Martin Hildebrand

A census of cusped hyperbolic 3-manifolds

The census provides a basic collection of noncompact hyperbolic 3-manifolds of finite volume. It contains descriptions of all hyperbolic 3-manifolds obtained by gluing the faces of at most seven ideal tetrahedra. Additionally, various geometric and topological invariants are calculated for these manifolds. The findings are summarized and a listing of all manifolds appears in the microfiche supplement.

Hyperbolic invariants of knots and links

Tables of values for the hyperbolic volume, number of symmetries, cusp volume and conformai invariants of the cusps are given for hyperbolic knots through ten crossings and hyperbolic links of 2, 3 and 4 components through 9 crossings. The horoball patterns and the canonical triangulations are displayed for knots through eight crossings and for particularly interesting additional examples of knots and links.

A computer generated census of cusped hyperbolic 3-manifolds

In this paper, we describe how we used a computer to produce a census of cusped hyperbolic 3-manifolds obtained from 5 or fewer ideal tetrahedra. We note some of the techniques involved in writing and debugging the programs and give a brief summary of the results.

Random random walks on the integers mod n

This paper considers typical random walks on the integers mod n such that the random walk is supported on constant k values. This paper extends a result of Hildebrand to show that for any integer n, roughly n2/(k-1) steps usually suffice to get the random walk close to uniformly distributed if the k values satisfy some conditions needed for the random walk to get close to uniformly distributed.

Random Processes of the Form Xn+1 = anXn + bn(mod p)where bn takes on a Single Value

This paper studies random processes of the form X n +1 = a n X n + b n (mod p) where b n has only one possible value and a n is o or 1 with probability l/2 each. For values of p satisfying certain constraints imposed by a n and b n , X n gets close to uniformly distributed on Z/pZ for large enough n. This paper explores how large n needs to be as a function of p. Adapting techniques used by Chung, Diaconis, and Graham and techniques previously developed by the author, this paper shows that if n > c 1 log p log log p then X n gets close to uniformly distributed as p approaches infinity and that if n > c 2 log p then X n approaches the uniform distribution for almost all p satisfying the constraints. Furthermore, if a = 2, this paper shows that for certain p, if X n gets close to uniformly distributed, then n > C3 log p log log p.

A note on various holding probabilities for random lazy random walks on finite groups

The author previously considered certain lazy random walks on arbitrary finite groups. Given a k-tuple (g1,...,gk) of elements of a finite group, one multiplies the previous position of the walk by gi[var epsilon] where i is uniform on {1,...,k} and [var epsilon] has a given distribution on {1,0,-1}. The previous work gave good bounds if P([var epsilon]=1)=P([var epsilon]=-1)=1/4 and P([var epsilon]=0)=1/2 or if P([var epsilon]=1)=P([var epsilon]=0)=1/2. The current paper develops some elementary comparison techniques which work for other distributions for [var epsilon] such as P([var epsilon]=1)=P([var epsilon]=0)=P([var epsilon]=-1)=1/3.

Random Lazy Random Walks on Arbitrary Finite Groups

This paper considers “lazy” random walks supported on a random subset of k elements of a finite group G with order n. If k=⌈a log2n⌉ where a>1 is constant, then most such walks take no more than a multiple of log2n steps to get close to uniformly distributed on G. If k=log2n+f(n) where f(n)→∞ and f(n)/log2n→0 as n→∞, then most such walks take no more than a multiple of (log2n) ln(log2n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdös and Rényi and of Pak.

A lower bound for the Chung-Diaconis-Graham random process

Chung, Diaconis, and Graham considered random processes of the form X n+1 = a n X n + b n (mod p) where p is odd, X 0 = 0, an = 2 always, and b n are i.i.d. for n = 0, 1, 2,.... In this paper, we show that if P(b n = -1) = P(b n = 0) = P(b n = 1) = 1/3, then there exists a constant c > 1 such that clog 2 p steps are not enough to make X n get close to being uniformly distributed on the integers mod p.

Log Concavity of a Sequence in a Conjecture of Simion

Simion presented a conjecture involving the unimodality of a sequence whose elements are the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. In this paper, the author uses ideas from an earlier paper where special cases of this conjecture were proved to prove log concavity and unimodality of the sequence.

The proof of a conjecture of Simion for certain partitions

Abstract Simion has a conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. The conjecture concerns the unimodality of a sequence of these numbers where the sum of the length and width of each rectangle is a constant and where the partition is constant. This paper demonstrates this unimodality if the partition is self-conjugate or if the Ferrers diagram of the partition has precisely one column or one row. This paper also shows log concavity for partitions of “staircase” shape via a Reflection Principle argument.