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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Laplace transform

\begin{equation} \begin{cases}\mathbb{D}_{t}^{\beta} u(x, y, t)=-a(x)\left(u_{x}(x, y, t)+u_{y}(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y&...
TUHOATAI's user avatar
10 votes
2 answers
780 views

Asymptotics of a strange oscillatory function

Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\sum_{n\geq 1}\sin(x/n^2)$. It is easy to see that $f(x) = O(\sqrt{x})$ for large real $x$. Is it true that $f(x)>0$ for $x>0$...
Satan's Minion's user avatar
2 votes
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Regularization for Newtonian n-body collisions in $\mathbb{R}^3$

In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
user12994's user avatar
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184 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
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Sets of integers "a little less dense" than the set of prime numbers

Given a set $A \subseteq \mathbb{N}$ of positive integers, put $$ S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \ N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}. $$ There are obvious ...
Stefan Kohl's user avatar
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Electrostatic potential energy of point-charges at primes up to $x$

Given a positive real (or integral) number $x$ we consider the electrostatic potential energy of equal point charges at all primes up to $x$ given by $$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$ ...
Roland Bacher's user avatar
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435 views

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$ $$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$ where $$ B_n=\sum_{k=0}^{\...
Faoler's user avatar
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The intersection of $ n $ cylinders in $ 3D$ space

I posted the question on here, but received no answer I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
user967210's user avatar
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Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
Muniain's user avatar
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Characterization of an integral operator with a Bessel kernel

I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
Didier Felbacq's user avatar
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0 answers
77 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
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An integral involving Riemann xi function

Let $z(t)$ and $w(t)$ be some complex functions of the real variable $t$ with $z(t)$ tending to some complex number as $t \rightarrow 1$ and $w(t)$ tending to $0$ as $t \rightarrow1$ I'm looking at ...
Alexis's user avatar
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1 answer
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Bound on $L^1$ norm of solution of two-point boundary value problem

This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
gmvh's user avatar
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How to derive a lower bound of a MinMax inequality?

Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$. The goal For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
M.K's user avatar
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Admissibility condition of wavelet functions

After a badly formulated question, I decided to make a new post searching for help. The basic problem is the follows: I have a wavelet function $\psi(t)$ (real or complex) and would like to compute (a)...
Luciano Magrini's user avatar
7 votes
2 answers
2k views

Solving 'impossible' integrals with a new (?) trick

The following identities have been suggested based on formulas in a previous question of mine. If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, ...
Emmanuel José García's user avatar
1 vote
1 answer
121 views

Functions for which $\lambda f(x)=f(\alpha_\lambda x + \beta_\lambda)$

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ containing an interval containing $1$ such that for any $\lambda\in S$ there exist $\...
gmvh's user avatar
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2 votes
1 answer
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Behaviour of the solution of a second order ODE

I am currently studying the following second order ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\ y(0) = 0\\ \dot y(T) = c \end{...
Falcon's user avatar
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0 answers
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The Hausdorff measure of intersection of annulus and conformal curve

Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...
mathematics is all's user avatar
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40 views

Godunov splitting convergence research

The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
Redsbefall's user avatar
5 votes
5 answers
967 views

What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$

FYI: I asked this question here couple of days ago but got no answer yet. $n$ is an integer We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
RajaKrishnappa's user avatar
2 votes
1 answer
143 views

Growth rate of elementary sequences

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
António Borges Santos's user avatar
1 vote
1 answer
213 views

Eigenvalues of a Schrödinger operator

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator $$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \...
JMK's user avatar
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2 answers
422 views

$\lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right]$

Find the limit \begin{equation*} \lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right] \end{equation*} where $\lambda > 0$. My guess is that ...
Vassilis Papanicolaou's user avatar
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1 answer
92 views

Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity

Consider the IVP $$ \left\{ \begin{aligned} \frac{d}{dt} \Phi_n(t,x) &= f_n(\Phi_n(t,x)) && \forall t \in \mathbf{R}_+ \\ \Phi_n(0,x) &= x && \forall x \in \mathbf{R} \end{...
zelda's user avatar
  • 73
1 vote
1 answer
118 views

Given a radial symmetric function $f$, the estimate of |$\Delta ^ {m/2}f$| in $R^{2m}$ by induction

This question might be a little strange; my order of Laplacian is related to the dimension of the space. Actually, I’m reading a result which is obtained by induction; it is the absolute value of ...
Elio Li's user avatar
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3 votes
2 answers
205 views

Extremum placement for two-variable function

While teaching Calculus 2, one of my students asked me the following Given 3 points $x_1$, $x_2$, $x_3$. Whether there exists one function $z = f(x,y)$ which has exactly 2 extremum and 1 saddle point:...
IscoBerlin's user avatar
4 votes
0 answers
160 views

Growth rate of a recursively defined sequence

For a side project with a friend (having to do with fractional iterates of the exponential function), we're looking at a sequence $a_n$ defined recursively by equations $a_1 = 1$ and $$a_n = \sum_{k=1}...
Todd Trimble's user avatar
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8 votes
1 answer
325 views

Special Schwartz function on the positive interval

Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following: $\int \zeta(t)\: dt=1$, $\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$, $\operatorname{supp}(\zeta)\subset (0,...
SnowRabbit's user avatar
43 votes
3 answers
2k views

Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$?

For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that $$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$ Let \begin{...
math110's user avatar
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7 votes
1 answer
450 views

On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau

To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series. The two papers the title ...
Daniele Tampieri's user avatar
2 votes
1 answer
131 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{...
G. Panel's user avatar
  • 557
8 votes
1 answer
552 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}...
jdhejw's user avatar
  • 317
2 votes
2 answers
130 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 ...
lming2's user avatar
  • 43
18 votes
1 answer
1k views

Does summing divergent series using cutoff functions give consistent results?

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\...
not all wrong's user avatar
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 ...
Nate River's user avatar
  • 4,604
3 votes
2 answers
597 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 ...
gdoug's user avatar
  • 149
1 vote
1 answer
146 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 $m$...
Leo McIntyre's user avatar
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 =...
Lakshmi Priya's user avatar
0 votes
1 answer
118 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^{\...
yannik0103's user avatar
0 votes
0 answers
66 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) ...
cortadisimo's user avatar
4 votes
2 answers
252 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 ...
Daisy_Duck's user avatar
2 votes
1 answer
174 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 ...
Romain Gicquaud's user avatar
5 votes
0 answers
227 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_{\...
Medo's user avatar
  • 656
1 vote
1 answer
104 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 ...
Samay Varjangbhay's user avatar
4 votes
1 answer
75 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 $\pm iz$: $$y''-(a-2q\cosh(2z))y=0.$$ ...
renphysics's user avatar
14 votes
1 answer
710 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 ...
Nomas2's user avatar
  • 303
5 votes
1 answer
205 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, $...
icecuber's user avatar
  • 320
3 votes
1 answer
142 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 ...
Nilotpal Kanti Sinha's user avatar
10 votes
1 answer
515 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 ...
Fergns Qian's user avatar

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