Jerome Malitz

An Introduction to Set Theory

Through the centuries mathematicians and philosophers have wondered if size comparisons between infinite collections of objects can be made in a meaningful way. Does it make sense to ask if there are as many even numbers as odd numbers? What does it mean to say that one infinite collection has greater magnitude than another? Can one speak of different sizes of infinity?

A bounded compactness theorem forL1-embeddability of metric spaces in the plane

LetM=(W, d) be a metric space. LetL1 denote theL1 metric. AnL1-embedding ofM into Cartesiank-space ℝk is a distance-preserving map from (W, d) into (ℝk,L1). Letc(k) be the smallest integer such that for every metric spaceM, M isL1-embeddable inRk iff everyc(k)-sized subspace ofM isL1-embeddable inRk. A special case of a theorem of Menger (see p. 94 of [5]) says thatc(1) exists and equals 4. We show thatc(2) exists and satisfies 6≦c(2)≦11. Whether or notc(k) exists for anyk≧3 is an open question.

Self-generating hexagonal cell division patterns

Consider the plane covered by regular hexagons. We investigate division patterns in which each hexagon is divided into two new regions, each new region has six neighbouring regions and each vertex in the new structure belongs to three new regions. These patterns are of interest for cell division processes in biology and are related to a certain class of hexagonal tilings of the plane.

An approach to hierarchical clustering via level surfaces and convexity

We consider the following approach to hierarchical data clustering. Let A = {p1, p2, …, Pn} be a set of n data points in Rd, d ≥ 2. Define f(z) = Σi=1n (¦z − pi¦q)−1, where z is in Rd, and ¦ · ¦q denotes the Lq-norm, q ≥ 1. The function f can be viewed as a “combined luminosity” formed by summing individual “point luminosities” located at the pi. The level surfaces of f define a hierarchical clustering of A in a natural way. We prove a general result on convexity that enables us to obtain this clustering for q = 1 by examining the values of f on the edges of a rectilinear grid induced by A. An algorithm is developed that is practical in several situations. For n < 1000, these include: (1) d small (say d ≤ 4) and the pi real; (2) d moderate (say d ≤ 25) and the pi binary.

Complete Theories with Countably many Rigid Nonisomorphic Models

Here we prove the following: Theorem. For every N ≤ ω there is a complete theory T n having exactly n nonisomorphic rigid models and no uncountable rigid models. Moreover, each non-rigid model admits a nontrivial automorphism . The T n are theories in the first-order predicate calculus and a rigid structure is a structure with no nontrivial endomorphisms, i.e., the only endomorphism of the structure into itself is the identity. The theorem answers a question of A. Ehrenfeucht. For the most part we use standard model theoretic notation with Th denoting the set of sentences true in and meaning . A complete set of sentences is one of the form Th for some . The universe of a structure may be denoted by ∣ ∣. An n -ary relation on X is a set of n tuples ( x 0 , …, x n −1 ) with each x i ∈ X . All the structures encounted here will be relational. If P is an n -ary relation then P ↾ Y , the restriction of P to Y , is {( x 0 , …, x n −1 ): ( x 0 , …, x n −1 ) ∈ P and x 0 , …, x n −1 Y }.

Addendum to: An Approach to Hierarchical Clustering via Level Surfaces and Convexity

This article is an addendum to the 2001 paper [1] which investigated an approach to hierarchical clustering based on the level sets of a density function induced on data points in a d-dimensional feature space. We refer to this as the “level-sets approach” to hierarchical clustering. The density functions considered in [1] were those formed as the sum of identical radial basis functions centered at the data points, each radial basis function assumed to be continuous, monotone decreasing, convex on every ray, and rising to positive infinity at its center point. Such a framework can be investigated with respect to both the Euclidean (L2) and Manhattan (L1) metrics. The addendum here puts forth some observations and questions about the level-sets approach that go beyond those in [1]. In particular, we detail and ask the following questions. How does the level-sets approach compare with other related approaches? How is the resulting hierarchical clustering affected by the choice of radial basis function? What are the structural properties of a function formed as the sum of radial basis functions? Can the levels-sets approach be theoretically validated? Is there an efficient algorithm to implement the level-sets approach?

The geometry of legal principles

We discuss several possible legal principles from the standpoint of Bayesian decision theory. In particular, we show that a compelling legal principle implies compatibility with decisions based on maximizing the expected utility.

An Introduction to Computability Theory

What are the capabilities and limitations of computers? Are they glorified adding machines capable of superfast arithmetic computations and nothing else? Can they outdo man in the variety of problems they can handle? Let’s narrow the question a bit. Consider the class of number theoretic functions that a computer can be programmed to compute or that a man can be instructed to compute. Are any of these functions computable by a computer but not by a man, or by a man but not by a computer? Is there a number theoretic function that is not computable by any computer, and if so, can such functions be described? Is there a computer that can be programmed to compute any function that any other computer can compute? Is man such a computer?

An Introduction to Model Theory

For the remainder of the text we turn our attention to that branch of logic called model theory. Here we consider formal languages with enough expressive power to formulate a large class of notions that arise in many diverse areas of mathematics. Within our idealized language we shall be able to describe different kinds of orderings, groups, rings, fields, and other commonly studied mathematical notions.