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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: ...
Jan Bohr's user avatar
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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?
Ali Taghavi's user avatar
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^...
Tian LAN's user avatar
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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 ...
cnikbesku's user avatar
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84 views

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 ...
Pitbull's user avatar
  • 131
3 votes
0 answers
86 views

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. ...
Luís Ferreira's user avatar
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(\...
Houa's user avatar
  • 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 $[0,\lambda]$-...
an_ordinary_mathematician's user avatar
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 $p=\pi/2$...
AlphaHarmonic's user avatar
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 ...
Fawen90's user avatar
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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 ...
Delio Mugnolo's user avatar
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{...
Ben Deitmar's user avatar
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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 ...
Wreck it Ralph's user avatar
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 $\sigma>0$ ...
NancyBoy's user avatar
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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}/\...
Isaac's user avatar
  • 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(...
Sam Sanders's user avatar
  • 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)\...
Thomas Kurbach's user avatar
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 ...
Sam Sanders's user avatar
  • 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)=...
J.Mayol's user avatar
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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....
Isaac's user avatar
  • 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 ...
Sam Sanders's user avatar
  • 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 ...
leo monsaingeon's user avatar
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 < \...
JZS's user avatar
  • 481
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 ...
newbie's user avatar
  • 53
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0 answers
156 views

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 $\...
Saúl RM's user avatar
  • 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&...
SoG's user avatar
  • 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) \\ &...
Dejv's user avatar
  • 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 $n=1$, ...
Pii_jhi's user avatar
  • 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-\...
Liviu Nicolaescu's user avatar
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 ...
tommy1996q's user avatar
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 ...
ABIM's user avatar
  • 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 ...
Sam Forster's user avatar
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 ...
AgnostMystic's user avatar
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 ...
Sam Sanders's user avatar
  • 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}...
node's user avatar
  • 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}^{\...
Jone Sweden's user avatar
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 $\mathbb{R}\flat$,...
Madeleine Birchfield's user avatar
3 votes
0 answers
122 views

Preconditioners for $Ax=y$ that rely on hierarchical statistical modeling

Solving $Ax=y$ exactly can be done as: fit a linear autoregressive model by treating rows of $A$ as data apply this model to $A^T y$ Imperfect predictive model corresponds to an approximate inverse ...
Yaroslav Bulatov's user avatar
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!...
T. Amdeberhan's user avatar
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 ...
B.Hueber's user avatar
  • 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\|\...
W.J.'s user avatar
  • 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 $n\in\mathbb{N}$. ...
Lorenzo's user avatar
  • 2,286
3 votes
0 answers
185 views

Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix

The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
Mark Wildon's user avatar
  • 11.2k
3 votes
0 answers
204 views

The inversion formula for the square root of a positive function

Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
ABB's user avatar
  • 4,058
3 votes
0 answers
289 views

Functional inverse of $z=1+w+\cdots+w^{n-1}$

Migrated from the MSE. I am interested in the functional inverse of $$ z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1. $$ This function is strictly increasing on $w\geq0$ and thus admits an inverse. By ...
Aaron Hendrickson's user avatar
3 votes
0 answers
137 views

Is there a characterization of tempered functions whose Fourier transforms are also tempered functions?

A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions. This MSE question asked ...
user141240's user avatar
3 votes
0 answers
120 views

If $u_n\rightharpoonup u$ in $L^2(0,T;L^2)$ then there is a subsequence such that $u_n(t)\rightharpoonup u(t)$ almost everywhere?

If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$? I'm not sure if this question is trivial or not,...
demlevi33's user avatar
  • 153
3 votes
0 answers
119 views

Question on the model completeness of the real field expanded by restricted Pfaffian functions

Currently I'm reading "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function" by Wilkie, and I do not ...
Bytegear's user avatar
  • 123
3 votes
0 answers
187 views

Analogue of Kolmogorov/Arnold superposition for general manifolds?

Previously asked and bountied at MSE with slightly different language: Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
Noah Schweber's user avatar
3 votes
0 answers
106 views

The behavior of an integral related to the inward normal vector near a point of the boundary of a domain

Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit $$ \lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz) $$ where $...
Daniele Tampieri's user avatar

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