**1**

vote

**0**answers

212 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

845 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

38 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

156 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

426 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

78 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

257 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

194 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

163 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

413 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

238 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

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 ...

**0**

votes

**1**answer

118 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

987 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

290 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

140 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

205 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

157 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

91 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

123 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

293 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

670 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

156 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

77 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

207 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

157 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

119 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 ...

**5**

votes

**1**answer

195 views

### Is Taylor expansion related to Helmholtz decomposition?

The Taylor expansion of a vector field $f(x)$ to the order of one is
$$f(x)=f(x_0)+Jf(x_0)\cdot\Delta x+o(\Delta x)$$
where $Jf$ is Jacobian of the vector field and $\Delta x=x-x_0$.
Suppose we ...

**6**

votes

**1**answer

339 views

### Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$.
It seems that the solution of linear ODE
$$\dot x= Ax$$
can not be written in a closed form using $\int$, $A$, $x(0)$ ...

**1**

vote

**1**answer

45 views

### A formula for the potential part in Riesz decomposition of simple subharmonic functions?

Consider a compact subset $E\subset\mathbb{C}$, holomorphic functions $f_j:V\to \mathbb{C}$, $1\leq j\leq k$, defined in a neighbourhood $V$ of $E$, and set $u:V\to\mathbb{R}\cup\{-\infty\}$, ...

**1**

vote

**0**answers

123 views

### Dalzel's integral for $\pi$ and the lemniscate constant

$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers
$$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$
and, for example, the Wallis product formula ...

**0**

votes

**1**answer

115 views

### Application of the inverse theorem function on singular points

This question is related to this one A question about the inverse theorem function in $\mathbb{R}^n$. In the response I received, I was told that the issue was related to the singularities, as I do ...

**-23**

votes

**1**answer

2k views

### Can Poisson Summation formula break?

The Poisson summation formula states if $f: \mathbb{R} \to \mathbb{R}$
then $\displaystyle \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n) $ where $$\hat{f}(\xi) = ...

**0**

votes

**2**answers

217 views

### A question about the inverse theorem function in $\mathbb{R}^n$

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ ...

**4**

votes

**2**answers

232 views

### Limit of a double integral

What is the $\varepsilon\to 0$ limit of the following double integral
$$\int\limits_{-1}^1d\tau\;\sqrt{1-\tau^2}\;\tau\int\limits_0^\infty dq\;q^2e^{iq(\tau+i\varepsilon)}\;?$$
I was asked about this ...

**3**

votes

**0**answers

91 views

### Meaning of fractional Fourier transform with imaginary iteration count

(I'm reposting this from math.stackexchange as I didn't get an answer there and thought it might be "advanced" enough for this site.)
As one may know, the Fourier Transform
$$F[f](\nu) = ...

**1**

vote

**2**answers

98 views

### Bound deg 3 partial differential operator on Laplace eigenfunction?

I am no expert on PDE and analysis but I am looking for certain technique from PDE.
Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...

**9**

votes

**3**answers

973 views

### Convergent subsequence of $\sin n$

It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$ there exists a sequence $(k_i)$ such that $\lim\sin k_i = \theta,$ but there appear to be no explicit such ...

**10**

votes

**2**answers

298 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 ...

**6**

votes

**1**answer

302 views

### Convergence and Closed Form of an Integral Involving Bell Numbers

1. Does the following integral converge ?
$$\int_0^\infty \frac{b(x)}{B(x)} dx$$
where
$$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$
2. ...

**4**

votes

**0**answers

210 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 ...

**1**

vote

**0**answers

297 views

### fourier decomposition of white noise

I found in some lecture notes that Brownian motion is defined by its Fourier series:
$$ B(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} \frac{a_m}{m} e^{2\pi i \,mt} $$
Then I would get that its ...

**0**

votes

**0**answers

120 views

### multivariate integral calculation in closed form

I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...

**6**

votes

**1**answer

240 views

### Certain asymptotics involving double infinite sum

Let $1<\alpha<\beta<3/2$. Set
$$
S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}.
$$
One can check that $S(n)$ is finite. My question is when $n\rightarrow ...

**2**

votes

**2**answers

168 views

### Ewald's generalized theta function

Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-(
Die Berechnung optischer und ...

**0**

votes

**1**answer

79 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 = ...