Ph.D. Recipients and their Thesis Abstracts

Interdisciplinary

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

A Diagrammatic Formal System for Euclidean Geometry

Abstract: It has long been commonly assumed that geometric diagrams can only be used as aids to human intuition and cannot be used in rigorous proofs of theorems of Euclidean geometry. This work gives a formal system FG whose basic syntactic objects are geometric diagrams and which is strong enough to formalize most if not all of what is contained in the first several books of Euclid's Elements. This formal system is much more natural than other formalizations of geometry have been. Most correct informal geometric proofs using diagrams can be translated fairly easily into this system, and formal proofs in this system are not significantly harder to understand than the corresponding informal proofs. It has also been adapted into a computer system called CDEG (Computerized Diagrammatic Euclidean Geometry) for giving formal geometric proofs using diagrams. The formal system FG is used here to prove meta mathematical and complexity theoretic results about the logical structure of Euclidean geometry and the uses of diagrams in geometry.

Essays in Mathematical Economics and Economic Theory

Abstract: This dissertation consists of two parts, one on the effect of open market operations and one on the existence of demand function in L^1 space, respectively. The first part of the thesis (Chapters 1–6) is to study how prices respond to open-market operations based on a transactions-based model of money demand with the use of homothetic utility function. It is shown that a monetary injection will lead to a gradual rise in prices. Of special interest is the determination of a stabilizing monetary injection policy corresponding to which the equilibrium prices are most stable.

The equilibrium price in this model is governed by a second order, nonlinear difference equation which can be reduced to a functional equation of the form f(f\circ f(x), f(x), x)= 0. The stable manifold theorem from dynamical systems theory is employed to establish the existence of f by relating it to the graph of a stable manifold of an auxiliary two-dimensional functional. Furthermore the contraction mapping theorem is established to prove the existence and design numerical algorithms for equilibrium prices based on some properly constructed nonlinear functionals. With some efficient numerical algorithms proposed and analyzed in this theses, the equilibria are computed very accurately and the behavior of the equilibria can be clearly observed through computer graphics.

The second part of the thesis (Chapter 7) is concerned with an optimization problem motivated by a generalized Cass-Shell sunspots model that allows for an infinite state of nature. The space for the demand function is L^1 which lacks the useful properties such as the reflexivity and weak compactness for closed bounded sets that are used in the conventional optimization theories. A new theory is developed to deal with these kinds of optimization problems. The proof of the main theorem depends on various results from real analysis and functional analysis, and particularly a new theorem from modern probability theory.

Last modified: April 7, 2003