The real-analysis tag has no wiki summary.

**2**

votes

**1**answer

227 views

### Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...

**14**

votes

**2**answers

927 views

### Generalization of Darboux's Theorem

Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let ...

**2**

votes

**2**answers

117 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 ...

**2**

votes

**1**answer

170 views

### Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds.
(Cf. discussion on p. 45.)
Definition
Let $E$ and $F$ be two Banach spaces together with a plain subset ...

**1**

vote

**0**answers

51 views

### How to test whether a distribution follows a power law? [on hold]

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 ...

**-4**

votes

**0**answers

27 views

### real analysis series and sequence [on hold]

If ∑(n=1)^∞▒an is absolutely convergent , then which of the following is not true?
(a)∑(m=n)^∞▒am →0 as n→∞
(b)∑(n=1)^∞▒ansinn is convergent
(c)∑(n=1)^∞▒e^an is divergent
(d∑_(n=1)^∞▒a_2^n ) is ...

**5**

votes

**1**answer

98 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
...

**-2**

votes

**0**answers

47 views

### Di Perna-Lions theory for transport equation [on hold]

Does someone know if some notes on the topic mentioned in the title are available online? I'm reading the paper "Ordinary differential equations, transport theory and Sobolev spaces" by Di Perna and ...

**8**

votes

**2**answers

560 views

### Mathematical equivalent to ladder operators?

A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...

**-2**

votes

**0**answers

58 views

### Reverse Fatou's Lemma [closed]

Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and ${E_{n \in ℕ}}$ be $\mathcal{F}$-measurable sets.
Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq \limsup_n ...

**1**

vote

**0**answers

119 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
...

**1**

vote

**0**answers

76 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 $f(0)=8$.

**8**

votes

**1**answer

309 views

### Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set

It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary ...

**0**

votes

**0**answers

48 views

### approximation of $L^{1}-$ function in the $L^{1}-$norm in some particular way [closed]

Fact: Suppose $f\in L^{1}(\mathbb R), \gamma_{0}\in \mathbb R, \hat{f}(\gamma_{0})=0, W$ is a some neighbourhood of $\gamma_{0},$ and $\epsilon >0.$ Then there exists $k\in L^{1}(\mathbb R)$ such ...

**0**

votes

**0**answers

78 views

### “Isoperimetric” inequality for Martin capacity [closed]

After some fruitless literature search, I was wondering if anyone has proved an "isoperimetric" inequality for Martin capacities (for transient Brownian motion with $d\geq 3$):
...

**1**

vote

**1**answer

92 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 ...

**4**

votes

**0**answers

194 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 ...

**2**

votes

**0**answers

39 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 ...

**18**

votes

**2**answers

1k views

### Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...

**7**

votes

**3**answers

1k views

### the dual space of C(X) (X is noncompact metric space)

It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow X \text{ is continuous and bounded} \}$ is $M(X)$, the space of Radon ...

**10**

votes

**1**answer

575 views

### Nondifferentiability set of an arbitrary real function

A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...

**9**

votes

**2**answers

555 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 $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...

**42**

votes

**2**answers

1k views

### Is a function with nowhere vanishing derivatives analytic?

My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to a ...

**1**

vote

**0**answers

60 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

**1**answer

294 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 ...

**1**

vote

**0**answers

102 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 $\mathbb{R}_+$ ...

**0**

votes

**1**answer

95 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 ...

**8**

votes

**1**answer

237 views

### When is this multiple integral finite?

Consider the following integral:
$$
I_k(\alpha)=\int_{[0,1]^k}|x_1-x_2|^{\alpha}|x_2-x_3|^{\alpha}\ldots|x_{k-1}-x_k|^{\alpha}|x_k-x_1|^{\alpha}d\mathbf{x}.
$$
where $k=2,3,4,\ldots$
The question is ...

**0**

votes

**0**answers

66 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 ...

**16**

votes

**1**answer

838 views

### An Entropy Inequality (generalized)

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability ...

**0**

votes

**0**answers

14 views

### Points in a general Cantor set [migrated]

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...

**-1**

votes

**1**answer

80 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
...

**2**

votes

**1**answer

104 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 ...

**0**

votes

**1**answer

148 views

### Limits of functions with converging zeros

What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros?
More precisely, suppose that $f_i: R^n \to R^m$ is a ...

**10**

votes

**3**answers

2k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

**11**

votes

**1**answer

235 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 ...

**0**

votes

**0**answers

39 views

### Help on step of “The real numbers are uncountable” proof [migrated]

I'm currently a student in a Real Analysis class, and we today we discussed a proof of why the real numbers are uncountable--however, I'm having a little trouble understanding why our proof is true. ...

**93**

votes

**11**answers

13k views

### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

**4**

votes

**1**answer

53 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

**0**answers

160 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 ...

**0**

votes

**0**answers

142 views

### An inequality on closed manifolds

Excuse me again, I am not sure what kind of question the following is. Any suggestion is appreciated!
Let $(M,g)$ be a closed Riemannian manifold, $g$ is real analytic. Let $u,v$ be two nonnegative ...

**0**

votes

**2**answers

97 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 ...

**3**

votes

**1**answer

108 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 ...

**12**

votes

**11**answers

1k views

### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...

**15**

votes

**4**answers

2k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

**0**

votes

**0**answers

40 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 ...

**4**

votes

**1**answer

118 views

### Estimates of Hausdorff dimension (and its derivatives)

For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...

**1**

vote

**0**answers

111 views

### Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...

**69**

votes

**17**answers

14k views

### Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...

**0**

votes

**0**answers

52 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 ...