All Questions
5,850 questions
3
votes
1
answer
167
views
Recovering residue using local real information
Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$.
Compute the residue of $f(z)$ at z = 0 using just the ...
- 31
15
votes
0
answers
511
views
Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
- 6,541
3
votes
1
answer
277
views
Nowhere dense set with high multiplicity
For a subset $S\subset[0,1]$ with $0<|S|<1$ ($|S|$ is the Lebesgue measure of $S$) we define the multiplicity function of order $n$ $m_{n,S}:[0,1] \rightarrow \{0,1,\ldots,n\}$ in the following ...
- 549
2
votes
0
answers
881
views
Why does a convex function have to have a convex domain? [closed]
Other than convenience in convex optimization, is there a reason that the definition of a convex function includes the requirement for the domain to be a convex set?
- 121
44
votes
7
answers
4k
views
The missing link: an inequality
I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:
Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then
$$F_n(x)=\...
- 43.2k
4
votes
0
answers
56
views
Is there an equivalent line time-invariant system for a linear time-varying system with specific properties? [closed]
Given a discrete-time linear time-varying system (LTV)
$$x(k+1) = A(k) x(k) + B(k) u(k)$$
where $A(k)$ and $B(k)$ are generated by a stationary random process. Is there an equivalent linear time-...
- 141
0
votes
0
answers
69
views
Quasi-stationary measure on a finite graph equals stationary measure?
Assume the simple random walk $X$ on the graph $G(V,E)$, s.t. $G$ is simple, undirected, finite, connected and let $B \subset V$, s.t. $V\setminus B$ is connected. Let $\sigma_B$ be the quasi-...
- 71
2
votes
1
answer
210
views
asymptotic estimate for log-tan sum
I am finding the following first order estimate.
Question. As $y\rightarrow\infty$,
$$\sum_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\,
\sim\,\,\frac{\pi}4\log^2y.$$
Is it true?
- 43.2k
9
votes
1
answer
2k
views
Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 263
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
1
vote
1
answer
211
views
Representation of Hilbert transform by a singular integral
Hilbert transform defines as follow:
$$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$
$$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$
Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...
- 371
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
2
votes
0
answers
139
views
Existence of solution of a variational inequality
Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \...
- 21
3
votes
0
answers
142
views
Probability of hitting two vectors
Call an element of $ \{-m,\dots,0,\dots,m\}^{2^n}$ a vector. Assume $m = O(2^{2^n})$.
Let $u_1,u_2$ be vectors.
Let $\{v_1,\dots,v_{2^n}\}$ and $\{w_1,\dots,w_{2^n}\}$ be linearly independent ...
- 13.9k
3
votes
0
answers
97
views
Dimension of a graph
Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?
If not, what extra condition do we ...
- 2,135
4
votes
1
answer
393
views
Locally doubling measures
Let us say that a measure $\mu$ on $\mathbb{R}^d$ is locally doubling if for each
$x\in\mathbb{R}^d$ there is a constant $C(x)$ such that for all $r>0$,
$\mu(B(x,2r)) \le C(x) \mu(B(x,r))$,
where $...
- 6,541
2
votes
1
answer
800
views
Interpolation in Sobolev spaces
Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...
- 843
1
vote
1
answer
186
views
Almost binomial sum limit
I got the following sum with which I want to prove one limit fact:
$$
f_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} (a^t)^{n-t}
$$
I want to prove that $f_n(a) \to 1$ while $n \to \infty$ for $a\...
- 342
4
votes
1
answer
264
views
Density of the max set of a non-differentiable function
For $f : [0;1] \to \mathbb{R}$, let $M_f := \{x \in [0;1] \mid f(x)$ is a local strict maximum of $f\}$. It is easy to see that for any $f$, $M_f$ is at most countable. It is also easy to see that ...
- 15.9k
30
votes
4
answers
3k
views
A counterexample for Sard's theorem in $C^1$ regularity
I can't seem to find an example of a function $f \colon \mathbb{R}^2\to \mathbb{R}$ which is $C^1$ and such that the set of its critical values is not of zero measure.
What examples are there?
$...
- 478
7
votes
2
answers
470
views
Continuous functions and infinity
Suppose $f(x)$ is continuous on $\mathbb{R}$, for $\forall \delta>0, \forall x\in\mathbb{R}, \lim_{n\rightarrow\infty}f(x+n\delta)=+\infty$. Is it correct that $\lim_{x\rightarrow+\infty}f(x)=+\...
- 183
3
votes
1
answer
331
views
Solving recurrent relation
I have the following recurrent relation and I want to find a close form of it if it exists at all.
$$
P_n = (1-p)^{n-1}P_{n-1} + \sum\limits_{k=2}^{n} \binom{n-1}{k-1} p^{\binom{k}{2}} (1-p)^{k(n-k)} ...
- 342
3
votes
2
answers
1k
views
Properties of matrix exponential without using Jordan normal forms
There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:
All eigenvalues of $A$ have negative ...
- 791
1
vote
1
answer
161
views
Proof of Convergence + Identifying Probability Distribution
I'm trying to prove that the series below converges to 1 and I noticed it looked strikingly similar to a probability distribution I once saw. My question is twofold:
Can anyone identify the ...
- 113
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k
4
votes
1
answer
753
views
Lebesgue-Besicovitch theorem for partition elements rather than balls
I'll state the classic result in its density (rather than the more general differentiation) version. Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ and $A\subset \mathbb{R}^n$ ...
- 6,541
0
votes
0
answers
93
views
What is the class of real sequences satisfying these conditions?
I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions:
$\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
16
votes
2
answers
528
views
Lipschitz constant for map between triangles
Let $T_1$ and $T_2$ be any two euclidean triangles with labeled sides. The sides are labeled respectively $e_1^1,e_2^1,e_3^1$ and $e_1^2,e_2^2,e_3^2$. Call $A:T_1\rightarrow T_2$ the affine map which ...
user99087
1
vote
1
answer
395
views
On rank of random $0/1$ matrices
It is known that a $0/1$ matrix picked from uniform distribution from $\{0,1\}^{n\times n}$ is non-singular with probability $1-o(1)$.
Fix an integer $t$.
Consider a random matrix formed the ...
user94040
9
votes
3
answers
398
views
Countable shifts of closed positive sets
Let $\mu$ be the Lebesgue measure, and $+$ be addition modulo $1$ in the interval $[0,1)$.
Question1: Is there a closed set $C\subset [0,1)$ of positive measure such that for any countable set $D\...
- 135
2
votes
0
answers
92
views
Estimating the size of a subset of $\mathbb{R}^N$
This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
- 2,480
2
votes
1
answer
290
views
Any viscosity solution must be the distance function?
Suppose $U \subseteq \mathbb{R}^d$ is open and bounded. Is it possible anybody could supply a simple proof that any viscosity solution of$$\begin{cases} |Du| = 1 & \text{in }U \\ u = 0 & \text{...
- 349
4
votes
1
answer
470
views
Covering measure one sets by closed null sets
(The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.)
For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval
$[0,1]$, define
$$\newcommand{\card}[1]{\...
- 135
1
vote
0
answers
63
views
Direct proof of fact $u \in C(U)$ satisfies $|Du| \ge 1$ in sense of viscosity if and only if property holds
Is it possible anybody could sketch me a direct proof of the fact that $u \in C(U)$ satisfies $|Du| \ge 1$ in the sense of viscosity if and only if the following property holds?
If $V \subseteq U$ is ...
- 349
3
votes
1
answer
239
views
Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying following
Suppose $U \subsetneq \mathbb{R}^d$ is open. How do I see that the distance function$$u(x) = \min_{y \in \mathbb{R}^d \setminus U} |x - y|$$is the unique nonnegative continuous function on $\mathbb{R}^...
- 349
5
votes
2
answers
341
views
a modification on an infinite Bernoulli convolution
The distribution $\nu_{\lambda}$ of the random series $\sum\pm\lambda^n$ is the infinite convolution product of $\frac12(\delta_{-\lambda^n}+\delta_{\lambda^n})$. This problem has been studied ...
- 43.2k
2
votes
1
answer
251
views
Automorphism on the unit interval compatible with a measure preserving set function
Cross-posting from math stack-exchange since it's not getting any visibility there.
I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \...
- 4,466
1
vote
1
answer
392
views
Integral kernel smooth
Assume that $Tf(x):=\int_{\mathbb{R}^n} K(x,y)f(y) dy$ is an operator such that $T \in L( H^{-k}, H^k)$ is continuous for any $k$, where $H^k$ is the $k-th$ order Sobolev space on $\mathbb{R}^n$.
...
- 11
3
votes
3
answers
233
views
sequencial shift on families =flipped powers. How?
Consider the following family of functions
$$f_n(w):=\sum_{k=0}^{\infty}\frac{(-1)^{k-1}}{k!}(k+n)^{k-1}w^k.$$
QUESTION 1. Does the following hold?
$$f_n(w)=-\frac1{n(f_{-1}(w))^n}.$$
Deeper ...
- 43.2k
7
votes
3
answers
369
views
Does a certain contractive mapping have a fixed point?
Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying
$$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$
where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...
- 169
2
votes
0
answers
100
views
Roots of a partially holomorphic function
Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\...
- 16.2k
2
votes
1
answer
497
views
Are the partial derivatives of a function increasing in both variables measurable?
Let $f$ be a function from $[0,1]\times[0,1]$ to $\mathbb{R}$ that is nondecreasing in both variables, i.e. $f(x_1,y_1)\le f(x_2,y_2)$ whenever
$x_1\le x_2$ and $y_1\le y_2$. It is known that the ...
2
votes
1
answer
404
views
Sturm Liouville problems for non-classical orthogonal polynomials
It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$
My ...
- 3,574
5
votes
2
answers
1k
views
Derivatives of $C^{\infty}$ non analytic function
Question: Given $f\in C^{\infty}$ which is not analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - ...
- 3,574
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
10
votes
1
answer
326
views
Partition into sets of positive outer measure
Let $\mu^{\star}$ denote Lebesgue outer measure. Suppose $X \subseteq [0, 1]$ and $\mu^{\star}(X) > 0$. Can we divide $X$ into uncountably many sets $\{X_i : i \in I\}$ such that for every $i \in I$...
- 121
8
votes
2
answers
980
views
Lebesgue outer measure
Denote the Lebesgue outer measure by $\mu^{\star}$. Is there a subset $X \subseteq [0, 1]$ such that $\mu^{\star}(X) > 0$ and $\mu^{\star} \upharpoonright \mathcal{P}(X)$ is a measure (countably ...
- 89
2
votes
1
answer
63
views
Decompose a function having antiderivatives into bounded components [closed]
Suppose $f:I\rightarrow\mathbb R$ has antiderivatives on an interval $I\subset\mathbb R$. Then $f$ can be decomposed as $f=g+h$, where both $g,h:I\rightarrow\mathbb R$ have antiderivatives. In ...
- 123
3
votes
1
answer
941
views
What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
- 8,071
2
votes
1
answer
337
views
Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
- 23