**4**

votes

**3**answers

395 views

### Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it ...

**1**

vote

**0**answers

37 views

### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...

**17**

votes

**21**answers

23k views

### Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

**3**

votes

**0**answers

53 views

### Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...

**1**

vote

**1**answer

100 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot ...

**5**

votes

**2**answers

244 views

### A result of Sierpiński on non-atomic measures

There is a classical result commonly attributed to W. Sierpiński that reads as follows:
Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ ...

**0**

votes

**0**answers

50 views

### Solution of the system of differential equations [on hold]

Consider that we are working in the polynomial ring $\mathbb{C}[x]$.
Suppose we have the following system of linear differential equations:
$$\left\{\begin{matrix}
L_1(y)=f_1\\
L_2(y)=f_2
...

**1**

vote

**0**answers

58 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...

**4**

votes

**1**answer

324 views

### All solutions to a set of integral equations

I would like a better understanding of the set of pairs $(f_1,f_2)$ of functions $[0,1] \times [0,1] \to [0,1]$ which satisfy the following conditions:
For all $y \in [0,1]$, $f_1(x,y) \geq ...

**4**

votes

**1**answer

205 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
...

**10**

votes

**1**answer

408 views

### A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ ...

**4**

votes

**2**answers

174 views

### Integrals involving trigonometric functions and polynomials

Can one describe all the real polynomials $P(x)$ such that the following integrals converge:
$$
\int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ?
$$
Among special cases are such ...

**8**

votes

**1**answer

270 views

### On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := ...

**22**

votes

**2**answers

7k views

### Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?

**6**

votes

**1**answer

331 views

### A property of the derivatives of a function

Suppose that $f,g_1,g_2,\dots$ are functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f'=f\,g_1$ and $g'_j=g_j^2-g_j g_{j+1}$. Here and in what follows, $j$ is any natural number. Then, by ...

**2**

votes

**0**answers

98 views

### An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - ...

**1**

vote

**0**answers

65 views

### Recurrence sequence

Is it possible to find a Recurrence sequence that Satisfying the following inequality
$ d_{n+k}\geq \alpha ^k d_n +\beta^k \delta(A,B),$
where $0<\alpha<1, \alpha ^k+\beta^k\geq ...

**2**

votes

**1**answer

326 views

### Convergence of a Trigonometric Series

After working with a Fourier series for a while, I realized that it would be of great help to me if I could prove that the following limit is zero:
...

**2**

votes

**1**answer

73 views

### Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any ...

**2**

votes

**0**answers

21 views

### Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let $n\ge 1$ be an integer, let
$$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$
for $x,y\ge 0$.
When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...

**0**

votes

**1**answer

91 views

### Wong-Zakai smooth approximation in probabilty for stochastic differential equations

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) ...

**3**

votes

**1**answer

214 views

### methods for situations where well-posedness criteria hold but global solutions do not exist

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...

**2**

votes

**1**answer

103 views

### numerical methods for discontinuous ODEs

Greetings,
what are state of art methods for numerical solution of ODEs with discontinuous right side?
I'm mostly interested piecewise-smooth right side functions, e.g. sign.

**5**

votes

**1**answer

287 views

### Multivariate Maximal Hilbert Transform

One way to define the maximal Hilbert transform of a function, $f$, is by
$$\mathcal{H}[f](x):=\sup_{\varepsilon>0} \left| \int_{|x-t|\geq\varepsilon} \frac{f(t)}{x-t} \, dt\right|, \quad ...

**1**

vote

**1**answer

91 views

### An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...

**2**

votes

**0**answers

26 views

### Uniform convergence of the best $L_1$ approximations by polynomials

Let $P_n$ be the vector space of real multivariate polynomials in $d$ variables of total degree no more than $n$ and let $f \in C(X)$, where $X \subset \mathbb{R}^d$ is compact. Let ...

**6**

votes

**2**answers

70 views

### Closed field lines in the plane

A dipole in the plane consists of a positive charge P and an equal and opposite negative charge N separated by a fixed distance . Almost all of the resulting electric field lines (which fill the ...

**8**

votes

**2**answers

245 views

### A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$

Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...

**6**

votes

**2**answers

675 views

### Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions.
I'm very interested in aspects of calculating or computing elementary operations and functions, by which I ...

**5**

votes

**4**answers

706 views

### An identity for the cosine function

Let $x = \pi/(2k+1)$, for $k>0$.
Prove that
$$
\cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k}
$$
I've confirmed this numerically for $n$ from $1$ to $30$.
I'm finding it surprisingly ...

**0**

votes

**1**answer

112 views

### growth of derivative

(Alexandre Eremenko gave the answer to the question I posted earlier. Then I realised the question didn't give what I wanted. Here is the updated question.)
Suppose that $f:[a,\infty)\to \mathbb{R}$ ...

**52**

votes

**2**answers

15k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**1**

vote

**1**answer

98 views

### Can a (non-measurable) autonomous flow have a non-trivial periodic orbit without a minimal period?

Given a set $X$, a function $x \colon \mathbb{R} \to X$ is periodic if there exists $\tau>0$ such that $x(t+\tau)=x(t)$ for all $t \in \mathbb{R}$; and if $\tau$ is the smallest positive number ...

**44**

votes

**22**answers

12k views

### Interesting Calculus Questions/Exercises

I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...

**0**

votes

**0**answers

29 views

### Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T

Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...

**2**

votes

**1**answer

151 views

### comparing Laplacian and gradient of function on boundary

Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$.
Suppose one knows that there is some $C_1,C_2 \in R$ such that
$ x ...

**1**

vote

**0**answers

75 views

### Maximal principle for stochastic heat equation

Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition
$u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$.
Then up to time $T$, the maximal value of $u$ should be ...

**1**

vote

**1**answer

256 views

### Colored noise in SDE

I want to numerically study the behavior of a system described by a set of differential equations in the presence of colored noise. It seems that the standard procedure is to use the Langevin ...

**2**

votes

**1**answer

144 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 ...

**2**

votes

**0**answers

87 views

### Does the following inequality hold under Zygmund condition?

Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, ...

**1**

vote

**0**answers

64 views

### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability ...

**0**

votes

**0**answers

61 views

### Weak convergence of SDE

Let $(X_t,Y_t)$ be the solution to the SDE
\begin{equation}
\begin{split}
dX_t &= f(X_t,Y_t)dt + \sigma_1 X_t dW^1_t\\
dY_t &= g(X_t,Y_t)dt + \sigma_2 Y_t dW^2_t
\end{split}
\end{equation}
...

**2**

votes

**1**answer

289 views

### Pseudo-differential evolution equation

I'm looking for results (or some ideas) on the following kind of pseudo-differential evolution equation:
$$
\frac{\partial u(t,x)}{\partial t} = \int_{-\infty}^{t} B(t-s,x)\, A(x,D_{x})u(s,x)\,ds \; ...

**0**

votes

**0**answers

66 views

### Heredity of harmonic functions

We know that if $u(x,y,z)$ is harmonic in
$\mathbb{R}^3\text{ or }\mathbb{R}^3-\{0\} $, then
$v(x,y):=u(x,y,1)$ may not be harmonic in $\mathbb{R}^2$.
($u = 1/r$, for example)
I wonder what ...

**0**

votes

**0**answers

43 views

### Deriving inequalities from a polynomially-bounded derivative

In this paper (p. 2, definition/remark) the following notion of ‘polynomial growth’ is defined for a non-negative real function $g(x)$ and a real constant $b\in(0;1)$:
There exist positive ...

**1**

vote

**0**answers

57 views

### Characterization of the maximizer of a function based on a parameter's value

Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter.
I have two optimization problems. ...

**0**

votes

**0**answers

90 views

### On isolated points of the approximate point spectrum of a bounded operator

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$.
Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively.
Let ...

**1**

vote

**0**answers

88 views

### On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...

**0**

votes

**1**answer

101 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**0**

votes

**0**answers

50 views

### Compact embedding for Bochner spaces

I was reading some topics about the Bochner integral, and this question arises:
Assume we have a sequence $u_n \in H^s(0,2\pi,L^2(\mathbb{T}))$ converging weakly to $v$ for all $s \in (\frac12, ...