Tools for quantitative reasoning and basic college level mathematical concepts for social science students. Mathematics of finance, linear equations and matrices, probability, game theory, derivative, integral, special functions: log, exp, trigonometric and function sketching techniques.

Sets; logic and implications; proof techniques with examples; mathematical induction and well-ordering; equivalence relations; functions; cardinality; countable and uncountable sets.

Vectors; matrices and systems of linear equations; vector spaces; linear maps; orthogonality; algebra of complex numbers; eigenvalue problems.

Functions of several variables; partial differentiation; directional derivatives; exact differentials; multiple integrals and their applications; vector analysis; line and surface integrals; Green?s, Divergence and Stoke?s theorems.

Natural numbers; modular arithmetic; introduction to groups; cyclic and permutation groups; homomorphisms and isomorphisms; normal; factor, simple and free groups; introduction to rings, integral domains, and fields; factor rings and ideals; extension fields; outline of Galois theory.

Functions and graphs; Mathematics of finance; linear algebra and matrix theory; logic, sets, counting; the fundamentals of probability theory.

Counting problems; combinatorial methods; integers, divisibility and primes; graphs and trees; combinatorics in geometry; introduction to complexity and cryptography.

Descriptive statistics; measures of association, correlation, simple regression; probability theory, conditional probability, independence; random variables and probability distributions; sampling distributions; estimation; inference (confidence intervals and hypothesis testing). Topics are supported by computer applications.

First order differential equations. Second order linear equations. Series solutions of ODE?s. The Laplace transform and applications. Systems of first order linear equations. Nonlinear equations and systems:existence, uniqueness and stability of solutions. Fourier series and partial differential equations.

Review of Unique Factorization Domains and Principal IdeaI Domains, Maximal and prime ideals, Nilradical, Local rings, Modules, Cayley-Hamilton theorem, Nakayama's lemma, Exact and split exact sequences, Noetherian rings, Noetherian modules, Hilbert basis theorem, Integral extensions, Integral closure, Non-singularity, Normal rings, Noether normalization, Hilbert nullstellensatz, Spec(A), Localization, Support of a module and the associated primes, Discrete valuation rings, Trace and separability, Completion, Artin-Rees Lemma, An overview of further topics: Dimension theory, Regular rings, Connections with geometric notions.