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

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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

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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

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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

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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

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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…

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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

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