All Questions
5,657 questions
7
votes
1
answer
463
views
Boundedness of total current in electrical network
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
- 1,495
7
votes
1
answer
504
views
Anisotropic perimeter and regularity of anisotropic minimal surfaces
1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...
- 980
7
votes
4
answers
3k
views
Upper bound of the expectation of sum of the absolute value pairs
We have two arrays $A,B$ of length $n$. All values are i.i.d drawn from same distribution on $[0,1]$. If we sort $A,B$ in non-decreasing order and let $A_{(i)},B_{(i)}$ denote the i-th value in the ...
user124861
7
votes
1
answer
386
views
A game-theoretical question in a political economy model
My research question in a dynamic model of political competition boils down to the following conjecture. I am confident that it holds (all simulations work), but I have not been able to prove it yet. ...
- 81
7
votes
2
answers
268
views
Meeting a set of lines in $\mathbb{R}^n$
Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?
$M$ intersects all the elements of ${\...
- 48.9k
7
votes
1
answer
317
views
Improving Baumgartner's result?
Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: ...
- 71
7
votes
1
answer
798
views
Intersection of connected components in $\mathbb{R}^n$
Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^* \in \mathbb{R}^n\setminus K$.
Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains $x^*$....
- 48.9k
7
votes
1
answer
1k
views
Is the absolutely continuous image of a nowhere dense set is also nowhere dense?
Let $f: [a, b] \subseteq \mathbb{R} \to \mathbb{R}$ be an absolutely continuous map. Does $f$ map a nowhere dense subset of $[a, b]$ to a nowhere dense set?
Remarks:
The answer is "no" if $f$ is ...
- 5,339
7
votes
1
answer
271
views
Sequential continuity and the Axiom of Choice
It is well-known that ZF cannot prove the following:
"for a function $f$ from reals to reals and any real $x$, $f$ is continuous at $x$ if and only if $f$ is sequentially continuous at $x$."...
- 4,359
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
$$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N} \sum_{...
- 6,213
7
votes
2
answers
706
views
Poisson binomial conjecture
Let $X_i\in\{0,1\}$
be mutually independent and distributed according to $\mathrm{Bernoulli}(p_i)$
and similarly, $Y_i\sim\mathrm{Bernoulli}(q_i)$,
for some parameters $p,q\in[0,1]^n$. Put $X:=\sum_{i=...
- 6,541
7
votes
1
answer
185
views
Question on ODE involving mollifiers from Taylor's book on PDEs
In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form
$$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$
with some initial condition $u(...
- 1,171
7
votes
1
answer
205
views
Comparing divergent and convergent sums
Let $(x_n)$ be a monotonically decreasing sequence of positive real numbers that is also summable.
Let $(y_n)$ be a sequence of positive real numbers such that
$\sum_n x_n y_n$ converges.
Let $(z_n)$ ...
- 124
7
votes
1
answer
396
views
Existence of complex function?
Motivated by a similar question Complex-doubly periodic function in two variables?, I would like to ask if there exists a non-zero function $(z_1,z_2) \mapsto f(z_1,z_2)$, where $z_1,z_2 \in \mathbb C$...
- 124
7
votes
2
answers
632
views
Determining if a quadratic form is non-negative if variables are non-negative
Let $f(x_1,\dots,x_n) = \sum_{1 \le i \le j \le n} c_{i,j}x_ix_j$ be a homogeneous quadratic form. Is there a quick-ish way to determine whether $f(x_1,\dots,x_n) \ge 0$ for all $x_1,\dots,x_n \ge 0$?
...
- 1,355
7
votes
1
answer
449
views
Stronger version of Besicovitch covering theorem
I'm wondering if the following strengthening of the Besicovitch covering theorem holds: Suppose $A\subset\mathbb R^n$ is a bounded subset and suppose $x\mapsto r_x$ is a function $A\to(0,\infty)$. Is ...
- 1,761
7
votes
2
answers
340
views
Sign-oscillations for power series with random coefficients
Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<...
- 19.7k
7
votes
1
answer
532
views
how wiggly is a generic level set?
Typical level sets of smooth real-valued functions are manifolds, so they cannot be fractals. If we coarse grain a bit though, sometimes we get space-filling behavior, eg. every point could be within ...
- 73
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 ...
7
votes
1
answer
274
views
Does this "local time" type limit exist a.e. for $C^2$ functions?
For $f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\...
- 6,213
7
votes
2
answers
533
views
A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited
Can we find a counterexample to the following assertion?
Assume that $f:[-1,1]\to [-1,1]$ an odd function of class $C^3$, and assume thaht $f$ is a concave increasing diffeomorphism of $[0,1]$ onto ...
- 333
7
votes
1
answer
625
views
Possible application of divergence Theorem?
suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e.
$$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$
then, for every ...
- 51
7
votes
1
answer
311
views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
- 71
7
votes
1
answer
277
views
Convergence in Lebesgue measure
It is well known that if $K_n$ are compact sets in $\mathbb{R}^n$ converging in Hausdorff distance to $K$ compact as well, then it does not follow that their Lebesgue measures converge (even if the ...
- 73
7
votes
2
answers
1k
views
Schauder basis $L^p(\mathbb{R})$
Let $\{e_{n}(x)\}_{n=0}^{\infty}$ be orthnormal basis of Hilbert space $L^2(\mathbb{R})$. If $\{e_{n}(x)\}_{n=0}^{\infty} \subset L^p(\mathbb{R})$ for some $p\geq 1$, is the $\{e_{n}(x)\}_{n=0}^{\...
- 259
7
votes
2
answers
257
views
$f$ locally (Lebesgue) integrable function on real line, $g(x):= \lim _{r\to \infty} \frac 1r \int_{x-r}^{x+r} f(t) dt$ exists for every real $x$
Let $f : \mathbb R \to \mathbb R$ be a function such that $f \in L^1[-a,a] , \forall a \in (0,\infty)$ and $g(x) : = \lim _{r\to \infty} \dfrac 1r \int_{x-r}^{x+r} f(t) dt$ exists in $\mathbb R$ for ...
user111492
7
votes
1
answer
503
views
Poincaré inequality for curl-integrable functions
Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
- 632
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
- 71
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 19.6k
7
votes
1
answer
736
views
Should coffee machines be deconcentrated?
We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the ...
- 1,399
7
votes
2
answers
419
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 217
7
votes
1
answer
166
views
Asymptotics of truncated logarithm on a cricle
Consider $f_n(x) = \min_{|z|=x} \Re \sum_{j=1}^{n} \frac{z^j}{j}$, a real function of positive variable $x>0$.
I am interested in lower bounds on $f_n(x)$. Specifically, I ask: what lower bounds ...
- 14.6k
7
votes
1
answer
414
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 536
7
votes
1
answer
1k
views
An equivalent condition for differentiability almost everywhere?
Given a function $f \in L^1 (\mathbb R)$, define the roughness $R_f$ of $f$ at $x \in \mathbb R$ by
$$\DeclareMathOperator{\esssup}{\operatorname{esssup}}
R_f (x) := \limsup_{r \to 0+}\dfrac{r \...
- 6,213
7
votes
1
answer
389
views
the fractional integration method of the proof of Stein-Tomas theorem?
In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in ...
- 429
7
votes
1
answer
282
views
Kolmogorov superposition on the Hilbert Cube
A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form
$$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...
- 12.5k
7
votes
2
answers
997
views
Uniform continuity of heat semigroup
I would like to illustrate my question with an example:
It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup.
It ...
- 536
7
votes
1
answer
624
views
Expectation involving maximum of Gaussian variables
Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
- 1,127
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
7
votes
1
answer
306
views
An indicator of a planar subset as an element of a tensor product
Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...
- 452
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 128k
7
votes
1
answer
364
views
Function of two sets
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
- 1,209
7
votes
1
answer
235
views
A criterion on a vector field for its flow to extend continuously at $t=\infty$
In my work in algebraic topology I need to build a special homotopy and I came up with a construction based on some ordinary differential equation in which I am not an expert. I miss some argument to ...
7
votes
1
answer
1k
views
Signed variant of the Flint Hills series
I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one:
$$
\sum_{n = 1}^\infty \...
- 17.2k
7
votes
1
answer
735
views
Conserved positive charge for a PDE
Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$:
\begin{equation}
\frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
- 642
7
votes
1
answer
374
views
Is each $G_\delta$-measurable map $\sigma$-continuous?
Definition. A function $f:X\to Y$ between topological spaces is called
$\bullet$ $G_\delta$-measurable if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G_\delta$ in $X$;
$\...
- 41.9k
7
votes
1
answer
310
views
Asymptotic behavior of a sequence of functions
For $n\in\mathbb{N}$ and $q\in(0,1)$, define
$$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$
...
- 2,188
7
votes
1
answer
234
views
When is this sum of perfect powers bounded
For any positive integers $n,d$, let
$$
A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)}
$$
It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is
a polynomial of degree $2d-2$. Then we ...
- 621
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
7
votes
1
answer
772
views
Maximal ideals of the rings of Baire-One Functions
A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all $x\...
- 1,788