here: Let's say that a metric space $X$ is Baire if every countable ...
fedja's user avatar
  • 61.9k
11 votes
0 answers
2k views

A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$ $$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
Jacob Lu's user avatar
  • 903
11 votes
0 answers
322 views

Does any real function have a Lipschitzian restriction on $D$?

Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
Dattier's user avatar
  • 4,074
11 votes
0 answers
381 views

Concerning Luzin-(N)-property

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
喻 良's user avatar
  • 4,201
11 votes
0 answers
320 views

Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...
Mason's user avatar
  • 211
11 votes
0 answers
137 views

Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...
Hatem's user avatar
  • 111
10 votes
0 answers
287 views

Coefficients of polynomials vs trigonometric product

Let's consider the family of sequences of coefficients in the expansion $$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$ Remark. Evidently, the RHS is a finite sum. Here is a ...
T. Amdeberhan's user avatar
10 votes
1 answer
518 views

Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions

Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
Tian LAN's user avatar
  • 435
10 votes
0 answers
268 views

On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$

I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well. For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
Lucio Tanzini's user avatar
10 votes
0 answers
172 views

Maximizing an integral w.r.t. a measure on the unit sphere

I would like to know if the answer to the following question is known. Let $d \ge 3$. What is the value of $$ \theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
Romeo's user avatar
  • 980
10 votes
0 answers
844 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
Torsten Schoeneberg's user avatar
10 votes
0 answers
722 views

Fractional Matching version of Hall's Marriage theorem

Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent: 1) there exist a perfect matching in $G$; 2) there exist non-negative weights on edges such that the sum of ...
Fedor Petrov's user avatar
10 votes
0 answers
315 views

Does antidifferentiability of continuous functions imply Dedekind completeness?

Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
James Propp's user avatar
  • 19.7k
10 votes
0 answers
439 views

Evaluating Shintani cone zeta functions

Hi everyone I am trying the evaluate sums of the form $$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$ ...
user3628's user avatar
  • 265
9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
pie's user avatar
  • 541
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 ...
SpectreDNZ's user avatar
9 votes
0 answers
522 views

Does the intersection of middle third and middle half Cantor sets contain an irrational number?

Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$. Likewise let $C_\frac{1}{2}$...
Dmitrii Korshunov's user avatar
9 votes
0 answers
165 views

Changing coordinate to smoothen a function

Let $U\subset \mathbb{R}^2$ be an open neighborhood of the origin $0$, and let $f:U\to \mathbb{R}$ be a continuous function which is smooth on $U\setminus\left\{0\right\}$. Let's say that $f$ is ...
user49822's user avatar
  • 2,178
9 votes
0 answers
347 views

Can one prove Rademacher’s theorem via the rising sun lemma?

The classical Rademacher’s theorem states that Lipschitz continuous functions on $\mathbb R^n$ are differentiable almost everywhere. In dimension one, a stronger result holds - it can be shown that ...
Nate River's user avatar
  • 6,155
9 votes
0 answers
509 views

On Riesz criteria for Riemann hypothesis:

Marcel Riesz defined a function : $R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$ The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$) For any $\varepsilon$ We have ...
TPC's user avatar
  • 784
9 votes
0 answers
180 views

Infinite series identities in search of a proof

This comes in relation to the Fishburn numbers. I stumbled on the following relation for which I ask a proof if true. Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then $$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
T. Amdeberhan's user avatar
9 votes
0 answers
569 views

A standard name for a function satisfying the intermediate value theorem?

Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property: $(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
Taras Banakh's user avatar
  • 41.8k
9 votes
0 answers
978 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
Mary's user avatar
  • 91
8 votes
0 answers
103 views

Sobolev embedding theorems in vector bundles on non-compact manifolds

Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
G. Blaickner's user avatar
  • 1,429
8 votes
0 answers
414 views

For $f$ Lipschitz with $|\nabla f| = 1$ a.e., what is the supremal Hausdorff dimension of the set on which $\varepsilon< |\nabla f| < 1-\varepsilon$?

Let $f$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere. Let $\varepsilon \geq 0$. What is the supremal Hausdorff dimension of the set on which $f$ is differentiable with $\varepsilon &...
Nate River's user avatar
  • 6,155
8 votes
1 answer
258 views

Sequential colimit of iterated quotients of Cauchy sequences

We work in constructive mathematics. The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
Madeleine Birchfield's user avatar
8 votes
0 answers
296 views

Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?

I have never seen a real-analytic approach to evaluate integrals of the form below $$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
TheSimpliFire's user avatar

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