All Questions
1,536 questions with no upvoted or accepted answers
3
votes
0
answers
100
views
How to compute the partial derivatives of this function?
For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
- 1,389
3
votes
0
answers
45
views
Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces
Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
3
votes
0
answers
95
views
Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?
We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality.
My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
- 31
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,155
3
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0
answers
318
views
The curse of dimensionality of the Kolmogorov–Arnold neural network
The Kolmogorov–Arnold neural networks (KAN), Ziming Liu et al., KAN: Kolmogorov–Arnold Networks is inspired by the Kolmogorov–Arnold representation theorem (KA theorem). Though it is not proved in the ...
- 2,239
3
votes
0
answers
138
views
What is the probability that the absolute value of the root of a polynomial is greater than $x$?
Note: This question was unanswered in MSE for a month so posting it in MO.
Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we ...
- 5,470
3
votes
0
answers
212
views
Differentiability along hyperplanes for rational functions
This is a follow up to my previous question.
Let $f\colon \mathbb R^3\to \mathbb R$ be a continuous function that is rational and differentiable along all planes through $0$, that is, we assume:
...
- 779
3
votes
0
answers
146
views
Two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number
Are there two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number?
- 356
3
votes
0
answers
141
views
Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$
Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
- 435
3
votes
0
answers
67
views
How powerful are sequences of Steiner symmetrizations?
I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
- 171
3
votes
0
answers
84
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About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
3
votes
0
answers
86
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Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
1
answer
198
views
Can gradient zero implies that a function is constant with Hörmander vector fields
Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
- 561
3
votes
0
answers
90
views
Upcrossing lemma and subharmonic functions
I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $
\lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-...
3
votes
0
answers
179
views
Maximum of an integral
Assume that $a>0$ and $r\in[0,1)$. How to prove that the function $$f(p)=\int_{-\pi}^\pi \left (1+r^2+2 r \cos x\right)^{a/2} |(2+a) \cos(x+p)-a r \cos(p)| \, dx$$ attains its maximum for $p=\pi/2$...
3
votes
0
answers
137
views
On the continuity with respect to the increasing convex order
For $p\ge 1$, let $\mathcal P_p(\mathbb R)$ be the set of probability measures on $\mathbb R$ of finite $p^{\rm th}$ moment. Denote by $W_p$ the Wasserstein metric of order $p$ and by $\preceq$ the ...
- 1,389
3
votes
0
answers
83
views
Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
3
votes
0
answers
125
views
Extracting moments of $\max(X_1,\ldots,X_k)$ from asymptotic behavior of $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$
For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions.
Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{...
- 1,295
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
0
answers
154
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 393
3
votes
0
answers
124
views
Leibniz rule bound for the inverse of the Laplacian?
Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\...
- 3,477
3
votes
0
answers
52
views
Closely related definitions with and without approximation built-in
Let us say that a (real) function class $A$ has 'approximation built-in' in case for every $f:\mathbb{R}\rightarrow\mathbb{R}$ in $A$ and any $x\in \mathbb{R}$, we can approximate $f(x)$ using only $f(...
- 4,359
3
votes
0
answers
75
views
Separate holomorphicity implies holomorphicity on analytic varieties
Suppose that $M$ and $N$ are two complex analytic varities and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
- 513
3
votes
0
answers
59
views
Generalisation of 'derivatives are Baire 1'
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable, then its derivative $f'$ is Baire 1 (which essentially follows by the definition of derivative).
Do functions differentiable almost ...
- 4,359
3
votes
0
answers
454
views
Surprisingly difficult limit of a sequence
Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$?
Of course $|a_n| \to \infty$, but we have
$$
\operatorname{Re}(a_n)=...
- 489
3
votes
0
answers
118
views
If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?
The question is as in the title.
Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
- 3,477
3
votes
0
answers
65
views
Representation of Baire 1 functions
Upper semi-continuous functions on the reals are Baire 1, which is readily observed by considering
$$ f_{n}(x):= \sup_{y\in [0,1]}(f(y)- n |x-y| ) \qquad (A).$$
Indeed $f_n$ as in (A) is continuous ...
- 4,359
3
votes
0
answers
176
views
A variant of the Laplace principle
$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
- 5,380
3
votes
0
answers
151
views
Is there a space of smooth functions dense in the domain of Coulomb-like potentials in dimension two?
Let $V : \mathbb{R}^2 \to \mathbb{R}$ be compactly supported, bounded away from the origin, and obey
$$ |V(x)| \lesssim r^{-\delta_0}, \qquad 0 < |x| \le 1, \qquad r : =|x|,$$
for some $0 < \...
- 481
3
votes
0
answers
105
views
Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
- 231
3
votes
0
answers
94
views
Question on an integral inequality
I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...
- 53
3
votes
0
answers
156
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Growth of the constants from the Stone-Weierstrass Theorem
The Stone Weierstrass theorem for $C([0,1])$ claims that for any continuous function $f:[0,1]\to\mathbb{R}$ and each $n\in\mathbb{N}$, there is a polynomial $p_{n,f}(x)=\sum_ia_{f,n,i}x^i$ such that $\...
- 10.6k
3
votes
0
answers
191
views
Does "Invariance of domain" hold true for injective Darboux function (instead of continuous injection)?
Let $f \colon U\subset \mathbb{R^n}\to\mathbb{R}^n$ be an injective Darboux map.
Does this imply that $f$ is an open map?
If $f$ is continuous then the result follows from "Invariance of domain&...
- 307
3
votes
0
answers
216
views
Harmonic polynomial of degree 3
Let $f:\Bbb R^3\to\Bbb R^3$ be a function defined by
$$
\begin{split}
f(x,y,z) = & \,\Big\{a_1 x y z +a_2\left(-x^3+3 x y^2\right) +a_3\left(3 x^2 y-y^3\right) +a_4\left(3 y^2 z-z^3\right) \\
&...
- 81
3
votes
0
answers
40
views
Bound of a regular function that cancels at some points
Let $K$ be a bounded convex set of $\mathbb{R}^n$ and $x_1,\ldots,x_k\in K$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function that cancels on points $x_1,\ldots,x_k$ . When $n=1$, ...
- 121
3
votes
0
answers
161
views
Distribution of harmonic sums mod 1
This is only to satisfy my curiosity. Consider the harmonic sums
$$ H_n =1+\frac{1}{2}+\cdots +\frac{1}{n},\;\;n=1,2,\dotsc, $$
and denote by $h_n$ their mod $1$ reductions,
$$ h_n=H_n\bmod 1=H_n-\...
- 34.7k
3
votes
0
answers
92
views
Questions about article "Ordinary differential equations, transport theory and Sobolev spaces" by DiPerna-Lions
I am reading the article, and I am more or less halfway through it. I have some questions though on some parts I am not understanding, so I wanted to ask about these here. I apologize for listing the ...
- 697
3
votes
0
answers
182
views
Rate of uniform approximation by piecewise constant functions
Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...
- 5,405
3
votes
0
answers
119
views
Is (the generalised) Sard's theorem optimal?
As mentioned in this question (https://math.stackexchange.com/questions/416607/show-that-fc-has-hausdorff-dimension-at-most-zero/446049#446049), in 1965 Sard proved the following result (paraphrased ...
- 700
3
votes
0
answers
105
views
Recursive differences of Cantor set
Let $C$ be the Cantor ternary set obtained by repeatedly deleting the middle thirds of the interval $[0,1].$ Now define
$$E_1=C$$
and $$E_{i+1}=\{ |x-y|,x \neq y\text{ and } \,x,y \in E_i \}$$
I ...
- 826
3
votes
0
answers
638
views
Complexity of modulus of convergence of Baire 1 function
A Baire 1 function on the reals is the pointwise limit of a sequence of continuous functions. Assuming a bounded Baire 1 function on the unit interval, can we say anything about the modulus of ...
- 4,359
3
votes
0
answers
146
views
A uniqueness result for the Neumann problem for the Laplace equation
Let $\Omega \subset \mathbb{R}^{3}$ be a $C^{1}$-domain, not necessarily bounded. Consider solutions $\phi : \overline{\Omega} \to \mathbb{R}$, $\phi \in C^{\infty}(\Omega) \cap C^{1}(\overline{\Omega}...
- 351
3
votes
0
answers
114
views
Choose a sub series of a random series, such that its expectation can be a given real number
Suppose $a>0$, and we have an infinite series of Bernoulli random variables $B_k$ with
$$\mathbb{Pr}{\large[}B_k=1{\large]} = \frac{1}{1+e^{a\cdot 2^k}}$$
Then
$$\text{E}\left[\sum_{k=-\infty}^{\...
- 43
3
votes
0
answers
74
views
Discreteness of the higher inductive-inductive Cauchy real numbers in real cohesive homotopy type theory
We work in cohesive homotopy type theory with propositional resizing, so that there is only one type of Dedekind real numbers $\mathbb{R}$ up to equivalence, and Mike Shulman's axiom $\mathbb{R}\flat$,...
- 2,624
3
votes
0
answers
315
views
When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
- 43.2k
3
votes
0
answers
99
views
Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
3
votes
0
answers
84
views
A weighted $W^{2,p}$ estimates
Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have
$$
\|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
- 379
3
votes
0
answers
205
views
Uniform limit of pointwise limits of continuous functions
Let $X$ be topological spaces, $Y$ a metric space and $(f_n)_{n\in\mathbb{N}}$ a sequence of functions, with $f_n:X\rightarrow Y$ pointwise limit of continuous functions for each $n\in\mathbb{N}$. ...
- 2,286