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5 votes
2 answers
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Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?

Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with $f_n \to f$ uniformly for some continuous $f$. $f'_n - g \to 0$ in $L^\infty$ for some measurable $g$, where we ...
Nate River's user avatar
  • 6,215
0 votes
0 answers
95 views

Functions representing all strings somewhere

Do there exist "nice" (maybe analytic?) functions $f_0,f_1:\mathbb R \to \mathbb R$ such that $\forall n\in\mathbb N,\forall \sigma\in\{0,1\}^n,\exists x\in\mathbb R, \forall \tau\in\{0,1\}^...
Bjørn Kjos-Hanssen's user avatar
2 votes
1 answer
203 views

Does this maximisation problem admit a finite upper bound?

Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
Fawen90's user avatar
  • 1,399
2 votes
2 answers
192 views

Upper/Lower bounds of real-analytic functions with infinite Taylor series

For example, in 1-D, given some positive increasing polynomial $p(x) = a_1x+\ldots+a_nx^n$, $p(0) = 0$, there exists constants $b_1,b_2$ such that for $x<\delta$, for some $\delta > 0$, we have ...
Doofenshmert's user avatar
0 votes
0 answers
120 views

Equality of two measures on functional spaces

It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
Anico's user avatar
  • 1
10 votes
1 answer
518 views

Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions

Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
Tian LAN's user avatar
  • 435
0 votes
0 answers
128 views

Lipschitz function approximated by smooth functions with zero a regular value

Consider a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$. Then I want a family of smooth functions $f_\epsilon : \mathbb{R}^n\to\mathbb{R}$, such that $f_\epsilon\to f$ uniformly on compact sets, ...
shadow10's user avatar
  • 1,090
2 votes
0 answers
259 views

Least number of circles required to cover a continuous function on $[a,b]$

I asked this question on MSE here. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of closed circles with fixed radius $r$ required to cover the graph of $f$? It is ...
pie's user avatar
  • 541
11 votes
2 answers
587 views

Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density

Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
confused's user avatar
  • 271
6 votes
1 answer
309 views

Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?

Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure. Let $f \in C^1 (\mathbb \Omega)$ for some open, connected, bounded subset $\Omega$ of $\mathbb R^n$. We consider for each $t \...
Nate River's user avatar
  • 6,215
0 votes
0 answers
48 views

First nonzero derivative bounded below (2 dimensions)

Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real-analytic, have only one zero (at $(0,0)$) and be strictly increasing ...
Doofenshmert's user avatar
0 votes
0 answers
96 views

Hilbert spaces that include algebraic polynomials

This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
FDK's user avatar
  • 1
-2 votes
1 answer
93 views

Express the connection between roots [closed]

$\DeclareMathOperator\elim{lim}\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\lmb{lmb}\DeclareMathOperator\Lmb{Lmb}\DeclareMathOperator\mts{mts}$There are two similar functions; they determine the ...
Luke's user avatar
  • 55
6 votes
2 answers
380 views

Proving convergence of solution of a fixed point equation

I encountered a nasty sequence $(x_n)_{n=1}^\infty $ defined as the smallest positive fixed point of the fixed point equation $ x_n = f_n(x_n) $, where $f_n$ is given by $$ f_n(x) = \sum_{k=0}^{\...
user24334'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
  • 435
5 votes
1 answer
349 views

Equilateral triangle in a Brownian path

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory ...
NancyBoy's user avatar
  • 393
3 votes
0 answers
68 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
  • 171
2 votes
2 answers
173 views

Gronwall-type inequality involving norms of distinct Lebesgue spaces

Let $d \geq 1$, $\Omega \subset \mathbb{R^d}$ be a bounded domain and let $\phi : [0,T]\times \Omega \mapsto \mathbb{R}$ be a measurable and bounded function. Assume that the following differential ...
Theleb's user avatar
  • 213
5 votes
1 answer
374 views

What is the length of an algebraic curve?

The following question seems to be somewhat standard, but I was unable to find any reference. I would be grateful for any pointers to relevant literature. We consider a real polynomial $p(x,y)$ of ...
user528052's user avatar
2 votes
1 answer
194 views

Functions with derivatives growing at rate $r>0$

Fix a non-empty closed subset $\Omega\subset\mathbb{R}$. Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and such that $\sup_{x\in \Omega}\,|\partial^k f(x)|\lesssim k^r$ for some $r\ge 0$ for all $k\in \...
Math_Newbie's user avatar
1 vote
0 answers
215 views

Computing a closed form representation for a Fourier series summation

I want to compute a closed form representation for the below given summation expession. $$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
Rajesh D's user avatar
  • 698
0 votes
0 answers
29 views

$ \sup_{\theta \in [0,2\pi)}\max_{r\leq \delta}\frac{\log\left(\frac{f(r,\theta)}{f(\delta,\theta)}\right)}{\log(r)}<\infty,$ $f$ real analytic

$\textbf{Conjecture.}$ Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $\delta <1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and suppose that $(0,0)$ is the only ...
Doofenshmert's user avatar
4 votes
2 answers
354 views

Injectivity of a convolution operator

Let $p,\mu,\nu$ be probability density functions on $\mathbb{R}$ such that $$ \int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x). $$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
Ribhu's user avatar
  • 407
0 votes
0 answers
106 views

How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin

Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
Rodrigo's user avatar
  • 51
0 votes
0 answers
49 views

ODE satisfied by a special function

Posted on MSE Context I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica . ...
NancyBoy's user avatar
  • 393
0 votes
0 answers
71 views

Minimum Slice of Real Analytic Function in Two Variables

Let $B\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and have only one zero, namely $(0,0)$. Moreover, assume that $...
Doofenshmert's user avatar
2 votes
2 answers
191 views

Gronwall's inequality in discretized time

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
  • 835
1 vote
0 answers
82 views

Counting the number of local minima of a function that is the sum of square roots of cosines

Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows $$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$ where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
loizuf's user avatar
  • 21
3 votes
1 answer
233 views

Analytic solutions to analytic differential equations

Let $U \subseteq \mathbb R^{n+2}$ be an open set for some $n \geq 0$, and let $f: U \to \mathbb R$ be an analytic function. Then we say the equation $f(x,y,y',\ldots,y^{(n)})=0$ is an analytic ...
cubicquartic's user avatar
6 votes
1 answer
309 views

Well distributed sets

Note: All integrals are taken with respect to Lebesgue measure. The symbol $\def\avint{\mathop{\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int}\nolimits} \avint$ denotes the average integral. We say ...
Nate River's user avatar
  • 6,215
1 vote
0 answers
67 views

Distribution of zeros for arbitrary Bessel functions

Consider the ODE $x^2 y''+x y' + (x^2-\alpha^2)y = 0$, where $\alpha$ is an arbitrary positive irrational number that is less than $ 2 \pi$. Let $J_{\alpha}(x)$ be a solution to the equation and ...
Literally an Orange's user avatar
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
9 votes
2 answers
424 views

Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?

This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here. For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
yummy's user avatar
  • 193
1 vote
1 answer
111 views

How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
10 votes
1 answer
668 views

On Pareto functions

The Pareto principle says that the top 20% of wealthy people people hold over 80% of the wealth. Suppose we had a non-negative function on $\mathbb R^n$ that satisfied this principle on every open ...
Nate River's user avatar
  • 6,215
2 votes
0 answers
207 views

Seeking alternative elementary proof instead of applying Lojaseiwicz's inequality for $f(x,y) \geq c (x^2+y^2)^{\frac{M}{2}}$

Let $B\subseteq \Bbb{R}^2$ be a closed ball centered on $(0,0)$ of radius $0<\delta<1$. Let $f:B\to \Bbb{R}_{\geq 0}$ be real analytic and contain only one zero in $A$, namely $(0,0)$. In other ...
Doofenshmert's user avatar
1 vote
1 answer
75 views

Lower bound of $\frac{f(x)}{x^{n+1}}$

Let $f:[0,a]\to \Bbb{R}_{\geq 0}$ be real analytic, $a<1$. Furthermore, $f(0) = 0$ and $f$ is strictly increasing on $[0,a]$. Let $n\in \Bbb{N}$ be the smallest positive integer such that $f^{(n)}(...
Doofenshmert's user avatar
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(...
B.Hueber's user avatar
  • 1,171
89 votes
1 answer
21k views

Is the largest root of a random polynomial more likely to be real than complex?

This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO. The number of real roots of a ...
Nilotpal Kanti Sinha's user avatar
1 vote
0 answers
64 views

Reference request for non-banded Toeplitz matrix

I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix. I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
hos's user avatar
  • 11
2 votes
1 answer
200 views

Subset in $[0,1]^k$ with positive density

Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?: For any $A\subseteq\left[0,1\right]^k$ with the measure ...
tom jerry's user avatar
  • 349
2 votes
0 answers
135 views

Estimating an integral of the Green function in the plane

Suppose $\Omega$ is a bounded, simply connected domain, $z_{0}\in{\Omega}$ and for any $z\in{\Omega}$, $d_{z}:=\text{dist}(z,\partial{\Omega})$. I am interested in understanding the behavior of ...
David Pechersky's user avatar
0 votes
0 answers
36 views

Sufficient condition for interpolation

If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying: $Z_0=X$, $...
mejopa's user avatar
  • 101
5 votes
1 answer
196 views

What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?

Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
AlpinistKitten's user avatar
0 votes
1 answer
80 views

Orthogonal space of polynomials

Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and $$ \int_0^{+\infty} f(t) t^n ...
henrysupercool's user avatar
0 votes
0 answers
89 views

Maximal function on mixed $L^{p}$

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
User091099's user avatar
4 votes
1 answer
201 views

How much can you improve a Hölder function by composing it with another?

Let $f: [0, 1] \to \mathbb R$ be a continuous function. Define the local Hölder exponent $H(f, x)$ of $f$ at $x \in [0, 1]$ by $$H(f, x) := \sup\left\{0 \leq \alpha \leq 1\mid\lim_{\delta \to 0_+} \...
Nate River's user avatar
  • 6,215
2 votes
0 answers
207 views

History of bump functions

When were the standard bump function examples such as $e^{-1/(1-x^2)}$ first understood, and what was the context or motivation at the time? As an upper bound I would guess that they must have been ...
Quarto Bendir's user avatar
2 votes
0 answers
188 views

A sharp version of a Tauberian theorem

The following Tauberian theorem is true (see Theorem I.11.1 of ''Tauberian theory: A century of developments''). Let $ a_n $ a sequence of real numbers. If $f(x) = \sum_{n=1}^\infty a_n x^n $ ...
an_ordinary_mathematician's user avatar
4 votes
1 answer
182 views

Extracting a subsequence Cesàro converging to the limsup of the Cesàro sums

Let $X_n$ be a sequence of uniformly bounded random variables — that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$. Write $\bar X_N := \frac{1}{N} \...
Nate River's user avatar
  • 6,215

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