91亚色

Education

B.A., Angelo State University
M.S., Texas Tech University
Ph.D., University of North Texas

Background

PhD Thesis: Quantization of Spin Direction for Solitary Waves in a Uniform Magnetic Field.

Some of my publications can be found on

Master's Thesis: Embeddings Of Tree-Like Continua In The Plane

My past work has been in continuum theory. Here we take continuum to mean a (nonempty) compact connected metric space. This was for my master’s thesis (). As the title indicates, the purpose was to study embedability properties of tree-like continua in the plane. Since such continua are one dimensional, they are all embeddable in R^3 due to the Embedding Theorem (Menger and Nobling).  Although some are also embeddable in the plane, it is not difficult to produce a skew tree-like continuum (that is, one that cannot be embedded in the plane). Several hypotheses are known to produce embeddability, such as being both tree-like and circle-like, or being an inverse limit of k-junctioned trees with arm preserving bonding maps. An interesting problem is then to try and find general necessary and sufficient conditions for embedding tree-like continua in R^2. In my thesis I provide a general sufficient condition using the Anderson-Choquet theorem and inverse limits. The corresponding result concerning a general necessary condition is nontrivial and is still an open question. My master's thesis advsior was Wayne Lewis whose research has revolved primarily around the pseudo-arc (the simplest nondegenerate hereditarily indecomposable continuum). It is believed that the pseudo-arc may occur as a (strange) attractor of some natural dynamical systems.

Research Interests

Multidimensional solitary waves (existence, stability, quantization and symmetry breaking phenomena), mathematical physics (including nonlinear optics, Bose-Einstein condensates,... , other nonlinear phenomena), dynamical systems, nonlinear partial differential equations,...

Localized (and in some cases stable) solitary wave solutions are known to exist for some nonlinear wave equations such as certain classes of Klein-Gordon and Schroedinger equations. A part of my current research involves vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation. The goal is ultimately to see if an “explanation” can be provided for the spins of elementary particles. It is currently accepted that charged particles have a fixed number of orientations of their intrinsic angular momentum, depending on the value of a parameter l (spin quantum number). This may only take values which are zero, integral, or half-integral. Experiments performed by Stern and Gerlach in 1922 and later by others provided for this interpretation. 

I have investigated the spin (vector) or intrinsic angular momentum of some solitary wave solutions when they are subjected to an externally applied uniform magnetic field (this was my PhD thesis and can be found in the Physica D paper mentioned above). The spin is the (Noether) conserved quantity which results from the rotational invariance of the equation. In the absence of the field, it is seen that there exist solutions with spin in any prescribed direction. However, under the influence of the magnetic field I found that the only stationary spinning solitary wave solutions have spin parallel or antiparallel to the magnetic field direction. These results have been shown for l=0, 1, 2, … and for solutions in space-time R^(3+1). I am currently in the process of furthering this work. The foundations for this research were laid by  and others.

It is important to note that the discussion in this research is not a quantum mechanical one. Although many of the constructions have analogues in quantum mechanics, the interpretations are different (classical).

Scholarly Works

Courses Taught

Western New England University

Mathematics Courses

• MATH 111     Analysis for Business and Economics I
• MATH 112     Analysis for Business and Economics II
• MATH 117     Mathematical Reasoning
• MATH 118     The Heart of Mathematics
• MATH 123     Calculus I For Management, Life, and Social Sciences
• MATH 124     Calculus II For Management, Life, and Social Sciences
• MATH 133     Calculus I
• MATH 134     Calculus II
• MATH 235     Calculus III
• MATH 236     Differential Equations
• MATH 306     Linear Algebra
• MATH 350     Engineering Analysis
• MATH 421     Analysis
• MATH 451-452   Senior Project I & II

Master of Arts in Mathematics for Teachers (MAMT) Courses

• MAMT 548     What Is Mathematics?
• MAMT 554     Number Theory
• MAMT 560     History of Mathematics
• MAMT 564     Analysis

Physics Courses

• PHYS 133     Mechanics
• PHYS 133     Mechanics Laboratory
• PHYS 134     Electricity and Magnetism
• PHYS 134     Electricity and Magnetism Laboratory

University of NorthTexas

Mathematics Courses

• MATH 1100     College Algebra
• MATH 1190     Business Calculus
• MATH 1400     College Math with Calculus
• MATH 1680     Elementary Probability and Statistics
• MATH 1710     Calculus I
• MATH 2770     Discrete Mathematical Structures

 Texas Tech University

Mathematics Courses

• MATH 1320     College Algebra
• MATH 1321     Trigonometry
• MATH 1330     Introductory Mathematical Analysis