Mathematics

This course develops the basic mathematical tools necessary for further study in economics and related disciplines. The course focuses on techniques of calculus (differentiation, partial differentiation, optimization, and integration), methods of linear algebra (use of matrices), and the solution of difference and differential equations. The ideas are taught systematically, with emphasis on their application to economic problems. Examples are used throughout the course for motivation and illustration. Specific topics are as follows: sets, functions, equations, and graphs; difference equations, sequences, and limits; differentiation, inverse functions, and exponential and logarithmic functions; partial differentiation, chain rule, and homogeneous functions; optimization in two variables (unconstrained and constrained); Lagrange multipliers; vector notation and convexity; matrix notation, systems of linear equations, and inverse matrices; integration; and differential equations.

This course covers the fundamental concepts of actuarial financial mathematics and how these concepts are applied in calculating present and accumulated values for various streams of cash flows. The topics covered include interest rates, present value, annuities valuation, loan repayment, bond and portfolio yield, bond valuation, rate of return, yield curve, term structure of interest rates, duration and convexity of general cash flows and portfolios, immunization, stock valuation, capital budgeting, dynamic cash flow processes, and asset and liability management.

This course covers the integral and differential properties of a complex variable. Complex differentiation, Cauchy-Riemann equations, harmonic functions, sequences and series, Taylor's and Laurent's series, and singularities and residues are examined. The class also focuses on complex integration, Cauchy's theorem and consequences, and Cauchy's integral formula and related theorems. Students study residue theorem and the applications used to evaluate integrals and summation of series, along with conformal transformations.

This course offers an introduction to analysis including: sets and relations; demonstrations by induction and contradiction; natural numbers; finite and infinite cardinalities, enumerability; rational and real numbers; limits and convergence of numerical sequences and series; straight topology-- open, closed, compact, connected, dense; Cantor ternary set; continuous functions-- Bolzano-Weierstrass Theorem, Intermediate Value Theorem, uniform continuity.

This course provides an introduction to databases, with a focus on mastering the skills and knowledge required to design and analyze a database and recover and store data in a fast and efficient manner. It reviews key concepts, including the difference between data and information; database design; database implementation (SQL; data manipulation language; normalization; storage and indexation); connectivity; introduction to NoSQL.

This course builds on the fundamentals of calculus, introducing students to hyperbolic functions and their inverses, and the complex exponential functions of two variables. Students are exposed to a wider class of differential equation models, both first and second order, to describe systems such as population models, electrical circuits, and mechanical oscillators. The course also covers convergence and divergence. Calculus topics include the intuitive idea of limits and continuity of functions of one variable, sequences, series, hyperbolic functions and their inverses, level curves, partial derivatives, chain rules for partial derivatives, directional derivative, tangent planes and extrema for functions of several variables. Complex exponential topics include definition, derivative, integral, and applications. Integration topics include techniques of integration and double integrals. Ordinary differential equations topics include first order (separable, linear via integrating factor) and applications, second order constant coefficient (particular solutions, complementary functions) and applications.

This course introduces the theory of functions of a complex variable, starting with an introduction to complex numbers and ending with applications of the Residue Theorem and conformal transformations.

Topics in this course include: vectors, dot products, conics and quadrics, matrices, determinants, and linear equations. Students solve systems of linear equations; prove theorems and identities using induction; discuss terms including determinant, eigenvalue, eigenvector, invertibility, kernel, image, and spectrum; find the eigenvectors and eigenvalues of a square matrix; find the determinant and inverse of a 3 x 3 matrix, and solve associated linear equations; deduce the solvability of a system of linear equations, without finding the solutions, via Gaussian elimination; deduce whether one vector is a linear combination of others, and by the same method deduce the dimension of the kernel and image of any matrix, using Gaussian elimination and examining pivots; and prove the equivalence of the dozen invertibility criteria of Strang's nutshell using the main theorems of linear algebra.

The notion of limit underlies the differential and integral calculus, a central topic in Mathematics. A good understanding of this concept was developed in the early nineteenth century, many years after the calculus was first used, and this is essential for more advanced calculus. The main purpose of this course is to provide a formal introduction to the concept of limit in its simplest setting: sequences and series.