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Chebyshev's Theorem

Hi, I´m looking for Chebyshev´s theorem which says that the inequality $|x(k)-y|<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\...
Rosendo's user avatar
  • 11
10 votes
2 answers
6k views

Who was the first to formulate the inverse function theorem?

Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$. ...
john's user avatar
  • 103
3 votes
2 answers
2k views

The extension of smooth function

If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
Adterram's user avatar
  • 1,441
2 votes
1 answer
689 views

Partitions of an interval

This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there. Specifically, consider "partitions" ...
Emilio Pisanty's user avatar
30 votes
4 answers
2k views

is f a polynomial provided that it is "partially" smooth?

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$. Suppose for each $n\in ...
Ben's user avatar
  • 407
1 vote
1 answer
771 views

A question about the tail of an absolutely integrable function

Assume $X$ is a measure space and $f : X \to [0,\infty]$ is an absolutely integrable function (that is $\int_X f \; d \mu < \infty$). This question is about the asymptotic behaviour of the function ...
Daniel Barter's user avatar
4 votes
2 answers
3k views

Chain rule for fractional laplacian

Does anyone know a formula of chain rule for fractional laplacian? say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...
Grant's user avatar
  • 41
4 votes
2 answers
977 views

Articles with examples of Darboux functions without fixed points

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \...
Beni Bogosel's user avatar
  • 2,222
5 votes
3 answers
718 views

Subsets of $\mathbb{R}^+$ closed under addition

No one's answered the question cumulant problem so here's a simpler question: Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition? In particular, ...
Michael Hardy's user avatar
0 votes
1 answer
265 views

H\"older spaces

In Gilbarg and Trudinger, they have an example where a function is in $C^1(\bar\Omega)$ but not in $C^\alpha(\bar\Omega)$ where $\alpha<1$. $\Omega$ is bounded and is defined as follows $\Omega:= ...
AAAA's user avatar
  • 3
2 votes
3 answers
549 views

Non-continuous representations of $\operatorname{SL}_2(\mathbf{R})$

How does one construct a non-continuous representation $\rho:\operatorname{SL}_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?
Hugo Chapdelaine's user avatar
2 votes
0 answers
114 views

Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.) ...
Joe's user avatar
  • 88
3 votes
0 answers
409 views

Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
user19172's user avatar
  • 529
2 votes
1 answer
4k views

Uniform $L_1$ convergence implies uniform convergence pointwise a.e.

Let $\Omega$ be a measure space (which can be assumed to be an interval with Lebesgue measure). It is well known that for a sequence $(f_n)$ in $L^1(\Omega)$ which converges to zero (in $L^1(\Omega)$,...
Florian's user avatar
  • 2,270
5 votes
1 answer
664 views

Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite. There are at least two ...
Pablo Lessa's user avatar
  • 4,304
23 votes
2 answers
651 views

Asymptotics of a Selberg-type integral

Let $\Delta(s_1,s_2,\ldots,s_n) := \prod_{i<j}(s_i-s_j)^2$. Is there a standard way to estimate the decay of the Selberg-type integral $$ I_n:= \frac{1}{n!^2}\int_0^1 \int_0^1\cdots\int_0^1 \...
Krishnan Rajkumar's user avatar
2 votes
0 answers
564 views

Young inequality in weighted spaces

Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$. Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$. Does ...
Samuele's user avatar
  • 1,205
7 votes
1 answer
876 views

A curious definite integral

I was playing around with $\mathcal{I}=\int_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that $$ \mathcal{I} = \begin{cases} \frac{1}{...
Koundinya Vajjha's user avatar
5 votes
0 answers
760 views

two versions of the nested interval property

There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...
James Propp's user avatar
  • 19.7k
7 votes
3 answers
4k views

Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous function. [Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable? If not, can one find a counter-example? Note that, for any $c$, ...
kenneth's user avatar
  • 1,399
1 vote
0 answers
163 views

On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest eigenvalue $\mu_{3}$ for the differential equation $\Delta f\left(...
user21990's user avatar
2 votes
3 answers
3k views

Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set

It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...
Joakim Arnlind's user avatar
4 votes
0 answers
213 views

The ring generated by measures

Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
David Spivak's user avatar
  • 8,659
9 votes
1 answer
782 views

Mean value property with fixed radius

Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e. $$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
Syang Chen's user avatar
4 votes
0 answers
273 views

Real Analytic Function and nth Prime

It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...
Marcus's user avatar
  • 153
1 vote
1 answer
342 views

Singular conformally-Euclidean metrics

Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance': $$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt{...
Beni Bogosel's user avatar
  • 2,222
5 votes
1 answer
1k views

Notions related to De Rham Cohomology

In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures. Closed forms/exact forms real parts of analytic functions/harmonic ...
Jeff's user avatar
  • 51
1 vote
1 answer
978 views

Concentration bound for weakly dependent random variables

Hi, Suppose we observe a sequence $R_1, ..., R_T$ of iid. random variables that equal $0$ with probability $p$ and with probability $1-p$ are sampled from a distribution with expected value $E(R) >...
Woland's user avatar
  • 53
3 votes
0 answers
181 views

Example showing that area is discontinuous in the 2-variation seminorm

The $p$-variation seminorm (where $p \ge 1$) of a continuous curve $\alpha: [0,1] \to \mathbb{R}^2$ is defined as the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_n = 1$ of: $\left(\...
Pablo Lessa's user avatar
  • 4,304
26 votes
3 answers
16k views

the dual space of C(X) (X is noncompact metric space)

It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f: X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
yaoxiao's user avatar
  • 1,706
0 votes
0 answers
345 views

Jacobian of the inversion map

Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\...
Diego Sulca's user avatar
5 votes
1 answer
540 views

Cosets of groups of functions

Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$. The set $\mathcal ...
Cristi Stoica's user avatar
4 votes
0 answers
109 views

rank of a C^1 map

I saw this three star problem in Hirsch .. If we have open sets $U \subset R^3$ ,$V \subset R^2$ and $f:U \to V$ is $C^1$ and onto...Prove there is at least one point in $U$ where $f$ has full rank ...
Marcus's user avatar
  • 153
5 votes
2 answers
541 views

Asymptotic behaviour of $\int f(t)^a\cos(at)dt$

Are there any known necessary or sufficient conditions such that $$\lim_{a\rightarrow \infty}\int_{-1}^1f(t)^a\cos(at)dt=0$$ where $f:[-1,1]\rightarrow[1,\infty)$ is an even smooth concave real ...
Roland Bacher's user avatar
28 votes
4 answers
3k views

"Converse" of Taylor's theorem

Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a_0,a_1,\ldots,a_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum_{i=0}^k a_i(c)t^i+o(t^k)$ (for ...
Mizar's user avatar
  • 3,146
2 votes
1 answer
297 views

A raceway problem

Let $f(x)=\sin x$, and $g(x)=\sin x + 1$. Consider a set $S=\{(x,y)| f(x)\leq y \leq g(x), x\in [0,2\pi]\}$. This set $S$ can be considered as "Raceway" My question is finding the shortest path in $S$...
Sungjin Kim's user avatar
  • 3,320
0 votes
0 answers
700 views

Sigma algebra generated

Let $\mathcal{L} \subset \mathbb{R}$ the Lebesgue sigma algebra and $\mathcal{B} \subset \mathbb{R}^{n}$ the Borel sigma algebra. I'll denotes by $\mathcal{L} \times \mathcal{B}$ the smallest sigma ...
Santos's user avatar
  • 11
6 votes
3 answers
2k views

Lipschitz continuity of singular values

How smooth are the singular values of a matrix $F$ in terms of entries of $F$? I am hoping for Lipschitz continuity, but was not able to find it.
Peter Bella's user avatar
4 votes
1 answer
977 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that $$\sum_{i=1}^...
Ktb's user avatar
  • 41
10 votes
2 answers
1k views

Does Rolle's Theorem imply Dedekind completeness?

I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...
James Propp's user avatar
  • 19.7k
6 votes
2 answers
4k views

Is there dual space of the distributions $\mathcal{D}'(R)$?

Dear MOs, Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
Anand's user avatar
  • 1,649
10 votes
0 answers
315 views

Does antidifferentiability of continuous functions imply Dedekind completeness?

Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
James Propp's user avatar
  • 19.7k
3 votes
3 answers
595 views

Every positive polynomial with rational coefficients is above a completely Q-factorized nonnegative polynomial ?

This question was originally asked in stackoverflow (https://math.stackexchange.com/questions/103941/every-positive-polynomial-is-above-a-completely-q-factorized-positive-polynomial) but as it has ...
Ewan Delanoy's user avatar
  • 3,595
1 vote
1 answer
334 views

Property Sigma Algebra [closed]

Is the set { $ \cup_{i \in \mathbb{N}} C_{i} \times D_{i} : C_{i} \in \mathcal{L} \ , D_{i} \in \mathcal{B}^{n} \ $ } a sigma algebra on $\mathbb{R} \times \mathbb{R}^{n}$ ?
Santos's user avatar
  • 11
10 votes
2 answers
3k views

Absolute continuity on $R^{n}$

I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$. I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
Nikita Evseev's user avatar
11 votes
2 answers
2k views

Multi-dimensional moment problem

Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment $$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
Kevin Walker's user avatar
  • 12.8k
1 vote
2 answers
307 views

Continuous family of Markov chains

Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition ...
Elena Yudovina's user avatar
0 votes
1 answer
1k views

surjective function from non-measurable sets

let $V$ be the vitali set and let $g:V\to\mathbb R$ be a surjective function. then the fuction $f:\mathbb R\to\mathbb R$ such that $f(x)=g([x])$ will be a function that is surjective in any interval ...
alberto.bosia'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
1 vote
4 answers
959 views

Does complete monotonicity of f imply log-concavity of f?

Let f be a completely monotonic function with $f(0)=1$, that is, $ f:[0, \infty) \rightarrow (0,1] $. My question is: Is f log concave, that is, is $(logf)''<0$ or equivalently $ f f''< f'^2 $....
J M del Castillo's user avatar

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