All Questions
5,672 questions
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
$$ I_n:= \frac{1}{n!^2}\int_0^1 \int_0^1\cdots\int_0^1 \...
2
votes
0
answers
564
views
Young inequality in weighted spaces
Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...
- 1,205
7
votes
1
answer
876
views
A curious definite integral
I was playing around with $\mathcal{I}=\int_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that
$$
\mathcal{I} =
\begin{cases}
\frac{1}{...
- 829
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 (...
- 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 $c$,
...
- 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(...
- 11
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 ...
- 777
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 ...
- 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.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
- 981
4
votes
0
answers
273
views
Real Analytic Function and nth Prime
It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...
- 153
1
vote
1
answer
342
views
Singular conformally-Euclidean metrics
Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance':
$$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt{...
- 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 ...
- 51
1
vote
1
answer
977
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) >...
- 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(\...
- 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 ...
- 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 $(\...
- 657
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 ...
- 4,312
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
...
- 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 ...
- 17.9k
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 ...
- 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 $S$...
- 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 ...
- 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.
- 63
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}^...
- 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 ...
- 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 *-...
- 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 ...
- 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 ...
- 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}$ ?
- 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}$ ...
- 399
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}...
- 12.8k
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 ...
- 188
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 ...
- 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 $....
1
vote
2
answers
1k
views
Is there a periodic function without minimum period such that all the possible periods are irrationals? [closed]
Let $f:\mathbb R\to\mathbb R$ be a periodic function. We say $f$ is without minimum period if, $\forall t$ such that $f(x+t)=f(x)\forall x$, there is a $t'$ such that $0<t'<t$ and $f(x+t')=f(x)\...
- 188
12
votes
5
answers
2k
views
analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 19.7k
12
votes
2
answers
2k
views
Implicit function theorem at a singular point?
Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...
- 123
0
votes
1
answer
606
views
Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]
Is there a difference between
$L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ?
Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue
$\sigma$-algebra $\...
user20515
5
votes
2
answers
1k
views
Stone-Weierstrass for monotone functions
Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$.
Does it follow that that there exists a real ...
user5810
0
votes
1
answer
659
views
Under what condition will this set contain a limit point of [0,1)?
Let $T_1,T_2,....T_n$ be numbers such that
$T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. ...
- 230
0
votes
1
answer
316
views
Modulo dynamics on [0,1)
For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\...
- 2,619
5
votes
2
answers
4k
views
Bounded sequences with divergent Cesàro mean
It is well known that there are bounded sequences with divergent Cesàro mean, i.e., a bounded $a_n$ for which given $$c_N := \frac{1}{N}\sum_{n=1}^N a_n,$$ the sequence $(c_N)_{N\geq1}$ has no limit. ...
- 702
2
votes
1
answer
1k
views
On an eigenvalue inequality
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
- 29
5
votes
3
answers
5k
views
Zeros of "exponential" function
Define ${f}_{i}(x) = \sum_{j=1}^{i} (-1)^{i-j}{i \choose j}j^x$, where $i=1,2,3,...$ and $x \in \mathbb{R}$.
For integer $x \geq i$, ${f}_{i}(x)$ reduces to ${f}_{i}(x)=i!S(x,i)$, where $S(x,i)$ is ...
- 2,619
6
votes
2
answers
1k
views
On the uncountability of zero sets
If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain.
I ...
- 8,512
1
vote
2
answers
450
views
A smoothness of $f(\sqrt[p] x)$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function let $p \in \mathbb{N}$, $p \geq 2$.
Assume that $f^{(k)}(0)=0$ for all $k \notin p \mathbb{N}$. Is it true that then $g(x)=f(\sqrt[p] x)$...
- 277
3
votes
1
answer
500
views
Hausdorff measure on product spaces of p-adic integers
This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
- 1,723
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
4
votes
2
answers
5k
views
Ratio of Sequences Sum Inequality
I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
- 1,182
2
votes
1
answer
255
views
Quotients of perfect powers separated by an integer
Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
- 3,595