Inspired by the problem of sensory coding in neuroscience, we study the maximum entropy distribution on weighted graphs with a given expected degree sequence. This distribution on graphs is characterized by independent edge weights parameterized by vertex potentials at each node. Using the general theory of exponential family distributions, we prove the existence and uniqueness of the maximum l...

We prove fractional analogs of the classical Sylvester-Gallai theorem. Our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set. Specifically, we show that if for every points v in a finite set V ⊂ C, there are at least δ|V | other points u ∈ V for which the line through v, u contains a third point in V , then V resides i...

Assessments of mathematical knowledge for teaching (MKT), which are often designed to measure specialized types of mathematical knowledge, typically include a representation of teaching practice in the assessment task. This analysis makes use of an existing, validated set of 10 assessment tasks to both describe and explore the function of the teaching contexts represented. We found that teachin...

Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) is defined to be minimum n such that every k-edge coloring of complete Kn contains either rainbow (all different colored) triangle or monochromatic copy H. In this paper, we study numbers for graphs with chromatic three as K̂m m≥2, where kipas m+1 vertices obtained from join K1 Pm, class five vertices, denoted by ℋ. We first gen...

Maximum Entropy Distributions on Graphs by Andre Yohannes Wibisono Master of Arts in Statistics University of California, Berkeley Professor Michael I. Jordan, Chair We study the maximum entropy distribution on weighted graphs with a given expected degree sequence. This distribution on graphs is characterized by independent edge weights parameterized by vertex potentials at each node. Using the...

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The research presented in this issue finds historical resonance in a classical scientific debate. Russian biologist Ilya Metchnikoff [1, 2] was awarded the Nobel Prize in 1908 for the development of the phagocyte theory, the central tenet of which was that the ‘swallowing and digestion’ of bacteria by circulating ‘white corpuscles’ (neutrophils and macrophages), enhanced by prior opsonization, ...