This book presents an overview of recent developments in biostatistics and bioinformatics. Written by active researchers in these emerging areas, it is intended to give graduate students and new researchers an idea of where the frontiers of biostatistics and bioinformatics are as well as a forum to learn common techniques in use, so that they can advance the fields via developing new techniques and new results. Extensive references are provided so that researchers can follow the threads to learn more comprehensively what the literature is and to conduct their own research. In particulars, the book covers three important and rapidly advancing topics in biostatistics: analysis of survival and longitudinal data, statistical methods for epidemiology, and bioinformatics.

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We introduce a triangular array of integers defined by a(n, k) = k(k + 1)n−k for n ≥ 1 and 1 ≤ k ≤ n, which we call the researcher’s triangle. The construction is multiplicative and differs from classical additive triangles such as Pascal’s. We give combinatorial interpretations using a leaders-followers model and a rooted hierarchical structure model. We identify several diagonal sequences as classical figurate numbers, including powers of two (A000079), natural numbers (A000027), and oblong numbers (A002378), and establish connections to triangular and tetrahedral numbers. We derive a bivariate generating function and a recurrence relation, and we discuss the row sums.

We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu.

In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.