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-4
votes
0answers
44 views

Proof : Limit of a sequence [on hold]

Prove from the definition of the limit of a sequence that $$\lim_{n\to\infty} \frac{2n^2+\cos(n)} {n^2+1} = 2 $$ (that is, for a given $\epsilon > 0$, find an explicit $N_\epsilon$) Please ...
-4
votes
0answers
43 views

Differentiability of a function [on hold]

Can someone help me out to give a bound of the first derivative w.r.t. $x$ of the function $\min_{x,y\in\Omega}\{1,\frac{d(x,\partial\Omega)d(y,\partial\Omega)}{|x-y|^2}\}$?. ...
-2
votes
0answers
24 views

Geometry of generalized Lipschitz functions [on hold]

Clearly, that geometry of Lipschitz function is a double cone whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone (image from wiki). I've found ...
0
votes
0answers
46 views

What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$ \begin{equation} V_{ij}^{kl}=V_{ji}^{lk} \end{equation} (So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...
5
votes
1answer
473 views

What is the value of the infinite product: $(1+ \frac{1}{1^1}) (1+ \frac{1}{2^2}) (1+ \frac{1}{3^3}) \cdots $? [on hold]

What is the value of the following infinite product? $$\left(1+ \frac{1}{1^1}\right) \left(1+ \frac{1}{2^2}\right) \left(1+ \frac{1}{3^3}\right) \cdots $$ Is the value known?
-1
votes
1answer
110 views

Analysis of Sobolev spaces [on hold]

I just wanted to know wthether the following is OK or not. Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
3
votes
0answers
104 views

Generalisation of Lebesgue differentiation theorem to Orlicz spaces

If $f\in L_{p}^{\rm loc}(\mathbb{R}^{n})$ and $1\leq p<\infty$, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow ...
2
votes
1answer
47 views

Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets

Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over ...
3
votes
1answer
66 views

Blow-Up for Semi-Linear Wave Equations

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
0
votes
0answers
73 views

if $u_\epsilon \rightarrow u$ weakly in $L^2$ then also $\partial_t u_\epsilon \rightarrow \partial_t u $ weakly in $L^2$

I am looking for a citation of the above claim, I am not sure under what conditions on the sequence $u_\epsilon$ exactly this applies. $u_\epsilon$ is a sequence of functions that depend on $x,t$ ...
36
votes
1answer
358 views

Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longmapsto \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in ...
0
votes
2answers
128 views

Are there dense sets of positive but not full measure? [closed]

This topic came out today during a discussion with a colleague. I realized that a counter-example to his claim could be constructed if there exits a subset $A \subset [0,1]$ such that $0 < \mu(A) ...
4
votes
1answer
76 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
7
votes
2answers
584 views

Stokes theorem with corners

I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19: Notation: for $1 \le n \le m$ $\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ ...
7
votes
2answers
520 views

Is it true that for each bounded continuous function we can find a set of analytic functions to uniformly converge it?

Is it true that for each bounded continuous function $f:\mathbb R \to \mathbb R$, we can find a set of analytic functions $g_i:\mathbb R \to \mathbb R, i=1,2,...$ such that $g_i$ uniformly converges ...
-2
votes
0answers
46 views

What special role plays the function $\pi^{\frac x\pi}$ in analysis? [migrated]

I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ...
1
vote
0answers
51 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
0
votes
0answers
7 views

Exterior power of multilinear functions applied to linearly dependent vectors is zero [migrated]

I'm working on a homework problem, and we are to show that if $T \in \wedge^p V^*$, and $v_1,\ldots,v_p$ are linearly dependent, then $T(v_1,\ldots,v_p) = 0$. What I've got so far: I understand that ...
0
votes
1answer
112 views

Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at $$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$ Now, I want to show that we don't have equality ...
0
votes
0answers
57 views

What does integrability of a strictly monotonic function imply about the tails of that function?

In particular, if $f:\mathbb{R}_{+}\rightarrow[0,1]$ is a strictly monotonic decreasing function and $f$ is integrable then does it necessarily hold that $f^{-1}(1/t)=o(t)$?
3
votes
1answer
125 views

Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla ...
6
votes
1answer
161 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 ...
2
votes
2answers
292 views

Bi-Lipschitz constant of arc-length parametrisation of convex curve

Assume that $f:[0,2\pi]\to [0,2\pi]$ is a an increasing diffeomorphism, and let $\underline f = \min f'$ and $\overline f = \max f'$ and define $$g(s) = \int_0^s e^{if(t)} dt.$$ Assume that ...
2
votes
0answers
95 views

About the small set expansion conjecture

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - ...
0
votes
0answers
29 views

A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...
4
votes
1answer
124 views

Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case. Question: Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
0
votes
1answer
85 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
1
vote
1answer
48 views

Graph of bounded continous functions with distance 1

Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$ We set $E = \{\{f,g\}: f,g \in ...
33
votes
6answers
1k views

On an example of an eventually oscillating function

For $x\in(0,1)$, put $$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$ This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...
-2
votes
0answers
22 views

Interchange summation and differentiation [migrated]

I asked this question already on math.stackexchange, but did not receive any answers see here Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that ...
0
votes
0answers
22 views

Deriving inequalities from a polynomially-bounded derivative

In this paper (p. 2, definition/remark) the following notion of ‘polynomial growth’ is defined for a non-negative real function $g(x)$ and a real constant $b\in(0;1)$: There exist positive ...
1
vote
1answer
61 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{ ...
1
vote
1answer
72 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
6
votes
1answer
96 views

Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that $$ ||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ? $$ where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
1
vote
1answer
66 views

“Schwarz symmetrization” on annulus

If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...
2
votes
1answer
153 views

Asymptotic behaviour of eigenvalues

If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
0
votes
1answer
122 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
vote
0answers
82 views

Summing a function at integer points

For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum $$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$ If $F_f(y)$ is defined for all $y$, it is periodic of period 1. ...
3
votes
1answer
95 views

Extension of Sobolev Functions

Let $\,D\subseteq\mathbb{R}^{n-1}$ be a convex bounded domain. Let$A:D\to(0,\infty)$ be a Lipschitz continuous function. Let $\,\Omega\,$ be bounded domain in $\,\mathbb{R}^{n}\,$ of the form ...
2
votes
1answer
117 views

Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?

I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that $$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...
0
votes
1answer
80 views

Number of critical points

Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and ...
1
vote
0answers
79 views

Set nor its compliment contain an uncountable closed set [closed]

Does there exist a set $X$ subset of the real numbers such that no uncountable closed set is contained in $X$ or $X^c$?
1
vote
1answer
78 views

What is the fractional derivative smoothness of functions from the Zygmund class?

Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional ...
3
votes
0answers
67 views

On Rényi entropy/divergence

The Rényi entropy for a probability density function $f$ with dominating measure $\mu$ of order $\alpha>0$ is defined as $$H_\alpha(f)={1 \over {\alpha-1}}\log\int f^\alpha d\mu.$$ If $f$ is ...
0
votes
0answers
35 views

minimal entropy approximation of a truncated discrete measure

Consider a measure $\mu$ on $\mathbb{N}$ given by the sequence $(\mu(n))_{n \geq 0}$ with $\mu(0)>0$. For example $\mu(n)=n^2+1$ on the figure below. For each $n$, let $X_n \sim \mu(\cdot \mid ...
0
votes
0answers
67 views

Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question: Given ...
1
vote
1answer
126 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
4
votes
1answer
122 views

Orlicz Norm and A result on expectation

I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined: Consider arbitrary, non-negative, convex function ...
7
votes
4answers
494 views

Inserting an open and simply-connected set between a compact set and an open set

In a paper I am reading, the following is considered obvious: Let $K$ be a compact and connected subset of $\,\mathbb R^2$, with $\mathbb R^2\smallsetminus K$ also connected, and $U\subset \mathbb ...
15
votes
1answer
359 views

Integrals of pullbacks and the Inverse function theorem(s?)

The usual story goes like this: Smooth picture (?): For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following is true: $\phi^{-1}$ is a local diffeomorphism a.e. ...