WILLIAM KEITH BROOKS

Keith Simmons

Computer scientists use formal verification to attempt to provide guarentees to the users of software about the run time characteristics of a given program. Verification techniques today often require a TCB or Trusted Code Base which due to time or effort constraints, the authors were not able to prove correct. When bugs occure in these pieces, they break down the strong guarentees of formal ve...

Glomerulonephritis Keith

Early diagnosis of glomerulonephritis (GN) in the adolescent is important in initiating appropriate treatment and controlling chronic glomerular injury that may eventually lead to end-stage renal disease (ESRD). The spectrum of GN in adolescents is more similar to that seen in young and middle-aged adults than to that observed in prepubertal children. In this article, the authors discuss the cl...

Counting Keith numbers

A Keith number is a positive integer N with the decimal representation a1a2 . . . an such that n ≥ 2 and N appears in the sequence (Km)m≥1 given by the recurrence K1 = a1, . . . ,Kn = an and Km = Km−1 + Km−2 + · · · + Km−n for m > n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a1 = a2 = · · · = an), and that the set of Keith numbers is of asympto...

Brooks' Theorem and Beyond

We collect some of our favorite proofs of Brooks’ Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon–Tarsi orientations, since analo...

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