Sharp Threshold Asymptotics for the Emergence of Additive Bases

A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1-alpha)n,(1+alpha)n] (or equivalently [alpha n, (2-alpha) n]) for some 0<alpha<1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.Comment: 22 page

Two recent p-adic approaches towards the (effective) Mordell conjecture

https://arxiv.org/pdf/1910.12755.pdfFirst author draf

Neuromatch Academy: a 3-week, online summer school in computational neuroscience

Neuromatch Academy (https://academy.neuromatch.io; (van Viegen et al., 2021)) was designed as an online summer school to cover the basics of computational neuroscience in three weeks. The materials cover dominant and emerging computational neuroscience tools, how they complement one another, and specifically focus on how they can help us to better understand how the brain functions. An original component of the materials is its focus on modeling choices, i.e. how do we choose the right approach, how do we build models, and how can we evaluate models to determine if they provide real (meaningful) insight. This meta-modeling component of the instructional materials asks what questions can be answered by different techniques, and how to apply them meaningfully to get insight about brain function