41
votes
7answers
1k views
How closed-form conjectures are made?
Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu( …
5
votes
0answers
112 views
Inertia group vs. differential equations
The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p- …
2
votes
1answer
121 views
Existence of Geodesics in continuous metrics
I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0, …
0
votes
2answers
145 views
An almost orthogonality principle for L^p
I recently asked this question on Math StackExchange and someone suggested that it would probably be more suited for Math Overflow. Since it still has not been answered, here it go …
2
votes
0answers
50 views
Fourier : from analyticity to exponential decay ; what prevents optimal decay ?
Hello everyone !
In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for s …
2
votes
0answers
65 views
First integrals of a 3D incompressible flow
Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\na …
10
votes
2answers
321 views
Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?
Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig ident …
2
votes
2answers
133 views
Elliptic Harnack inequality for 1D Schrodinger operator?
For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:
There exist $C_{H …
-1
votes
0answers
71 views
Question about resonance problem
I have this :
I don't understand the last sentence , what is the relation between the eigenvalues of $(P_0)$ and resonance ?
Please help me
Thank you .
3
votes
1answer
84 views
An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$
Consider the following series:
$$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
$$ …
2
votes
2answers
123 views
Bruhat-Schwartz functions and derivatives in p-adic numbers
First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:
Let $\varphi:\mathbb{Q}_p\rightarrow\mat …
0
votes
3answers
393 views
I have this linear PDE…
Hi,
The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$
where subscript $y,z$ indicates derivatives and $A,B,C,D$ a …
1
vote
1answer
347 views
New differintegral formula: how is it related to other differintegral formulas?
Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x …
12
votes
1answer
224 views
Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$
I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^ …
5
votes
2answers
267 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 a …

