8
votes
2answers
458 views

Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...
2
votes
2answers
117 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 ...
3
votes
0answers
76 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
9
votes
2answers
564 views

Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...
7
votes
2answers
469 views

Survey of the history of calculus?

Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has: Mathematics knows no minimum interval of ...
1
vote
1answer
113 views

On the geometry of roots of a sum of complex linear fractions

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex ...
1
vote
0answers
83 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, ...
2
votes
1answer
63 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 $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke: Teorem. Let $f : U ⊂ \mathbb{R}^n ...
3
votes
1answer
340 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 ...
7
votes
1answer
474 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 ( ...
4
votes
2answers
108 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
vote
1answer
118 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 ...
0
votes
0answers
79 views

Detailed Taxonomy of Multivariate Real Functions: $\mathbb{R}^n\rightarrow\mathbb{R}$

I want to classify the following functions: $$ f:(x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\sqrt{(x+1)^2+y^2}+\sqrt{(x-1)^2+y^2}-2$$ $$ ...
4
votes
1answer
156 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 ...
6
votes
1answer
223 views

Alternative proof of Lojasiewicz inequality

is there a "brute force proof" of the Lojasiewicz inequality? By "brute force" i mean without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e. standard ...
1
vote
1answer
172 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
1answer
141 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 ...
4
votes
1answer
230 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 ...
4
votes
1answer
154 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 ...
1
vote
0answers
120 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 ...
2
votes
2answers
166 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 ...
6
votes
0answers
152 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 ...
6
votes
2answers
429 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 ...
8
votes
2answers
778 views

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 ...
6
votes
5answers
720 views

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 ...
5
votes
2answers
633 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 ...
1
vote
3answers
237 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 ...
2
votes
5answers
217 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.
4
votes
2answers
394 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 ...
3
votes
3answers
219 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 ...
5
votes
1answer
759 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 ...
2
votes
0answers
74 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 $z$ ...
7
votes
2answers
1k 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 ...
2
votes
3answers
445 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 ...
5
votes
2answers
293 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 ...
3
votes
1answer
178 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 ...
11
votes
2answers
1k 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 ...
4
votes
2answers
219 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
vote
2answers
166 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 ...
11
votes
6answers
2k 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, ...
9
votes
0answers
202 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 ...
14
votes
5answers
1k 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 ...
0
votes
1answer
127 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 ...
1
vote
1answer
382 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 ...
7
votes
1answer
421 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 ...
1
vote
3answers
503 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 ...
2
votes
0answers
312 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 ...
2
votes
0answers
200 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] ...
17
votes
4answers
4k 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 ...
17
votes
2answers
1k 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 ...