## Interests in Mathematics: Probability, stochastic processes, real analysis, point set topology

.

## Interests in Applied Computer Science/IT: Operations research, machine learning, knowledge discovery in databases, discrete mathematics applications, numerical computing, applied cryptography

## Some Musical Favorites

Organ works by J. S. Bach (especially performed by E. Power Biggs or Albert Sweitzer), Goldberg Variations (especially performed by Glen Gould), English and French Suites for Keyboard, St. Matthew Passion. Also Beethoven's piano sonatas, string quartets, symphonies 1, 3, 5, 6, 7, 9, Bruckner's symphonies, Mahler, Max Reger organ works, Mendelssohn organ works, Brahms symphonies 1--4, the two cello sonatas, variations on a theme by Haydn. This list could go on indefinitely so I will stop here...

## Some favorite Literary Works/Authors

The Shakespeare dramas, poetry by John Milton, John Dryden, Byron, Poe, Ezra Pound, William Butler Yeats, T. S. Eliot, Robert Lowell, Robinson Jeffers, Carl Sandburg. Fiction by Dickens, Melville, Joyce, Conrad, Faulkner, Mailer, Ayn Rand, H. G. Wells, R. Heinlein, A. Clarke, I. Asimov, Ted Sturgeon, Robert Silverberg, Ursula K. LeGuin, Greg Bear.

## Favorite Philosophers

F. Nietzsche, A. Schopenhauer.

## CURRICULUM VITAE

Degrees and Certificates:

M. Sc., Information Technology

B. A., Mathematics

Cert., UNIX Operating System

Degree Goals:

PhD, Computer Information Systems

PhD, Mathematics

Education:

My graduate studies in computer science actually began at UMASS Lowell but they were completed in information technology at Southern New Hampshire University (est. 1932). In the US computer science and information technology are two different but related disciplines. I suppose one could claim CS and IT are like two cousins rather than like identical twins.

Although my Masters degree is in IT, most of my undergraduate and graduate computer science and information technology courses were taught by professors with PhDs in computer science or in mathematics, at UMASS Boston/UMASS Lowell and at SNHU and who taught many courses common to both CS (UMASS) and IT (SNHU) programs.

(My graduate studies in pure mathematics are on hold right now due to reasons of poor physical health that prevent me from on campus attendance for classes. But Srinivasa Ramanujan was an absolute genius who got a much worse break than I did. Who said life was fair? It's not)

Southern New Hampshire University

Graduate Department of Information Technology

2500 North River Road

Manchester, NH 03106

Graduate Studies: Information Technology

University of Massachusetts Lowell

Department of Mathematics and the Division of Continuing Studies

One University Avenue

Lowell, Massachusetts

Certificate Studies: Unix Operating System (This included courses Discrete Structures I and II)

University Of Massachusetts Boston

Department of Mathematics and Science

100 Morrissey Boulevard

Boston, Massachusetts 02125

Undergraduate Studies: Mathematics

Undergraduate Major: Mathematics

Undergraduate Minor: Computer Science

Various Subjects I Covered in Mathematics:

Groups, left and right cosets and factor groups, group homomorphisms, group isomorphisms, Sylow p-Groups, permutation groups, isometry groups, skew fields, modules (left and right), ring theory, factor rings, ring homomorphisms, ring isomorphisms, integral domains, unique factorization domains, Euclidean rings, commutative field theory, extension fields, algebraic extensions, algebraic closure, finite fields, ideals, prime ideal rings, maximal ideals, principal ideal rings, valuations, Noetherian rings, Galois theory (The material covered mostly was from Fraleigh's textbook).

Also completed graduate level and postgraduate level academic work (2005-2006, 2008-2009) in random processes, design of experiments and hypothesis testing, probability and statistics (at UMASS Lowell), differential equations/existence and uniqueness of solution theorems, linear/nonlinear dynamics, phase space, the Hamiltonian dynamics of conservative systems, chaos, bifurcation, fractals, catastrophe on smooth potential energy manifolds and Lyapunov stability. Some of the topics we covered also included Poincare sets, the Hopf bifurcation, the iterated Smale horseshoe map for the Duffing equation, the Cantor set and invariant tori.

Postgraduate textbook used:

Various Subjects/Topics I Covered in Computer Science

At the University of Massachusetts Lowell I completed graduate computer science courses (2012--2013) in relational algebra, data mining and knowledge discovery, SQL, computer network security and symmetric/asymmetric encryption, wireless security, digital forensics, cloud computing and virtual systems. I completed my graduate studies to fulfill the requirements for the M.Sc. at the Graduate Department of Information Technology at Southern New Hampshire University where I covered additional CS material on relational database design and software development, more SQL, Java, systems analysis and design, data communications, software engineering (including CASE and UML modeling), operating systems, computer simulation and animation and IT project management which covered some techniques from operations research like queues/waiting line modeling, network optimization, CPM, PERT and decision trees.

I have covered DES/AES/RSA encryption, block and stream ciphers, the discrete logarithm problem, the Diffie--Hellman algorithm, Dijkstra's algorithm, Hasse diagrams, message hash algorithms, knapsack ciphers, discrete mathematics which covered combinatorics, algorithms, the inclusion-exclusion principle, graphs, weighted graphs, bit complexity, Boolean rings, fuzzy sets, finite state machines, some algebraic coding theory over finite fields with the main application to linear Hamming codes.

Some of the undergraduate and graduate textbooks used (UMASS Lowell):

A. Doerr. K. Levasseur, SRA/Pergamon, 1989.

Pearson/Prentice Hall, 2008.

Online mathematics tutor (volunteer) in undergraduate mathematics

Tutoring Subjects: Precalculus, College Algebra, Analytic Geometry, Differential and Integral Calculus, Advanced Calculus, Ordinary Differential Equations, Linear Algebra, Abstract Algebra, Complex Analysis, General Topology, Elementary Number Theory.

Researcher in elementary/analytic number theory, discrete mathematics (bit complexity, graphs and algorithms), stochastic processes, machine learning, mathematical physics. From 2002--2006 my mathematical physics research mentor was a physicist working in the field of theoretical quantum gravity/quantum field theory. Our joint refereed/published paper can be found in

After I completed my undergraduate studies in mathematics, I was seeking for five years, a mathematics professor in number theory as a research mentor. Instead I wound up with a quantum gravity theorist as my research mentor. My input into our joint quantum gravity paper (I was the junior author) mainly had to do with fiber bundle maps (There is a great introduction to fiber bundle maps in the famous Munkres textbook on topology), regular polytopes, quaternions and octonions. I certainly appreciated his guidance and for refereed research he was a great guy to get to collaborate with, but since I had wanted a number theorist as a mentor to start out with, life never seems to take you where you want to go, does it?

“Using Bonse’s inequality to find upper bounds on prime gaps,” (author: R. Betts,

“Spacetime Holography and the Hopf Fibration,” (authors: R. Betts, coauthor and L. Crowell, principal author,

A Radix Representation for each van der Waerden number $W(r, k)$ with $r$ colors: Why \(\log_{r}W(r, k) < k^{2}\) is true whenever $k$ is the number of terms in the arithmetic progression.

http://arxiv.org/abs/1604.07036

Two Upper Bounds for both the van der Waerden Numbers $W(r, k + 1)$ and $W(r + 1, k)$, that show the existence of a recurrence relation.

http://arxiv.org/abs/1603.01831

The Asymptotic Behaviors of \log_{r}W(r, k) and \log_{k}W(r, k), when W(r, k) is a van der Waerden Number.

http://arxiv.org/abs/1601.04697

"How to find the least upper bound on the van der Waerden Number W(r, k) that is some integer Power of the coloring Integer r."

http://arxiv.org/abs/1512.03631

"The Expansion of each van der Waerden number W(r, k) into Powers of r, when r is the Number of Integer Colorings, determines a greatest lower Bound for all k such that W(r, k) < r^k^2."

http://arxiv.org/abs/1208.4268

"A uniformly convergent Series for zeta(s) and closed Formulas, that include Catalan Numbers"

http://arxiv.org/abs/1008.0387v1

(A partial listing follows) Among those successful mathematical problem solvers of open problems appearing in the

(See the

1.

2.

3.

4.

5.

6.

Other Writing Experience:

Published author of news commentary and poetry in small literary press journals and community newspapers across the United States. Published literary criticism, poetry and unconventional creative fiction in online ezines (small press) in the United Kingdom, in France and in the United States. Received Honorable Mention in

C, Java, Common Lisp, PL/SQL, Oracle Database 11g, MySQL, SQLite, Perl/Python/PHP (for the LAMP stack), Unix Bash, Unix Korn shell, Awk, Flexsim, MATLAB, Octave, Mathematica, MINITAB, NumPy, SciPy, FORTRAN 77, Fortran 90, PARI (developed by Henri Cohen), Wampserver.

University of Massachusetts Boston Alumni Association

Mathematical Association of America (www.maa.org )

Number Theory Email List (since 2002).

French, Spanish (University of Massachusetts), Classical Latin, written German and Technical/Scientific Russian (University of Wisconsin--Madison).

Honors:

Contest Winner: Ole Miss Problem of the Week Contests (at least three weekly math problem contests won, name citations, 2002--2004)

Member: Sigma Alpha Pi Honor Society/National Society of Leadership and Success

M. Sc., Information Technology

B. A., Mathematics

Cert., UNIX Operating System

Degree Goals:

PhD, Computer Information Systems

PhD, Mathematics

Education:

My graduate studies in computer science actually began at UMASS Lowell but they were completed in information technology at Southern New Hampshire University (est. 1932). In the US computer science and information technology are two different but related disciplines. I suppose one could claim CS and IT are like two cousins rather than like identical twins.

Although my Masters degree is in IT, most of my undergraduate and graduate computer science and information technology courses were taught by professors with PhDs in computer science or in mathematics, at UMASS Boston/UMASS Lowell and at SNHU and who taught many courses common to both CS (UMASS) and IT (SNHU) programs.

(My graduate studies in pure mathematics are on hold right now due to reasons of poor physical health that prevent me from on campus attendance for classes. But Srinivasa Ramanujan was an absolute genius who got a much worse break than I did. Who said life was fair? It's not)

Southern New Hampshire University

Graduate Department of Information Technology

2500 North River Road

Manchester, NH 03106

Graduate Studies: Information Technology

University of Massachusetts Lowell

Department of Mathematics and the Division of Continuing Studies

One University Avenue

Lowell, Massachusetts

Certificate Studies: Unix Operating System (This included courses Discrete Structures I and II)

University Of Massachusetts Boston

Department of Mathematics and Science

100 Morrissey Boulevard

Boston, Massachusetts 02125

Undergraduate Studies: Mathematics

Undergraduate Major: Mathematics

Undergraduate Minor: Computer Science

Various Subjects I Covered in Mathematics:

Groups, left and right cosets and factor groups, group homomorphisms, group isomorphisms, Sylow p-Groups, permutation groups, isometry groups, skew fields, modules (left and right), ring theory, factor rings, ring homomorphisms, ring isomorphisms, integral domains, unique factorization domains, Euclidean rings, commutative field theory, extension fields, algebraic extensions, algebraic closure, finite fields, ideals, prime ideal rings, maximal ideals, principal ideal rings, valuations, Noetherian rings, Galois theory (The material covered mostly was from Fraleigh's textbook).

Also completed graduate level and postgraduate level academic work (2005-2006, 2008-2009) in random processes, design of experiments and hypothesis testing, probability and statistics (at UMASS Lowell), differential equations/existence and uniqueness of solution theorems, linear/nonlinear dynamics, phase space, the Hamiltonian dynamics of conservative systems, chaos, bifurcation, fractals, catastrophe on smooth potential energy manifolds and Lyapunov stability. Some of the topics we covered also included Poincare sets, the Hopf bifurcation, the iterated Smale horseshoe map for the Duffing equation, the Cantor set and invariant tori.

Postgraduate textbook used:

*Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems*, Jordan. D. W., Smith, P., Oxford, 1999).Various Subjects/Topics I Covered in Computer Science

At the University of Massachusetts Lowell I completed graduate computer science courses (2012--2013) in relational algebra, data mining and knowledge discovery, SQL, computer network security and symmetric/asymmetric encryption, wireless security, digital forensics, cloud computing and virtual systems. I completed my graduate studies to fulfill the requirements for the M.Sc. at the Graduate Department of Information Technology at Southern New Hampshire University where I covered additional CS material on relational database design and software development, more SQL, Java, systems analysis and design, data communications, software engineering (including CASE and UML modeling), operating systems, computer simulation and animation and IT project management which covered some techniques from operations research like queues/waiting line modeling, network optimization, CPM, PERT and decision trees.

I have covered DES/AES/RSA encryption, block and stream ciphers, the discrete logarithm problem, the Diffie--Hellman algorithm, Dijkstra's algorithm, Hasse diagrams, message hash algorithms, knapsack ciphers, discrete mathematics which covered combinatorics, algorithms, the inclusion-exclusion principle, graphs, weighted graphs, bit complexity, Boolean rings, fuzzy sets, finite state machines, some algebraic coding theory over finite fields with the main application to linear Hamming codes.

Some of the undergraduate and graduate textbooks used (UMASS Lowell):

*Discrete Mathematics and its Applications*, Sixth Ed., K. H. Rosen, McGraw Hill, 2007.*Applied Discrete Structures for Computer Science*, Second Ed.,A. Doerr. K. Levasseur, SRA/Pergamon, 1989.

*Computer Network Security: Theory and Practice*, Jie Wang, Springer, 2009.*Computer Security: Principles and Practice*, W. Stallings, L. Brown,Pearson/Prentice Hall, 2008.

**Research/Tutoring Experience (2002--):**Online mathematics tutor (volunteer) in undergraduate mathematics

Tutoring Subjects: Precalculus, College Algebra, Analytic Geometry, Differential and Integral Calculus, Advanced Calculus, Ordinary Differential Equations, Linear Algebra, Abstract Algebra, Complex Analysis, General Topology, Elementary Number Theory.

Researcher in elementary/analytic number theory, discrete mathematics (bit complexity, graphs and algorithms), stochastic processes, machine learning, mathematical physics. From 2002--2006 my mathematical physics research mentor was a physicist working in the field of theoretical quantum gravity/quantum field theory. Our joint refereed/published paper can be found in

*Foundations of Physics Letters*(Kluwer), April 2005.After I completed my undergraduate studies in mathematics, I was seeking for five years, a mathematics professor in number theory as a research mentor. Instead I wound up with a quantum gravity theorist as my research mentor. My input into our joint quantum gravity paper (I was the junior author) mainly had to do with fiber bundle maps (There is a great introduction to fiber bundle maps in the famous Munkres textbook on topology), regular polytopes, quaternions and octonions. I certainly appreciated his guidance and for refereed research he was a great guy to get to collaborate with, but since I had wanted a number theorist as a mentor to start out with, life never seems to take you where you want to go, does it?

**:-)****Published Papers:**“Using Bonse’s inequality to find upper bounds on prime gaps,” (author: R. Betts,

*The Journal of Integer Sequences*, Vol. 10, (2007), Article 07.3.8).“Spacetime Holography and the Hopf Fibration,” (authors: R. Betts, coauthor and L. Crowell, principal author,

*Foundations of Physics Letters*, Vol. 18, (2005).**Preprints:**A Radix Representation for each van der Waerden number $W(r, k)$ with $r$ colors: Why \(\log_{r}W(r, k) < k^{2}\) is true whenever $k$ is the number of terms in the arithmetic progression.

http://arxiv.org/abs/1604.07036

Two Upper Bounds for both the van der Waerden Numbers $W(r, k + 1)$ and $W(r + 1, k)$, that show the existence of a recurrence relation.

http://arxiv.org/abs/1603.01831

The Asymptotic Behaviors of \log_{r}W(r, k) and \log_{k}W(r, k), when W(r, k) is a van der Waerden Number.

http://arxiv.org/abs/1601.04697

"How to find the least upper bound on the van der Waerden Number W(r, k) that is some integer Power of the coloring Integer r."

http://arxiv.org/abs/1512.03631

"The Expansion of each van der Waerden number W(r, k) into Powers of r, when r is the Number of Integer Colorings, determines a greatest lower Bound for all k such that W(r, k) < r^k^2."

http://arxiv.org/abs/1208.4268

"A uniformly convergent Series for zeta(s) and closed Formulas, that include Catalan Numbers"

http://arxiv.org/abs/1008.0387v1

__Peer reviewed Journal Citations__(A partial listing follows) Among those successful mathematical problem solvers of open problems appearing in the

*American Mathematical Monthly*(*AMM*, from 1989--2006):(See the

*AMM*“Problems And Solutions” headings “Solved Also By…”)1.

*AMM*Problem #10784 (“Problems And Solutions,” August--September 2002, vol. 109 #7, pages 665--666): “Nonsingular Sums of Matrices.”2.

*AMM*Problem #10746 (“Problems And Solutions, January 2002, vol. 109 #1, pages 79--80): “Shrinking Lattice Curves.”3.

*AMM*Problem #6672 (“Problems And Solutions,” October 1993, vol. 100 #8, pages 803--806): “An Identity Related to the Landen Transformation.” (This problem was a mathematical identity which involved the use of elliptic integrals).4.

*AMM*Problem #E3431 (“Problems And Solutions,” November 1992, vol. 99 #9, page 877): “A Common Least Multiple.”5.

*AMM*Problem #E3432 (“Problems And Solutions,” August--September 1992, vol. 99 #7, pages 684--685): “Asymptotics of the Harmonic Sum.”6.

*AMM*Problem #E3410 (“Problems And Solutions,” March 1992, vol. 99 #3, pages 278--279): “The Period of Fibonacci Sequences Modulo m.”Other Writing Experience:

Published author of news commentary and poetry in small literary press journals and community newspapers across the United States. Published literary criticism, poetry and unconventional creative fiction in online ezines (small press) in the United Kingdom, in France and in the United States. Received Honorable Mention in

*Byline*’s writing/poetry contest, November, 2000. I also studied expository science writing at the University of Minnesota.**Computer Skills/Programming Languages:**C, Java, Common Lisp, PL/SQL, Oracle Database 11g, MySQL, SQLite, Perl/Python/PHP (for the LAMP stack), Unix Bash, Unix Korn shell, Awk, Flexsim, MATLAB, Octave, Mathematica, MINITAB, NumPy, SciPy, FORTRAN 77, Fortran 90, PARI (developed by Henri Cohen), Wampserver.

**Organizational Affiliations:**University of Massachusetts Boston Alumni Association

Mathematical Association of America (www.maa.org )

__[email protected]__Number Theory Email List (since 2002).

**Languages (read, write, experience in written translation):**French, Spanish (University of Massachusetts), Classical Latin, written German and Technical/Scientific Russian (University of Wisconsin--Madison).

Honors:

Contest Winner: Ole Miss Problem of the Week Contests (at least three weekly math problem contests won, name citations, 2002--2004)

Member: Sigma Alpha Pi Honor Society/National Society of Leadership and Success