It is important the all incoming graduate students have a good foundation in the following areas: Mathematical maturity - Familiarity with the axiomatic approach, induction, and proof by contradiction Set theory - Subsets; unions and intersections; countability and uncountabilty; de Morgan's laws; Cartesian products; equivalence relations and equivalence classes Algebra - Definition of a group and a normal subgroup; some familiarity with homomorphisms, their kernels and cokernels Linear algebra - Vector spaces, matrices, determinants, linear transformations, eigenvalues, eigenvectors and eigenspaces ?Analysis - Limits and continuity; lim sup and lim inf; the Riemann integral; epsilon-delta proofs; open, closed, and compact subset of Euclidean space; uniform convergence and uniform continuity; working knowledge of inequalities; partial derivatives and multiple integrals

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