All Questions
5,628 questions
9
votes
2
answers
519
views
The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-...
- 91
8
votes
2
answers
2k
views
Expression for the sum of square roots of zeros of a polynomial
Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$.
General question. Does there exist a simple expression for the ...
- 34.3k
1
vote
3
answers
362
views
Estimating L1 functions over the ball with radius 2r
Let $ f $ be in $ L^1(\Omega) $ where $
\Omega $ is an open subset of $ \mathbb{R}^n $. Also, assume that $ B(x_i,r_i) $ is a collection of disjoint open balls in $ \Omega $ such that $ B(x_i,2r_i) \...
- 520
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, ...
- 27.7k
3
votes
0
answers
3k
views
Derivative of the regularized upper incomplete gamma function
I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function
\begin{eqnarray}
\Gamma(x)&=&\int_0^\...
- 607
11
votes
8
answers
3k
views
Almost-converses to the AM-GM inequality
Let us consider the Arithmetic Mean -- Geometric Mean inequality for nonnegative real numbers:
$$ GM := (a_1 a_2 \ldots a_n)^{1/n} \le \frac{1}{n} \left( a_1 + a_2 + \ldots + a_n \right) =: AM. $$
...
- 531
2
votes
1
answer
239
views
Volumes of families of semialgebraic sets
Let $f_0(T),f_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f_1(T)X+f_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is $\sqrt{\max\{0,f_1(T)...
- 23
0
votes
1
answer
302
views
An interpolation inequality.
For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such ...
- 45
8
votes
3
answers
837
views
Second order difference implies differentiability
Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies
$|f'(x+h)-f'(x)|\le c |h|^...
- 83
2
votes
3
answers
1k
views
on the set of numbers generated by integer linear combination of two real numbers.
Let $b > a > 0$ be two real numbers. I am interested in the set of numbers
$X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$.
What ...
- 29
6
votes
1
answer
808
views
Must the Minkowski sum of a Borel set and a *closed* ball be Borel?
Let A be a Borel set in R^n. Must then A + B(0,1) be Borel?
Here B(0,1) is the closed ball centered at 0 of radius 1.
I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a ...
- 63
-1
votes
1
answer
349
views
A question about approximation of Real analytic functions
Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$
for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in ...
- 105
0
votes
0
answers
92
views
Class of integrable 0/1-functions "with no null sets."
I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable.
Let $f$ be in this class and $E$ be the set of all points where $f$ is equal to ...
- 567
3
votes
1
answer
121
views
Second difference
Is there an elementary example of a function f, such that $|f(x+t)+f(x-t)-2f(x)|/|t|^a\le C$, where $a>1$, such that $f$ is not $C^1$?
- 95
1
vote
1
answer
868
views
Limit of functions and asymptotic behaviour
Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$.
I denote the polynomials $p_l(t) = \sum_{k=0}^{m(l)} ...
- 11
1
vote
0
answers
1k
views
Properties of a rational function of multiple variables
Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...
1
vote
0
answers
133
views
Extension of a function
Hello,
Given a $\mathcal{C}^\infty$ function $\varphi$ defined on a portion of a surface $\Sigma^-$ and let $\Sigma$ be a closed surface or union of surfaces bounding a compact volume $\Omega \...
user16974
2
votes
1
answer
760
views
Apollonian gasket and the degree of convergence
Let $r_1,r_2\dots$ be the radii of Apollonian gasket.
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$
I know that if three circles $A$, $B$ and $C$ ...
- 293
12
votes
1
answer
1k
views
A generalization of intermediate value theorem on R^k
Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...
- 207
2
votes
0
answers
160
views
Radius of convergence to be proved more precisely (differential equation)
There is a differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
It is possible to get a solution which is a power series (see below). However, I am looking for an ...
4
votes
0
answers
340
views
Viscosity solution of the PDE
Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and
$$|Du|-f(x,u)=0$$
where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
- 61
1
vote
1
answer
193
views
If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative?
Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the ...
- 3,245
2
votes
1
answer
351
views
Operation on measurable sets in lines, containing an interval?
Question 1: In $\mathbb{R}^2$, let $l_1$,$l_2$ be two parallel lines and $l_3$ another line which is not parallel to $l_1$. Given two measurable sets $E_1$ and $E_2$ in $l_1$ and $l_2$ respectively, ...
7
votes
2
answers
2k
views
Tails of sums of Weibull random variables
Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
- 1,791
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 19.6k
2
votes
2
answers
6k
views
Derivative indicator function
I am wondering what is the derivative of the following function with respect to $x(t)$ in sense of distributions.
$$
I\left(\int_0^t x(\tau)d\tau \leq c\right)
$$
where $I$ is the indicator function ...
- 175
13
votes
1
answer
1k
views
Is it necessary to use AC to solve this problem ?
Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, $A\...
user30230
15
votes
1
answer
858
views
Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
- 34.7k
3
votes
1
answer
399
views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\...
7
votes
2
answers
518
views
Morse lemma with least amount of regularity.
I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
- 1,211
1
vote
1
answer
383
views
Solution of a PDE and its uniqueness
Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
$\...
- 339
1
vote
1
answer
297
views
Injective with finite discontinuities mapping from $\mathbb R^n$ to $[0,1]$
As a continuation to the fully answered question:
Injective and Integrable Mapping from $\mathbb R^3$ to $\mathbb R$
Does there exist an injective mapping $f:\mathbb R^n\rightarrow[0,1]$ that has only ...
- 310
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 ...
- 1,704
6
votes
2
answers
3k
views
Approximating erf by tanh
It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...
- 6,541
3
votes
1
answer
6k
views
Classical Derivative, Weak Derivative and Integration by Parts
Hello,
While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.
QUESTION
Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $...
- 895
4
votes
1
answer
4k
views
Why can't I interchange integration and differentiation here?
I think my questions relates to this other: "counterexamples to differentiation under integral sign"
In fact, it provides a counterexample
Consider $f(x,y)=y^3e^{-y^2x}$ and define $F(y) =\...
- 53
1
vote
1
answer
168
views
Mollification with prescribed boundary values
Suppose you are given a $C^1$-function $f:\mathbb R^n\to\mathbb R$ which restricts to a smooth function $f|_{\partial B}:\partial B\to \mathbb R$, where $B$ is the unit ball in $\mathbb R^n$. Can one ...
- 11
6
votes
2
answers
425
views
orderings of the field R((x, y))
I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the ...
- 620
1
vote
1
answer
1k
views
Calculating the Lebesgue decomposition of a measure [closed]
How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
- 13
8
votes
3
answers
2k
views
Does a weaker condition than vanishing derivative imply a function being constant?
I learned this question from math.stackexchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does
$$g(x):=\lim_{\epsilon\to 0}\frac{f(x+...
- 183
2
votes
3
answers
365
views
Construct a fixed-point set operator
How to find an uncountable set $S$, and construct an function $f : 2^S
\longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in
T$?
for example, let $S =\mathbb{R}$, how can I ...
- 73
1
vote
1
answer
620
views
Smallest Lipschitz constant on non-convex domains
It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...
- 959
0
votes
1
answer
135
views
Example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition
I'm looking for the example of a concave function $g \colon [0,1] \mapsto \mathbb{R}$, with $g(0)=0$, for which
$\lim\limits_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$, and
$\lim\limits_{x\to 0^+}\frac{\...
- 51
4
votes
1
answer
319
views
Concerning strata in $C^\infty(M)$
The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf ...
- 17.5k
8
votes
1
answer
726
views
Multiplication of Cauchy and Dedekind real numbers
In Michael Dummett's book "Elements of Intuitionism", the product of real numbers is defined as follow:
$x\cdot y= \{ \langle r_n\rangle \cdot \langle s_n\rangle$ | $\langle r_n\rangle\in x , \langle ...
- 81
1
vote
0
answers
127
views
Name for class of functions satisfying $\lim_{x\to 0^+}\lambda g(x)/g(\lambda x)>1$
I would like to ask whether is used some name for functions $g:A\to\mathbb{R}$, $A\subset \mathbb{R}$, for which $$\exists \lambda>1:\;\; \lim_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}>1.$$
- 51
1
vote
2
answers
576
views
Dini condition and integrability condition
Assume that $A$ is an arbitrary positive integrable function on $[0,1]$. Whether exists a convex function $f_A(x)=x g(x)$ of $(0,+\infty)$ into itself (depending on $A$) such that $\lim_{x\to +\...
- 303
2
votes
3
answers
557
views
A consequence of convexity
Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$.
Studying the behaviour of the difference quotient, it is clear that
$$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$
...
- 293
1
vote
1
answer
137
views
Concave functions of different behaviour in the neighbourhood of $0$ from the Shannon function
I'm looking for an example of a concave function $g \colon [0,1] \to \mathbb{R}$, $g(0)=0$ such that:
$$\liminf_{x\to 0^+}\frac{g(x)}{-x\ln x}\neq \limsup_{x\to 0^+}\frac{g(x)}{-x\ln x}.$$
Moreover, ...
- 51
11
votes
1
answer
2k
views
Transcendentality of all irrationals in the Cantor set
Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic ...
- 113