Questions tagged [ca.classical-analysis-and-odes]

Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

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17 votes
1 answer
930 views

Catalan's constant fast convergent series

NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known. Working with some conjectured continued fractions that were published here, I have found ...
0 votes
0 answers
151 views

How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its ...
1 vote
0 answers
74 views

Analytical solution to coupled ODEs arising in molecular evolution

The following system of coupled ODEs arises in the study of DNA sequence evolution: \begin{eqnarray*} \frac{da}{dt} & = & \frac{\mu (1-y) b u}{S - y(S-b-v)} - (\lambda +\mu ) a \\ \frac{db}...
5 votes
5 answers
3k views

Generalized Gauss-Green theorem

I am looking for a generalized version of the Gauss-Green theorem also known as the divergence theorem: Divergence theorem - Wikipedia A quick search on MathSciNet suggests that there are ...
96 votes
28 answers
14k views

Probabilistic proofs of analytic facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
3 votes
2 answers
621 views

Simultaneous extensions of strongly convex functions

21/03/2017: I have decided to accept Denis Serre's answer, even though it does not exactly answer my question, however I like its simplicity and I'd say it is close enough to the desired claim. Of ...
0 votes
0 answers
96 views

Find an explicit solution to $ -u''= u^2\log |u|$

It is known that $u=e^{-|x|^2+\frac{n}{2}}$ is a solution to the equation $-\Delta u= u\log |u|~~ \text{in}~ \mathbb{R}^n$, which can be obtain by a limiting process ($p\to 1^+$) of $-\Delta u=\frac{u|...
0 votes
2 answers
334 views

How can I derive functional properties of (the solutions of) this simple functional differential equation?

I've not yet finished a course in functional analysis so I'm unsure how to go about this, but I've always been fascinated by a simple functional differential equation I concocted for almost no reason. ...
10 votes
2 answers
596 views

Bi-Lipschitz extension

Given a bi-Lipschitz homeomorphism $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
1 vote
1 answer
144 views

On the compact embedding of Sobolev space

In dimension three, we know that the Sobolev space $W^{\frac{13}{11},11}(D)$ is compactly embedded into $W^{1,11}(D)$, where $D$ is a bounded domain in $R^3$ with smooth boundary. My question is: Does ...
4 votes
0 answers
167 views

References for derivative w.r.t. initial condition of an ODE

Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ be measurable such that for all $n \in \mathbb N$ we have $$ \sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, ...
2 votes
2 answers
232 views

What can be said about the roots of the polynomial $x^{n+1} - (1 - x)^n ( n + x )$?

I have encountered the polynomial equation $$x^{n+1} = (1 - x)^n ( n + x )$$ where $n \geq 0$, and I am interested in its real roots. The number $n$ can be an integer or, more generally, any positive ...
2 votes
1 answer
147 views

Hölderness of the inverse to a $W^{1,p}$-homeomorphism (with additional conditions) of a Lipschitz domain

Let $\Omega_1 \subset \mathbb R^2$ be a bounded simply-connected Lipschitz domain, and $f: \bar \Omega_1 \rightarrow \bar \Omega_2$ be a homeomorphism, which is a diffeomorphism on $\Omega_1$ such ...
85 votes
12 answers
87k views

Why is the gradient normal? [closed]

This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
5 votes
1 answer
239 views

Does convolution of a compactly supported function with Gaussian need to have fraction of the $L_1$ mass in the original interval?

Let $f \in L_1(\mathbb{R})$ be such that $\operatorname{supp} f \subset [0,1]$, and let $K$ be the gaussian kernel $K(t) := \frac{1}{\sigma \sqrt{2 \pi}} \exp(-t^2/2\sigma^2)$, with some small $\sigma ...
1 vote
0 answers
150 views

A generalization of polynomials in one variable

Let us consider the space of polynomials $P^N$ of degree $\le N$. If $f\in P^N$ vanishes in $>N$ points, then $f\equiv 0$, but for any $N$ points, or fewer, there exists $f\neq 0$ vanishing at ...
3 votes
1 answer
617 views

What is the Fourier series of $\sin(1/x)$ in $[-\pi,\pi]$?

What is the Fourier series of $\sin(1/x)$ (or $x^k\sin(1/x)$, where $k$ is a positive integer) in $[-\pi,\pi]$? This function evidently does not satisfy Dirichlet's conditions. However, Dirichlet's ...
54 votes
8 answers
9k views

Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
1 vote
1 answer
128 views

Time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2}\exp u(x) \, dx< +\infty$

I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2} \exp u(x) \, d x< +\infty$. LEMMA 1.1 (...
3 votes
0 answers
103 views

Regularity of solutions to an integral ODE

I'm trying to figure out the regularity for solutions to the following integral equation: \begin{align} \begin{cases} \displaystyle{\frac{d}{dt}}u(t,x)&=\displaystyle{\frac{1}{\sigma_d\epsilon^{d+...
2 votes
1 answer
489 views

Radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in On the inverse function ...
6 votes
1 answer
732 views

Twisted Riemann sums

Let $f(x)$ be a real-valued Riemann integrable function supported in $[0,1]$ with range in $[0,1]$. Let $\alpha$ be irrational. Consider the weighted Riemann sum $$S_N:=\frac{1}{N}\sum_{k=1}^Nf\left(\...
8 votes
3 answers
1k views

Are all positive eigenfunctions principal eigenfunctions?

In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$? Also, more generally, does this also apply for $...
3 votes
1 answer
134 views

Recover an $L^1$ integrand by partial differentiation

Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by $$F(x,y) = ...
1 vote
0 answers
58 views

$L^p$ norm of Fourier transform of function composed with a diffeomorphism

Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
3 votes
0 answers
148 views

Number of positive roots for an exponential sum

Given $n\geq 3$ distinct constants $c_1, c_2, ..., c_n \in\mathbb{C}$, I want to bound/estimate the number of positive real roots for the equation $$f(x):=\sum_{i=1}^{n}\dfrac{c_i^n}{\prod_{j\neq i}(...
2 votes
1 answer
194 views

Asymptotics for oscillatory integral

Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$ $$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
5 votes
1 answer
980 views

Analytic functions where all derivatives vanish at infinity and which are bounded

Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$. I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
-4 votes
1 answer
103 views

Uncountable Cantor's diagonal argument on $S^2$ [closed]

Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a unit vector $v \in \mathbb{R}^2$ and a continuous function $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $...
1 vote
0 answers
66 views

Estimating commutator of Fourier integral

Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by $$ s_n(g)(x) = \frac{nf(x)}{4\pi i} \int_{\mathbb R^2} e^{\frac{in\vert x-y\...
2 votes
0 answers
73 views

Verify the explicit solution formula for a degenerate Fokker-Planck equation $\partial_t u = \nabla\cdot(D\,\nabla u + Cxu)$

Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$ $$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$ where $D \in \mathbb{R}^{d \times d}$ ...
0 votes
2 answers
236 views

Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$

How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$? Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
4 votes
1 answer
154 views

Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians

Consider a macroscopic free energy functional of the form $$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
2 votes
1 answer
2k views

Uniqueness of the logarithm function

In my Analysis class, we started to prove a theorem that said: Let a > 1. So there is a unique increasing function $f:(0,\infty)\to\mathbb{R}$ so that: $f(a) = 1$ $f(xy) = f(x) + f(y)\quad\forall ...
3 votes
1 answer
211 views

Reference request: analysis of a nonlinear Fokker-Planck type equation

It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
33 votes
6 answers
2k views

Is there a topology on growth rates of functions?

I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions $(0,\infty) \to (0,\infty)$, where two ...
1 vote
0 answers
94 views

Regularity of solution to heat equation

If $u$ is solution of $u_{t}=\Delta u$ in a bounded domain $\Omega$, is it true that: $\sum_{\lvert \alpha\rvert=2k}\|D^{\alpha}u\|_{L^{2}(\Omega)}^{2}\leq |\Delta^{k}(u^{(l))}\|_{L^{2}(\Omega)}^{2}=\...
3 votes
1 answer
227 views

Computing global maximum

For $\lambda\in\mathbb{R}$, I want to find the expression of $f(\lambda)$: $$f(\lambda)=\max_{E\in\mathbb{C}}arccosh{\frac{|E^2+i\lambda E|+|E^2+i\lambda E-4|}{4}}-arccosh{\frac{|E^2-i\lambda E|+|E^2-...
7 votes
5 answers
1k views

Generalizations of the Euler–Maclaurin Summation Formula

I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me ...
0 votes
0 answers
36 views

Conditions for an ODE with convolution term to have a probability distribution solution

Suppose we have a simple ODE like: $$ y''(x) + 2ay'(x) + by(x) = 0 $$ with the condition $y(0)=0$ for $x\leq 0$. Then the solution on $(0,\infty)$ will be of the form $Axe^{-ax}$ when $a^2=b$, $Ae^{-...
0 votes
0 answers
47 views

When is the derivative of an ODE solution wrt the initial condition a simple exponential?

Suppose we are given an autonomous ODE $y' = f(y)$ such that $y$ is smooth with respect to the initial condition. We then have $$(\nabla y)' = \nabla f(y) \nabla y.$$ If $\nabla f(y(t)) $ commutes ...
5 votes
2 answers
286 views

Euler–Maclaurin formula in $\mathbb{Z}^d$

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \...
11 votes
2 answers
1k views

Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?

The expected distance $d$ of randomly selected points within a unit square to the square's center is $d = \frac{1}{6} P$ where P is the universal parabolic constant $P = \sqrt{2} + \ln{\left(1+\...
142 votes
7 answers
14k views

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ where the ...
121 votes
12 answers
27k views

How to solve $f(f(x)) = \cos(x)$?

I found the following equation on some web page I cannot remember, and found it interesting: $$f(f(x))=\cos(x)$$ Out of curiosity I tried to solve it, but realized that I do not have a clue how to ...
0 votes
1 answer
83 views

On the validity of a certain Grönwall-type inequality

Assume that $u~ \colon \mathbb{R}_+ \to [-M,M]$ is a bounded continuously differentiable function such that $u(0) = 0$ and $$u(t) \leq \int_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{1}\tag{1}$$ where ...
21 votes
4 answers
6k views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $$f(x) =...
2 votes
1 answer
326 views

What are the best definitions for smoothness of a 2D curve (real-valued function)?

Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps, some with sharp ...
10 votes
0 answers
199 views

Bi-Lipschitz mappings

Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...
0 votes
1 answer
135 views

How are the Legendre Polynomials of second kind for negative degrees defined?

For a script I have to evaluate the associated Legendre polynomial of second kind $Q^0_{n}(z)$. Until now, I was using an implementation that is based on the definition in equation 8.702 in the Book &...

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