**-5**

votes

**1**answer

122 views

### Solution of $y'' - ax y' - y = 0$ [closed]

I am looking for solutions $y(x)$, $x$ positive, for the ODE
$y''(x) - a x y'(x) - y(x) = 0$,
where $a$ is any real number. It seems not to be any of the usual ODEs (i.e. those listet on Wikipedia), ...

**2**

votes

**0**answers

76 views

### From Selberg integral to Dyson integral

My question is from the drivation from Slberg integral to Dyson integral in this paper:
Selberg integral :
$$ S_n(\alpha,\beta,\gamma) =
\int_0 ^1 \cdots \int_0 ^1
\prod_{i=1}^n ...

**7**

votes

**2**answers

90 views

### Bounds on coefficients of factors of a multivariate polynomial

Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the ...

**0**

votes

**1**answer

158 views

### Problem with understanding an equation

I have read the article Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets and I don't know how to interpret Equation (10) on
page 630, because this equation ...

**4**

votes

**1**answer

319 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 ...

**5**

votes

**2**answers

305 views

### Thurston-Cannon $S^2$-filling curves

I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers:
Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry
...

**2**

votes

**1**answer

144 views

### Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...

**2**

votes

**1**answer

77 views

### Gradient of Ronkin function

I have a complex curve $P(z,w)=z+w-1=0$. I get the amoeba map
$$(z,w)\rightarrow (\log|z|,\log|w|)$$ of this curve. It's look like this http://en.wikipedia.org/wiki/Amoeba_(mathematics) (the first ...

**4**

votes

**1**answer

345 views

### Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...

**2**

votes

**1**answer

228 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 ...

**3**

votes

**1**answer

46 views

### Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3
$$x'(t)=f(x(t),d)$$
where the dynamics f is homogeneous of degree 1 and there is exactly one
line of equilibrium points. This line is independent of the ...

**2**

votes

**0**answers

127 views

### How many ways we have to prove that a topologically (or analytically) nice mapping is injective?

I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and ...

**1**

vote

**1**answer

70 views

### variational problem related to an integral

Recently I came up with a type of variational problem in stochastic process.
It can be stated in the following way:
Given $a$ and $b$ positive, and an increasing function $f$ on $(0,1)$ (may be not ...

**1**

vote

**0**answers

155 views

### matrix Khintchine inequality

The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( ...

**3**

votes

**1**answer

201 views

### Asymptotic behavior for the solution of a nonlinear ODE

In a nutshell, if $u$ is a solution to
$$
\partial_r^2 u(r)+ \frac{1}{r} \partial_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r_0>0\\
\lim_{r \to \infty} u(r) = 0,
\quad \text{and} ...

**8**

votes

**1**answer

863 views

### If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...

**1**

vote

**0**answers

57 views

### Sufficient conditions for sums of Laguerre polynomials to be non-negative

I am interested in sufficient conditions on non-negative sequences of coefficients $\{c_{2n}\}_{n\ge 0}$ guaranteeing that
$$%\begin{equation}\label{cond}
\sum_{n=0}^\infty c_{2n} L_{2n}^{(1)}(x)\ge ...

**3**

votes

**1**answer

189 views

### Non-linear first order ODE

This is a two part question. On one hand, I am trying to find positive solutions of the following equation:
$$1-\frac{d}{dx}f=\frac{f\log(f)}{x}$$
for $x>1$.
If that is not possible, I would at ...

**3**

votes

**1**answer

146 views

### Infinite series - analytical solution

Analytical Solution is required for:
$$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty ...

**1**

vote

**0**answers

213 views

### Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$
where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...

**6**

votes

**0**answers

891 views

### Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...

**2**

votes

**1**answer

39 views

### Prescribing finitely many unparameterised planar geodesics

Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing ...

**3**

votes

**2**answers

158 views

### Identifying a special function from its power series

Here is a power series, which looks a bit like a Hypergeometric function series, but I don't think that it is. Has anyone any idea what it is? Here $n,p,r$ are integers with $n\ge 0$ and $p\ge r\ge ...

**5**

votes

**2**answers

447 views

### What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$

It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...

**1**

vote

**0**answers

79 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 ...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**1**

vote

**3**answers

263 views

### What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$,
for numerical purpose, what are the possible basis function for $X$?
In finite element method, the basis functions are tooth functions, or polynomial functions.
Is ...

**5**

votes

**1**answer

199 views

### Integral Identity Involving Bell Numbers

Is the following identity true ?
$$\int_0^\infty \frac{b(x)}{B(x)} dx \quad \overset{?}{=} \quad \int_0^\infty \frac{x!}{x^x} dx$$
where
$$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad \text{and} ...

**2**

votes

**2**answers

165 views

### Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...

**7**

votes

**1**answer

414 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 ...

**4**

votes

**1**answer

265 views

### Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ...

**1**

vote

**0**answers

82 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 ...

**0**

votes

**1**answer

119 views

### Stability analysis of ODE

My questions concerns the stability analysis of the following dynamical system :
$\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} ...

**11**

votes

**1**answer

1k views

### Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$
runs over the integers?
The existence of the limes inferior follows from Dirichlet's approximation theorem,
but the ...

**2**

votes

**0**answers

61 views

### Closed form for a simple hypergeometric q series

I've run across an interesting hypergeometric q-series that I feel must have been found before:
$\sum_{n=0}^{\infty}(-1)^n$$\frac{e^{n b y}}{\prod_{k=1}^{n}(e^{\pi k b^2}-e^{\pi k b^{-2}})} = ...

**10**

votes

**2**answers

291 views

### Simultaneous zero set of two equations in $\mathbb R^3$

Can we have positive reals $x,y,z$ with
$$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutation, other than the line $x=y=z$?
I put this at ...

**2**

votes

**2**answers

145 views

### Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that ...

**1**

vote

**2**answers

210 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} ...

**4**

votes

**1**answer

163 views

### Asymptotic solution of the integral equation

What is the asymptotic solution (for $s\gg 1$) of the following integral equation $$z(s)=1+\gamma\int\limits_{-\infty}^s ds_1\int\limits_{-\infty}^{s_1}ds_2
\cos{(s_1^2-s_2^2)}z(s_2)\;?$$
In fact I ...

**1**

vote

**0**answers

92 views

### growth bound for solution of an ordinary integro-differential equation

I am considering the following ordinary integro-differential equation
$$
A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2
$$
where
$$
A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2
+ ...

**2**

votes

**1**answer

128 views

### elliptic integral with singularities

I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle ...

**2**

votes

**0**answers

93 views

### Positiveness of the double integral

How it can be proved that the following double integral (which emerged in a physics project) $$\int\limits_{-1}^1d\tau\;\tau \int\limits_0^\infty dk\frac{k\sin{(kr\tau)}}{(1+\beta ...

**5**

votes

**1**answer

297 views

### Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO ...

**5**

votes

**2**answers

741 views

### Integral with Dirac delta (me or wolfram mathematica?) [closed]

I asked the question on math.stackexchange but didn't get an answer so I came here.
I tried to compute with Wolfram Mathematica the following integral
$$I=\int_0^\pi\int_{-\infty}^\infty x e^{-\mathrm ...

**2**

votes

**0**answers

158 views

### A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE
$$\frac{d}{dt}D^0_t(\cdot) = ...

**1**

vote

**0**answers

78 views

### The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion:
\begin{equation*}
B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}.
...

**1**

vote

**2**answers

220 views

### Construction of a convex function nondifferentiable on a countable set [closed]

Let $H$ be a countable subset of $[0,1]$. Construct a convex function $f:[0,1]\rightarrow\mathbb{R}$ such that $f$ is nondifferentiable on $H$ and differentiable in the rest.

**0**

votes

**2**answers

163 views

### looking for f(x) in f(x) = a.exp((x-f(x))/b) [closed]

not a math expert, but this problem is really bugging me. whichever way i turn it, i cant find an expression for f(x) which satisfies
f(x) = exp(( x - f(x)) /b ).
I can also express the problem as ...

**0**

votes

**1**answer

120 views

### Ulam stability of homogeneous polynomials

Let $P$ be a homogenous polynomial with real coefficients in several variable(at least three variable)
Is the following statement true:
For every $\epsilon$ there is a $\delta$ such that for ...

**-1**

votes

**1**answer

134 views

### derivative of a real function without open domain [closed]

I am trying currently why we (the mathematicians) imply that whenever the derivative of a real function $f$ is involved, always the domain $D_f$ of $f$ should be open interval or at least semi-open ...