# Volume II, Issue 2

# A brief introduction to measurable cardinals

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We introduce the notion of a measurable cardinal, motivated by examples from measure theory. We then develop some
initial inaccessibility results for such cardinals and summarize Solovay's results regarding the consistency
of set theory with the hypothesis that there is an accessible measurable cardinal. These results lead us to a question
that appears to be open (it is certainly open ended) regarding the consistency of weaker forms of choice and the existence
of measurable cardinals. This paper aims to be self contained and accessible to an advanced undergraduate; however, its
motivation rests in measure theory so previous exposure to the subject will be helpful.

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# Atomic Realizations of Chemical Reaction Networks

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When considering chemical reaction networks (CRNs), we often ignore the atomic compositions of the species involved (atom-free stoichiometry). Thus, a reasonable question is, "When can a CRN support an atomic realization?" That is, when can we assign each species an atomic structure which is distinct from that of the other species? After briefly covering definitions, we consider the main tool which we use, an algorithm from Schuster and Hofer , and draw out a proof that it obtains all extreme vectors. We then give necessary and sufficient conditions for atomic realizations, and discuss the implications, with reference to Famili and Palsson.

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# Asymptotic eigenvalue distribution of random lifts

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A random n-lift L_n(G) of a base graph G is obtained by replacing each vertex v_i of G by a set V_i of n vertices, and generating a random matching between V_i and V_j for each edge (v_i,v_j) in G. We show that the spectral density of a random lift L_n(G) approaches that of a tree as we increase $n$ by showing that the expected number of short cycles of length k in L_n(G) (denoted Z_k(G)) tends to a constant lambda_k. Moreover, we show that Z_k(G) is Poisson distributed with parameter lambda_k. We also give experimental results of the level spacing distributions and compare them to the Gaussian Orthogonal Ensemble of random matrix theory.

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Volume I, Issue 2 of The Waterloo Mathematics Review is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Canada License.