Date of Award
Doctor of Philosophy (PhD)
Calkin, Neil J
The study of random objects is a useful one in many applications and areas of mathematics. The Probabilistic Method, introduced by Paul Erdos and his many collaborators, was first used to study the behavior of random graphs and later to study properties of random objects. It has developed as a powerful tool in combinatorics as well as finding applications in linear algebra, number theory, and many other areas. In this dissertation, we will consider random vectors, in particular, dependency among random vectors. We will randomly choose vectors according to a specified probability distribution. We wish to determine how many vectors must be generated before the vectors are almost surely dependent, that is, there is a high probability that a subset of the vectors is linearly dependent.
In Chapter 1, we will review previous work done in this area. A typical result in the study of random objects is a threshold function that describes the behavior of a given property of the objects. We will discuss previous threshold functions and methods used to find them. The results found for this problem before now have been for vectors of bounded or fixed weight. In Chapter 2, we will develop the methods we will use later on vectors of fixed weight. We will then use these methods in Chapter 3 to vary the probability model under which the vectors are generated. Instead of considering vectors of fixed weight, we will consider a general probability model for choosing the vectors: each position in a vector will be assigned a probability of containing a nonzero entry. Finally, in Chapter 4 we will specify a function for this probability. We will then find a threshold result for the specified probability model. This result will give a lower bound for the number of vectors needed before they are almost surely dependent.
Lockard, Shannon, "Random Vectors Over Finite Fields" (2007). All Dissertations. 54.