Mathematical Sciences Major

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MATHEMATICS (Math)
Offered by the Department of Mathematical Sciences.

Math 130. Basic Math – Algebra and Trigonometry. This course is designed to help reinforce algebraic and trigonometric skills necessary for success in the technical core. Basic graphing, algebraic manipulation, and trigonometric calculations are covered. Elementary functions, function manipulation, and some function applications are also discussed. This course may be used as an Academy option to fulfill graduation requirements. This course does not fulfill the requirements of any major.

Math 141. Calculus I. A study of differential calculus. Topics include functions and their applications to physical systems; limits and continuity: a formal treatment of derivatives; numeric estimation of derivatives at a point; basic differentiation formulas for elementary functions; product, quotient and chain rules; implicit differentiation; and mathematical and physical applications of the derivative, to include extrema, concavity and optimization. Significant emphasis is placed on using technology to solve and investigate mathematical problems.

Math 142. Calculus II. A study of integral calculus with a focus on the Fundamental Theorems and their application. Topics include: estimating area under a curve; accumulation and total change; numeric integration methods; antiderivative formulas for the elementary functions; integration by substitution, parts and tables; improper integrals; differential equations; exponential growth and decay; an introduction to Taylor Series; and mathematical and physical applications of the Fundamental Theorems. Physical applications include area and volume problems and the concept of work.

Math 152. Advanced Placed Calculus II. A more intense study of integral calculus for advanced-placed Fourth-Class cadets. Content is similar to Math 142, but with more in-depth treatment. Additional emphasis is placed on the mathematical and physical applications in preparation for cadets interested in pursuing a technical major or minor.

Math 243. Calculus III. Multivariate calculus, including vector functions, partial differentiation, directional derivatives, line integrals and multiple integration. Maxima and minima in multiple dimensions and the method of Lagrange Multipliers. Sequences and series. Solid analytical geometry to include lines, planes and surfaces in 3-space. Designed for cadets who indicate an interest in a technical major.

Math 245. Differential Equations. Modeling with and analysis of linear and non-linear ordinary differential equations. Includes matrix algebra and matrix inverses, first-order ordinary differential equations (numerical methods, separation of variables, integrating factors, and method of undetermined coefficients),  and second-order linear differential equations/first-order linear systems (Laplace transforms, determinants, eigenevalues, eigenvectors, and stability), and second-order non-linear differential equations/first-order non-linear systems. Applications may include population growth, predator/prey and mass-spring system modeling.

Math 253. Advanced Placed Calculus III. A more intense study of multivariate calculus for advanced-placed fourth class cadets. Content is similar to Math 243. Additional emphasis is placed on mathematical and physical applications in preparation for cadets interested in pursuing a technical major.

Math 300. Introduction to Statistics Descriptive statistics emphasizing graphical displays; basic probability  and probability distributions; sampling distribution of the mean and the Central Limit Theorem; statistical inference including confidence intervals, hypothesis testing; correlation; and regression. Math 300 is designed primarily for social science and humanity majors. It emphasizes the elements of statistical thinking; focuses on concepts, automates most computations, and has less mathematical rigor than Math 356.

Math 310. Mathematical Modeling. An introductory course in mathematical modeling. Students model various aspects of real-world situations chosen from Air Force applications and from across academic disciplines, including military sciences, operations research, economics, management, and life sciences. Topics include: the modeling process, graphical models, proportionality, model fitting, optimization and dynamical systems. Several class periods are devoted to in class work on small projects.

Math 320. Foundations of Mathematics. Course emphasizes exploration, conjecture, methods of proof, ability to read, write, speak, and think in mathematical terms. Includes an introduction to the theory of sets, relations, and functions. Topics from algebra, analysis, or discrete mathematics may be introduced.  

Math 340. Discrete Mathematics. Useful for cadets interested in applications of mathematics to computer science and electrical engineering. Propositions and logic; sets and operations on sets; functions, recursion and induction; graphs, trees and their applications; discrete counting and combinatorics.

Math 342. Numerical Analysis. An introductory numerical analysis course. Specific topics include roundoff, truncation, and propogated error, root finding, fixed-point iteration, interpolating polynomials, and numerical differentiation and integration The approach is a balance between the theoretical and applied perspectives with some computer programming required.

Math 344. Matrices and Differential Equations. Properties, types, and operation of matrices; solution of linear systems; Euclidean vector spaces, linear independence, and bases; eigenvalues and eigenvectors. Computational aspects. Applications to differential equations. First- and second-order differential equations and systems. Models may include population growth, warfare, and economic growth.

Math 346. Engineering Math.  Provides advanced mathematical concepts and skills necessary for technical disciplines. Topics include differential and integral vector calculus (gradient, directional derivative, divergence, curl, Divergence Theorem, Stokes’ Theorem), Fourier series, orthogonal functions, and partial differential equations (separation of variables, transform methods, numerical techniques).

Math 356. Probability and Statistics for Engineers and Scientists. Classical discrete and continuous probability distributions. Generalized univariate and bivariate distributions with associated joint, conditional and marginal distributions. Expectations of random variables. Central Limit Theorem with applications in confidence intervals, hypothesis testing, regression and analysis of variance. Designed for cadets in engineering, sciences or other technical disciplines.

Math 359. Design and Analysis of Experiments. An introduction to the philosophy of experimentation and the study of statistical designs. The course requires a knowledge of statistics at the Math 300 level. Topics include design and analysis of single-factor and many factor studies. A valuable course for all science and engineering majors.

Math 360. Linear Algebra. A first course in linear algebra focusing on Euclidean vector spaces and their bases.  Using matrices to represent linear transformations, and to solve systems of equations, is a central theme. Emphasizes theoretical foundation; computational aspects are covered in Math 344.

Math 366. Real Analysis I. A theoretical study of functions of one variable focused on proving results related to concepts first introduced in differential and integral calculus.An essential prerequisite for graduate work in mathematical analysis, differential equations, optimization and numerical analysis.

Math 370. Introduction to Point-Set Topology. Review of set theory; topology on the real line and on the real plane; metric spaces; abstract topological spaces with emphasis on bases; connectedness and compactness. Other topics such as quotient spaces and the separation axioms may be included. A valuable course for all majors in the graduate school option.

Math 372. Introduction to Number Theory. Basic facts about integers, the Euclidean algorithm, prime numbers, congruencies and modular arithmetic, perfect numbers and Legendre symbol will be covered and used as tools for the proof of quadratic reciprocity. Special topics such as public key cryptography and the Reimann Zeta function will be covered as time allows.

Math 374. Combinatorics and Graph Theory. Permutations, combination, recurrence relations, inclusion-exclusion, connectedness in graphs, colorings, and planarity. Theory and proofs, as well as applications to area such as logistics, transportation, scheduling, communication, biology, circuit design, and theoretical computer science.

Math 377. Applied Probability and Statistics. Descriptive statistics. Classical discrete and continuous random variable and probability distributions. Generalized univariate and bivariate distributions with associated conditional and marginal distributions. Central Limit Theorem. Single sample inferential statistics with applications using confidence intervals and hypothesis testing. Simple and multiple linear regression.

Math 378. Advanced Probability and Statistics. Advanced discrete and continuous random variables and probability distributions. Point estimation, joint, conditional and marginal distributions. Functions of random variable and the Central Limit Theorem. Inferential statistics for categorical data and multiple samples including ANOVA. Nonparametric statistics. Design and analysis of experiments, including factorial, fractional and response surface designs.

Math 405. Math Seminar.  A problem-solving course reviewing major areas and concepts of undergraduate mathematics. An assessment exam may be administered.

Math 420.  Mathematics Capstone I.  The first semester of the mathematics capstone experience.  Students will decide on a topic of independent research in, or related to, the mathematical sciences and begin work with a faculty advisor.  Significant progress toward a thesis will be made during the semester.

Math 421.  Mathematics Capstone II.  The second semester of the mathematics capstone experience.  Students will complete work on their independent research project and produce a thesis to present their findings.

Math 451. Complex Variables. A valuable course for cadets intending to pursue graduate work in mathematics or its applications, particularly in areas involving partial differential equations. Analytic functions; integration, the Cauchy Integral Theorem and applications; power and Laurent series, residues and poles; conformal mapping with applications to potential theory and fluid flows.

Math 465. Modern Algebra. A valuable course for cadets intending to pursue graduate work in mathematics or its applications. Focuses on the study of algebraic structures and functions between these structures. Topics include: cyclic groups, permutation groups, normal subgroups and quotient groups; rings, ideals, polynomial rings and fields. Depending on instructor and student preferences, applications to coding theory, crystallography or combinatorics are explored.

Math 467. Real Analysis II. A theoretical study of functions of several variables to include topology of cartesian spaces, compact and connected sets, convergence of sequences and functions, continuous functions, fixed-point theorems, contractions, Stone-Weierstrass approximation theorems, differentiation, partial differentiation, mapping theorems and Implicit Function Theorem.

Math 468. Dynamical Systems. A study and application of linear and nonlinear differential equations to physical systems from both computational and analytical points of view. Topics vary; typical choices include systems of differential equations, stability analysis , bifurcations, maps and chaos.

Math 469. Partial Differential Equations.  Solutions of boundary value problems and applications to heat flow, wave motion, and potential theory. Methods of solution include separation of variables and eigenfunction expansion, including Fourier series. Topics typically include the method of characteristics generalizations to higher dimensions, and the use of non-Cartesian coordinate systems. Additional topics may include numerical methods, nonlinear equations, and transform methods.

Math 470. Mathematical Physics. An introduction to various mathematical topics needed in graduate level physics and applied mathematics courses. Topics vary; typical choices include special functions (Legendre polynomials, Bessel functions, etc.), calculus of variations, complex functions (Laurent series, contour integration and the Residue Theorem), Fourier series and their convergence properties, integral transform concepts (Fourier and Laplace transforms, Green’s functions), dynamical systems.

Math 495. Special Topics. Selected advanced topics in mathematics.

Math 499. Independent Study and Research. Individual study and/or research under the direction of a faculty member.