# Liberty International University

## ACADEMICS - Mathematics, Bachelor

**Program Description **

This option is designed for students who plan to pursue graduate studies in the mathematical sciences. A strong foundation in Analysis and Algebra can be complemented with electives in Geometry and Number Theory to provide a well rounded curriculum. The electives can also be directed towards Applied Mathematics, Differential Equations, and Statistics.

**V63.0009 Algebra and Calculus**

4 points. Fall and spring terms.

Prerequisite: High school math or permission of the department.

Intensive course in intermediate algebra and trigonometry. Topics include algebraic, exponential, logarithmic, trigonometric functions and their graphs.

**V63.0017 Calculus for the Social Sciences**

4 points.

Prerequisite:

a. SAT score of 630 or higher.

b. ACT score of 25 or higher.

c. Completion of Algebra and Calculus V63.0009) with a grade of C or higher

This course is appropriate for students who would like to learn something of Calculus without the intention of going further. A student in CAS will not receive credit for both V63.0017 and V63.0121. Other students may receive credit for both V63.0017 and V63.0121 only when V63.0017 is taken before V63.0121.

Derivatives, antiderivatives, and integrals of functions of one real variable. Integration, Logarithmic and exponential functions. Maxima and minima. Introduction to Probability Applications.

**V63.0120 Discrete Mathematics**

4 points. Fall and spring term.

A first course in discrete mathematics. Sets, algorithms, induction. Combinatorics. Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.

**V63.0121 Calculus I**

4 points. Fall and spring terms.

Prerequisite:

a. SAT score of 750 or higher

b. ACT/ACTE Math score of 34 or higher.

c. AB 4 or higher.

d. BC 3 or higher.

e. Completion of Algebra and Calculus (V63.0009) with a grade of C or higher.

d. passing placement exam.

Derivatives, antiderivatives, and integrals of functions of one real variable. Trigonometric, inverse trigonometric, logarithmic and exponential functions. Applications, including graphing, maximizing and minimizing functions. Areas and volumes.

**V63.0122 Calculus II**

4 points. Fall and spring terms.

Prerequisite: Passing V63.0121 Calculus I with a grade of C or better or a BC of 4 or higher or passing placement test..

Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

**V63.0123 Calculus III**

4 points. Fall and spring terms.

Prerequisite: Passing V63.0122 Calculus II a grade of C or higher or passing placement test.

Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals.

**V63.0140 Linear Algebra**

4 points. Fall and spring term.

Prerequisite: A grade of C or better in V63.0121 Calculus I or the equivalent.

Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer’s rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms.

**V63.0141 Honors Linear Algebra I - identical to G63.2110**

4 points. Fall term.

Prerequisite: A grade of B or better in V63.0325 Analysis Ior V63.0343 Algebra 1 or the permission of the instructor.

Linear spaces, subspaces, and quotient spaces; linear dependence and independence; basis and dimensions. Linear transformation and matrices; dual spaces and transposition. Solving linear equations. Determinants. Quadratic forms and their relation to local extrema of multivariable functions.

**V63.0142 Honors Linear Algebra II - identical to G63.2120**

4 points. Spring term.

Prerequisite: V63.0141 Intensive Linear Algebra I.

Special theory, eigenvalues and eigenvectors; Jordan canonical forms. Inner product and orthogonality. Self-adjoint mappings, matrix inequalities. Normal linear spaces and linear transformation between them positive matrices. Applications.

**V63.0221-0222 Honors Calculus I, II**

5 points each term. Fall and spring terms. Includes a recitation section.

Prerequisites for Honors I: A score of 5 on the Advanced Placement Calculus AB exam or 3 on BC, a B+ or better in V63.0121 Calculus I, placement exam, or permission of the instructor.

Prerequisites for Honors II: Honors I or permission of the instructor.

This sequence provides an enriched version of V63.0122 Calculus II and V63.0123 Calculus III, for serious students who have already mastered the basics of integration and differentiation for functions of one variable. Students must be able to differentiate reasonable-looking functions and evaluate standard integrals; they should also know the basic properties and applications of differentiation and integration (e.g. solving max/min problems and finding areas). Honors Calculus I discusses differentiation and integration more deeply, including Taylor expansion, convergence and divergence of series, sums, and integrals, an introduction to differential equations, and additional topics not normally included in Calculus II. Honors Calculus II provides a similarly enriched introduction to multivariable calculus, going deeper and further than Calculus III. This sequence is intended for well-prepared students who plan to major in mathematics or a closely related subject such as economics, physics, or computer science.

**V63.0224 Vector Analysis - identical to G63.1002**

4 points. Spring term.

Prerequisite: Passing V63.0122 Calculus II, V63.0123 Calculus III and V63.0140 Linear Algebra with a grade of C or better.

Functions of several variables. Partial derivatives, drain rule, change of variables. Lagrange multipliers. Inverse and implicit function theorems. Vector calculus; divergence, gradient and curl; Theorems of Gauss, Green, and Stokes with applications to fluids, gravity, electromagnetism and the like. Introduction to differential forms. Degree and fixed points of mappings with applications. Additional topics depending on the interests of the class, as time permits.

**V63.0233 Theory of Probability**

4 points. Fall term.

Prerequisite: V63.0122 Calculus II and V63.0123 with a grade of C or better and/or the equivalent.

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.

**V63.0234 Mathematical Statistics**

4 points. Spring term.

Prerequisite: V63.0233 Theory of Probability with a grade of C or better and/or the equivalent. Not open to students who have taken V63.0235 Probability and Statistics.

An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences.

**V63.0235 Probability and Statistics**

4 points. Spring term.

Prerequisite: V63.0122 Calculus II with a grade of C or better and/or the equivalent.

A combination of V63.0233 Theory of Probability and V63.0234 Mathematical Statistics at a more elementary level, so as to afford the student some acquaintance with both probability and statistics in a single term. In probability: mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal approximation; application to coin-tossing, radioactive decay, etc. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.

**V63.0240 Combinatorics**

4 points. Spring term of even years.

Prerequisite: V63.0122 Calculus II with a grade of C or better and/or the equivalent.

Techniques for counting and enumeration including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph-theoretic problems.

**V63.0243 Intro to Cryptography**

4 points. Introduction to Cryptography V63.0243 Identical to V22.0480 (offered spring term) Prerequisite: V22.0310 with a grade of C or better or permission of the instructor.

An introduction to both the principles and practice of cryptography and its application to network ecurity. Topics include: symmetric-key encryption (block ciphers, modes of operations, AES); message

authentication (pseudorandom functions, CBC-MAC); public-key encryption (RSA, EIGamal); digital signatures (RSA, Fiat-Shamir); authentication applications (identification, zero-knowledge) and others time permitting.

**V63.0245 Logic**

4 points. Spring term of odd years.

Prerequisite: V63.0122 Calculus II with a grade of C or better and/or the equivalent.

Propositional calculus, quantification theory, properties of axiomatic systems, Henkin’s theorem. Introduction to set theory, interpretations, models. Computability and its applications to the incompleteness theorem.

**V63.0246 Abstract Algebra**

4 points. Spring term

Prerequisite: V63.0122 Calculus II and V63.0140 Linear Algebra with a grade of C or better.

An introduction to the main concepts, constructs, and applications of modern algebra. Groups, transformation groups, Sylow theorems and structure theory; rings, polynomial rings and unique factorization; introduction to fields and Galois theory.

NOTES: This course does not count toward the math major because of its considerable overlap with the more intensive Algebra I, V63.0243, required as part of the majors program in Mathematics. It is, however, accepted toward the math minor, and is a strongly recommended course in the Steinhardt Math Ed major.

**V63.0248 Theory of Numbers**

4 points. Fall term.

Prerequisite: V63.0122 Calculus II with a grade of C or better and/or the equivalent.

Divisibility theory and prime numbers. Linear and quadratic congruences. The classical number-theoretic functions. Continued fractions. Diophantine equations.

**V63.0250 Mathematics of Finance**

4 points. Fall term.

Prerequisite: V63.0123 Calculus III, (and an introductory course in probability or statistics, V63.0233 Theory of Probability, V63.0235 Probability and Statistics, V31.0018 Statistics, V31.0120 Analytical Statistics or equivalent) with a grade of C+ or better.

Introduction to the mathematics of finance. Topics include: Linear programming with application pricing and quadratic. Interest rates and present value. Basic probability: random walks, central limit theorem, Brownian motion, lognormal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization.

**V63.0251 Introduction to Mathematical Modeling**

4 points. Spring term.

Prerequisite: V63.0121 Calculus I, V63.0122 Calculus II and V63.0123 Calculus III with a grade of C or better or permission of the instructor.

Formulation and analysis of mathematical models. Mathematical tool include dimensional analysis, optimization, simulation, probability, and elementary differential equations. Applications to biology, sports, economics, and other areas of science. The necessary mathematical and scientific background will be developed as needed. Students will participate in formulating models as well as in analyzing them.

**V63.0252 Numerical Analysis**

4 points.

Prerequisite: V63.0123 Calculus III, V63.0140 Linear Algebra with a grade of C or better.

In numerical analysis one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course gives an introduction to this subject for mathematics majors. Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method.

**V63.0255 Mathematics in Medicine and Biology - identical to G23.1501**

4 points. Fall term.

Prerequisite: V63.0121 Calculus I and V23.0011 Principles of Biology I or permission of the instructor.

Intended primarily for premedical students with interest and ability in mathematics. Topics of medical importance using mathematics as a tool: control of the heart, optimal principles in the lung, cell membranes, electrophysiology, countercurrent exchange in the kidney, acid-base balance, muscle, cardiac catheterization, computer diagnosis. Material from the physical sciences and mathematics is introduced as needed and developed within the course.

**V63.0256 Computers in Medicine and Biology - identical to G23.1502**

4 points. Spring term.

Prerequisite: V63.0255 Mathematics in Medicine and Biology or permission of the instructor. Familiarity with a programming language such as Pascal, FORTRAN, or BASIC is recommended.

Introduces the students of biology or mathematics to the use of the computer as a tool for modeling physiological phenomena. Each student constructs two computer models selected from the following list: circulation, gas exchange in the lung, control of cell volume, and the renal countercurrent mechanism. The student uses the models to conduct simulated physiological experiments.

**V63.0262 Ordinary Differential Equations**

4 points. Fall and spring terms.

Prerequisite: V63.0122 Calculus II, V63.0123 Calculus III and V63.0140 Linear Algebra with a grade of C or better or the equivalent.

First and second order equations. Series solutions. Laplace transforms. Introduction to partial differential equations and Fourier series.

**V63.0263 Partial Differential Equations**

4 points. Spring term.

Prerequisite: V63.0262 Ordinary Differential Equations with a grade of C or better or the equivalent.

Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Non-linear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.

**V63.0264 Chaos and Dynamical Systems**

4 points. Fall term.

Prerequisite: V63.0121 Calculus I with a grade of B or better or the equivalent.

Topics will include dynamics of maps and of first order and second-order differential equations: stability, bifurcations, limit cycles, dissection of systems with fast and slow time scales. Geometric viewpoint, including phase planes, will be stressed. Chaotic behavior will be introduced in the context of one-variable maps (the logistic), fractal sets, etc. Applications will be drawn from physics and biology. There will be homework and projects, and a few computer lab sessions (programming experience is not a prerequisite).

**V63.0270 Transformations and Geometries**

4 points. Fall term of odd years.

Prerequisite: V63.0123 Calculus III with a grade of C or better or the equivalent. Also, V63.0140 Linear Algebra with the grade of C or better is strongly suggested.

An axiomatic and algebraic study of Euclidean, non-Euclidean, affine, and projectile geometries. Special attention to group theoretic methods.

**V63.0375 Topology**

4 points. Offered on request.

Prerequisite: V63.0325 Analysis I or permission of the department.

Set-theoretic preliminaries. Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy groups.

**V63.0282 Functions of a Complex Variable**

4 points. Spring term.

Prerequisite: V63.0123 Calculus III plus one higher level course such as V63.0140 Linear Algebra with the grade of C or better.

Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy’s theorem and the Cauchy integral formula. Singularities, residues, and Laurent series. Fractional Linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow etc.

**V63.0325 Analysis I**

4 points. Fall and spring term.

Prerequisite: V63.0123 Calculus III and V63.0140 Linear Algebra or the equivalent.

The real number system. Convergence of sequences and series. Rigorous study of functions of one real variable: continuity, connectedness, compactness, metric spaces, power series, uniform convergence and continuity.

**V63.0326 Analysis II**

4 points. Spring term.

Prerequisite: V63.0325 Analysis I or permission of the department.

Functions of several variables. Limits and continuity. Partial derivatives. The implicit function theorem. Transformation of multiple integrals. The Riemann integral and its extensions.

**V63.0343 Algebra 1**

4 points. Fall term and Spring terms

Prerequisite: V63.0123 and V63.0140 Linear Algebra with a grade of C or better and/or the equivalent. Additionally, it is suggested for students to have taken V63.0325 Analysis I as a prerequisite.

Groups, homomorphisms, automorphisms, permutation groups. Rings, ideals and quotient rings, Euclidean rings, polynomial rings.

**V63.0344 Algebra 2**

4 points. Spring term.

Prerequisite: V63.0343 Algebra I with a grade of C or better.

Extension fields, roots of polynomials. Construction with straight-edge and compass. Elements of Galois theory.

**V63.0377 Differential Geometry**

4 points. Spring term of odd years.

Prerequisite: V63.0326 Analysis II or permission of the department.

The differential properties of curves and surfaces. Introduction to differential manifolds and Riemannian geometry.

**V63.0393-0394 Honors I, II**

4 points each term. Fall and spring term.

Prerequisite: Approval of the director of the honors program.

A lecture seminar course on advanced topics selected by the instructor and the audience, alternating between pure and applied, fall and spring. Topics vary yearly. Detailed course descriptions are available during preregistration.

**V63.0394 Honors II**

4 points.

The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem. Nehari, Conformal Mapping; Ahlfors, Complex Analysis.

**V63.0395-0396 Special Topics I, II**

4 points each term. Offered on request.

Prerequisite: Permission of the department.

Covers topics not offered regularly; experimental courses and courses offered on student demand. Detailed course descriptions are available during preregistration.

**V63.0997-0998 Independent Study**

2 or 4 points each term. Fall and spring terms.

Prerequisite: Permission of the department.

To register for this course a student must complete an application form for Independent Study and have the approval of a faculty sponsor and the Director of Undergraduate Studies.