# Tagged Questions

**9**

votes

**2**answers

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

**2**

votes

**1**answer

102 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 ...

**9**

votes

**0**answers

154 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 ...

**1**

vote

**1**answer

132 views

### Differentiability of partially defined functions?

This problem arose while reading Lee's Introduction to Smooth Manifolds.
Given Banach space: $E,F$
with a plain subset: $A\subseteq E$
and a function: $f:A\to F$
Call a function differentiable at ...

**1**

vote

**0**answers

82 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/$C^\infty$ ...

**9**

votes

**1**answer

240 views

### 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 ...

**2**

votes

**1**answer

215 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 ...

**1**

vote

**2**answers

225 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) = ...

**0**

votes

**0**answers

50 views

### Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$.
We fix the value of one variable $q_i \in (0,1)$, then maximize ...

**-2**

votes

**1**answer

101 views

### a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...

**0**

votes

**1**answer

199 views

### Theorem with an example [closed]

i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?

**9**

votes

**1**answer

799 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

**0**answers

112 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 ...

**3**

votes

**1**answer

171 views

### About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...

**5**

votes

**0**answers

81 views

### Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$.
By this I mean that there exists a polynomial $P$
with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!)
such that $P(\phi) = 0$
Let ...

**2**

votes

**1**answer

63 views

### radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke:
Teorem. Let $f : U ⊂ \mathbb{R}^n ...

**0**

votes

**0**answers

52 views

### How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define,
$$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$
where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...

**9**

votes

**1**answer

518 views

### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence
\begin{equation}
a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),
\end{equation}
with $a(1,1)=1$ and ...

**26**

votes

**0**answers

890 views

### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
...

**0**

votes

**1**answer

303 views

### Pros and cons of probability model for permutations

I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by ...

**0**

votes

**0**answers

79 views

### Detailed Taxonomy of Multivariate Real Functions: $\mathbb{R}^n\rightarrow\mathbb{R}$

I want to classify the following functions:
$$ f:(x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\sqrt{(x+1)^2+y^2}+\sqrt{(x-1)^2+y^2}-2$$
$$ ...

**2**

votes

**1**answer

117 views

### Is there a dense rational sequence of positive separation?

Let us consider the set $\ell_\neq$ of bounded sequences of unequal real terms. We use the following descriptions. A sequence $x=(x_0,x_1,...)\in\ell_\neq$ is dense if, for all $\varepsilon>0$, ...

**0**

votes

**1**answer

102 views

### Positive kernel property

Let $k:[0,1]^2\rightarrow (0,+\infty)$ be a continuous function and let
$f,g:[0,1]\rightarrow (0,+\infty)$ be measurable functions. We assume that
$$\forall x\in [0,1],\quad
f(x)=\int_0^1 k(x,y) g(y) ...

**10**

votes

**3**answers

303 views

### The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...

**9**

votes

**1**answer

195 views

### Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...

**1**

vote

**1**answer

122 views

### Pohozaev result for equations with weights

I am interested in nonnegative solutions of
$-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$.
Or instead the equation $ -\Delta u + ...

**9**

votes

**3**answers

504 views

### Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...

**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 ...

**14**

votes

**2**answers

889 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 ...

**6**

votes

**1**answer

204 views

### Alternative proof of Lojasiewicz inequality

is there a "brute force proof" of the Lojasiewicz inequality? By "brute force" i mean without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e. standard ...

**1**

vote

**1**answer

94 views

### Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...

**2**

votes

**1**answer

63 views

### Expressions in “continued” monotone functions

Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction
Now take a look at this question: ...

**4**

votes

**1**answer

264 views

### Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?

From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...

**2**

votes

**1**answer

186 views

### A special case of the Divergence theorem

I am interested in the following statement:
Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
domain $U$, continuous up to the boundary $\partial U$, and vanishing on ...

**1**

vote

**0**answers

77 views

### Representing quasianalytic functions in several variables

For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...

**7**

votes

**1**answer

407 views

### Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...

**1**

vote

**0**answers

81 views

### Does the difference quotient of an absolut cont. funct. converge in L^1?

Assume that $\mu$ is a finite Radon measure on the real line and $f$ is integrable wrt. $\mu$. Define
$F(x)=\int_{]\infty;t]}f(y)d\mu(y) $
Is the following statement true?
The functions ...

**1**

vote

**2**answers

190 views

### specific improper integral involving erf

I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:
$$
\int_{1}^{\infty} ...

**10**

votes

**2**answers

287 views

### Is it possible to have the set $f^{-1}(\lbrace x \rbrace)$ perfect for every $x$?

There are examples of functions $f \colon [0,1] \longrightarrow [0,1]$ such that
for any $\alpha $, $f^{-1}(\lbrace \alpha \rbrace)$ is uncountable. My favorite example is $$f(r) = \limsup_n ...

**4**

votes

**0**answers

187 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

**0**

votes

**1**answer

72 views

### An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:
For $x \in \mathbb{R}^{n}$ and $\alpha = ...

**7**

votes

**1**answer

264 views

### Kakeya and Nikodym maximal functions

I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more ...

**1**

vote

**1**answer

118 views

### Where find proof of such theorem about uniform convergence of differences

Where to find a proof of theorem which says that:
if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesque measure or on the set of second category with Baire propert ...

**2**

votes

**1**answer

213 views

### bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
...

**3**

votes

**0**answers

197 views

### The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...

**2**

votes

**2**answers

201 views

### “Then obviously…” (a bound on strongly differential functions on an interval)

On the fourth page of their 1967 paper Local Behavior of Solutions of Quasilinear Parabolic Equations, Aaronson and Serrin comment: "Consider a strongly differentiable function $w$ of the real ...

**2**

votes

**1**answer

123 views

### Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.
In the book "Understanding Analysis" by Abbott is stated in page 128 ...

**1**

vote

**1**answer

414 views

### New differintegral formula: how is it related to other differintegral formulas?

Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x)= \lim_{t\to s} ...

**14**

votes

**2**answers

467 views

### Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here:
Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate
$$I_n
:=
...

**2**

votes

**1**answer

414 views

### Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac{1}{b-a} \int_a^b ...