# Tagged Questions

**4**

votes

**1**answer

32 views

### Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on ...

**1**

vote

**1**answer

204 views

### Sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$

Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish ...

**0**

votes

**0**answers

23 views

### Calculation of characteristic strip, solutions of a geometric PDE [on hold]

Say I have a family of spheres of radius $1$ with centers in the $xy$-plane$$u = G(x, y, \lambda, \mu) = \sqrt{1 - (x - \lambda)^2 - (y - \mu)^2}.$$I have a few questions.
What are all the ...

**0**

votes

**0**answers

62 views

+50

### Closed-Form solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...

**38**

votes

**1**answer

940 views

### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...

**1**

vote

**0**answers

40 views

### A sufficient condition of the following integral to be positive

Suppose $G=(A,B,E)$ is a bipartite graph whose partition has the parts $U=\{u_1,\cdots,u_m\}$ and $V=\{v_1,\cdots,v_n\}$. Consider the following integral
...

**2**

votes

**0**answers

39 views

### On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$
\begin{equation}
...

**0**

votes

**0**answers

23 views

### Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, ...

**-3**

votes

**0**answers

34 views

### the hessian and the gradient [on hold]

$A\in{\mathbb{R^{m\times n}}}$, $b\in{R^m}$. For $ x\in{\mathbb{R^n} }$, we define q(x)=f(Ax+b), with $f:\mathbb{R^m}\longrightarrow{\mathbb{R}}$. Calculate the hessian matrix and the gradient of f .

**-2**

votes

**0**answers

64 views

### Prove a function is entire

Let $f$ function such that $\int_{R^m} (1+|y|)^N|f(y)|dy <\infty$.
Consider a function $g(z) = \int h(y,z) f(y) dy,$
where $|h(y,z)| \leq C e^{|z||y|}$ with $z\in C^m$ and ...

**0**

votes

**0**answers

34 views

### Limit probability of a complete bipartite random graph $G(n,n,p)$ is connected [on hold]

I need to calculate the following probability limit for a complete bipartite random graph $G(n,n,p)$ in the Erdos-Renyi model:
\begin{equation}
\lim_{n\rightarrow\infty}\mathbb{P}[G(n,n,p) \text{ is ...

**1**

vote

**1**answer

119 views

### Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...

**1**

vote

**0**answers

44 views

### Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq ...

**0**

votes

**1**answer

49 views

### Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions?
Or are there restrictions on the function space over which this can be defined?
For example is the Rademacher complexity defined or has ...

**2**

votes

**1**answer

167 views

### Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let ...

**5**

votes

**1**answer

74 views

### Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set ...

**2**

votes

**0**answers

46 views

### How much must a curve bend to intersect another curve twice?

Suppose $c_1$ and $c_2$ are segments of smooth plane curves. To be concrete, say $c_1$ and $c_2$ are graphs of smooth functions $f_i:[a_i,b_i]\to \mathbb R$, $i=1,2$. If the curves were lines, then ...

**0**

votes

**0**answers

16 views

### Comparison of lengths: Theodorus spiral and Archimedes' Spiral

It is known that the Spiral of Theodorus approximates the Archimedean spiral.
Let's consider the arc-length of the Spiral of Theodorus $L_T(\phi)$ and the Archimedean one: $L_A(\phi)$. Can we ...

**3**

votes

**1**answer

318 views

### About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...

**1**

vote

**0**answers

44 views

### Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694
Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...

**0**

votes

**0**answers

84 views

### local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...

**1**

vote

**0**answers

27 views

### Optimizing sum of approximate and exact functions

This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers.
I have two real values functions, where one ...

**-4**

votes

**1**answer

276 views

### Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...

**5**

votes

**1**answer

178 views

### Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...

**5**

votes

**0**answers

148 views

### Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra.
Suppose $\Omega$ is ...

**3**

votes

**1**answer

75 views

### Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...

**19**

votes

**12**answers

2k 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 ...

**2**

votes

**1**answer

937 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**153**

votes

**11**answers

45k 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 ...

**1**

vote

**1**answer

101 views

### An analytic family of in fact non-existent improper Riemann integrals

Question:
Are there any useful interpretations or "applications" of the formula
$$\intop_0^\infty e^{-ax}\frac{\sin x}{x}\,dx=\frac{\pi}{2}-\arctan(a)\qquad \forall\;a\in \mathbb{R},
$$
in which the ...

**-3**

votes

**0**answers

71 views

### Bernoulli's Inequality when $-2≤x<-1$ [migrated]

Why is it that Bernoulli inequality $(1+x)^r>1+rx$ is said to be true for every integer $r≥0$ and every real $x≥-1$; why the range $-2≤x<-1$ is not included?
It seems that, by induction (or ...

**3**

votes

**2**answers

276 views

### A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...

**2**

votes

**1**answer

130 views

### Uniqueness from orthogonality relation?

This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...

**1**

vote

**0**answers

43 views

### Quotient of two smooth functions extension

Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...

**1**

vote

**1**answer

102 views

### A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...

**1**

vote

**0**answers

75 views

### extension for a complex operator

Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial ...

**2**

votes

**1**answer

94 views

### Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.
Is there possible an extension of real/complex numbers in which logarithms and ...

**29**

votes

**4**answers

7k views

### Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable?
I ask for pedagogical reasons. Results in ...

**18**

votes

**3**answers

460 views

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...

**1**

vote

**1**answer

55 views

### Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations:
$$
k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right|
$$
$$
...

**2**

votes

**3**answers

235 views

### Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty ...

**3**

votes

**1**answer

380 views

### On the embedding of a function space $X$ into $L^2\cap L^4$

It is well-known that if $\Omega\in \mathbb{R}^n$ is a bounded domain, then we have the embedding
$$
L^4({\Omega})\subset L^2({\Omega})
$$
since $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ ...

**2**

votes

**1**answer

89 views

### Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? [closed]

I have a question about Sobolev space.
Let $\Omega$ be an open subset of $\mathbb{R}^{d}$,
we consider the Sobolev space
$H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...

**1**

vote

**0**answers

37 views

### The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...

**2**

votes

**2**answers

171 views

### Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

This question is an expansion of another question that I asked over at Math Stack Exchange.
In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ ...

**2**

votes

**1**answer

78 views

### A continuity/bootstrap argument

I am trying to understand how one can prove the following assertion using a continuity argument:
Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to ...

**1**

vote

**0**answers

59 views

### Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$
Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...

**3**

votes

**1**answer

180 views

### Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...

**0**

votes

**0**answers

48 views

### Bounds on the the spherical harmonics on $S^{p-1}$

The only reference I could find in this regard is for upper bounding the n-homogeneous spherical harmonics on $S^{p-1}$ as in equation 4.29 here, ...

**1**

vote

**1**answer

55 views

### linear recurrence inequality of positive terms

This is a follow up on my previous linear recurrence inequality question.
I have some matrices which satisfy a linear recurrence formula of the form
$$
A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad ...