Publications - 2010

Advanced Number Theory with Applications

This book is designed as a second course in number theory at the senior undergraduate/junior graduate level to follow a course in elementary methods, such as that given in my book Fundamental Number Theory with Applications, Second edition, published in 2008 and listed in this profile. Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that students can easily cross-reference and find the appropriate data. With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an overview of Fermat’s Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue–Siegel–Roth theorem, Hall’s conjecture, the Erdös–Mollin-–Walsh conjecture, and the Granville–Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes’, Selberg’s, Linnik’s, and Bombieri’s sieves. He also discusses recent results on gaps between primes and the use of sieves in factoring. By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at the graduate level.

A fourth-order finite difference method for solving the system of two-dimensional Burgers' equations

A Note on Bartlett's M Test for Homogeneity of Variances

A note on the negative Pell equation

A review of canonical forms for selfadjoint matrix polynomials.

Central norms: Applications to Pell's equation

Classical Topics in Discrete Geometry

Construction of equally entangled bases in arbitrary dimensions via quadratic Gauss sums and graph states

Covering large balls with convex sets in spherical space

Distinguishing Number of Countable Homogeneous Relational Structures

Estimation of Q-factor and phase velocity using the recovered stress-strain relaxation spectrum, Proceedings of the GeoCanada 2010 meeting

Geochemical studies on the contamination and dispersion of heavy metals in intertidal sediments around a military air weapons shooting rage

Hermitian quadratic matrix polynomials: solvents and inverse problems.

Local cloning of entangled states

Minimum Hellinger distance estimation in a two-sample semiparametric model

Morphic groups

On covering numbers of sublevel sets of analytic functions

On polynomial inequalities on exponential curves in C^n

Optimal component selection for component-based systems

In Component-based Software (CBS) development, it is desirable to choose software components that provide all necessary functionalities and at the same time optimize certain nonfunctional attributes of the system (for example, system cost). In this paper we investigate the problem of selecting software components to optimize one or more nonfunctional attributes of a CBS. We approach the problem through the lexicographic multi-objective optimization perspective and develop a scheme that produces Pareto-optimal solutions. Furthermore we show that the Component Selection Problem (CSP) can be solved in polynomial time if the components are connected by serial interfaces and all the objectives are to be minimized, whereas the corresponding maximization problem is NP-hard.

Parametrizing structure preserving transformations of matrix polynomials.

Projective free algebras of continuous functions on compact abelian groups

Quantitative bounds for Markov Chain Convergence: Wasserstein and Total Variation distances

Random generation of graphs with a given degree sequence

Robust empirical Bayes tests for continuous distributions

Security estimates for quadratic field based cryptosystems

Some algebraic aspects of the center problem for ordinary differential equations

Study behavior in a preparatory mathematics program: a case of bilingual Arab pre-University students

The T(5) property of families of overlapping unit disks

Toward a Mackey formula for compact restriction of character sheaves

Tree composition condition and moments vanishing

Weighted Profile Least Squares Estimation for a Panel Data Varying-coefficient Partially Linear Model