Two UC-sets whose union is not a UC-set

UC Irvine UC Irvine

Sets Whose Difference Set Is Square-free

The purpose of this note is to give an exposition of the best-known bound on the density of sets whose difference set contains no squares which was first derived by Pintz, Steiger and Szemerédi in [PSS88]. We show how their method can be brought in line with the modern view of the energy increment strategy employed in problems such as Szemerédi’s Theorem on arithmetic progressions, and explore ...

UC Irvine UC Irvine

UC Irvine UC

The motor neuron disease amyotrophic lateral sclerosis (ALS) typically begins with localized muscle weakness. Progressive, widespread paralysis often follows over a few years. Does the disease begin with local changes in a small piece of neural tissue and then spread? Or does neural decay happen independently across diverse spatial locations? The distinction matters, because local initiation ma...

UC Santa Cruz UC

The conditions and properties of hydrothermal vents and hydrothermal fields are compared in terms of their ability to support processes related to the origin of life. The two sites can be considered as alternative hypotheses, and from this comparison we propose a series of experimental tests to distinguish between them, focusing on those that involve concentration of solutes, self-assembly of m...