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 $f(\phi(x,y))=f(x)+f(y)$....
- 716
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)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
- 457
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 1,724
0
votes
0
answers
33
views
Non-positive definite solution for differential Riccati equation
Consider the matrix-valued differential Riccati equation (DRE):
$$
\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,
$$
where all coefficients are continuous.
...
- 503
1
vote
0
answers
52
views
Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
- 562
2
votes
1
answer
317
views
Recommendation for books on boundary-value problems that include perturbed boundaries and many solved problems
I am looking for a book or resource that contains applied math analytical methods and a lot of solved problems in Boundary-Value Problems for second-order PDEs, and if it could be related to wave-...
CommunityBot
- 1
3
votes
1
answer
488
views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
CommunityBot
- 1
2
votes
1
answer
91
views
References for Green's functions right focal boundary-value problem
Could you please give me some references for the computation of a Green's function for a second-order right focal difference equation?
For this problem:
\begin{gather*}
\Delta^2 u(t)=f(t), \; t\in\{0,...
CommunityBot
- 1
43
votes
3
answers
7k
views
Could the Riemann zeta function be a solution for a known differential equation?
Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
- 3,738
7
votes
1
answer
352
views
Tight upper bounds on trigonometric polynomials
According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
- 13k
2
votes
1
answer
197
views
Seeking articles on closed-form formulas for specific partial fraction expansions
I'm currently researching a general closed-form formula, in terms of elementary functions, for functions that have the following type of partial fraction expansion:
$$\frac{1}{x^{p}}+\sum_{n=1}^{+\...
- 188k
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 ...
4
votes
1
answer
442
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 $(0,2\pi)$ ...
- 6,476
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 ...
- 5,624
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
$$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
- 153
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
2
votes
2
answers
242
views
A Inequality in the paper by Kenig, Ponce and Vega
I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle",
...
- 9,007
10
votes
1
answer
580
views
About certain infinite products with the property $f(a)=f(1/a)$
In the paper Transformations of infinite series, Bryden Cais gives the following transformations of infinite products
Theorem 4. If
$$
f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1}{...
- 35.5k
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, ...
- 63.9k
18
votes
11
answers
5k
views
Applications of measure, integration and Banach spaces to combinatorics
I'm going to be teaching a Master's level analysis course (measure theory, Lebesgue integration, Banach and Hilbert spaces, and if there's time, some spectral or PDE stuff) in the fall. My problem is ...
- 63.9k
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 $u:[0,1]\to\mathbb{R}$ ...
- 24.2k
7
votes
2
answers
660
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.
...
- 1,440
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 ...
- 63.9k
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 ...
CommunityBot
- 1
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
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 $\...
- 7,817
15
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.
...
CommunityBot
- 1
10
votes
1
answer
571
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 ...
- 3,312
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
$$J(e^{2\pi i/3})...
- 4,713
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 128k
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\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{ ?}$$
- 141
1
vote
0
answers
122
views
When is there an inclusion between regular Orlicz Spaces?
It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and ...
- 191
17
votes
1
answer
1k
views
Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
4
votes
0
answers
233
views
References for derivative w.r.t. initial condition of an ODE
Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ be measurable such that for all $n \in \mathbb N$ we have
$$
\sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, ...
- 835
2
votes
1
answer
531
views
Radius of the ball where the inverse of Lipschitz maps exists
I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $\delta_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke in On the inverse function ...
- 12.9k
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
$$f(x):=\sum_{i=1}^{n}\dfrac{c_i^n}{\prod_{j\neq i}(...
- 1,538
3
votes
1
answer
252
views
Reference request: analysis of a nonlinear Fokker-Planck type equation
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 1,081
7
votes
5
answers
1k
views
Generalizations of the Euler–Maclaurin Summation Formula
I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me ...
- 2,689
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 ...
CommunityBot
- 1
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 ...
- 128k
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 ...
- 128k
18
votes
2
answers
5k
views
Nonvanishing of Jacobians implies global injectivity?
I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.
In 1953, Samuelson asked the following:
If the ...
- 1,446
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 $$u(t)=a_0+\sum_{k=1}^{\infty}...
- 2,494
1
vote
1
answer
387
views
SDE with non-degenerate diffusion visits every point
I am asking an extension of the question here for SDEs of the Ito form.
Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...
CommunityBot
- 1
0
votes
2
answers
163
views
Convergence of solutions to parametrized ODE when no limiting ODE exists
There is plenty of literature on the convergence of the solutions to the real ODE, parametrized by $N \in (0;\infty)$,
\begin{equation}
f_N' (x)
=
a_N (x) \cdot f_N (x)
+ b_N (x)
\end{equation}
to the ...
- 34.7k
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 188k
11
votes
1
answer
468
views
References on infinite series involving the tetration operator, like $ \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } $
I wonder whether there are any references on infinite series involving the tetration operator, including:
\begin{align} S_{1} &:= \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } \\
&= \frac{1}{2} + \...
- 4,829
3
votes
0
answers
141
views
Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
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 ...
- 6,367