All Questions
Tagged with ca.classical-analysis-and-odes reference-request
323 questions
1
vote
0
answers
216
views
Periodic solution of first order ODE
There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system
$$
\left\{
\begin{array}{l}
\dot{x}=f(x), \\
x(t_0)=x_0,
\...
1
vote
0
answers
65
views
Equivalence of $L^p$ harmonic functions on the ball and a representation by harmonic homogeneous polynomials
In Harmonic function theory, there is a theorem which says that if $u$ is an harmonic function on $B\left(a,r\right)$, then there exist homogeneous harmonic polynomials $p_{m}$ in $\mathbb{R}^{n}$ ...
- 325
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 ...
- 203
5
votes
2
answers
732
views
The Weyl algebra modules which are also rings
Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...
- 389
10
votes
1
answer
955
views
Ramanujan's problem 754 still open?
In addition to the MO question The Ramanujan Problems. , I would like to ask the following.
Problem 754 from the list of the Ramanujan's problems ( http://www.imsc.res.in/~rao/ramanujan/...
- 16.5k
5
votes
2
answers
454
views
Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th ...
- 1,601
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
7
votes
2
answers
697
views
Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)
According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin (1919)...
- 1,371
13
votes
2
answers
2k
views
An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
- 1,727
1
vote
1
answer
207
views
Number of solutions of a system of equation!
Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
4
votes
1
answer
902
views
Exact Differential Equations of Order n via Pfaffian Differential Equations?
I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ...
- 93
4
votes
1
answer
375
views
Asymptotic solution of the integral equation
What is the asymptotic solution (for $s\gg 1$) of the following integral equation $$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)\;?$$
In fact I ...
- 16.5k
2
votes
0
answers
270
views
Dalzel's integral for $\pi$ and the lemniscate constant
$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers
$$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$
and, for example, the Wallis product formula $$\pi=4\prod_{n=1}^\...
- 16.5k
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
6
votes
0
answers
200
views
Spectral theory for Dirac Laplacian on a funnel
I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
- 313
7
votes
2
answers
1k
views
For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?
(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
user14166
12
votes
2
answers
2k
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Reference for a nice proof of "undetermined coefficients"
I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
- 7,273
12
votes
1
answer
1k
views
A generalization of intermediate value theorem on R^k
Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...
- 207
10
votes
5
answers
2k
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Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$
Reposted from math.stackexchange where my question received only five views and no answers...
I'm trying to learn the Selberg trace formula, but have very little background in harmonic analysis. I ...
- 7,062
7
votes
2
answers
2k
views
Cantor Sets Inside Cantor Sets
(Or: "I heard you liked Cantor Sets...")
I'm working on a student project, and the following construction came up very naturally: If $C$ is the usual Cantor Set, build a countable union of copies of ...
- 4,586
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
3
votes
5
answers
518
views
Good reference for the construction of a Greens functions fur the Sturm-Liouville
Does anyone know a good reference for the constructions of a Greens functions fur the Sturm-Liouville Boundary Value Problem.
- 1,256
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
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
1
answer
1k
views
Request for the proof of a result from Ramanujan's letter to Hardy.
Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link seems interesting :
If $$\int\limits_{0}^{\infty} ...
- 4,795
2
votes
0
answers
145
views
Reference Request: Holomorphic Dependence on Parameters to Solutions of Complex Linear ODEs Near an Irregular Singularity
I'm looking for a reference which discusses the following:
Consider the ODE
$\frac{d^2H}{dz^2} + f(z,x)\frac{dH}{dz} + g(z,x)H(z) = 0$
where
1) $f$ and $g$ depend holomorphically on $x$ and $z$
2)...
8
votes
2
answers
2k
views
Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
3
votes
3
answers
2k
views
How do we use an Ehresmann connection to define a semispray?
Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
- 8,512
6
votes
2
answers
390
views
Solution uniqueness for ODE
I have a vectorial, non-linear second order ordinary differential equation
$$Z''=f(Z)$$
for which I have a solution $Z^0$ on $[0,1]$ with $Z^0_i(0)=0$ and $Z^0_i(1)=1$. I would like to know under ...
- 14.4k
4
votes
1
answer
265
views
a limit by Gosper involving a product of arctan and $4^{1/\pi}$
On the Wolfram page about pi formulas, there is this curious limit by R. W. Gosper (130) $$\lim\limits_{n\to\infty}\prod\limits_{k=n}^{2n}\dfrac{\pi}{2\arctan k}=4^{1/\pi}.$$
The only reference given ...
- 13.4k
18
votes
2
answers
5k
views
Nonvanishing of Jacobians implies global injectivity?
I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.
In 1953, Samuelson asked the following:
If the ...
- 981
4
votes
2
answers
283
views
Bounding the series of the geometric means of the terms of a given positive series
Let $ \{ a _ k \} _{k\in\mathbb{N} _ +} $ be a sequence of non-negative numbers, and let $MG(a_1,\dots,a_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality
$$ \sum _ {n\ge 1}...
- 60.5k
1
vote
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
26
votes
9
answers
14k
views
Graduate ODE textbook
Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, ...
10
votes
0
answers
245
views
A slightly generalized existence and uniqueness theorem for integral equations (reference request)
I want to use the following statement without including the proof, which is completely straightforward but rather tedious:
Let $G_0:\mathbb R\times\mathbb R^m\to\mathbb R^m$, $\Theta_0:\mathbb R\...
- 61.9k
18
votes
5
answers
3k
views
Smoothness of $f(\sqrt x)$
I found that I need to use the following facts in a paper that I am writing.
Let $f\in C^\infty(\mathbb R)$, then
If $f(0)=0$, then $f(x)=x g(x)$ for some $g\in C^\infty(\mathbb R)$.
If $f$ is even, ...
- 32.4k
0
votes
1
answer
200
views
How are epidemic models simulated in case of mobility?
I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
- 439
2
votes
1
answer
547
views
Equivalent references for Schwartz's book of the distribution theory
Hello,
It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like
$$
\dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \...
- 1,649
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
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
5
votes
0
answers
855
views
Extension operator for Lipschitz domain for fractional Sobolev spaces
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, with Lipschitz smooth boundary. Then a well known result by Stein gives that there exists an extension operator $E: H^k(\Omega)\rightarrow H^k(\...
- 3,217
2
votes
0
answers
228
views
Functions with "gradients of bounded variation"
Dear all,
I would like to know whether the following concept is one that is commonly studied, or has a name, or if there are any textbooks that make reference to it:
We say that a function $f:[a,b] \...
- 708
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
19
votes
2
answers
2k
views
Integral representation of higher order derivatives
I'm quite curious about the following phenomena, that still puzzle me although I have a proof, and I'd be really glad if someone may shred some light, showing an interpretation or a generalization. I ...
- 60.5k
1
vote
2
answers
561
views
Convergence of squares of the moduli of partial sums of Fourier series
Let $\mu$ be a complex measure on the unit circle. The Wiener theorem says that the sequence of the Cesaro means of $|\hat\mu_n|$ has a limit. Define $p_n(z)=\sum_{k=0}^n \hat\mu_k z^k$. Then the Abel ...
- 11
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 ...
- 39k
7
votes
4
answers
4k
views
Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 8,512
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
0
votes
2
answers
200
views
Good probability measues on $S^1$ reprented by a kernel
I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
- 1,471