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.
3,444
questions
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Eigenvalues of a Schrödinger operator
I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator
$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$
$$\varphi(0) = \...
3
votes
2
answers
459
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$\lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right]$
Find the limit
\begin{equation*}
\lim_{n \to \infty} \frac{2^n}{n} \left[
1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k}
\right]
\end{equation*}
where $\lambda > 0$.
My guess is that ...
1
vote
1
answer
94
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Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity
Consider the IVP
$$
\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{...
1
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1
answer
119
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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 ...
3
votes
2
answers
208
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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:...
4
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0
answers
163
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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
$$a_n = \sum_{k=1}...
8
votes
1
answer
338
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Special Schwartz function on the positive interval
Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following:
$\int \zeta(t)\: dt=1$,
$\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$,
$\operatorname{supp}(\zeta)\subset (0,...
43
votes
3
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Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$?
For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$
Let
\begin{...
7
votes
1
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459
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On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau
To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...
2
votes
1
answer
132
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Boundedness of an exit time from a campact set
Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of
\begin{align*}
& x(0)=x_0 \\
& \dot{x}=v(x).
\end{...
8
votes
1
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560
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What is the minimum of this functional?
Recently I encountered an inequality from mathematical analysis.
Let $f(x)$ be twice continuously differentiable in $[0,1]$ with
$f(0)=f(1)=0$, then for all $x\in(0,1),f(x)\neq 0$, show that:$$\int_{0}...
2
votes
2
answers
134
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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 ...
18
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1
answer
1k
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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 $\...
23
votes
5
answers
3k
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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
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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 ...
1
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1
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155
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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 $m$...
4
votes
2
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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 =...
0
votes
1
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130
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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:
$$
\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...
0
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0
answers
72
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Inverse Laplace of the Complex conjugate of the Laplace transform
Let the Laplace transform of f(t) be F(s) and let the inverse Laplace transform of F(s*) be g(t). is there a theorem relating f(t) and g(t)? Basically, looking for a way to calculate g(t) from f(t) ...
4
votes
2
answers
265
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Implicit function theorem without uniqueness?
Imagine you are given $f(x,y) := y^2-\sin(x)^2$
and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$.
One idea that comes to mind is the ...
2
votes
1
answer
187
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Fourier coefficients of the logarithm of a given function
Let $f$ be a $1$-periodic real function that I know is bounded away from zero:
$$
f(x) = \sum_{n = -\infty}^\infty c_n e^{2\pi i n x}
$$
Let me also assume that $f$ is analytic with Fourier ...
5
votes
0
answers
232
views
Is there a way to solve this integral on the sphere explicitly?
Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that
$k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral
$$f(y):=\int_{\...
1
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1
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105
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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 ...
4
votes
1
answer
87
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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
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715
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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
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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
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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
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525
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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 ...
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^...
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:
$$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$
$$\...
6
votes
1
answer
374
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How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
0
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1
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165
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If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $\|\cdot \|$ its induced norm. Let $a: H \times H \to \mathbb R$ be a bilinear form. We say that
$a$ is coercive IFF there is $C>...
25
votes
2
answers
2k
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Writing a function on $\mathbb{R}$ as a sum of two injections
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
0
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0
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111
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Are $\zeta'(0)$ and $\beta'(0)$ algebraic numbers?
Let $\zeta$ be the Riemann zeta function and $\beta$ the Dirichlet beta function. We know that $\zeta (0)=-1/2$ and $\beta (0)=1/2$ are algebraic numbers over $\mathbb{Q}$. This led me to the ...
5
votes
0
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486
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Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?
Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacobian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill,
The integrals
$$\operatorname{sn}^2u,\operatorname{...
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 ...
0
votes
1
answer
177
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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 ...
2
votes
0
answers
120
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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 ...
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 ...
0
votes
0
answers
85
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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 $T(x_0)$ ...
0
votes
1
answer
124
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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 ...
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
$$J(e^{2\pi i/3})...
2
votes
0
answers
62
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The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product
$$
\langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)}
$$
...
1
vote
1
answer
171
views
Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?
Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system?
that is, does ...
1
vote
2
answers
111
views
Computation of tangent functional
In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows.
If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as
\begin{equation}
\...
-2
votes
1
answer
158
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
11
votes
0
answers
2k
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Eric T. Sawyer's proof of Fourier restriction conjecture
Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
3
votes
4
answers
440
views
Asymptotic for Ramanujan's $\tau$-function
The Ramanujan's $\tau$-function is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$.
Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
6
votes
1
answer
759
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\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{ ?}$$
1
vote
0
answers
66
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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 ...