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

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
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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
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1 answer
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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
208 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
163 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
338 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
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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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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
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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
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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
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2 answers
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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
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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
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3 votes
2 answers
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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
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1 answer
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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
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2 answers
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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
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1 answer
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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
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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) ...
cortadisimo's user avatar
4 votes
2 answers
265 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
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 ...
Romain Gicquaud's user avatar
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_{\...
Medo's user avatar
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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
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 $\pm iz$: $$y''-(a-2q\cosh(2z))y=0.$$ ...
renphysics's user avatar
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 ...
Nomas2's user avatar
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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, $...
icecuber's user avatar
  • 330
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 ...
Nilotpal Kanti Sinha's user avatar
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 ...
Fergns Qian's user avatar
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^...
rikhavshah's user avatar
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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: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\...
Nomas2's user avatar
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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 ...
Euler-Masceroni's user avatar
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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>...
Akira's user avatar
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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: \...
Burak's user avatar
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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 ...
Nomas2's user avatar
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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{...
Nomas2's user avatar
  • 303
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 ...
li ang Duan's user avatar
0 votes
1 answer
177 views

Can we integrate arbitrary rational functions of Jacobian elliptic functions?

We can integrate arbitrary rational functions of the trigonometric functions because of the tangent half-angle substitution (https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution). This led ...
Nomas2's user avatar
  • 303
2 votes
0 answers
120 views

Uniqueness of the solution to systems of first-order linear PDEs

Context: Let $\Omega \subset \mathbb{R}^p$ be an domain. For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
Paruru's user avatar
  • 51
0 votes
1 answer
113 views

When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, it is claimed on page 323 that for an arbitrary Banach space $E$, if $\pi$ is the topology on $E^*$ of uniform convergence on compact subsets of ...
i like math's user avatar
0 votes
0 answers
85 views

When can an affine functional on the dual be represented as an element of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
i like math's user avatar
0 votes
1 answer
124 views

Matrices and vectors of intervals

I'm working on a project and think that matrices and vectors of intervals will be useful. I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
Paul R's user avatar
  • 39
0 votes
1 answer
285 views

Uniqueness of the $J$ invariant

It seems that The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that $$J(e^{2\pi i/3})...
Nomas2's user avatar
  • 303
2 votes
0 answers
62 views

The $n$-th reproducing kernel of orthogonal polynomial

Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product $$ \langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)} $$ ...
Karim's user avatar
  • 21
1 vote
1 answer
171 views

Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system? that is, does ...
li ang Duan's user avatar
1 vote
2 answers
111 views

Computation of tangent functional

In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows. If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as \begin{equation} \...
i like math's user avatar
-2 votes
1 answer
158 views

If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]

This seems like it should be true but I was wondering if anyone could prove it. Thanks!
li ang Duan's user avatar
11 votes
0 answers
2k views

Eric T. Sawyer's proof of Fourier restriction conjecture

Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
a curious fellow's user avatar
3 votes
4 answers
440 views

Asymptotic for Ramanujan's $\tau$-function

The Ramanujan's $\tau$-function is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$. Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
Nomas2's user avatar
  • 303
6 votes
1 answer
759 views

A Poincaré-like inequality

Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have $$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx \le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
Iosif Pinelis's user avatar
1 vote
0 answers
66 views

Are analytic solutions for the Navier-Stokes equations sufficient?

Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space. However, I am wondering, whether it is possible to consider just analytic ...
tobias's user avatar
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