Author Archives: Matt R.
Signing Off
Dear All, You may have noticed that this blog has not been updated in several months. I thought I should announce that I will no longer be updating and/or maintaining this blog (this includes responding to comments). For the time … Continue reading
Open Question on Sufficiency To Be a Density Basis
This post is adapted from a question I posted on Math.StackExchange approximately two weeks ago, which has not received a satisfactory answer. If you have a solution or a counterexample, please add it to Math.SE instead of here. Let be … Continue reading
A Differentiation Basis without the Vitali Covering Property
— 1. Introduction — A differentiation basis is a fine collection of bounded measurable sets ; i.e. for every there exists a sequence with diameter tending to zero. Classical examples of differentiation bases are the collections of balls and cubes. … Continue reading
Kuran’s Theorem
It is well-known that a harmonic function satisfies the mean value property for all balls and spheres contained in . (We abuse notation and use to denote both the -dimensional Lebesgue measure and the -dimensional surface measure.) This mean value … Continue reading
UMOM Champions
I’m very proud to share that my dear mother, Mrs. Kathy Rosenzweig, has been selected as a UMOM Champion finalist as part of the organization’s celebration of its 50th anniversary. For those unfamiliar with the organization, UMOM is Arizona’s largest … Continue reading
A Banach algebra proof of the prime number theorem
Originally posted on What's new:
The prime number theorem can be expressed as the assertion as , where is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to…
Heat Ball and Heat Sphere Mean Value Property
Like solutions to the Laplace equation, (classical) solutions to the heat equation satisfy a mean value property. But instead of integrals over balls or spheres, the heat mean value property involves integrals over a heat ball. The heat ball of … Continue reading
An Extension of a Theorem of Marczewski
In [3], Marczewski proved that convergence almost surely (a.s.) is equivalent to convergence in probability precisely when is the (at most) countable union of disjoint atoms (i.e. purely atomic). In particular, convergence a.s. is defined by a topology when is … Continue reading
On the Normability of the Space of Measurable Functions
Suppose is a finite measure space, and let denote the space of real- or complex-valued measurable functions on . We can topologize by saying a sequence in if and only if We say that converges in measure to . It … Continue reading
Harmonic Functions: Weyl Lemma
In this post, we prove a generalization of the result that any (clasically) harmonic function defined on a bounded, open set is analytic. If is weakly harmonic, i.e. for any compactly supported function (the space of such functions is denoted … Continue reading
Math by Matt 