BASIC Exam: A Basic Exam is given upon arrival to every new graduate student in mathematics. Prospective graduate students are strongly encouraged to try the past exams and prepare thoroughly for it. Passing the Basic Exam is a requirement for both the Master's and Ph.D. degrees.
Basic Exam Syllabus
Algebra:
Introduction. Basic notions. Sets, relations, functions, binary operations. Algebraic systems: integers, integers modulo a given integer, rational, real, and complex number systems, matrices. Isomorphism and examples. Equivalence classes.
Groups. Definitions and examples: cyclic groups, abelian groups, matrix groups, groups of permutations; homomorphism, kernel of a homomorphism, isomorphism, subgroup, normal subgroup, quotient group. Fundamental Homomorphism Theorem for groups.
Rings. Definition and examples: integers, the integers modulo a given integer, matrix and polynomial rings, commutative and non-commutative rings. Homomorphism, kernel of a homomorphism, ideal, quotient ring, isomorphism. Fundamental Homomorphism Theorem for rings.
Fields. Definitions and examples: the rational, real, and complex fields, the field of integers modulo a prime. Constructions via a commutative ring modulo a maximal ideal.
Linear Algebra. Vector spaces, linear dependence and independence of vectors, basis of a vector space. Linear transformation, matrix representation of a linear transfomation. Rank-Nullity Theorem. Determinant of a linear transformation.
REFERENCES
Gallian, J.A., Contemporary Abstract Algebra, Fourth edition, D.C. Heath and Company, Lexington, 1998.
Herstein, I.N., Abstract Algebra, Prentice-Hall, Upper Saddle River, New Jersey, 1996.
Analysis:
Functions, sequences, limits, continuity. Open and closed sets in R. Bounded sets, supremum, infimum. Cluster points. Accumulation points. Limit superior and limit inferior. Sequences. Functions. Limits, operations with limits of sequences and functions. Monotone sequences. Monotone functions. The Cauchy criterion for convergence. Continuity. Operations with continuous functions. The intermediate value property. Properties of continuous functions on closed intervals. Inverses of strictly monotone, continuous functions. Uniform continuity.
Differentiation of real-valued functions. The derivative. Chain rule. The mean value theorem. Cauchy's mean value theorem. L'Hospital's rule. Taylor's theorem. Estimation of the remainder. Extremes, concavity.
The Riemann Integral. Properties of definite integrals. The Fundamental Theorem of Calculus. Integration by substitution. Integration by parts. Mean value theorems for integrals. Integrals of discontinuous functions.
Numerical series. Absolute and conditional convergence. Alternating series. Series of nonnegative terms: comparison, ratio, and root tests.
Sequences and series of functions. Uniform convergence. Consequences of uniform convergence. Convergence tests. Power series. Interval of convergence. Taylor and Mclaurin series. Arithmetic of power series.
Improper integrals. Conditional and absolute convergence. Improper integrals with nonnegative integrands. Principal value. Convergence tests.
The level of difficulty corresponds to that of W. Fulks, Advanced Calculus, and Introduction to Analysis, third edition, John Wiley and Sons, 1978. (See especially chapters 1-6 and 13-17.)
Computational Mathematics:
College algebra. Fundamental concepts of algebra; algebraic equations and inequalities; functions and graphs; zeros of polynomial functions; exponential and logarithmic functions; systems of equations and inequalities.
Trigonometry. Trigonometric and inverse trigonometric functions; trigonometric identities and equations; applications involving right triangles and applications illustrating the laws of sines and cosines.
Calculus. Standard material from a three-semester calculus sequence. Analytic geometry, differential and integral calculus of algebraic, trigonometric, and transcendental functions. Techniqes of integration, infinite series, and functions of more than one variable.
Differential equations. Elementary techniques for solving ordinary differential equations and applications to solutions of problems in science and engineering.
Linear algebra. Matrix algebra, solutions to systems of linear equations, determinants, vector spaces, linear transformations, eigenvalues, linear programming, approximation techniques.