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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2 votes
1 answer
444 views

Flow induced by differentiable velocity field is differentiable

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
2 votes
1 answer
668 views

Transfinite sums related to a sequence

Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products of all $...
2 votes
1 answer
132 views

Boundedness of an exit time from a campact set

Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of \begin{align*} & x(0)=x_0 \\ & \dot{x}=v(x). \end{...
8 votes
1 answer
560 views

What is the minimum of this functional?

Recently I encountered an inequality from mathematical analysis. Let $f(x)$ be twice continuously differentiable in $[0,1]$ with $f(0)=f(1)=0$, then for all $x\in(0,1),f(x)\neq 0$, show that:$$\int_{0}...
2 votes
2 answers
3k views

Multiple outliers for two variable linear regression

Problem Visually, the "extreme" outliers in the following graph are somewhat obvious: Question Given: T - Set of all temperatures Y - Set of all years ΣT - Sum of temperatures. ΣY - Sum of years. ...
2 votes
2 answers
134 views

Upper bound estimation for second-order variable-coefficient ODE

I'm tackling a second-order linear ordinary differential equation with variable coefficients $a(t)$ and seek advice on estimating the upper bound of $y(t)$ s.t $|y(t)|\le M$. The equation in question ...
1 vote
1 answer
420 views

Sufficient and necessary conditions for decomposing the sum of random variables

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
6 votes
1 answer
669 views

Fourier series of smooth functions in infinitely many variables

Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
20 votes
1 answer
1k views

Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
1 vote
1 answer
151 views

Existence for a singular Sturm-Liouville "eigenvalue" problem with non-homogeneous boundary condition

Consider the following singular Sturm-Liouville problem: $$ -(r^{N - 1}h')' - r^{N - 1}c(r)h = \lambda r^{N - 3} h \text{ in } (0, 1), \qquad h(1) = \alpha $$ where $N \in \mathbb N$, $N \geq 3$; $c(...
23 votes
5 answers
3k views

What phenomena are better modelled by SDE instead of ODE?

Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
3 votes
2 answers
600 views

Are there any functions $f$ beyond trivial examples where $\int f(x +f(x + f(x +\dotsb)))\,dx$ $= F(x+F(x+F(x +\dotsb))) + C$ for some function $F$?

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
2 votes
1 answer
434 views

Correction terms in the asymptotic expansion of hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of $\rho$ below), $$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\frac{1}...
3 votes
1 answer
401 views

Positive definiteness of a matrix-valued function

This question is a repost from math.se, where I didn't receive an answer. Are there simple conditions on an $d \times d$ matrix B under which $$ f(t, s) = \begin{cases} \exp(-B |t - s|^\alpha), &...
5 votes
0 answers
232 views

Is there a way to solve this integral on the sphere explicitly?

Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that $k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral $$f(y):=\int_{\...
3 votes
1 answer
263 views

Measuring how suboptimal control is

Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I ...
1 vote
1 answer
155 views

Derivation of indefinite integral involving hypergeometric function

I am doing a project on projectile motion and I ended up with this integral: $$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$ where $g, c,$ and $m$...
13 votes
5 answers
12k views

Beginners text on calculus of variations

I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options. I work on Machine Learning, and that where I intend to apply this. ...
4 votes
2 answers
264 views

Implicit function theorem without uniqueness?

Imagine you are given $f(x,y) := y^2-\sin(x)^2$ and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$. One idea that comes to mind is the ...
4 votes
2 answers
1k views

Does a function exist which is not Riemann integrable and satisfies the given condition:

I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that $$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...
0 votes
0 answers
72 views

Inverse Laplace of the Complex conjugate of the Laplace transform

Let the Laplace transform of f(t) be F(s) and let the inverse Laplace transform of F(s*) be g(t). is there a theorem relating f(t) and g(t)? Basically, looking for a way to calculate g(t) from f(t) ...
0 votes
1 answer
130 views

Asymptotic behavior of the polylogarithm function and generalisation

So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper: $$ \sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...
36 votes
6 answers
4k views

Taylor's theorem and the symmetric group

Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem. Question: ...
0 votes
1 answer
132 views

Example of a concave function with $\lim_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$ which fullfills some additional condition

I'm looking for the example of a concave function $g \colon [0,1] \mapsto \mathbb{R}$, with $g(0)=0$, for which $\lim\limits_{x\to 0^+}\frac{g(x)}{-x\ln x}=\infty$, and $\lim\limits_{x\to 0^+}\frac{\...
1 vote
1 answer
135 views

Concave functions of different behaviour in the neighbourhood of $0$ from the Shannon function

I'm looking for an example of a concave function $g \colon [0,1] \to \mathbb{R}$, $g(0)=0$ such that: $$\liminf_{x\to 0^+}\frac{g(x)}{-x\ln x}\neq \limsup_{x\to 0^+}\frac{g(x)}{-x\ln x}.$$ Moreover, ...
2 votes
1 answer
187 views

Fourier coefficients of the logarithm of a given function

Let $f$ be a $1$-periodic real function that I know is bounded away from zero: $$ f(x) = \sum_{n = -\infty}^\infty c_n e^{2\pi i n x} $$ Let me also assume that $f$ is analytic with Fourier ...
1 vote
1 answer
105 views

Can I accurately approximate solutions for m for any k being an integer : $\sum_{n=1}^{k+1} \frac{k ! m^{(k-n+1)}}{(k-n+1) !}-\frac{k !}{2} e^m = 0$

I had noticed that when approximating solutions for $m$ to the above equation for a given $k$, as $k$ grows larger, the solutions $m$ takes the form $m\approx k+c$ where $c$ is some constant. I'm ...
14 votes
4 answers
1k views

$L^p$ norm means

Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
1 vote
0 answers
127 views

Name for class of functions satisfying $\lim_{x\to 0^+}\lambda g(x)/g(\lambda x)>1$

I would like to ask whether is used some name for functions $g:A\to\mathbb{R}$, $A\subset \mathbb{R}$, for which $$\exists \lambda>1:\;\; \lim_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}>1.$$
4 votes
1 answer
866 views

Interchanging min and max for a continuous function of two variables

Let $f:[0,1]\times[0,1]\to\mathbb{R}$ be a continuous function. Define $$ M_x:=\max\limits_{0\leq y\leq 1} f(x,y), \qquad m_y:=\min\limits_{0\leq x\leq 1} f(x,y). $$ Is there a useful set of ...
4 votes
1 answer
87 views

Eigenvalues of the modified Mathieu equation with normalizable solution

The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$: $$y''-(a-2q\cosh(2z))y=0.$$ ...
14 votes
1 answer
715 views

Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
5 votes
1 answer
208 views

Can a solution to this parameterized ODE converge to zero?

Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$? $$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$ We are also given y(0) = 2/3, $...
3 votes
1 answer
160 views

What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?

This question was posted in MSE but is still open hence posting in MO. The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
10 votes
1 answer
525 views

Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let $$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$ and $$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$ It ...
1 vote
1 answer
53 views

Upper bounds for the spatial differential of the inverse of a flux

It is well known that given a regular velocity field $b: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ (say, continuous in time and uniformly Lipshitz in space), the flux $X$ associated to $b$ is a ...
2 votes
0 answers
49 views

Quadratic surjective map between spheres

The quadratic function $f:\mathbb R^4\to\mathbb R^3$ $$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$ surjectively maps the sphere $S^3$ to the sphere $S^...
25 votes
2 answers
2k views

Writing a function on $\mathbb{R}$ as a sum of two injections

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
2 votes
0 answers
73 views

How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?

Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\...
4 votes
0 answers
270 views

The convention of Fourier transform on symmetric spaces

When trying to understand the Plancherel formula of reductive symmetric space of Harish-Chandra class, I get confused on the convention of Fourier and related transforms. $\newcommand{\H}{\mathcal{H}} ...
6 votes
1 answer
374 views

How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]

I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
5 votes
0 answers
486 views

Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?

Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacobian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill, The integrals $$\operatorname{sn}^2u,\operatorname{...
25 votes
3 answers
2k views

Slick proof of Stirling's Formula?

In Upper Limit on the Central Binomial Coefficient, Noam Elkies and David Speyer have given a nice proof that the central binomial coefficient $\binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}$. This can ...
2 votes
1 answer
589 views

Solution to difference differential equation with constant coefficients

This problem arose when solving a continuous Markov chain exercise from a book I am studying. Given a set of positive $q_i$ with $i \in \mathbb{Z} $, and non-negative $\lambda$ and $\mu$ that add to 1,...
0 votes
1 answer
165 views

If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?

Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $\|\cdot \|$ its induced norm. Let $a: H \times H \to \mathbb R$ be a bilinear form. We say that $a$ is coercive IFF there is $C>...
9 votes
2 answers
1k views

Recurrence of Legendre polynomial roots/ quadrature points

Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$. We know that these roots are distinct, and ...
1 vote
1 answer
295 views

Injective with finite discontinuities mapping from $\mathbb R^n$ to $[0,1]$

As a continuation to the fully answered question: Injective and Integrable Mapping from $\mathbb R^3$ to $\mathbb R$ Does there exist an injective mapping $f:\mathbb R^n\rightarrow[0,1]$ that has only ...
0 votes
0 answers
111 views

Are $\zeta'(0)$ and $\beta'(0)$ algebraic numbers?

Let $\zeta$ be the Riemann zeta function and $\beta$ the Dirichlet beta function. We know that $\zeta (0)=-1/2$ and $\beta (0)=1/2$ are algebraic numbers over $\mathbb{Q}$. This led me to the ...
18 votes
0 answers
697 views

Are these continued fractions of integrals known?

Simplified repost of Are these continued fractions of integrals known? on MSE EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
0 votes
1 answer
103 views

Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems? (time-varying case)

Because flowmaps are homeomorphic maps on a compact domain $\Omega$, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain ...

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