All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
8
votes
1
answer
437
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
2
votes
1
answer
104
views
Looking for review of delay differential equations involving $f(x)$ and $f(x/k)$
A research problem unexpectedly leads me to a delay differential equation of the form
$$
f(x)=\alpha(f(x),f(x/2))\,f'(x)+\beta(f(x),f(x/2))\,f'(x/2)+\gamma(f(x),f(x/2))
$$
For special cases of $\alpha,...
- 3,065
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
1
vote
0
answers
52
views
Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
- 562
2
votes
1
answer
197
views
Seeking articles on closed-form formulas for specific partial fraction expansions
I'm currently researching a general closed-form formula, in terms of elementary functions, for functions that have the following type of partial fraction expansion:
$$\frac{1}{x^{p}}+\sum_{n=1}^{+\...
- 463
4
votes
1
answer
443
views
Reference or proof of a theorem of L. Fejér on summability of Fourier series
In the article "Ensembles exceptionnels" by A. Beurling, the author cites the following theorem of Fejér:
Suppose that a $2\pi$ periodic function $ f $, Lebesgue integrable in $(0,2\pi)$ ...
0
votes
0
answers
33
views
Non-positive definite solution for differential Riccati equation
Consider the matrix-valued differential Riccati equation (DRE):
$$
\dot P_t +PA+A^\top P+Q-(B^\top P+S)^\top (B^\top P+S)=0, \quad t\in [0,T];\quad P(T)=G,
$$
where all coefficients are continuous.
...
- 503
5
votes
2
answers
644
views
On the derivative of the Bernstein polynomial
$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that
$$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
- 128k
5
votes
2
answers
272
views
Integral involving Legendre polynomial
In a physics problem the following integral shows up $$\int\limits_0^{2\pi}P_m(\cos{(\theta-\alpha)})\,\cos^{m+2}{(n\alpha)}\;d\alpha,$$ where $P_m$ is the Legendre polynomial and $n,m$ are integer ...
- 16.5k
71
votes
3
answers
5k
views
Does iterating the derivative infinitely many times give a smooth function whenever it converges?
I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
- 1,763
21
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 $\...
- 313
13
votes
1
answer
1k
views
Apéry's constant $\zeta(3)$ fastest convergent series
UPDATE Feb.02.2024
The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of ...
- 2,836
2
votes
2
answers
242
views
A Inequality in the paper by Kenig, Ponce and Vega
I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle",
...
- 87
7
votes
1
answer
488
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 ...
- 6,367
6
votes
1
answer
796
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
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 128k
12
votes
6
answers
2k
views
Can the positive root of this polynomial be expressed elementarily?
For each real $A>0$, let $x_A$ denote the positive root $x$ of the polynomial $x^5-3x-A$. Is the function $(0,\infty)\ni A\mapsto x_A$ elementary?
[I am using this definition of elementary ...
- 128k
10
votes
1
answer
572
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 ...
- 299
7
votes
2
answers
661
views
For a manual evaluation of a definite integral
I note that Mathematica could yield the identity
$$\int_0^1\frac{\log(1+x^2(x-1)/2)}{x^2(x-1)}dx=\frac{\pi(\pi-4)-12\log^22+24\log2}{16}.\tag{1}\label{1}$$
But I don't know how Mathematica got this.
...
- 15.6k
1
vote
1
answer
109
views
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 $u:[0,1]\to\mathbb{R}$ ...
- 3,065
0
votes
1
answer
301
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
$$J(e^{2\pi i/3})...
- 317
15
votes
1
answer
2k
views
How did Fermi calculate this integral?
In his 1926 paper Fermi states without further explanation that it follows from the Thomas-Fermi equation
$$\frac{d^2\psi(x)}{dx^2}=\frac{\psi(x)^{3/2}}{\sqrt{x}},\label{1} \tag{1}$$
and boundary ...
- 16.5k
2
votes
0
answers
57
views
Regularization for Newtonian n-body collisions in $\mathbb{R}^3$
In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
- 46
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
67
votes
3
answers
12k
views
Is this differential identity known?
Recently I discovered the differential identity
$$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$
valid for any odd natural number $k$; for ...
- 114k
43
votes
3
answers
7k
views
Could the Riemann zeta function be a solution for a known differential equation?
Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
7
votes
1
answer
494
views
Carleson's lectures at UCLA
It seems that Professor Lennart Carleson gave a series of Lectures at UCLA in 1985. For example, one could find several mentions about these lectures in the book by Garnett & Marshal (see for ...
- 747
17
votes
1
answer
1k
views
Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
3
votes
1
answer
252
views
Reference request: analysis of a nonlinear Fokker-Planck type equation
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 730
4
votes
0
answers
233
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
$$
\sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, ...
- 835
7
votes
2
answers
2k
views
The source of the Integral
Wolfram alpha calculates the integral
$$\int\limits_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$
However, I need to cite the source of this identity (the table of ...
- 16.5k
2
votes
2
answers
451
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
0
votes
0
answers
56
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 ...
- 101
54
votes
13
answers
90k
views
Good differential equations text for undergraduates who want to become pure mathematicians
Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate ...
0
votes
2
answers
163
views
Convergence of solutions to parametrized ODE when no limiting ODE exists
There is plenty of literature on the convergence of the solutions to the real ODE, parametrized by $N \in (0;\infty)$,
\begin{equation}
f_N' (x)
=
a_N (x) \cdot f_N (x)
+ b_N (x)
\end{equation}
to the ...
- 335
3
votes
0
answers
163
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
$$f(x):=\sum_{i=1}^{n}\dfrac{c_i^n}{\prod_{j\neq i}(...
- 1,538
2
votes
1
answer
107
views
If $b\in C^1(E, \mathbb{R})$ and $b'$ is compact, then $b$ is weakly continuous — a reference request
While reading the well known book Minimax Methods in Critical Point Theory with Applications to Differential Equations by Paul Rabinowitz, in the proof of a generalisation of the Mountain Pass Theorem ...
1
vote
0
answers
122
views
When is there an inclusion between regular Orlicz Spaces?
It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and ...
- 191
1
vote
1
answer
387
views
SDE with non-degenerate diffusion visits every point
I am asking an extension of the question here for SDEs of the Ito form.
Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...
- 503
7
votes
1
answer
352
views
Tight upper bounds on trigonometric polynomials
According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
- 10.4k
4
votes
1
answer
232
views
Name of a Frobenius-like method for ODEs
Mike McNulty, who is a postdoc working with me, showed me the following trick for looking at asymptotic behavior of ODEs near singular points that he found; my question: does it have a well-known name ...
- 39k
4
votes
2
answers
610
views
Unit ball of the sum space
Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$.
Let $\|\cdot\|_+$ be given by
$$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$
It is well-known that $\|\...
- 39k
4
votes
5
answers
2k
views
Reference request: importance of Lipschitz continuity
I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc.
Could you point me in the direction of some literature that discusses why Lipschitz ...
- 161
2
votes
1
answer
91
views
References for Green's functions right focal boundary-value problem
Could you please give me some references for the computation of a Green's function for a second-order right focal difference equation?
For this problem:
\begin{gather*}
\Delta^2 u(t)=f(t), \; t\in\{0,...
- 27
11
votes
1
answer
468
views
References on infinite series involving the tetration operator, like $ \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } $
I wonder whether there are any references on infinite series involving the tetration operator, including:
\begin{align} S_{1} &:= \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } \\
&= \frac{1}{2} + \...
- 4,829
3
votes
2
answers
809
views
Growth of $L^p$ norms as $p \to \infty$
Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
- 1,211
4
votes
1
answer
487
views
Nonsmooth version of Hopf boundary point lemma
Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\...
- 5,380
1
vote
1
answer
123
views
Where is the maximum of the product of two logistic curves?
I've got an asymmetric peak-like function $y(x) = y_1(x)y_2(x)$,
where $y_1(x) = 1 / (1 + f_1(x)) = 1 / ( 1 + e^{( -r_1(x-x_1))})$ is an increasing logistic function
and $y_2(x) = 1 / (1 + f_2(x)) ...
- 11
6
votes
2
answers
513
views
Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
- 335
6
votes
2
answers
921
views
Has the "partial Sophomore's Dream function" been studied before?
We can consider the generalized Harmonic numbers $$H_{n,m} := \sum_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim_{n \to \infty} H_{n,m} = \zeta(m). $$
...
- 4,829