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PROGRAMS OF STUDY & COURSE DESCRIPTIONS
Mathematics (MAT)

Major: Yeshiva College

General Track:
MAT 1412, 1413, 1510, 1520, 1521, 2105, and 12 additional credits in advanced MAT courses; also three correlate courses approved by the senior professor (PHY 1041–1042 R&L and COM 1300 are strongly recommended).

Students interested in actuarial mathematics are advised to take MAT 2461, 2462. For such students, ECO 1011, 1021 and COM 1300 are recommended correlates.

Computer Track:
MAT 1412, 1413, 1510, 2105, and one of the following sequences approved by the senior professor: 1) MAT 1520, 1521; 2) MAT 2215, 2216; 3) MAT 2461, 2462. Also COM 1300, 1320, 1504, 1621, 2113, 2545 and one of COM 3563, 3610, 3640.

Minor: Yeshiva College
MAT 1412, 1413, 1510, 2105 and 6 additional MAT credits approved by the senior professor.

The normal sequence of courses in the first two years is 1412, 1413, 1510, and 2105. A mathematics placement examination is administered during the period of orientation. Students must take this test before registering for MAT 1160, 1412.

Courses in statistics are listed under STA and STB.

1160 Precalculus. Three hours of lecture. Two hours of recitation. 4 credits.
Number systems, functions, equations, and inequalities; algebra of polynomials, exponentials, and logarithms; analytic geometry of lines and circles; vectors, trigonometry, and complex numbers.
Prerequisite: two years of high school mathematics and placement by examination.

1412, 1413 or 1413H Calculus I, II. Three hours of lecture. Two hours of recitation. 4 credits.
First semester: limits, continuity, derivatives; applications to graphing, maxima and minima, and related rates; mean value theorem; integration, fundamental theorem of the calculus, integration by substitution. Second semester: applications of integration in geometry and physics; methods of integration; improper integrals; indeterminate forms; numerical integration; sequences, power series and Taylor series, polar coordinates; parametric equations.
Prerequisite: three years of high school mathematics and placement by examination, or MAT 1160.

1422 or 1422H Great Proofs in Mathematics. 3 credits.
Landmark theorems from ancient to modern times. Geometry: Pythagorean theorem, Euler’s formula, platonic polyhedra. Number Theory: irrational numbers, infinitude of primes, unique factorization, Pythagorean triples. Set theory: sets, relations, functions, infinite cardinal numbers, Schroeder-Bernstein theorem. Probability: expectation, conditioning, independence, the law of averages. Theory of computation: Turing machines, decidable and undecidable problems.
Prerequisite: Permission of the instructor.

1504 Discrete Structures. Three hours of lecture. Two hours of lab. 4 credits.
Boolean algebra and predicate calculus; proof methods; sets, functions and relations; combinatorics; graph theory and algorithms; mathematical induction and recursion; probability and average case analysis of algorithms. (See COM 1504).
Prerequisite: three years of high school mathematics.

1510 Multivariable Calculus. 3 credits.
Vectors, vector functions and curves; functions of several variables, partial derivatives; multiple integrals, Jacobians; vector fields, line and surface integrals; theorems of Green, Gauss, and Stokes.
Prerequisite: MAT 1413.

1520, 1521 Advanced Calculus I, II. 3 credits.
Real numbers, limits, intrinsic properties of continuous functions, differentiability and integrability, uniform convergence, implicit and inverse function theorems, point-set topology, metric spaces, curves and surfaces.
Prerequisite: MAT 1510.

1540, 1541 Functions of a Complex Variable I, II. 3 credits.
Analytic functions, Cauchy-Riemann equations, Cauchy integral formula, residue theory; conformal mappings, normal families, Riemann mapping theorem, Weierstrass theorem, applications.
Prerequisite: MAT 1510.

2105, 2106 Linear Algebra I, II. 3 credits.
Systems of linear equations, Gaussian elimination, matrices, matrix algebra; vector spaces, linear transformations, similarity; inner product spaces; determinants; eigenvalues and eigenvectors, diagonalization; quadratic forms; canonical forms; complex vector spaces, spectral theory; applications.
Prerequisite: MAT 1412.

2215, 2216 Modern Algebra I, II. 3 credits.
Basic concepts of modern abstract algebra: groups, rings, and fields, with illustrations and applications, particularly in elementary number theory and the theory of equations.
Prerequisite (with permission, corequisite): MAT 2105.

2251 Theory of Numbers. 3 credits.
Divisibility, prime factorization, distribution of primes, linear and quadratic congruences, primitive roots, quadratic residues, quadratic reciprocity, Diophantine equations.
Prerequisite: MAT 1413.

2461 Probability Theory. 3 credits.
Probability spaces; combinatorics; conditional probability; discrete and continuous random variables; examples; density and distribution functions; independence; expectation and variance; moment-generating functions; law of large numbers; central limit theorem; applications. (See STA 2461).
Prerequisite: MAT 1510.

2462 Mathematical Statistics. 3 credits.
Hypothesis testing, confidence intervals, regression analysis, correlation, t-distribution, time series analysis, analysis of variance, F-distribution. (See STA 2462).
Prerequisite: MAT 2461.

2471 Queuing Theory. 3 credits.
Classification of queues; systems without memory; systems with losses; queues as birth-and-death processes; embedded Markov chains; networks; diffusion and Monte Carlo approximations.
Prerequisite: MAT 2462.

2481 Topics in Actuarial Mathematics. 3 credits.
Prerequisites: MAT 2461, MAT 2462.

2601 Differential Equations. 3 credits.
Classification of differential equations; existence and uniqueness of solutions; initial-value problems, boundary value problems; power series methods, integral transforms; numerical algorithms and error estimation; topological methods.
Prerequisite: MAT 1413.

2611 Partial Differential Equations. 3 credits.
Solution of parabolic, hyperbolic, and elliptic equations; initial and boundary value problems arising in physical situations such as heat conduction, wave propagation and gravitational potential; method of characteristics, separation of variables, Laplace and Fourier transforms.
Prerequisites: MAT 1510, MAT 2601.

2651 Numerical Analysis. 3 credits.
Computer-assisted analysis of differential and integral equations; applications of polynomial interpolation and numerical linear algebra to differential equations; error estimation, rates of convergence, extrapolation, multistep and implicit methods, stability, stiffness, ill-posed problems, chaos, and difference methods. Special attention is given to the analysis of boundary value problems by finite-element methods.
Prerequisites: MAT 1510, MAT 2105 and familiarity with a programming language.

2701 Geometry. 3 credits.
Embedded hypersurfaces; metrics, connections, and curvature; hyperbolic and spherical geometries; projective geometries; algebraic curves; applications to physics.
Prerequisite: MAT 1510.

2901 Mathematics of Finance. 3 credits.
Discrete models for options, pricing derivatives, continuous stock price models, Brownian motion, the Black-Scholes formula, the Black-Scholes differential equation, hedging options, dynamic programming, bond price models, yield curves, forwards and futures, Keynes interest rate parity formula.
Prerequisite: familiarity with differential equations and probability.

3301, 3302, 3303, 3304 Topics in Modern Mathematics. 3 credits.
Selected subjects in analysis, algebra, geometry, topology, and applied mathematics. Students may register for up to four semesters with permission of the senior professor.
Prerequisite: permission of the instructor.

4901 Independent Study.

4911 Guided Project.
Meet with the Yeshiva College academic dean.

4931, 4932 Seminar. 1–3 credits.
Seminar in current problems and literature of mathematics.
Prerequisite: permission of the instructor.

4933, 4934 Problem Seminar. 1–3 credits.
Techniques for solving problems in mathematics. Recommended for mathematics majors.
Prerequisite: permission of the instructor.