The real-analysis tag has no wiki summary.

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### 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 ...

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141 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 ...

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114 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 ...

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202 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 ...

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202 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 ...

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59 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 $ ...

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112 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,\; ...

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55 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 ...

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254 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 ...

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120 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
...

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90 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 $f(0)=8$.

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131 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
...

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216 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 ...

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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 ...

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82 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 ...

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314 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 ...

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77 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 ...

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

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102 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
...

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258 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 ...

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122 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 ...

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

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70 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 $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, ...

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230 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 ...

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### 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 ...

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62 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 ...

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135 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 ...

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### 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 ...

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161 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 ...

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300 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 ...

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154 views

### Legendre differential equation with additional term

In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( ...

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257 views

### Cofinality of a $\sigma$-ideal of $\mathbb{R}$

The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...

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### Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...

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145 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 $T(I_i)=[0,1]$. ...

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148 views

### A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...

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### Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...

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### Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$.
My ...

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### Interpolation and embeddings for parabolic function spaces

I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...

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283 views

### A question on null sequences

Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero?
In case the answer is ...

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142 views

### Natural integration constant for normal and discrete integration: is there a connection?

It is often assumed that integration, unlike differentiation is defined only to an arbitrary constant. So the antiderivative function is often left undefined or postulated to be zero in zero.
But I ...

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235 views

### What does this ODE have to do with the associated Legendre polynomials?

I am currently struggeling with the following differential equation:
$$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$
where $a \in \mathbb{R}$ constant, $\phi ...

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128 views

### Uniform boundedness in $L^1[0,1]$ implies finite $\limsup$ almost everywhere for a subsequence? [closed]

Given a sequence of functions $f_k \in L^1([0,1])$ such that $||f_k||_{L^1(0,1)}\leq C$.
Is there a subsequence $\{k_l\,|\,l\in \mathbb N\}\subseteq \mathbb{N}$ such that for $\mathcal{L}^1$-almost ...

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### formula for repeated finite differences

I am looking for a proof of a well-known fact, whose proof must be very easy, though I've been struggling to find it. Let $\Delta$ be the map from real-valued functions of a real variable, given by ...

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101 views

### Metric density theorem in most general setting?

It's a consequence of Lebesgue's theorem that every measurable $E\subset\mathbb{R}^n$ has a metric density that's $1$ a.e. on $E$ and $0$ a.e. on $\mathbb{R}^n\setminus E$. What are the most general ...

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385 views

### Spectrum of Mathieu equation

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...

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153 views

### Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?

I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim_{n \rightarrow \infty}\sum_{k=0}^{n} ...

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137 views

### Question about the derivative of a fuctional

I have this lemma+proof and i dont understand why it follows from $J'(u_n)\rightarrow 0$ that $-\Delta_p u_n- f(x,u_n)\rightarrow 0$ such that
$J(u)=\frac1p\int_{\Omega} |\bigtriangledown u|^p ...

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80 views

### Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...

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655 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 ...

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233 views

### Equicontinuity and $L^2$ convergence imply uniform convergence

I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real ...