**16**

votes

**3**answers

2k views

### Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...

**5**

votes

**0**answers

65 views

### Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only):
Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then
$$
\left| \int_{\mathbb{R}^3} f - ...

**2**

votes

**1**answer

76 views

### Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$.
I assume that my ODE ...

**0**

votes

**0**answers

22 views

### diffusion and potentials in several dimensions

In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ...

**-1**

votes

**0**answers

75 views

### how in can to extend the adjoint

Let $T_{a},a\in C$ be a closed operator defined on $D$ subspace of $L^{2}(R)$ onto $L^{2}(R)$ $(T_{a}: D\rightarrow L^{2}(R) )$ with
$D$ contains a Schwartz space $S$.
...

**-5**

votes

**0**answers

61 views

### how to prove the inequality on hold? [closed]

Question:How to prove the following inequality?
Is there anyone who can tell me ?thank you
$$(x-y)^{\gamma}\leq x^{\gamma}- y^{\gamma}, \forall x\geq y\geq 0$$
where $\gamma>3/2$

**2**

votes

**0**answers

152 views

### Euler-Maclaurin Formula Review

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely
Generalizations to broader ...

**-3**

votes

**0**answers

24 views

### unable to solve this differential equation [migrated]

$\frac{dy}{dt}=(1-y)(1+6y)$
How can I solve this, please help.
I tried using Classical Runge-Kutta , but the results are not satisfying.
Can anyone suggest some other method?

**28**

votes

**3**answers

3k views

### What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...

**2**

votes

**1**answer

169 views

### Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com
http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe
Now I ...

**6**

votes

**1**answer

205 views

### Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that
$$
\|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 ...

**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 ...

**2**

votes

**1**answer

143 views

### Ramanujan-type sum

Could you show that
$$\sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi ...

**7**

votes

**1**answer

149 views

### $C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a ...

**9**

votes

**1**answer

480 views

### Find functions such that f(f(x))=f(x)e^x

Are there monotonically increasing functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = e^x f(x)$?

**3**

votes

**1**answer

164 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 ...

**1**

vote

**0**answers

95 views

### Ramanujan sum type

I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi ...

**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 ...

**6**

votes

**1**answer

317 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
...

**7**

votes

**4**answers

4k views

### Applications of Euler-Cauchy ODEs

The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...

**1**

vote

**0**answers

54 views

### History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -
$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall ...

**3**

votes

**1**answer

76 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

3k 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 ...

**4**

votes

**5**answers

811 views

### A graduate course on Sturm Liouville theory?

I have some general questions on Sturm-Liouville theory. We are planning to introduce a graduate course on Sturm-Liouville theory and every one has been asked to propose topics which might be suitable ...

**3**

votes

**0**answers

59 views

### Factoring Bessel functions into an amplitude and a phase

Take some $\nu>0$. Let $J_\nu(x)$ be the Bessel function of the first kind. Let's restrict its domain to $\mathbb R^+$. Is it possible to find a pair of functions $A_\nu(x), \phi_\nu(x):\mathbb ...

**7**

votes

**2**answers

214 views

### differential equation of conics

Function $y(x)$ on the plane defines a conic (strictly speaking, at most second order algebraic curve) if and only if $\frac{d^3}{dx^3} (y'')^{-2/3}=0$. What is intuition behind this? How may I see ...

**1**

vote

**1**answer

140 views

### What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper,
http://arxiv.org/pdf/math/0401310.pdf
Has this been improved?

**0**

votes

**1**answer

71 views

### Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps.
In the decomposition
$$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...

**0**

votes

**1**answer

79 views

### Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...

**0**

votes

**2**answers

53 views

### Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix.
I've seen a similar ...

**0**

votes

**0**answers

28 views

### Fourier tranform of the Euclidean norm [migrated]

where can I find the Fourier transform of the power of the Euclidean norm?, that is:
$$\mathcal{F}[\|x\|^{p}](\omega) = \int_{\mathbb{R}^{d}}\exp(-2\pi i \langle\omega, x\rangle) \|x\|^{p} dx$$
...

**7**

votes

**1**answer

252 views

### Rotation invariance of an integral

Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...

**2**

votes

**3**answers

238 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

**2**answers

234 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**2**

votes

**0**answers

111 views

### Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...

**1**

vote

**0**answers

34 views

### convergence of ODE [closed]

I have 2 coupled linear ODEs. I used Mathematica to solve for analytical solution. But the analytical solution looks too complicated. I only need to derive some monotonicity property of the solution. ...

**2**

votes

**2**answers

89 views

### Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain:
$$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$
where $m>0$ is fixed.
Question. To ...

**8**

votes

**2**answers

906 views

### Wasserstein distance in R^d from one dimensional marginals

This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...

**2**

votes

**1**answer

70 views

### Challenging problems concerning Jacobian elliptic functions with complex modulus

I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since ...

**14**

votes

**4**answers

477 views

### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

**2**

votes

**0**answers

45 views

### Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$.
My question concerns solutions to $\triangle_g u =0$ that are say ...

**5**

votes

**2**answers

129 views

### Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...

**0**

votes

**0**answers

28 views

### Uniqueness and Properties of Nonlinear 2nd Order ODE with Asymptotically Constant Coefficients

I have the following differential equation:
$$ V(z) = e^z [1-\varphi(\gamma)]+(\gamma-g)V'(z)+\frac{1}{2}\kappa^2 V''(z)$$ with
$$ e^z \varphi'(\gamma) = V'(z)$$ where $\varphi(\cdot)$ is a ...

**21**

votes

**4**answers

703 views

### When are some products of gamma functions algebraic numbers?

I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...

**5**

votes

**4**answers

394 views

### How do the roots of a polynomial change when another polynomial is added?

I need to obtain an analytical solution to an equation of the following form:
$$
(x-a)(x-b)(x-c)=d(x-e)(x-f),
$$
where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable.
Of ...

**2**

votes

**0**answers

42 views

### 1D inhomogeneous linear Schrodinger equation

I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm ...

**1**

vote

**1**answer

154 views

### find solution of complex number recurrence equation

I have the following recurrence equation:
$$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$
for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex ...

**0**

votes

**0**answers

88 views

### Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers.
Does
$$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in ...

**2**

votes

**0**answers

77 views

### Is a one-dimensional unstable manifold of an ODE a union of the associated equilibrium point and two full orbits? [closed]

Consider an ordinary differential equation (ODE) system \begin{align}
\frac{dx}{dt} = f(x)
\end{align} where $x \in \mathbb{R}^n$ ($n \geq 2$) and the vector field $f$ is defined on an open subset $X$ ...

**2**

votes

**0**answers

27 views

### A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...