All Questions
5,858 questions
4
votes
0
answers
141
views
Level sets of function of inner products of vectors on hypercube
Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
- 1,127
3
votes
1
answer
369
views
Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]
Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n.
We know that $f\equiv 0$. It's call Hausdorff theorem.
This theorem is wrong on $\mathbb{R^+}$, a ...
user46896
0
votes
1
answer
247
views
Proving a complicated inequality with powers of logarithms
I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...
- 1,241
2
votes
1
answer
951
views
A special case of the Divergence theorem
I am interested in the following statement:
Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
domain $U$, continuous up to the boundary $\partial U$, and vanishing on $\...
4
votes
2
answers
206
views
How to find an ODE with prescribed terminal values?
Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
2
votes
1
answer
125
views
Regularized integral and asymptotic expansion
Let $f:(0, \infty) \longrightarrow (0, \infty)$ be a monotonously increasing function (in fact, a step function) and let $P$ be a polynomial of degree $N$. Suppose I know that for some $k$, the limit
$...
- 9,853
3
votes
0
answers
77
views
Elliptic operator applied to the distance function
Let $\Omega$ and open subset of $\mathbb{R}^n$. Let us consider the following operator:
$$
\Delta_A (u)\, \, \colon= \text{div}(A \nabla u ), \qquad u \in C^{\infty}(\Omega)
$$
where $A(x)$ is a ...
- 823
5
votes
0
answers
365
views
Version of Stone Weierstrass for functions not vanishing at infinity
I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
- 51
2
votes
1
answer
761
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
2
votes
1
answer
169
views
Approximation of the cumulative normal distribution
As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...
- 2,390
4
votes
1
answer
132
views
Integral Expression in Complex Dynamics
Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...
- 41
1
vote
0
answers
99
views
simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
- 43.2k
3
votes
1
answer
390
views
An elementary inequality: reference request
Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$.
Now ...
- 6,962
0
votes
2
answers
319
views
Fixed point theorem that does not require the hemi-continuity of the set valued map?
All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...
- 383
1
vote
1
answer
189
views
Precompactness of a sequence of convex functions
Suppose we have a bounded convex open set $\Omega$ in $\mathbf{R}^n$,and a sequence of convex functions $P_n$ such that $||P_n||_{L^2(\Omega)}\leq C\forall n$.Is it possible to find a subsequence ...
- 151
3
votes
0
answers
97
views
Dimension of a graph
Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?
If not, what extra condition do we ...
- 2,135
1
vote
0
answers
91
views
Modulus of continuity of the Dirichlet Laplacian problem
I remember the following statement is correct but I cannot find a reference for that, can anybody help me to give one?
Let $\Omega\subseteq\mathbb{R}^{n}$ be an open, bounded, smooth domain,
$\varphi\...
- 31
8
votes
0
answers
433
views
Heisenberg group: function without vertical derivative
Let $\mathbb H$ be Heisenberg group with vector fields
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
and $U\subset\mathbb H$ is an open set.
I am ...
- 399
1
vote
0
answers
190
views
Roots of generalized homogeneous polynomials
A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha \...
1
vote
0
answers
200
views
Differentiability criterion in the Zygmund class
Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous
function, $ \omega $ be a modulus of continuity and assume
$$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$
whenever $ x,h \in \...
- 541
2
votes
0
answers
60
views
Finding a function in contour integration involving Riemann mapping
Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
2
votes
1
answer
157
views
Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?
Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?
user75246
9
votes
2
answers
1k
views
Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
- 1,723
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
0
votes
0
answers
471
views
Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
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
3
votes
2
answers
1k
views
Function with all but mixed second partial derivatives twice differentiable?
Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
- 107
1
vote
1
answer
2k
views
An injective smooth function with injective differential must have a continuous inverse?
Let $U \subset \mathbb R^n$ be an open subset and let $f \colon U \to \mathbb R^m$ be a $C^\infty$ function. We suppose that $f$ is injective and that the differential $Df(x)$ is injective for all $x \...
- 121
1
vote
1
answer
918
views
Pros and cons of probability model for permutations
I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by L....
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 $\...
1
vote
1
answer
166
views
Question abouth Skorokhod representation of random variables (II)
This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...
- 1,835
7
votes
1
answer
772
views
Maximal ideals of the rings of Baire-One Functions
A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all $x\...
- 1,788
4
votes
1
answer
127
views
Algorithm for definite integral of rational functions of x and exp(-x)
I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...
- 1,150
1
vote
0
answers
74
views
Is the vanishing on boundary condition for the eigenvalue problem of the $p$-Laplacian important?
Consider the eigenvalue problem of the $p$-Laplacian, $$-\Delta _p u=\lambda |u|^{p-2}u,\ u\in W_0^{1,p}(\Omega)$$
In most of the literature I saw, an extra condition is mentioned that $u$ vanish on ...
- 698
3
votes
1
answer
363
views
"exchange" of real analyticity and integration
Sorry for the impreciseness of the title. It is merely meant for an analogy.
Exchange of limiting operations and integrations are basically derived from Lebesgue's dominated convergence theorem. For ...
- 1,839
7
votes
1
answer
463
views
Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions
This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f \...
- 882
3
votes
1
answer
284
views
Measure of the same set in different models of ZF
Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.
Is $\mu^\mathcal{M}(A^\mathcal{...
- 409
3
votes
0
answers
511
views
Two (strictly related) proofs by induction of inequalities
This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
1
vote
1
answer
859
views
Continuous and dense embeddings and the density of sets in Hilbert space
Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose $B$...
- 113
4
votes
1
answer
766
views
Preimage of a smooth function
Suppose we are given a smooth function $f\colon \mathbb{R}^n \rightarrow \mathbb{R}$ and some number $c$. What can be said about the preimage $f^{-1}(c)$.
There's the theorem on regular preimages, ...
- 355
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 ...
- 4,304
13
votes
1
answer
1k
views
Which functions are Wiener-integrable?
I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them.
Background
The Wiener integral is an analytic tool to define certain "...
- 54.6k
0
votes
1
answer
237
views
Hadwiger-Nelson problem in higher dimensions
Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by
$V(\text{HN}_n) = \mathbb{R}^n$;
$E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ and ...
- 48.9k
0
votes
1
answer
1k
views
Global Implicit Function Theorem
Let $F:\mathbb R^2\rightarrow\mathbb R$ be a measurable function. Under what conditions on $F$ does there exist a function $\theta:\mathbb R^2\rightarrow\mathbb R$ such that
$F(x,\theta(z,x))=z$ for ...
- 13
1
vote
0
answers
99
views
Existence of a viscosity solution
Setup
I'm trying to find sufficient conditions for the existence of a viscosity solution to the following PDE,
$$
f(t,s,z) + \partial_sf(t,s,z) \\
+ \sum_{i=1}^{\infty} \left[
\partial_{z_i} f(t,s,z)...
- 5,405
2
votes
0
answers
73
views
A question on groupoids and measurable fields of Hilbert spaces
Suppose that we have the following data:
$ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and
range maps denoted by $ s $ and $ r $ respectively.
$ (\lambda^{x})_{x \in \...
- 1,396
3
votes
1
answer
435
views
Is the countable intersection of residual sets in [0,1] with Hausdorff dimension 1 of full Hausdorff dimension?
Let $E_k\subset [0,1]$ be residual subsets (i.e. containing dense $G_\delta $ set) with $E_{k+1}\subset E_k$ and $\dim_HE_k=1, \forall k.$ My question is : $\dim_H\bigcap_k E_k=1?$ Thanks.
- 31
5
votes
0
answers
1k
views
Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
1
vote
2
answers
259
views
Markov-Bernstein like inequalities for monotone polynomials
Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...
- 311
1
vote
1
answer
200
views
Is regularity closed under products?
Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - \frac{1-...
- 11.3k