Bachelor of Science in Mathematics

B

Areas (as follows):

1: Algebra

3: Calculus and Analysis

4: Differential Equations

5: Geometry

7: Applied Mathematics

8: Statistics

9: Projects and Selected Topics

C

Course sequence in area

Functions, domain and range, examples of functions. Limits and continuity. Derivatives, applications of derivatives in optimization, linearization and graphing, the Mean Value Theorem. Integration, the Fundamental Theorem of Calculus, areas, volumes of solids of revolution, arc length. Conic sections.

Prerequisite: None.

Functions, Inverse functions. Transcendental functions. L'Hopital's rule. Techniques of integration. Improper integrals. Sequence and infinite series of real numbers. Polar coordinates. Parametric curves in the plane.

Prerequisite: 1440131.

Systems of linear equations, Gauss and Gauss-Jordan elimination processes. Matrix algebra, determinants, Cramer's rule. Vector spaces, subspaces, basis and dimension, rank, change of basis. Characteristic polynomial, eigenvalues and eigenvectors of square matrices, diagonalization. Inner product spaces, orthogonal projections, Gram-Schmidt process. Computer applications. Introduction to linear transformation.

Prerequisite: 1440131, 1440132.

Vectors and analytic geometry in space. Graphing surfaces in three dimensions. Vector–valued functions and motion in space. Functions of several variables. Extreme values and Lagrange multipliers. Multiple integrals. Areas and volumes.

Prerequisite: 1440131, 1440132.

Integration in vector fields. Line integrals, circulation and flux, path independence and conservative fields. Green's Theorem in the plane. Surface area and surface integrals. Parameterized surfaces. Stokes' and Divergence Theorems. Curvilinear coordinates. Transformation of coordinates. Introduction to Cartesian tensors.

Prerequisite: 1440231.

Logic, propositional logic, truth tables, propositional formulas, logical implication and equivalence, tautologies and contradictions, quantifiers. Methods of proof. Sets, applications of sets, Venn diagrams, Cartesian product, the power set. Cardinality. Mathematical Induction. Relations and partitions, functions. Zorn's Lemma and Axiom of Choice.

Prerequisite: 1440131.

This course is an introduction to the necessary software used for scientific programming such as MATLAB and MATHEMATICA or Maple. It is designed for science and engineering students. The main concern is the learning of advanced techniques for solving and graphing basic problems of Calculus and Linear algebra. Moreover, this course focuses on advanced scientific writing using LATEX packages.

Prerequisite: 1440131 and 1440211.

This course covers first and higher order ordinary differential equations (ODE) with applications in various fields. It contains: Basic concepts. First order ODE's, initial value problems, an existence and uniqueness theorem. Higher order ODE's with constant coefficients. Laplace transform and inverse. Power series solutions, Frobenius theorem. Introduction to Linear systems of ODE's.

Prerequisite: 1440132.

The axiomatic Systems, Finite geometry. Finite Projective Plane, Non-Euclidean geometry. Hyperbolic geometry (Sensed Parallels, Asymptotic Triangles. Saccheri Quadrilaterals, Area of Triangles, Ultraparallels, Transformation of the Euclidean Plane.

Prerequisite: 1440233.

Descriptive statistics; Axiomatic probability; Random variables and their moments; Special discrete and continuous distributions; Sampling distributions; Estimation; Hypothesis testing; Linear regression; Analysis of variance.

Prerequisite: 1440131.

Review of basic concepts of probability, random variables and distribution theory. Distribution of functions of random variables. Expectation and moment generating functions. Unbiased and Sufficient estimators. Point estimation, optimal properties of estimators. Interval estimation. Hypotheses testing.

Prerequisite: 1440281.

Groups. Subgroups. Quotient groups and homomorphisms. Introduction to rings and fields. Ideals. Ring homomorphisms and quotient rings. Applications.

Prerequisite: 144023.

Linear transformations. Change of basis, transition matrix and similarity. Nilpotent linear transformations and matrices. Canonical representation of matrices, Jordan canonical forms. Linear functionals and the dual space. Bilinear forms. Quadratic forms and real symmetric bilinear forms. Complex inner product spaces. Normal operators. Unitary operators. The spectral theorem. Theorems on normal and unitary operators.

Prerequisite: 1440211.

Formal systems, syntax, semantics, formal proofs, completeness, and decidability. Theories of computability, Gödel’s Incompleteness Theorems. First order logic, properties of formal systems.

Prerequisite: 1440233.

Solutions of linear systems. Numerical solutions of nonlinear equations. Interpolation, least squares, polynomial approximation, spline approximation. Numerical differentiation and integration. Numerical solutions of ordinary differential equations. Computer implementation.

Prerequisite: 1440231, 1440232.

Complex numbers and functions. Analytic functions and Cauchy-Riemann equations. Elementary functions. Contour integration. Cauchy's theorem and Cauchy’s integral formula. Laurent series. Residues and residue theorem. Applications to real integrals. Analytic continuation.

Prerequisite: 1440132.

Mathematical techniques and methods to solve real-world problems. Model formulation, approximation, and validation. Optimization and simulation. Applications in physics, engineering, and other fields.

Prerequisite: 1440231, 1440232.

Partial differential equations (PDEs), their classification and solution methods. Boundary and initial value problems. Separation of variables, Fourier series, and transform methods. Applications to physical and engineering problems.

Prerequisite: 1440241.

Basic concepts of number theory, including divisibility, prime numbers, congruences, and Diophantine equations. Fundamental Theorem of Arithmetic, modular arithmetic, Chinese Remainder Theorem, and applications.

Prerequisite: 1440233.