All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
9
votes
2
answers
2k
views
Interpolation of Sobolev spaces
I know quite a bit about the abstract theory of Interpolation of Banach spaces. Today I had a colleague from Environmental sciences (who used to be in our Applied Maths department) come and ask me ...
- 18.7k
3
votes
0
answers
111
views
When does the constant term in the following expansion is nonzero?
Dyson's Theorem
The constant term in the expansion of
$$\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_i}$$
is the multinomial coefficient
$$\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!},$$
...
- 1,997
10
votes
1
answer
1k
views
What would the best treatment of Gehring's lemma look like?
In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
- 2,041
3
votes
1
answer
278
views
Lyapunov stability of linear system
Consider a linear ODE system
$$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$
Proposition. Suppose that $$\sup_{t\ge 0}\Big\{\int_0^t\Big(a_{kk}(...
- 31
3
votes
1
answer
334
views
Does this function have any exponential growth?
Has anyone seen any function of the following type?
$$
g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant $c>...
- 1,649
4
votes
5
answers
891
views
Analytic hypoellipticity of linear ordinary differential operators
Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...
- 1,412
3
votes
0
answers
375
views
An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only
The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
- 128k
3
votes
3
answers
2k
views
How do we use an Ehresmann connection to define a semispray?
Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
- 8,512
0
votes
1
answer
125
views
A slight generalization of triconfluent Heun equation: what is known?
I have recently come across an ODE of the form
$$y''+(a+b x^2)y'+(c+dx+h/x^2)y=0 \hspace{30mm} (*)$$
where $y=y(x)$ and $a,b,c,d,h$ are arbitrary constants.
As far as I understand (please correct ...
- 312
3
votes
0
answers
113
views
the topological equivalence of linear autonomous system
N. Ladis and Kuiper gave the classification of the topological equivalence of linear autonomous system. More precisely, they proved if two linear autonomous system $\dot{X}= AX, \dot{X}= BX$ are ...
- 799
8
votes
1
answer
5k
views
How to "globalize" the inverse function theorem?
Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function ...
- 81
5
votes
1
answer
136
views
Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'
Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
- 1,771
0
votes
0
answers
490
views
The following ODE global existence theorem reference?
There is an ODE existence theorem of the form:
Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function.
Suppose that there is a constant $c$ such that if $y$ is a solution, then $...
- 251
5
votes
2
answers
774
views
Can we calculate the inner product of a semicontinous function with the Dirac delta function?
Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
- 1,649
3
votes
1
answer
329
views
methods for situations where well-posedness criteria hold but global solutions do not exist
I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...
- 1,051
1
vote
1
answer
403
views
Derivative of a time evolution operator w.r.t. a parameter
Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...
- 8,086
9
votes
2
answers
1k
views
Hilbert transforms of measures
Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
- 91
1
vote
2
answers
215
views
vector valued BVP for ODE's
I am dealing with a vector valued second order homogeneous BVP:
$\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$
where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and $...
- 1,211
4
votes
1
answer
902
views
Exact Differential Equations of Order n via Pfaffian Differential Equations?
I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ...
- 93
11
votes
1
answer
660
views
"A sea-side town where every house can see the sea"
This is a reference request.
The phrase in the title is, if I remember correctly, how Eli Stein described the following set (the definition may be faulty, but I think it is right):
There exists ...
- 39.1k
1
vote
3
answers
1k
views
Interpretation of the two-dimensional de-Rham complex
The de-Rham complex in one dimension describes phenomena that can be described in terms of ordinary differential equations. The de-Rham complex in three dimensions can be used to describe classical ...
- 5,327
6
votes
2
answers
390
views
Solution uniqueness for ODE
I have a vectorial, non-linear second order ordinary differential equation
$$Z''=f(Z)$$
for which I have a solution $Z^0$ on $[0,1]$ with $Z^0_i(0)=0$ and $Z^0_i(1)=1$. I would like to know under ...
- 14.4k
4
votes
1
answer
354
views
Reference request: Invariant sets of dynamical systems
(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...
- 185
2
votes
2
answers
213
views
Caratheodory equations
Ok, I am reading Fillipov book on discontinuous right hand side differential equations (the red book).
He states the next lemma:
"
Let the function $f(t,x)$ satisfy the Caratheodory conditions and ...
- 1,594
5
votes
2
answers
952
views
Good references for analytic solutions to nonlinear ordinary differential equations?
I am faced with a non-autonomous initial value problem for a function $x:[0,\infty) \to \mathbb{R}^2$ of the form
$$ x'(t) = f(t,x(t)) $$
for $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}^2$ with ...
- 33.3k
4
votes
0
answers
122
views
Complex L^1 spaces; reference request
I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. ...
user78249
5
votes
3
answers
566
views
Reference/Introduction to partial difference(NOT differential!) equations
The title says it all. Despite heavy googling I have not been able to find anything. What I am interested in, is theory (maybe modelling), not for the moment finite difference methods as ...
- 2,600
1
vote
0
answers
120
views
Interpolation functional for BV spaces?
Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
- 121
4
votes
2
answers
283
views
Bounding the series of the geometric means of the terms of a given positive series
Let $ \{ a _ k \} _{k\in\mathbb{N} _ +} $ be a sequence of non-negative numbers, and let $MG(a_1,\dots,a_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality
$$ \sum _ {n\ge 1}...
- 60.5k
1
vote
1
answer
448
views
Absolute convergence of multi-dimensional Fourier series
For a Lipschitz function $f$ defined in $[0,2\pi]^d$ for $d>1$, is that true
that the multi-dimensional Fourier series converges absolutely?
In other words, $\sum_{k\in \mathbb{Z}^d}|\hat{f}(k)|<...
- 401
2
votes
0
answers
283
views
Uniqueness of analytic center manifold
In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
- 21
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
4
votes
1
answer
265
views
a limit by Gosper involving a product of arctan and $4^{1/\pi}$
On the Wolfram page about pi formulas, there is this curious limit by R. W. Gosper (130) $$\lim\limits_{n\to\infty}\prod\limits_{k=n}^{2n}\dfrac{\pi}{2\arctan k}=4^{1/\pi}.$$
The only reference given ...
- 13.4k
1
vote
3
answers
376
views
Extension of lipschitz functions along a curve
Given a curve $\gamma$ in a Banach space $X$ and a function f defined along the curve s.t.
$$\big\Vert f(\gamma(t))-f(\gamma(s))\big\Vert\\leq L\big\Vert\gamma(t)-\gamma(s)\big\Vert$$
is it possible ...
- 1,256
4
votes
1
answer
375
views
Asymptotic solution of the integral equation
What is the asymptotic solution (for $s\gg 1$) of the following integral equation $$z(s)=1+\gamma\int\limits_{-\infty}^s ds_1\int\limits_{-\infty}^{s_1}ds_2
\cos{(s_1^2-s_2^2)}z(s_2)\;?$$
In fact I ...
- 16.5k
5
votes
2
answers
917
views
Is the inclusion of Lebesgue spaces compact?
[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-...
- 39.1k
6
votes
3
answers
423
views
Infinite electrical networks and possible connections with LERW
I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
Given a lattice L, we turn it into a ...
- 85.6k
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795
5
votes
0
answers
224
views
Unit eigenvalue of the linearized Poincare return map
Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...
- 51
3
votes
0
answers
133
views
Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals
Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...
- 10.5k
3
votes
3
answers
285
views
Limit connected with a periodic function
I am posting the following question from Math.Stackexchange:
Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula
$$
f(x)=2x-1.
$$
For a real ...
- 217
5
votes
0
answers
79
views
Some questions about the Lévy monoid of certain densities
Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
- 10.5k
2
votes
2
answers
467
views
Ewald's generalized theta function
Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-(
Die Berechnung optischer und ...
- 75
3
votes
0
answers
132
views
Uniqueness of solution of elliptic equation with exponential nonlinearity
Consider the following equation
$$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$
where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...
- 31
3
votes
0
answers
135
views
Motivation for the existence of periodic solutions [closed]
I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
$$\ddot{...
- 41
4
votes
2
answers
736
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
- 8,512
1
vote
0
answers
176
views
Lyapunov stability, nonlinear system
Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article.
...
- 21
2
votes
0
answers
99
views
Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?
Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
- 10.5k
2
votes
0
answers
73
views
Reference request on a notion of independence for families of [real-valued] functions
This is basically another reference request.
Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that ...
- 10.5k
1
vote
0
answers
66
views
"Harmonic oscillator" with $p$-Laplacian
I wonder if there is any literature on the eigenvalue problem for the "$p$-harmonic oscillator" $$-(|u'|^{p-2}u')'(x)+(x^2-\lambda) |u(x)|^{p-2} u(x)=0$$ in $L^p(\mathbb R)$, $p\in(1,\infty)$. Are ...
- 175