**1**

vote

**3**answers

179 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

**5**answers

470 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

**1**answer

84 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

**1**answer

72 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

**0**answers

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) + ...

**2**

votes

**1**answer

90 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

**1**answer

78 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

**1**answer

181 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

**1**answer

387 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

**1**answer

180 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

**1**answer

182 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

**2**answers

309 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 ...

**14**

votes

**0**answers

261 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 ...

**2**

votes

**1**answer

95 views

### ODE system has zero as the only solution?

Let $V \subset H$ be a continuous, compact and dense embedding with $V$ and $H$ Hilbert spaces.
Let $\beta_j:[0,T] \to \mathbb{R}$ be functions for each $j$, and let $v_j$ be a basis of $V_0$.
...

**1**

vote

**1**answer

35 views

### Convergence in distribution and ODE

Assuming we have an ODE $y'_n(x) = f_n(x) y_n(x)$
with $f_n$ be Gauß-densities with mean value 0 and variance $\frac{1}{n}$, then we have that they converge in distribution to a delta peak $δ(x)$. ...

**1**

vote

**1**answer

165 views

### Generalization of the triple tangent identity

It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$
I came across the following generalization of this equality:
$$\sqrt{1-k^2} {\rm ...

**1**

vote

**1**answer

34 views

### Can we implicitly fit a system of linear ODEs by reduced information?

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...

**2**

votes

**1**answer

61 views

### The asymptotic distribution of a subset of Bessel function zeroes

For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J_n(\alpha)=0 \textrm{ ...

**0**

votes

**0**answers

80 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.
...

**3**

votes

**0**answers

101 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**

votes

**2**answers

162 views

### How to study analytically this ODE?

I have tried to find some known ODEs before posting on this forum, but I did not find anything about this kind of ODE:
$y'(x)^2 + a(x)*y(x)^2 = 1$
with $a\in C^∞(\mathbb{R},\mathbb{R})$ and $y\in ...

**1**

vote

**0**answers

64 views

### How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...

**0**

votes

**1**answer

155 views

### Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R.
Is it possible to extend this function to a smooth function on R²?
...

**3**

votes

**1**answer

136 views

### Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...

**1**

vote

**0**answers

66 views

### A question about the existence of a solution for an ODE

Let $\gamma_+$, $\gamma_-:\mathbb{R}_+\to\mathbb{R}$ be two given functions. Assume that $\gamma_+$ ($\gamma_-$) is smooth, strictly increasing (decreasing) and $\gamma_{+}(+\infty)=+\infty$ ...

**0**

votes

**0**answers

137 views

### path integral and index theorem

I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int ...

**0**

votes

**1**answer

63 views

### Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue

Given the Sturm-Liouville type (time independent Schroedinger) equation
\begin{equation}
\frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R}
\end{equation}
where ...

**6**

votes

**1**answer

185 views

### Generalizing “variation of parameters”

I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here ...

**1**

vote

**0**answers

97 views

### A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...

**9**

votes

**2**answers

496 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 ...

**3**

votes

**0**answers

83 views

### Operator theory of initial-value ODE problems

The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces.
In contrast to that, the theory of ...

**3**

votes

**1**answer

398 views

### the existence of a real polynomial satisfying the following property

It is easy to verify that
$$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2}
\quad \quad 0<t\leq1$$
I want to ask if there exist a real polynomial $h(t)$ such ...

**1**

vote

**0**answers

35 views

### Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question:
Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that
$F(x,\cdot)$ is convex with respect to the second variable.
$F(\cdot,v)$ ...

**3**

votes

**2**answers

233 views

### Uniqueness of solutions to an ODE system

For each $i$ (up to infinity), let $u_i \in C^1(0,T)$ satisfy
$$\frac{d}{dt}u_i(t) + \sum_{j=1}^\infty b(t;w_j,w_i)u_j(t) = 0$$
$$u_i(0) = u_i(T)$$
where $b(t;\cdot,\cdot)$ is an inner product on some ...

**13**

votes

**3**answers

863 views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...

**9**

votes

**3**answers

321 views

### $L^p$ norm means

Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...

**0**

votes

**2**answers

201 views

### A book about almost periodic functions [closed]

Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.

**1**

vote

**0**answers

177 views

### Solving polynomials of arbitrary degree

Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5?
More specifically: does there exist ...

**0**

votes

**1**answer

57 views

### Extending point-wise bound to uniform bound

Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$
$$ C(t):= \sup_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$
Does it follow that $f$ is bounded?
Note ...

**11**

votes

**2**answers

1k views

### How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...

**7**

votes

**1**answer

246 views

### A conjecture about the measure estimates of a trigonometric polynomial

Formulation of the Conjecture
Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( ...

**4**

votes

**1**answer

136 views

### Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...

**0**

votes

**0**answers

64 views

### The trivility of Besov space for large parameter

For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define
$$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$
and
...

**5**

votes

**0**answers

219 views

### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...

**1**

vote

**0**answers

88 views

### Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< ...

**1**

vote

**1**answer

92 views

### Controling mixed derivatives

This is a cross-post from Math.SE since the question got nothing (but upvotes) even after offering a decent bounty. If it is too trivial or in other ways not suited for this site, please let me know ...

**2**

votes

**2**answers

124 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 ...

**2**

votes

**2**answers

139 views

### Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...

**1**

vote

**0**answers

81 views

### Estimating convolutions of powers

I would like an asymptotic estimate of
$$
\sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}}
$$
that does not involve any infinite summation. In order to lighten the notation, I ...

**1**

vote

**0**answers

178 views

### Is there an asymptotic bound for this oscillatory integral?

I have an oscillatory integral:
$$ \int u(x,y) e^{i\lambda f(x,y)} dx $$
with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying:
$$ \text{Im} f \geq ...