PhD Thesis: Quantization of Spin Direction for Solitary Waves in a Uniform Magnetic Field.
Some of my publications can be found on arXiv.
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 Henry Warchall 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).
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 (Embeddings Of Tree-Like Continua In The Plane). 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.
Western New England University
- 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
- PHYS 133 Mechanics
- PHYS 133 Mechanics Laboratory
- PHYS 134 Electricity and Magnetism
- PHYS 134 Electricity and Magnetism Laboratory
University of NorthTexas
- 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
- MATH 1320 College Algebra
- MATH 1321 Trigonometry
- MATH 1330 Introductory Mathematical Analysis