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3
votes
2answers
683 views

Is it meaningful to work on convergencies, integration, etc. on the Zariski topology?

Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want ...
2
votes
1answer
72 views

Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that $$ \frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...
0
votes
1answer
175 views

Theorem with an example [on hold]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
-5
votes
0answers
90 views

The problem of Reimann zeta function [closed]

$\zeta(2)=\sum_0^\infty 1/n^{2}<\pi^2/6=1.644934<2$ From the popular knowledge $\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$ but $\int_0^\infty x/(e^x-1)dx=\int_0^\infty ...
-3
votes
0answers
105 views

The problem of Riemann zeta function [closed]

$\zeta(2)=\sum_0^\infty 1/n^{2}<1.8$ From the popular knowledge $\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$ but $\int_0^\infty x/(e^x-1)dx=\int_0^\infty xe^{-x}/(1-e^{-x})dx$ $=\int_0^\infty ...
13
votes
0answers
190 views

A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$? This question is motivated ...
0
votes
1answer
61 views

Equivalent of Stirling-like numbers

let $b_{n,k}$ be the numbers defined formally by $$X^n=\sum_{k=0}^n b_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod_{k=0}^{n-1}(X-k)$. I am looking for an equivalent of $b_{n,k}$ when $k$ ...
1
vote
1answer
52 views

The class of bounded uniformly continuous functions in viscosity solution theory for Hamilton-Jacobi equations

Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class $BUC(\mathbb{R}^n)$ or ...
28
votes
1answer
570 views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ...
0
votes
0answers
52 views

Under condition of Zygmund is the following inequality true?

Let $f:R\rightarrow R$ be a continuous function and satisfies the following Zygmund condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, ...
-2
votes
0answers
87 views

Is it possible to list $\mathbb{Q}$ so that the result set to be a monotoic sequence? [migrated]

Let $\mathbb{Q}$ be the set of rational numbers. Is it possible, relabeling if needed, to list $\mathbb{Q}$ such that the result set to be a monotonic sequence? If not, why? If it is true, where is ...
0
votes
0answers
85 views

Can we affirm that this function is absolutely continuous?

Suppose you have a sequence of functions $g_{\tau_n}$ ($0<\tau_n \rightarrow0$) from $]0; +\infty[$ in $\mathbb{R}$ that converges uniformly to a function $g$ in every compact interval. Suppose ...
2
votes
2answers
110 views

Regarding sub-additive sequences and Fekete's lemma

A non-negative sequence $\{a_n\}$ is sub-additive if $a_{m+n}\leq a_m + a_n.$ Fekete's lemma says that for any non-negative sub-additive sequence: $$\lim_{n\to\infty} \frac{a_n}{n} = \inf_{n} ...
0
votes
0answers
41 views

minimum distance between sets and relation with functions [closed]

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
5
votes
2answers
107 views

Smooth convex extensibility of combination of two line segments

This is a refined version of my earlier question Convex extensibility of combination of two lines. Is there a smooth function $f:[0,1]\times [0,1]\rightarrow\mathbb R$ such that for all $x\in ...
2
votes
1answer
91 views

Countable vs. ultra-negligible sets [duplicate]

A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\sum_n \vert I_n \vert \leq \epsilon$. Let us ...
14
votes
2answers
402 views

Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...
2
votes
5answers
191 views

Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
3
votes
1answer
79 views

Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions: (i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$; (ii) $f$ is non-degenerate, in the sense that there isn't a ...
0
votes
0answers
16 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
0
votes
2answers
97 views

reference needed for sobolev type estimates

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
0
votes
0answers
6 views

Looking for a trigonometric function with known shape [migrated]

I am looking for a trigonometric function where I know the shape of and also have some function values. I almost have it but my formula is not precise. If you plot 3/4-1/4*sin(3*t)-3/4*cos(6*t) you ...
1
vote
1answer
129 views

Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional Morse Theory and Multiple Solution Problems": if i have then $(4.1)$ is formal : it means that EDIT1: $(4.1)$ tel us that ...
0
votes
0answers
32 views

The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation \begin{equation*} x\,y'' + (1 - x)\,y' + n\,y = 0. \end{equation*} The associated Laguerre polynomial ...
1
vote
0answers
56 views

Multivariate monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...
3
votes
0answers
137 views

Is there such a matrix in $SO(n)$?

Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and $$ \frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = ...
0
votes
1answer
61 views

k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...
0
votes
2answers
110 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
1
vote
0answers
65 views

Geometric interpretation of Euler's identity for homogeneous functions [closed]

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is called homogeneous of degree $d \geq 0$ if $$f(\lambda x_1, \ldots, \lambda x_n ) = \lambda^d f(x_1, \ldots, x_n)$$ Differentiating both sides ...
1
vote
1answer
92 views

Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ [closed]

Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant. My question is: Find sufficient and necessary ...
1
vote
0answers
67 views

Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...
3
votes
1answer
57 views

Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. I would like to know how many extrema $p$ has on the standard simplex ...
3
votes
1answer
70 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.
1
vote
0answers
65 views

Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
1
vote
0answers
58 views

Tubular neighbourhood which is nowhere piecewise linear

I recently asked this question. I think, if the following were true, then I would solve my problem. Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
7
votes
1answer
493 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
1
vote
0answers
85 views

Relationship between weak Lp and strong Lq topologies for q<p

Specificaly: Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence? Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If ...
3
votes
1answer
155 views

Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set. \begin{equation}\label{main12} C= \{x\in \mathbb{R}^d ~|~ ...
3
votes
0answers
113 views

derivatives of composite function [closed]

There's a formula for the $n$th derivative of a composite function $f(g(x))$ - it's called Faa di Bruno's formula - but I'm not really interested in the formula but in the proof given in the book of ...
16
votes
2answers
435 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
1
vote
0answers
110 views

Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
5
votes
1answer
151 views

Smallest positive zero of Weierstrass nowhere differentiable function

Consider the Weierstrass nowhere differentiable function $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos(4^n \pi x)$. It seems that the smallest positive zero of $f(x)$ occurs at $x=\frac{1}{5}$, but I ...
8
votes
1answer
230 views

Can there be a measurable set that integrals have the same given value if their integral on $\mathbb{R}$ are the same?

We know for an integrable function $f$, if $\int_\mathbb{R} f=1$, then $\forall \lambda\in [0,1] $, there exists a measurable set $E$ that $\int_E f=\lambda$. Now consider integrable functions $f$ ...
1
vote
1answer
210 views

measure zero in R but not in R^2

I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?
1
vote
1answer
130 views

Level sets of a Weierstrass nowhere-differentiable function

Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known ...
4
votes
1answer
80 views

General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
0
votes
0answers
28 views

Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded?

I have the following question: Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...
0
votes
3answers
194 views

Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
5
votes
1answer
192 views

Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by $T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$, where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$. $T(x)$ has its period $1$, so ...
1
vote
2answers
70 views

a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...