Credit Hours - 3

This course builds on the elementary number theory introduced in MATH 224 Topics include: the Fundamental theorem of Arithmetic, Proof and Application: GCD,

LCM. Asymptotic notations,Congruences: Introduction to Congruences, Residue systems and Euler Phi-function, Linear Congruence, Chinese Remainder theorem, Theorems of Euler, Fermat and Wilson Arithmetic functions and Dirichlet Multiplication: Mobius, Euler Phi, Mangoldt, Sum of divisors etc functions, Dirichlet’s product and Mobius inversion formula, averages of arithmetical functions Quadratic Residues and Quadratic Reciprocity Law: Quadratic Residues, Legendre’s symbol and its properties, The quadratic reciprocity law and applications, the Jacobi symbol. Prime Number distribution.

Reading List:

• Apostol, T. M. (1998). Introduction to Analytical number theory.  Springer
• Chandrasekharan, K. (2012).  Introduction to Analytical number theory. Springer
• Tenenbaum, G. (2015). Introduction to Analytical and Probabalistic number theory. Springer.
• Jones, G., & Jones, J. (1998). Elementary Number Theory. Springer
• Ireland, K., & Rosen, M. (1998). A Classical Introduction to Modern Number Theory (2nd Edition). Springer

Credit Hours - 3

This course is a sequel to Math220. In this course, we continue the solution of linear systems by treating matrices with special structures. We also continue with data fitting using polynomials. Several high order methods for discretizing the derivative and definite integral are also treated. The course ends with approximations of eigenvalues for large matrices. We explain the concept of the dominant eigenvalue and its eigenvector. We also look at simultaneous approximation of eigenvalues.

Reading List:

• Burden, R. L. & Faires, J. D. (2008). Numerical analysis. Cengage Learning, (9th Edition).
• Chapra, S. (2008). Applied numerical methods with Matlab for engineers and scientists (3rd Edition). McGraw Hill.
• Epperson, J. F. (2013). An introduction to numerical methods and analysis (2nd Edition).Wiley.
• Matthews, J.H. & Fink, K.D. (2014). Numerical methods using Matlab. Pearson (5th Edition).
• Sauer, T. (2006).  Numerical Analysis. Pearson.

Credit Hours - 3

This is a first course in point set topology. Students will be introduced to 
topological spaces and be able to identify open and closed sets with respect to the given topology. Other aspects to be discussed are basis for a topological space. Separation and countability properties. Limit points. Connectedness. Subspace topology. Homeomorphism.Continuity. Metrizability. Continuity via convergent sequences. Compactness. 

Reading List:

• Davis, S. W. (2004 ). Topology. McGraw-Hill Higher Education.
• Lipschultz, S. (1965). Schaum's outline of theory and problems of general topology. Schaums Outlines.
• McIntyre, M. (2009). Topology. Departmental Lecture Notes
• Chirgwin, B.H, Plumpton, C., & Kilmister, C.W. (1972).  Elementary Electromagnetic Theory. Vols. II.  and III.     Pergamon Press.      
• Griffiths, D.J. (2014). Introduction to Electrodynamics. Pearson Educational
• Jackson, J. D. (1962). Classical Electrodynamics. Wiley and Sons.
• Reitz, J.R., Milford. F. J., & Christy, R.W. (1979). Foundations of Electromagnetic Theory (3rd  Edition). Narosa Pub. House.
• Morris, S. (2011). Topology. without tears ebook.

Credit Hours - 3

This is a continuation of MATH 353. We now consider vector spaces of functions and discuss convergence of sequences of functions; pointwise and uniform convergence. Other topics discussed include; power series, the contraction mapping theorem and applications. We examine the definition of  the Riemann integral and conditions for integrability. We give a proof of the fundamental theorem of calculus and major
basic results involved in its proof . We finish with some point set topology in R.

Reading List:

• Davidson, K. R. & Donsig, A. P. (2010). Real Analyis and its Applications. Springer
• Lang, S. (2015). Undergraduate Analysis. Springer
• McIntyre, M. (2016). Analysis notes. Departmental Lecture Notes.
• Royden, H. & Fitzpatrick, P. (2010). Real Analysis (4th Edition).
• http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf
• Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill Higher Education.

Credit Hours - 3

The primary aim of Math 354 is to study groups and their properties. We shall develop the foundations of group theory and study some notable groups like cyclic groups, permutation groups, finite Abelian groups and their characterization. Other ideas include: subgroups, cyclic groups.The Stabilizer-Orbit theorem.Lagrange's theorem. Classifying groups. Structural properties of a group. Cayley's theorem. Generating sets. Direct products. Finite abelian groups. Cosets and the proof of Lagrange's theorem. Proof of the Stabilizer-Orbit theorem. 

Reading List:

• Dummit, D. S. & Foote, R. M. (2003). Abstract Algebra (3rd Edition). Wiley
• Fraleign, J. B. (2013). A First Course in Abstract Algebra (8th Edition). Addison Wesley.
• Gallian, J. A. (2013).  Contemporary Abstract Algebra (8th Edition). Brooks/Cole.
• Judson, T. (2015). Abstract Algebra: Theory and Applications. Open Source available at
• http://abstract.ups.edu/index.html 
• Pinter, C. C. (2010).  A Book of Abstract Algebra (2nd Edition). (Dover Books on Mathematics).

Credit Hours - 3

This course introduces the basic methods applied in financial mathematics. We will discuss probability functions, stochastic processes, random walks and martingales; Ito's lemma and stochastic calculus. Students will understand the stochastic differential equations for a geometric Brownian motion process. We will study mean reverting models such as the Ornstein- Uhlembeck process, as well as stochastic volatility models such as the Heston Model. Stochastic models for stock pricing are also discussed;  we study a binomial option pricing model, the Black-Scholes model and the capital asset pricing model.

Reading List:

• Bass, R. (2003). The basics of financial mathematics. Springer.
• Doob, J. L. (2014). Stochastic processes. Wiley Interscience.
• Wilmott, P. & Howison, S. (1995). The Mathematics of Financial Derivatives: A Student    Introduction. Cambridge University Press.
• Oksendal, B. (2010). Stochastic Differential Equations (5th Edition). Universitext
• Parzen, E. (2010). Modern probability theory and applications. John Wiley, Canada

Credit Hours - 3

The methods of classical mechanics have evolved into a broad theory of dynamical systems and therefore there are many applications outside of Physics; for example to biological systems.  Topics to be discusses will include1-dimensional dynamics: damped and forced oscillations. Motion in a plane: projectiles, circular motion, use of polar coordinates and intrinsic coordinates. Two-body problems, variable mass. Motion under a central, non-inertial frame. Dynamics of a system of particles. 

Reading List:

• Corben, H. & Stehle, P. (1994). Classical Mechanics (2nd edition). Dover
• Kibble, T. W. B. & Berkshire, F. H. (2011). Classical mechanics (5th Edition). Imperial College Press.
• Marsden, J. & Abraham, R. (2012). Foundations of mechanics. Westview Press.
• Morin, D. (2008). Introduction to Classical Mechanics. Cambridge University Press. 
• Susskind, L. (2014). Classical mechanics.  Penguin Books Ltd

Credit Hours - 3

This course is a study of discrete rather than continuous mathematical structures. Topics include: asymptotic analysis and analysis of algorithms, recurrence relations and equations, Counting techniques (examples include: Inclusion-exclusion and pigeon-hole

principles and applications, Multinomial Theorem, generating functions),

Elementary Number Theory and Cryptography, Graph Theory, Discrete probability

theory. Planarity, Euler circuits, shortest-path algorithm. Network flows. Modelling computation: languages and grammars, models, finite state machines, Turing machines 

Reading List:

• Gossett, E. (2008). Discrete Mathematics with Proof.  Wiley.
• Levin, O. (2013). Discrete Mathematics. http://discretetext.oscarlevin.com/home.php
• Lipschultz, S. (2007). Schaum's outline of discrete mathematics. Schaums Outlines.
  Rajagopalan, S. P. & Sattanathan, R. (2015). Discrete mathematics. Margham Publications.
• Rosen, K. H. (2012). Discrete mathematics and its applications. McGraw-Hill

Credit Hours - 3

In this course the student is introduced to  a collection of closely related alternative formulations of classical mechanics. It   provides a detailed introduction to the key analytical techniques of classical mechanics. Topics discussed include
rigid body motion, rotation about a fixed axis. General motion in a plane, rigid bodies in contact, impulse. General motion of a rigid body. Euler-Lagrange equations of motion.

Reading List:

• Finch, J. D. & Hand, L.N. (1998). Analytical mechanics. Cambridge University Press
• Fowles, G. R. & Cassiday, G. L. (2004). Analytical Mechanics (7th Edition). Brooks/Cole
• Lanczos, C. (2011). The variational principles of mechanics. Dover
• Merches, I. & Radu, D. (2014). Analytical Mechanics: Solutions to problems in Classical Physics. CRC press.
• Helrich, C. (2017). Analytical Mechanics. Springer

Credit Hours - 3

This course develops the mathematical foundations for the application of the electromagnetic model to various problems. Mathematics discussed includes scalar and vector fields, grad, div and curl operators. Orthogonal curvilinear coordinates. Electrostatics: charge, Coulomb's law, the electric field and electrostatic potential, Gauss's law, Laplace's and Poisson's equations. Conductors in the electrostatic field. Potential theory.

• Reading List:
• Chirgwin, B.H., Plumpton, C., & Kilmister, C.W. (1972).  Elementary Electromagnetic Theory. Vols. II.  and III.     Pergamon Press.        
• Friedriches, K. O. (2014). Mathematical Methods of Electromagnetic Theory. AMS 
• Griffiths, D. J. (2014). Introduction to Electrodynamics. Pearson Educational.
• Jackson, J. D. (1962). Classical Electrodynamics. Wiley and Sons
• Reitz, J. R., Milford. F. J., & Christy, R.W. (1979). Foundations of Electromagnetic Theory, (3rd Edition). Narosa Pub. House. 

Credit Hours - 3

Differential equations can be studied analytically, numerically and qualitatively. The focus of this course is to find solutions to differential equations using analytic techniques. Differential forms of 2 and 3 variables. Exactness and integrability conditions. Existence and uniqueness of solution. Second order differential equations with variable coefficients. Reduction of order, variation of parameters. Series solution. Ordinary and regular singular points. Orthogonal sets of functions. Partial differential equations. 

Reading List:

• Agarwal, R. P., &  O’Regan, D. (2009). Ordinary and Partial Differential Equations. Springer, New York.
• Collatz, L. (2013).  Differential Equations : An Introduction and Applications. John Wiley and Sons Ltd.  
• Edwards, C. H.,  & Penny, D. E. (2015). Elementary Differential Equations, (7th Edition). Pearson Education Ltd. 
• Goodwine, B. (2011). Engineering Differential Equations - Theory and Applications. Springer, New York.
• Zill, D. G. (2014). A first course in Differential Equations with Modelling Applications
• (7th Edition). Brooks/Cole.

Credit Hours - 3

The major goal for this course is to understand and apply the concepts of

differentiation and integration to functions of several variables.
Functions of several variables, partial derivative. Directional derivative, gradient. Local extema, constrained extrema. Lagrange multipliers. The gradient, divergence and curl operators. Line, surface and volume integrals. Green's theorem, divergence theorem, Stokes' theorem. 

Reading List

• Lang, S. (2016). Calculus of Several Variables. Undergraduate Texts in Mathematics, Springer.
• Marsden, J. & Tromba, A. (2003). Vector Calculus. W H Freeman.
• Stewart, J. (2014). Multivariable Calculus (6th edition). Brooks/Cole
  Strang, G. ( 2012). Calculus(http://ocw.mit.edu/resources/res-18-001-calculus-online- textbook-spring-2005/textbook/).
• Thomas, G. & Weir, M. (2013). Calculus: Early Transcendentals (13th Edition). Pearson

Credit Hours - 3

This is the first rigorous analysis course. Topics to be discussed include: normed vector spaces, limits and continuity of maps between normed vector spaces. Students will be expected to produce proofs to justify their claims.
We study the algebra of continuous functions. Bounded sets of real numbers. Limit of a sequence. Subsequences. Series with positive terms.Convergence tests. Absolute convergence. Alternating series. Cauchy sequences and complete spaces.

Reading List:

• Davidson, K. R. & Donsig, A. P. (2010). Real Analyis and its Applications. Springer.
• Lang, S. (2015). Undergraduate Analysis. Springer.
• McIntyre, M (2016). Analysis notes. Departmental Lecture Notes.
• Royden, H Fitzpatrick, P, (2016) Real Analysis, (4th edition). 
• http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden- fitzpatrick.pdf
• Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill Higher Education.

Credit Hours - 3

We will develop a core of material called linear algebra by introducing

definitions and procedures for determining properties and proving theorems about matrices and linear transformations, with applications. Topics to be discussed include: spanning sets; subspaces, solution spaces. Bases. Linear maps and their matrices. Inverse maps. Range space, rank and kernel. Eigenvalues and eigenvectors. Diagonalization of a linear operator. Change of basis. Diagonalizing matrices. Diagonalization theorem. Bases of eigenvectors. Symmetric maps, matrices and quadratic forms.

Reading List:

• Hefferon, J. (2014).  Linear Algebra http://joshua.smcvt.edu/linearalgebra 
• Kolman, B. (2003.) Linear Algebra (8th Edition).
• Lang, S. (2014). Linear Algebra. Undergraduate Texts in Mathematics, Springer.
• Lipshultz, S. (2008). Schaum’s outline of Linear Algebra. 
• Robinson, D.J.S. (2012). A course in linear algebra with applications. World Scientific Publishing Co. Pty Ltd.