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0
votes
0answers
45 views

Is there any nonnegative bounded function satisfying the following property? [on hold]

Is there a smooth funtion $f(r)$, $r\geq 0$, satisfying the following property: $0\leq f(r) \leq c$, $\int^{\infty}_{r_0}\frac{f(r)}{r}dr<\infty$ for some $r_0>0$, and there exists an sequence ...
3
votes
1answer
87 views

Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
4
votes
0answers
64 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
6
votes
0answers
94 views

Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that $||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...
-1
votes
2answers
111 views

Is real analytic function good enough (see problem)? [closed]

Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then ...
0
votes
0answers
122 views

Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
0
votes
0answers
96 views

how to solve f(f(x))=x^2+x [duplicate]

Now I just know the equation f(f(x))=x^2+x, how can I find the f(x)? I have already tired many times,but I found it is difficult to solve it by any way I knew.So please help me solve the problem,and ...
3
votes
3answers
324 views

Non-zero smooth functions vanishing on a Cantor set

It is easy to give examples of continuous functions $f:[0,1]\to \mathbb R_+\cup\{0\}$ non-zero but vanishing on a Cantor set (ex: Can Cantor set be the zero set of a continuous function?). It is ...
0
votes
0answers
11 views

Hello, everyone, I want to ask you a question about a proof in the Terence Tao's Real Analysis notes [migrated]

everyone. I am using Terence Tao's Real Analysis notes to self learn Analysis 1. There is one thing in the proof of Theorem 27 (Least Upper Bound Property) in “week 2 note” that I don’t understand ...
4
votes
0answers
109 views
+50

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
0
votes
1answer
197 views

Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details). The first one is based on ...
-2
votes
1answer
395 views

Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
15
votes
7answers
916 views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
12
votes
4answers
459 views

Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers $$ A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3 = \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 . $$ In an email, physicist Alan Sokal ...
0
votes
0answers
28 views

Integrating over the Intersection of Convex Regions

Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)? The ...
1
vote
2answers
194 views

Set of distinct real numbers such that all combination of sums are distinct

Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers. For a given $p \in \mathbb N$, consider further the set $$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, ...
2
votes
1answer
133 views

Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
0
votes
0answers
58 views

what is the best estimation for the following

Suppose a continuous $2\pi$-periodic function $f:R\rightarrow R$ satisfies $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \mathrm{const}\frac{\delta}{\Big(\log\frac{1}{\delta}\Big)^{\gamma}}, \,\,\, ...
4
votes
2answers
193 views

What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in ...
1
vote
0answers
70 views

Estimating convolutions of powers

I would like an asymptotic estimate of $$ \sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}} $$ that does not involve any infinite summation. In order to lighten the notation, I ...
6
votes
1answer
160 views

Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...
0
votes
1answer
55 views

Continuous real function on germs

Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider ...
4
votes
1answer
105 views

On a.e. approximate differentiability of certain continuous real functions

I have the following question: If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on ...
2
votes
2answers
105 views

Name of a generalized version of semi-continuity

I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references. We will say a ...
6
votes
1answer
186 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...
2
votes
1answer
189 views

Measure of the same set in different models of ZF

Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models. Is ...
2
votes
1answer
192 views

The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math. Let's consider two examples of curves: 1)The curve defined by the ...
1
vote
0answers
52 views

A Density Problem

Let $ \mathscr{D}=\mathscr{D}(\mathbb{R}^n - {0}) $ be the space of all smooth functions with compact support in $ \mathbb{R}^n - {0} $ topologized by the standard Schwartz topology and let $ ...
2
votes
2answers
97 views

Boundedness of ratio of linear functions

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
0answers
51 views

How to test whether a distribution follows a power law? [closed]

I have the data of how many users post how many questions. For example, [UserCount, QuestionCount] [2, 100] [9, 10] [3, 80] ... ... it means each of the 2 users posts 100 questions, each of the 9 ...
2
votes
2answers
138 views

Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
5
votes
1answer
107 views

Is this graph of reciprocal power means always convex?

Let $$ p = (p_1, \ldots, p_n) $$ be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$. Is the function ...
1
vote
0answers
83 views

Inequality for an integral [closed]

How to prove that the function $$f(r)=(1 - r^2) \int_0^{2\pi}|Re[\frac{e^{i a} (2 - e^{-i s} r)}{(-e^{i s} + r)^2}]|ds$$ for real $a$ and $r\in[0,1]$ attains its maximum for $r=0$ with $f(0)=8$.
1
vote
0answers
127 views

On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$

Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property $$ \left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert ...
4
votes
0answers
203 views

Why does it seem that $rca=rba$? [closed]

The following paradox has got me stumped. I'm hoping someone can point out the error. Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous ...
3
votes
0answers
64 views

Almost everywhere in a rectangle [duplicate]

I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't. Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
1
vote
0answers
64 views

Structure of the zero set of analytic maps (Lojasiewicz’s Structure Theorem for Varieties)

The Lojasiewicz’s Structure Theorem for Varieties states that for $\Phi:\mathbb{R}^n\rightarrow \mathbb{R}$ real analytic with $\Phi(0)=0$ the set $\Phi^{-1}(0)$ is locally a union of subvarieties of ...
1
vote
1answer
299 views

What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]

I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? Only I've calculated the following identity: $$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln ...
0
votes
0answers
67 views

Asymptotic analysis involving a circular multiple integral

Let $t_1,\ldots,t_m>0$, and $m\ge 4$ be an even integer. Consider the function: $$ f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m} |x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots ...
9
votes
2answers
564 views

Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...
-1
votes
1answer
82 views

derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has ...
11
votes
1answer
244 views

Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
2
votes
1answer
106 views

A question on existence of solutions of a linear ODE system

I am working on a problem of harmonic functions on surfaces, and in one step I got the following system of ODEs with prescribed asymptotes. I was wondering what methods could give us the existence or ...
1
vote
0answers
104 views

Uniform bound for an alternating series of functions

I have mainly two questions, the first one being motivated by the second one. 1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
4
votes
1answer
56 views

Reference: Hardy space regularity of the Jacobian determinant

I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes. Theorem: For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ...
9
votes
0answers
173 views

A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos ...
3
votes
1answer
112 views

For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1

This is related to a question asked on mathstackexchange http://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...
0
votes
0answers
45 views

Approximate of binary function by indicator functions of one variable

Let $(X,F,u)$ and $(Y,G,v)$ be two probability spaces. I know a result that the indicator functions of product measurable sets can be approximated by indicator functions of one variable. That is, for ...
0
votes
2answers
105 views

Markov-Bernstein like inequalities for monotone polynomials

Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...
0
votes
0answers
53 views

About approximate eigenvalue

I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4. Suppose $X$ is a real Banach Space, $M$ is a ...