All Questions
7,280 questions
64
votes
8
answers
6k
views
Two (probably) equal real numbers which are not proved to be equal?
Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal?
I never really understood the assertion that "the reals do not have decidable equality"...
- 39.9k
2
votes
1
answer
103
views
Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
- 281
2
votes
1
answer
246
views
Inequality with Hermite polynomials
Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by
$$\sqrt{\sqrt{\pi} 2^n n!}$$
for the purpose of normalization.
These are orthogonal with respect to the weight function $e^{...
- 4,034
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 ...
- 677
2
votes
0
answers
97
views
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
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
0
votes
0
answers
109
views
A Lipschitz function induced by the infimum of the length of curves
Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of ...
- 69
2
votes
1
answer
203
views
Does this maximisation problem admit a finite upper bound?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 61.9k
5
votes
1
answer
282
views
Is there a singular function that is Hölder continuous of every order less than $1$?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
Does there exist a singular function that is Hölder ...
- 10.6k
8
votes
3
answers
1k
views
Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
- 10.5k
5
votes
2
answers
297
views
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with
$f_n \to f$ uniformly for some continuous $f$.
$f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,
where we ...
- 2,504
6
votes
1
answer
309
views
Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $f \in C^1 (\mathbb \Omega)$ for some open, connected, bounded subset $\Omega$ of $\mathbb R^n$. We consider for each $t \...
2
votes
0
answers
259
views
Least number of circles required to cover a continuous function on $[a,b]$
I asked this question on MSE here.
Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of closed circles with fixed radius $r$ required to cover the graph of $f$?
It is ...
- 541
0
votes
0
answers
95
views
Functions representing all strings somewhere
Do there exist "nice" (maybe analytic?) functions $f_0,f_1:\mathbb R \to \mathbb R$ such that
$\forall n\in\mathbb N,\forall \sigma\in\{0,1\}^n,\exists x\in\mathbb R, \forall \tau\in\{0,1\}^...
- 24.8k
0
votes
0
answers
128
views
Lipschitz function approximated by smooth functions with zero a regular value
Consider a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$. Then I want a family of smooth functions $f_\epsilon : \mathbb{R}^n\to\mathbb{R}$, such that $f_\epsilon\to f$ uniformly on compact sets, ...
- 63.9k
0
votes
0
answers
120
views
Equality of two measures on functional spaces
It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
CommunityBot
- 1
6
votes
3
answers
256
views
Determine unknown matrix function of particular form from known points
I encountered the following problem recently in a practical context.
Fix $n \ge 1$.
Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form
$$ X \mapsto ...
CommunityBot
- 1
7
votes
3
answers
547
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 10.6k
2
votes
2
answers
191
views
Gronwall's inequality in discretized time
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 31
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 ...
- 171
1
vote
1
answer
168
views
About the sigma algebra generated by the Hausdorff measure on $\mathbb R^n$
Let $\mathcal{H}^k$ be the $k-$dimensional Hausdorff measure on $\mathbb R^n$, with $k \in \{1, \ldots n\}$. By Carathéodory's theorem we know that there exists a sigma algebra $\mu(\mathcal{H}^k)$ of ...
CommunityBot
- 1
0
votes
0
answers
96
views
Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 1
-2
votes
1
answer
93
views
Express the connection between roots [closed]
$\DeclareMathOperator\elim{lim}\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\lmb{lmb}\DeclareMathOperator\Lmb{Lmb}\DeclareMathOperator\mts{mts}$There are two similar functions; they determine the ...
- 55
2
votes
2
answers
173
views
Gronwall-type inequality involving norms of distinct Lebesgue spaces
Let $d \geq 1$, $\Omega \subset \mathbb{R^d}$ be a bounded domain and let $\phi : [0,T]\times \Omega \mapsto \mathbb{R}$ be a measurable and bounded function. Assume that the following differential ...
- 213
3
votes
0
answers
141
views
Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$
Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
- 435
5
votes
1
answer
349
views
Equilateral triangle in a Brownian path
I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory ...
4
votes
2
answers
354
views
Injectivity of a convolution operator
Let $p,\mu,\nu$ be probability density functions on
$\mathbb{R}$ such that
$$
\int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x).
$$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
- 133
1
vote
0
answers
215
views
Computing a closed form representation for a Fourier series summation
I want to compute a closed form representation for the below given summation expession.
$$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
CommunityBot
- 1
0
votes
0
answers
72
views
Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
3
votes
0
answers
67
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 ...
- 171
5
votes
1
answer
196
views
What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?
Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
- 6,367
5
votes
1
answer
374
views
What is the length of an algebraic curve?
The following question seems to be somewhat standard, but I was unable to find any reference. I would be grateful for any pointers to relevant literature.
We consider a real polynomial $p(x,y)$ of ...
- 53
2
votes
1
answer
194
views
Functions with derivatives growing at rate $r>0$
Fix a non-empty closed subset $\Omega\subset\mathbb{R}$.
Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and such that $\sup_{x\in \Omega}\,|\partial^k f(x)|\lesssim k^r$ for some $r\ge 0$ for all $k\in \...
- 63.9k
0
votes
0
answers
29
views
$ \sup_{\theta \in [0,2\pi)}\max_{r\leq \delta}\frac{\log\left(\frac{f(r,\theta)}{f(\delta,\theta)}\right)}{\log(r)}<\infty,$ $f$ real analytic
$\textbf{Conjecture.}$
Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $\delta <1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and suppose that $(0,0)$ is the only ...
- 171
0
votes
0
answers
106
views
How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin
Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
CommunityBot
- 1
0
votes
0
answers
49
views
ODE satisfied by a special function
Posted on MSE
Context
I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica . ...
- 393
9
votes
2
answers
616
views
construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
- 1,446
3
votes
1
answer
233
views
Analytic solutions to analytic differential equations
Let $U \subseteq \mathbb R^{n+2}$ be an open set for some $n \geq 0$, and let $f: U \to \mathbb R$ be an analytic function. Then we say the equation $f(x,y,y',\ldots,y^{(n)})=0$ is an analytic ...
- 39k
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 $...
- 171
89
votes
1
answer
21k
views
Is the largest root of a random polynomial more likely to be real than complex?
This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO.
The number of real roots of a ...
- 341
9
votes
2
answers
758
views
Number of critical points of smooth functions on $S^1$
Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
- 18.7k
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-...
- 21
9
votes
2
answers
424
views
Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
- 10.6k
1
vote
1
answer
141
views
Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
CommunityBot
- 1
6
votes
1
answer
309
views
Well distributed sets
Note: All integrals are taken with respect to Lebesgue measure. The symbol $\def\avint{\mathop{\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int}\nolimits} \avint$ denotes the average integral.
We say ...
- 109k
7
votes
1
answer
185
views
Question on ODE involving mollifiers from Taylor's book on PDEs
In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form
$$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$
with some initial condition $u(...
- 3,584
20
votes
3
answers
2k
views
Do convex and decreasing functions preserve the semimartingale property?
Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...
- 1,446
7
votes
1
answer
834
views
Representing $\Gamma(a-x)$ in terms of $\Gamma(kx)$ and $\Gamma(a)$ and elementary functions
I asked this question on MSE here.
I wonder if it is possible to represent $\Gamma(a-x)$ in terms of powers of $\Gamma(a)$, powers of $\Gamma(kx)$, and elementary functions. I am not looking for any ...
- 114k
1
vote
1
answer
111
views
How to show such result for generalized $ O(|x|^{-1/2}) $ function?
Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
- 24.2k
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 ...
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 \...
- 63.9k