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17 votes
2 answers
905 views

Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...
TaQ's user avatar
  • 3,584
17 votes
1 answer
1k views

Continuous functions of three variables as superpositions of two variable functions

Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's ...
KhashF's user avatar
  • 3,599
17 votes
2 answers
750 views

Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?

Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism? More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d&...
qp10's user avatar
  • 173
17 votes
3 answers
1k views

Decoupling a double integral

I came across this question while making some calculations. QUESTION. Can you find some transformation to "decouple" the double integral as follows? $$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
T. Amdeberhan's user avatar
17 votes
2 answers
1k views

Kolmogorov superposition for smooth functions

Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as $$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$ for ...
O.R.'s user avatar
  • 807
17 votes
2 answers
2k views

"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

Recently, I encountered this problem: "Given a sequence of positive number $(x_n)$ such that for all $n$, $$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$ Find the limit $\lim_{n \rightarrow \infty} \...
Paresseux Nguyen's user avatar
17 votes
3 answers
2k views

The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$

Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum: $$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$ ...
TOM's user avatar
  • 2,288
17 votes
3 answers
975 views

Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds

For real variable $x$, the function \begin{equation} f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n \end{equation} clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
Isaac's user avatar
  • 3,477
17 votes
2 answers
3k views

The Riemann hypothesis as a problem in analysis

The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the ...
Alex Gavrilov's user avatar
17 votes
1 answer
794 views

Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?

Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
Taras Banakh's user avatar
  • 41.8k
17 votes
1 answer
986 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
Mohammad Golshani's user avatar
17 votes
2 answers
2k views

Explicit and fast error bounds for polynomial approximation

Main Question This question is about finding explicit, calculable, and fast error bounds when approximating continuous functions with polynomials to a user-specified error tolerance. EDIT (Apr. 23): ...
Peter O.'s user avatar
  • 697
17 votes
1 answer
580 views

Aperiodic monotile in $\mathbb{R}$

Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
Dominic van der Zypen's user avatar
17 votes
4 answers
1k views

In choiceless constructivism: If $f'=0$ then is $f$ constant?

Prove, without any Choice principles or Excluded Middle, that if a pointwise differentiable function has derivative $0$ everywhere, then it is constant. The function in this case maps $\mathbb R$ to $\...
wlad's user avatar
  • 4,943
17 votes
1 answer
3k views

Integrals of pullbacks and the Inverse function theorem(s?)

The usual story goes like this: Smooth picture (?): For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following is true: $\phi^{-1}$ is a local diffeomorphism a.e. ...
Saal Hardali's user avatar
  • 7,789
17 votes
2 answers
4k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
Rajesh D's user avatar
  • 698
16 votes
6 answers
2k views

Alternative proofs sought after for a certain identity

Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so QUESTION. can you provide another verification for the problem below? Problem. Prove ...
T. Amdeberhan's user avatar
16 votes
3 answers
1k views

Can integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function $$ g:(a,b)\to\mathbb{R},\qquad ...
H. Berbeleque's user avatar
16 votes
2 answers
2k views

An analogue of the exponential function by replacing infinite series with improper integral

For every positive real number $x$ we define $$E(x)= \int_0^{\infty} x^t/t!\,\mathrm dt$$ where $t!=\Gamma(t+1)$. This is motivated by classical exponential function. Is this function well defined (...
Ali Taghavi's user avatar
16 votes
3 answers
4k views

Which functions have all derivatives everywhere positive?

Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$. The only examples I can ...
Will Sawin's user avatar
  • 148k
16 votes
7 answers
6k views

Understanding Gibbs's inequality

Short version Gibbs's inequality is a simple inequality for real numbers, usually understood information-theoretically. In the jargon, it states that for two probability measures on a finite set, ...
Tom Leinster's user avatar
  • 27.7k
16 votes
2 answers
1k views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
YuerWu's user avatar
  • 415
16 votes
3 answers
1k views

A kernel 'more analytic' than $\exp(-x^2)$

I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits_{k=0}^{\infty} |c_k| \varepsilon^k (2k)! < \...
Ben Deitmar's user avatar
  • 1,295
16 votes
4 answers
2k views

Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?

Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with $f_n \to f$ uniformly for some (necessarily) continuous $f$. $f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$. Is it true ...
Nate River's user avatar
  • 6,155
16 votes
1 answer
686 views

Fourier's proof of reality of all roots of Bessel function $J_0(x)$

In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real. I want to ask if there is a modern version of this proof exist in literature? If someone ...
TPC's user avatar
  • 784
16 votes
3 answers
1k views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
Bogdan's user avatar
  • 161
16 votes
4 answers
1k views

Proof of complete monotonicity of a binomial function

By plotting the function and its derivatives, one can easily be convinced that the function $$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and ...
MCH's user avatar
  • 1,324
16 votes
1 answer
3k views

Did Euler know (unconsciously) to integrate by differentiating?

Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based ...
Andreas Rüdinger's user avatar
16 votes
1 answer
888 views

Kakeya crossed-needles problem

The Kakeya needle problem asks for the minimum area planar region in which one can completely turn around a line segment through a series of translations and rotations. There is no minimum: There are &...
Joseph O'Rourke's user avatar
16 votes
2 answers
2k views

Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory

(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$: $$ \int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i} $$ We can generalize this ...
annie marie cœur's user avatar
16 votes
6 answers
3k views

A normal distribution inequality

Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
Hans's user avatar
  • 2,239
16 votes
2 answers
1k views

Is there always a way up?

I am trying to find a simple criterion for a real continuous function $f$ on a connected, open subset $U$ of $\mathbb R^n$ that would imply the following property (P) For any $x, y \in U$ such that $f(...
Pluviophile's user avatar
  • 1,608
16 votes
1 answer
487 views

Bull's-eye Riemann sum

Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane. Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$ In other ...
Jairo Bochi's user avatar
  • 2,479
16 votes
1 answer
2k views

Are continuous functions almost completely determined by their modulus of continuity?

Given a function $f: \mathbb{R}\to\mathbb{R}$, we define its left modulus of continuity, $L(f): \mathbb{R} \times (0, \infty)\to [0,\infty]$ by $$L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)) \...
James Baxter's user avatar
  • 2,069
16 votes
2 answers
528 views

Lipschitz constant for map between triangles

Let $T_1$ and $T_2$ be any two euclidean triangles with labeled sides. The sides are labeled respectively $e_1^1,e_2^1,e_3^1$ and $e_1^2,e_2^2,e_3^2$. Call $A:T_1\rightarrow T_2$ the affine map which ...
user avatar
16 votes
1 answer
661 views

Does every real function have this weak derivation property?

After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
Dattier's user avatar
  • 4,074
15 votes
4 answers
3k views

No Tonelli or Fubini

Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
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}}{...
Trax's user avatar
  • 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 ...
coudy's user avatar
  • 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 ...
Jonathan's user avatar
  • 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 ...
Stanley Yao Xiao's user avatar
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 \...
Sascha's user avatar
  • 536
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 ...
Gerald Edgar's user avatar
  • 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 ...
Rami's user avatar
  • 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 ...
freedome321's user avatar
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+\...
Nate River's user avatar
  • 6,155
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....
user 1357113's user avatar
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 ...
wonderich's user avatar
  • 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 $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ ...
Right's user avatar
  • 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 ...
Serge the Toaster's user avatar

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