Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics.

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0
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1answer
86 views

Number of critical points

Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and ...
0
votes
0answers
72 views

Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1. Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...
0
votes
0answers
50 views

Maximum modulus principle and restriction

Suppose I have a complex multivariate polynomial $f\in \mathbb{C}[z_1,\ldots,z_n]$ and a connected, bounded open set $U$. The maximum modulus principle says $f$ achieves its maximum value on the ...
51
votes
2answers
1k views

History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught: 1º that $$ \frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2}, \tag1 $$ 2º that, via the fundamental theorem of calculus, this is ...
3
votes
0answers
101 views

Homogeneous polynomial

Can a homogeneous harmonic polynomial mapping $p(x)=(p_1(x),p_2(x),p_3(x))$, $p(0)=0$, of odd degree $m\ge 3$, of the unit ball $B^3$ into $R^3$ be injective. This means that $p_i$ are harmonic ...
2
votes
0answers
96 views

Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...
0
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0answers
83 views

Positive curvature of the boundary away from a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact: Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...
2
votes
2answers
127 views

Limiting Ratio of Solutions to Ordinary Differential Equations

I'm trying to find the limit of the ratio of two functions $ \lim_{t \rightarrow \infty} \frac{f(t)}{g(t)} $ but only have the initial conditions and the differential equations they solve, but the ...
0
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0answers
37 views

Comparing Calculation Error in Divergent Numerical Methods

I'm not an expert in numerical methods, but I'm doing a simulation based on non-linear differential equations (General Relativity), there solutions has singularities, thus at some points numerical ...
0
votes
0answers
295 views

What is the Fourier transform of this function?

Consider the function $$ f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du. $$ It is known that $f(x_1,x_2)\in ...
12
votes
2answers
485 views

Surjectivity of curl

Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that $$ \int_{\mathbb R^3} v(x) dx=0. $$ Is it true that there exists a ...
1
vote
0answers
74 views

variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...
4
votes
1answer
450 views

Elementary Proof of the Uniqueness of Smooth Structures on R

Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm ...
0
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0answers
73 views

Alternative definition of the Lagrange Inversion formula

While reading this paper, the author provides an alternative definition of the Lagrange inversion formula. Call me crazy, but my intuition tells me that there's something wrong with his derivation. ...
4
votes
0answers
101 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 ...
3
votes
0answers
150 views

Derivative of Wronskian

In the proof of Theorem 2 in this paper here on arxiv on page 10 for $k=2$ it is claimed that if the Wronskian of two solutions $y_1,y_2$ to the differential equation $$-y''_i(x) + q(x) y_i(x) = ...
5
votes
1answer
181 views

Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
2
votes
1answer
214 views

Complete solution set of a Convolutional Equation?

Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best.. Setup: In what ...
1
vote
2answers
82 views

Finding conditions to guarantee existence of solutions to IVP [closed]

Consider the following IVP: $x'=f(t,x)$ and $x(0)=x_0$, where $x\in \mathbb{R}^n$ and $t\in \mathbb{R}$. Suppose that for all $(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$. In order for the ...
1
vote
0answers
26 views

Taking the potential of a super additive measure

In recent research of my coauthors and me, it has become necessary to consider the Riesz potential of a superadditive measure. Recall that the $s$-dimensional Riesz potential of a finite Borel ...
1
vote
0answers
99 views

How to find a invariant surface of a diffeomorphism

Recently, I read a paper about discrete Schrödinger operator. There is a map related to trace map from $C^3$ to $C^3$ as follows: $$T(x,y,z)=(y,z,yz-x).$$ We can calculated that $T$ has the folliwng ...
0
votes
0answers
34 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
4
votes
1answer
144 views

PDEs on torus $\mathbb T$

(Hope this question is o.k. for MO) I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular ...
0
votes
0answers
120 views

Notion of solution of pde

Let's consider the following Schrodinger equation $$iu_t+\Delta u+F(u)=0$$ in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...
3
votes
1answer
271 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 ...
0
votes
0answers
97 views

Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum: $$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$ as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...
-2
votes
2answers
62 views

Systems of ODEs that fulfill a matrix relationship at steady state [closed]

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$ with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...
3
votes
1answer
187 views

Relationship between Laplacian and Hessian on compact Lie groups

If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has $$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$ where $\Delta$ ...
0
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0answers
109 views

A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...
1
vote
1answer
59 views

Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability: One factor would be the condition number of the approximation operator. The other factor ...
1
vote
0answers
103 views

Is there a unique solution? [closed]

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...
1
vote
1answer
132 views

Another type of derivative, and the associated primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that: $ ...
0
votes
0answers
20 views

Composition of Lossless Systems from Delay and Mixing regarding junction admittance

Given $m_1, \dots m_N \in \mathbb{N}$ and matrix $\mathbf{A} \in \mathbb{C}^{N\times N}$. Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{-m_1}, \dots, z^{-m_N})$ with $z\in\mathbb{C}$. ...
6
votes
1answer
158 views

Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which $$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$ and $$x_1+\ldots+x_k\leq 1.$$ I am ...
2
votes
2answers
234 views

Is this infinite series related to some well-known special functions?

Please allow me to resort once again to the expertise of the MathOverflow community : During research I encoutered the following infinite series : $$\sum_{n=-\infty}^{+\infty} ...
0
votes
0answers
82 views

How to solve a couple of ODEs

Let $\phi_+ (\phi_-)$ be a strictly increasing (decreasing) function defined on $R_+$ such that $\phi_+(\phi_-)\in\mathcal{C}^0(R_+)\cap\mathcal{C}^1(R_+^{\ast})$ and $\phi_+(0)=0(\phi_-(0)=0)$. ...
6
votes
2answers
307 views

Regularity of random Fourier series

The following two statements appear to be true (but do correct me if I am wrong): The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm ...
1
vote
3answers
180 views

Estimating a sum [closed]

Good morning everyone, I would like to make a question about estimating a sum. Consider the following sum $$S_n:=\sum_{k=0}^{n-1} \frac{k^2}{(n-k)^2 (n+k)^2} $$ It is easy to see that this sum is ...
7
votes
5answers
485 views

Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum: $$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$ as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...
4
votes
1answer
89 views

Estimate on sum of $J_n^4$

If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$ What is known about $$ ...
1
vote
1answer
77 views

Maximal minimum of Bessel functions

This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question ...
0
votes
0answers
60 views

Nonlinear ODE system

Let $\Psi(a) = \frac{a}{2}$ if $a>0$ and $0$ if $a\le 0$. Now we consider the following coupled system of nonlinear ODEs: $$\begin{aligned}&\frac{1}{2}\sigma_1^2 u_1''(x) + \mu_1 u_1'(x) + ...
1
vote
1answer
98 views

Limit-circle and limit-point at endpoints

I was wondering if the following holds: If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
1
vote
1answer
80 views

Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge: $$ \int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ? $$ Among special cases are such ...
4
votes
1answer
190 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 ...
2
votes
1answer
401 views

A proof from Lang's undergraduate analysis

This is from P.580 of Serge Lang's undergraduate analysis (2nd edition). $\textbf {Proposition 2.3.}$ Let $A$ be an admissible set in $\mathbb R^n$ and assume that its closure $\bar{A}$ is contained ...
-2
votes
1answer
181 views

A calculus question [closed]

Fix $q>1$. Define the function $$ f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r. $$ The problem is whether the following is true, $$ \lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
1
vote
1answer
188 views

Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$. Let $k\ge0$ be a nonnegative integer. If we add another factorial ...
3
votes
2answers
317 views

Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem. Let ...
15
votes
0answers
266 views

Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series $\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely. Then there is a partition of the symmetric group ${\rm ...