All Questions
5,850 questions
13
votes
2
answers
1k
views
Probability vector $p$ majorizes its normalized entropy vector $\small \frac{-p\log p}{H(p)}$
I guess the following inequality
$$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$
holds for any continuous convex function $g$ and any probability ...
- 303
5
votes
0
answers
156
views
What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,155
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
10
votes
1
answer
1k
views
A strange Lipschitz function
Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold?
The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
- 6,155
1
vote
2
answers
102
views
About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$
Suppose we have a non-decreasing sequence of positive real numbers that tend to infinity: $0<w_1\leq w_2\leq w_3\leq...$ It is known that:
For every $n$ and $p\geq 2n$, we have $w_p \geq (1-\frac {...
- 525
3
votes
1
answer
240
views
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and
$$
[f]_{\frac{2}{\...
- 1,219
7
votes
1
answer
179
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 128k
2
votes
1
answer
133
views
Points of differentiability of convex functions
Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
- 2,998
9
votes
1
answer
366
views
Can the canonical Eudoxus-real representatives be defined easily?
(See e.g. here for background on the Eudoxus reals, which motivates this question.)
Let $\mathcal{Z}=(\mathbb{Z};+,<)$. Say that a Eudoxus function is an $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such ...
- 21.2k
0
votes
1
answer
139
views
Proving negativeness of function involving $-\log t$
I have been trying to solve the following function is non-increasing with respect $\theta$
\begin{equation}
h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
- 23
7
votes
1
answer
346
views
Mean Cauchy sequences
Let $X$ be a complete metric space. Suppose a sequence of elements $x_n$ is Cauchy in mean, in the sense that
$$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N} \sum_{...
- 6,155
0
votes
0
answers
121
views
Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?
Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
- 7,127
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
20
votes
1
answer
2k
views
Is $1/F$ Schwartz if $F$ is "reverse Schwartz"?
Let's call a positive function $F:\mathbb{R}\to\mathbb{R}$ "reverse Schwartz" if $F$ is smooth and
$$\forall n \forall k,\quad\lim_{x\to\infty}\frac{|x|^n}{|\partial_x^k F(x)|}=0\quad .$$
In ...
- 23.3k
2
votes
1
answer
315
views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions?
I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 311
7
votes
1
answer
224
views
Does the decomposability of $\mathbb{R}$ imply analytic LLPO?
By "BISH" I mean constructive mathematics without axiom of countable choice.
By $\mathbb{R}^f$ I mean real numbers as fundamental sequences of rational numbers and by $\mathbb{R}^d$ I mean ...
4
votes
1
answer
249
views
Does this functional admit an absolute minimizer?
This is a close relative of the following problem.
Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
- 6,155
7
votes
2
answers
706
views
Poisson binomial conjecture
Let $X_i\in\{0,1\}$
be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$
and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$,
for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=...
- 6,541
0
votes
0
answers
55
views
Compactness and Leray-Schauder degree
What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
- 471
17
votes
2
answers
1k
views
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
Let $P$ denote the following proposition:
There exists a set $S$ of subsets of $\mathbb{R}$ such that
$S$ is totally ordered by inclusion;
each member of $S$ has no accumulation points;
the union of ...
- 708
1
vote
1
answer
50
views
Increasing function of $\theta$ for the Ali-Mikhail-Haq Survival Copula
I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$)
\begin{equation}
f(\theta)= \frac{h(t,\...
- 23
4
votes
1
answer
110
views
Scaling of stopped Hölder norm of Brownian motion
I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.
For fixed $T>0$, self similarity ...
- 153
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 ...
- 5,470
-1
votes
1
answer
122
views
Divergent summation [closed]
Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series:
$$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
- 21
9
votes
1
answer
553
views
Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
- 541
3
votes
1
answer
224
views
Extension of Sobolev function defined on unit cube
Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
- 93
4
votes
1
answer
96
views
On the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$.
Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$?
Remark 1: A numerical experiment suggests that $...
- 128k
2
votes
1
answer
231
views
Is Boltzmann entropy well-defined for arbitrary probability density function?
$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by
$$
\varphi (s) :=
\begin{cases}
0 &\text{if} \quad s =0 , \\
s \...
- 835
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
0
votes
1
answer
153
views
Lebesgue measure of the level set of sum of two nonnegative functions
Let $f, g:\mathbb{R}^n\to \mathbb{R}$ be nonnegative functions such that $g$ is a strictly positive homogeneous function. As commented by Fedor Petrov below, one may not have that for any $\lambda>...
- 407
2
votes
1
answer
474
views
Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?
Let
$$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$
be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
-4
votes
1
answer
302
views
A Question in Fourier Analysis proposing a conjecture
Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
- 698
0
votes
0
answers
60
views
Criteria for log-absolute-monotonicity
Consider a function $f: [0,1] \rightarrow \mathbb R$ defined by a power series $f(x) = a_0 + a_1 x + a_2 x^2 + \dots$, where all $a_i$ are positive.
Is there are any criterion in terms of the ...
- 3,475
16
votes
4
answers
2k
views
Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with
$f_n \to f$ uniformly for some (necessarily) continuous $f$.
$f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
Is it true ...
- 6,155
7
votes
0
answers
313
views
Did Lebesgue like non-measurable set or not?
I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:
Vitali's nonmeasurable set, ...
- 877
6
votes
1
answer
388
views
Decimal expansion definition of real numbers, constructively
The two most common definitions of $\mathbb{R}$ are as Dedekind cuts or Cauchy sequences of rational numbers.
A real analysis student of mine is working out of the book Real Analysis and Applications ...
- 10.1k
2
votes
0
answers
99
views
Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$
Do you know if there exists a closed form for the integral :
$$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$
where $p$, $q$, $r$ are natural integers such as this integral ...
- 69
2
votes
0
answers
85
views
Higher cohomology groups for the trivial action of the reals on themselves
For a freely generated countable abelian group $A$ with the trivial action on itself ($a\cdot b = b$) the resulting cohomology groups are well-known and eventually vanish (see e.g. here). Coming from ...
- 1,411
8
votes
1
answer
449
views
What do smooth signatures give you?
My background is in rough paths theory.
In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are ...
- 1,904
0
votes
0
answers
22
views
Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 708
7
votes
0
answers
249
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
5
votes
1
answer
355
views
Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty$
I want to know whether or not
$$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}} \frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty.$$
Here $D $ denotes the ...
- 61
3
votes
1
answer
187
views
Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
1
vote
0
answers
125
views
Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$
Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
user538396
2
votes
1
answer
127
views
Density of smooth functions in weighted Sobolev space
Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
- 23
0
votes
0
answers
73
views
An example of a groupoid that satisfy the following hypothesis
In the paper titled, 'Tannaka–Krein duality for compact groupoids I, Representation theory', the author proves the Peter Weyl theorem on compact groupoids. In the statement, he gives the hypothesis ...
5
votes
1
answer
174
views
Do the zeroes of some hypergeometric functions interlace?
Confluent hypergeometric functions differing from $F={}_1F_1(a,b,z)$ by $\pm1$ in either parameter $a$ or $b$ are called contiguous to $F$. For rational $a, b$, assume I know $z_0$ is a zero of $F$. ...
7
votes
5
answers
513
views
Probability of $\operatorname{Bin}(n,p)=\operatorname{Bin}(n,q)$ is decreasing when $n$ increases
$\newcommand{\Bin}{\operatorname{Bin}}$I would like to show that $\mathbb P(\operatorname{Binomial}(n,p) = \operatorname{Binomial}(n,q))$ decreases when $n$ increases for a fixed pair $(p,q)$. This ...
- 243
2
votes
1
answer
836
views
Does $\int_{\mathbb R^d} (1+|x|^{1 + \alpha}) \ell (x) \, d x < \infty$ imply $\int_{\mathbb R^d} (1+|x|) |\ell (x)|^{1-\alpha} \, d x < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$. Let $\ell : \bR^d \to \bR_+$ be a continuous function such that
$$
\|...
- 835
6
votes
3
answers
851
views
Almost everywhere-periodic functions with many periods
Let $f : \mathbb{R} \to \mathbb{R}$ be a Lebesgue measurable function and $D$ be a countable dense subset of $\mathbb{R}$.
Suppose that for a.e. $x \in \mathbb{R}$ we have
\begin{equation*}
f(x + d) = ...