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2 votes
0 answers
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On the second order analog of the upper 1-Lipschitz envelope of a function

Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope $$ \hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
Castoro Moro's user avatar
102 votes
21 answers
15k views

Proofs of the uncountability of the reals

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem (Wayback Machine). At first, I was excited to see a variant proof (as it did not ...
Unknown's user avatar
  • 2,855
2 votes
0 answers
135 views

Estimating an integral of the Green function in the plane

Suppose $\Omega$ is a bounded, simply connected domain, $z_{0}\in{\Omega}$ and for any $z\in{\Omega}$, $d_{z}:=\text{dist}(z,\partial{\Omega})$. I am interested in understanding the behavior of ...
David Pechersky's user avatar
2 votes
1 answer
146 views

Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane

Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral: $$I(v) := \int_0^1 \det(v(t),v'(t))dt$$ tells us about $v$, where $\det(v(t)...
stupid_question_bot's user avatar
1 vote
1 answer
79 views

PDF of the difference of two Beta Prime distribution

I am struggling to find the PDF of the difference of two Beta Prime distribution. Definition A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...
NancyBoy's user avatar
  • 393
0 votes
0 answers
56 views

How explicit the optimiser of this optimisation problem can be?

Provided the given parameters as follows : $\mu\in\mathbb R, \sigma\in\mathbb R_+$ are constant, $\kappa, r, \alpha, \beta: \mathbb R_+\to\mathbb R_+ $ are measurable functions such that $\kappa(y)\...
GJC20's user avatar
  • 1,334
238 votes
10 answers
43k views

If $f$ is infinitely differentiable then $f$ coincides with a polynomial

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ ...
C.S.'s user avatar
  • 4,795
0 votes
2 answers
531 views

Any idea of solving an optimization problem with cubic constraints?

I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem? $$ \begin{array}{ll} \underset {y, z} {\...
Erik's user avatar
  • 21
2 votes
1 answer
272 views

Decompose a function into a bounded part and a Lipschitz part

Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that $$ \sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty. $$ Are there functions $g,h: \mathbb R^d \...
Akira's user avatar
  • 835
2 votes
1 answer
154 views

Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$

Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that $$ f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
Akira's user avatar
  • 835
11 votes
2 answers
1k views

Twice continuously differentiable implied by existence of limit

I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that $$ \frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x) $$ for all $x\in X$ when ...
Sonam Idowu's user avatar
10 votes
1 answer
817 views

Can a nowhere locally Hölder function be differentiable almost everywhere?

Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$-Hölder continuous - that is, it is not $\alpha$-Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...
Nate River's user avatar
  • 6,215
13 votes
4 answers
2k views

Is there an increasing function on $[a, b]$ which is differentiable, but not absolutely continuous?

Is there an increasing function on $[a, b]$ which is differentiable, but not absolutely continuous?
LMP's user avatar
  • 577
2 votes
1 answer
183 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
Kanydo Mat's user avatar
2 votes
1 answer
320 views

Fourier series but different waveform

Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
Zhang Yuhan's user avatar
2 votes
2 answers
365 views

Is there a compactly supported differentiable function whose Fourier transform is not in L1?

In my MSE answer here, I discussed the example of compactly supported continuous function $$g(x)= \begin{cases} \dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\ 0,&\text{otherwise} \end{cases}$$ ...
D.R.'s user avatar
  • 833
4 votes
1 answer
279 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
António Borges Santos's user avatar
6 votes
0 answers
431 views

How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
Jorge Zuniga's user avatar
  • 2,836
3 votes
0 answers
68 views

How powerful are sequences of Steiner symmetrizations?

I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
cnikbesku's user avatar
  • 171
8 votes
1 answer
594 views

What is the minimum of this functional?

Recently I encountered an inequality from mathematical analysis. Let $f(x)$ be twice continuously differentiable in $[0,1]$ with $f(0)=f(1)=0$, then for all $x\in(0,1),f(x)\neq 0$, show that:$$\int_{0}...
Fate Lie's user avatar
  • 505
6 votes
1 answer
289 views

Archimedean ordered fields without maxima and minima in constructive mathematics

In constructive mathematics, let us define an ordered (Heyting) field $F$ to be a commutative ring with a binary relation $<$ which is irreflexive, where for all $x$, $\neg (x < x)$ asymmetric, ...
Madeleine Birchfield's user avatar
1 vote
0 answers
82 views

Counting the number of local minima of a function that is the sum of square roots of cosines

Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows $$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$ where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
loizuf's user avatar
  • 21
0 votes
1 answer
127 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
5 votes
2 answers
422 views

$C^1$ harmonic functions on a dense open set are globally harmonic

In a paper I am studying, at a certain point the authors introduce a function $u\in C^1(B_1,\mathbb{R})$ which is harmonic in a dense open subset $U$ of $B_1$. From this, they seem to conclude that $u$...
No-one's user avatar
  • 1,149
0 votes
0 answers
86 views

Solve equation with three square roots

I am trying to solve a more general question and I have the following subproblem: Find $x>0$ that satisfies for fixed $ i \geq 3$, $$\left(1 + \frac{1}{b^2}\right) x = \frac{\sum_i a_i^2} {b^2} + \...
Margot.'s user avatar
  • 49
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
0 votes
1 answer
117 views

How to understand the unique continuation result

Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm $$ \|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}. $$ Suppose $K(x) \in C^1\left(\mathbf{R}^...
Davidi Cone's user avatar
0 votes
1 answer
235 views

Does this property implies Lipschitz continuity?

Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be such that, for $x,y,z \in \mathbb{R}^{n}$, we have that $$|f(z) - f(x)| \leq |f(z) - f(y)| \Rightarrow \|z-x\| \leq \|z-y\|$$ Can I say that this ...
aureliano_buendia's user avatar
0 votes
2 answers
159 views

Cauchy's functional multiplicative equation on the unit interval

This question might be trivial, but I didn't find a clean reference and have not attempted to prove it myself yet: Let $f:[0,1]\rightarrow [0,1]$ be a continuous and monotonic function such that $f(0)=...
Pedram's user avatar
  • 97
0 votes
0 answers
63 views

Arrangements of fixed $k$-polyplets in a $n\times n$ matrix

Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
Cardstdani's user avatar
0 votes
0 answers
48 views

First nonzero derivative bounded below (2 dimensions)

Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real-analytic, have only one zero (at $(0,0)$) and be strictly increasing ...
Doofenshmert's user avatar
6 votes
2 answers
333 views

Attainment of maximum

A basic result in real analysis is that a continuous function $f:[0,1]\rightarrow \mathbb{R}$ attains its maximum on $[0,1]$, i.e. there is $x\in [0,1]$ such that $f(x)=\sup_{y\in [0,1]} f(y)$. A ...
Sam Sanders's user avatar
  • 4,359
0 votes
0 answers
71 views

Minimum Slice of Real Analytic Function in Two Variables

Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and have only one zero, namely $(0,0)$. Moreover, assume that $...
Doofenshmert's user avatar
0 votes
0 answers
95 views

Functions representing all strings somewhere

Do there exist "nice" (maybe analytic?) functions $f_0,f_1:\mathbb R \to \mathbb R$ such that $\forall n\in\mathbb N,\forall \sigma\in\{0,1\}^n,\exists x\in\mathbb R, \forall \tau\in\{0,1\}^...
Bjørn Kjos-Hanssen's user avatar
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
141 votes
17 answers
38k views

Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/...
vonjd's user avatar
  • 5,935
10 votes
1 answer
571 views

Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let $$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$ and $$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$ It ...
Fergns Qian's user avatar
6 votes
1 answer
392 views

How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]

I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
Euler-Masceroni's user avatar
0 votes
1 answer
102 views

On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
Ali's user avatar
  • 4,145
2 votes
0 answers
75 views

Regularity of solutions to an elliptic boundary value problem

Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
Laithy's user avatar
  • 969
2 votes
1 answer
154 views

Are these two norms on localized versions of $L^p_q$ equivalent?

$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$. Let $E$ be the space of all real-valued ...
Akira's user avatar
  • 835
1 vote
0 answers
67 views

Distribution of zeros for arbitrary Bessel functions

Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
Literally an Orange's user avatar
2 votes
1 answer
112 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
Ali's user avatar
  • 4,145
1 vote
1 answer
133 views

A question about the maximal function

Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
Xin Qian's user avatar
  • 155
1 vote
0 answers
95 views

Distance between two convex sets

Setting If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive. In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
Anthony's user avatar
  • 125
2 votes
0 answers
43 views

Good Polynomial lower estimates for beta function

I'm looking for polynomial lower estimates for beta function, and what I've found so far is this, which can be found in proposition 2.3 in this paper Proposition 2.3 1. If $0<𝑞<1$ and $𝑝 \geq ...
Ilovemath's user avatar
  • 677
0 votes
0 answers
136 views

Antiderivatives via Taylor series and the FT of Calculus

If $f$ is a real function on an interval $[a,b]$ such that $f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
Joe Shipman's user avatar
3 votes
2 answers
613 views

A problem about how dominated convergence is used in the analysis of variation

I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
Elio Li's user avatar
  • 809
0 votes
0 answers
21 views

Unimodality of distribution from Lévy symbol

Also posted in MSE. Assume that one want to study a distribution $f$ on $\mathbb{R}$ for which the Lévy symboln, i.e.: $$ \forall u\in\mathbb{R},\quad\psi(u) := \log \mathbb{E}\left[e^{iuX}\right] $$ ...
NancyBoy's user avatar
  • 393
1 vote
2 answers
90 views

Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?

Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
Akira's user avatar
  • 835

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