CS Curricula

Courses tagged with math

COSC-121: Thinking Like a Computer Scientist (1) math

Analytical thinking is inherent in every aspect of computer science. We need to be able to answer questions such as: how do I know that my program works correctly? How efficient is my approach to solving a problem? How does human-readable code get translated into something that can run on physical hardware? What problems are even solvable by computers? In order to study such questions, computer scientists must be able to communicate with one another using a common language, express ideas formally and precisely, and reason logically about these ideas. This course will introduce mathematics as the primary analytical tool used by computer scientists. Topics may include but are not limited to set notation, symbolic logic, proof techniques such as induction and contradiction, and applications of these topics in computer science. Much more important than any individual topic, however, is the experience that students will gain with formal reasoning. (Amherst)

COSC-223: Probability and Computing (1) math

Probability is everywhere in computer science. In networks and systems, it is a key tool that allows us to predict performance, to understand how delay changes with the system parameters, and more. In algorithms, randomization is used to design faster and simpler algorithms than their deterministic counterparts. In machine learning, probability is central to the underlying theory. This course provides an introduction to probability with a focus on computer science applications. We will discuss elementary probability theory, including topics such as discrete random variables and distributions and Markov chains, and settings in which these are used in computer science (e.g., modeling real-world workload distributions, analyzing computer system performance, and designing and analyzing randomized algorithms). (Amherst)

COSC-351: Information Theory (1) math

Information Theory formally studies how to efficiently transmit and store digital information. (Amherst)

CPI 200: Mathematical Foundations of Informatics (3) math

Practical introduction to the mathematics necessary for studies in informatics. Topics include discrete math, analytic geometry, calculus and linear algebra. (ASU)

CSE 259: Logic in Computer Science (3) math

Logic has been called the calculus of computer science. The argument is that logic plays a fundamental role in computer science, similar to that played by calculus in physical sciences and other engineering disciplines. Indeed, logic plays an important role in computer architecture (Boolean logic, digital gates, hardware verification); software engineering (specification, verification); programming languages (semantics, type theory, logic programming); databases (relational algebra, database query language); artificial intelligence (automated theorem proving, knowledge representation); algorithms and theory of computation (complexity, computability, expressiveness). This course is a mathematically solid introduction to propositional logic, first order logic, logic programming, and their applications in computer science. (ASU)

MAT 210: Brief Calculus (3) math

Differential and integral calculus of elementary functions with applications. (ASU)

STP 226: Elements of Statistics (3) math

Basic concepts and methods of statistics, including descriptive statistics, significance tests, estimation, sampling, and correlation. (ASU)

STP 231: Statistics for Life Science (3) math

Concepts and methods of statistics; display and summary of data, interval estimation, hypothesis testing, correlation, regression. Applications to biological sciences. (ASU)

MAT 242: Elementary Linear Algebra (2) math

Introduces matrices, systems of linear equations, determinants, vector spaces, linear transformations, and eigenvalues. Emphasizes development of computational skills. (ASU)

MAT 243: Discrete Mathematical Structures (3) math

Logic, sets, functions, elementary number theory and combinatorics, recursive algorithms, and mathematical reasoning, including induction. Emphasizes connections to computer science. (ASU)

MAT 265: Calculus for Engineers I (3) math

Limits and continuity, differential calculus of functions of one variable, introduction to integration. (ASU)

MAT 266: Calculus for Engineers II (3) math

Methods of integration, applications of calculus, elements of analytic geometry, improper integrals, Taylor series. (ASU)

MAT 267: Calculus for Engineers III (3) math

Vector-valued functions of several variables, partial derivatives, multiple integration. (ASU)

MAT 343: Applied Linear Algebra (3) math

Solving linear systems, matrices, determinants, vector spaces, bases, linear transformations, eigenvectors, norms, inner products, decompositions, applications. Problem solving using MATLAB. (ASU)

STP 420: Introductory Applied Statistics (3) math

Introductory probability, descriptive statistics, sampling distributions, parameter estimation, tests of hypotheses, chi-square tests, regression analysis, analysis of variance, and nonparametric tests. (ASU)

IEE 380: Probability and Statistics for Engineering Problem Solving (3) math

Applications-oriented course with computer-based experience using statistical software for formulating and solving engineering problems. (ASU)

COMP 3240: Discrete Structures (3) math

Characterization of computer science data structures and algorithms in terms of sets and relations, functions, recurrence relations. Use of propositional and predicate calculus to describe algorithms. Proving correctness and running time bounds for algorithms by induction and structural induction. (Auburn)

MATH 1610: Calculus I (4) math

Limits, the derivative of algebraic, trigonometric, exponential, logarithmic functions. Applications of the derivative, antiderivatives, the definite integral and applications to area problems, the fundamental theorem of calculus. (Auburn)

MATH 1620: Calculus II (4) math

Techniques of integration, applications of the integral, parametric equations, polar coordinates. Vectors, lines and planes in space. Infinite sequences and series. Students may receive credit for only one of MATH 1620 or MATH 1627. (Auburn)

MATH 2660: Topics In Linear Algebra (3) math

Matrices, row-reduction, systems of linear equations, (finite-dimensional) vector spaces, subspaces, bases, dimension, change of basis, linear transformations, kernels, orthogonality, Gram-Schmidt. (Auburn)

STAT 3010: Statistics For Engineers And Scientists (3) math

Introduction to statistical methods and analysis used in engineering and science. (Auburn)

CSCI 3030: Mathematical Structures for Computer Science (3) math

The course prepares computer science majors for advanced study by emphasizing components of discrete mathematics related to computer science. The topics include sets, functions and relations, logic, Boolean algebra, graph theory, proof techniques and matrices. Examples will emphasize computer science applications. (Augusta)

MATH 1111: College Algebra (3) math

A symbolically intensive functional approach to algebra that incorporates the use of appropriate technology. Emphasis will be placed on the study of functions and their graphs, inequalities, and linear, quadratic, piece-wise defined, rational, polynomial, exponential, and logarithmic functions. Appropriate applications will be included. (Augusta)

MATH 1113: Precalculus Mathematics (3) math

A rigorous study of polynomial, exponential, logarithmic, and trigonometric functions, primarily intended to prepare science and mathematics majors for calculus. (Augusta)

MATH 1401: Elementary Statistics (3) math

A study of frequency distributions of data, graphical and numerical presentations of data, probability, discrete and continuous distributions, sampling distributions, estimation, hypothesis testing, simple linear regression and correlation and goodness of fit. (Augusta)

MATH 2011: Calculus and Analytical Geometry I (4) math

An introduction to calculus including limits and continuity, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential, and logarithmic functions, applications of derivatives, and basic integration. (Augusta)

MATH 2011H: Honors: Calculus and Analytical Geometry I (4) math

An introduction to calculus including limits and continuity, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential, and logarithmic functions, applications of derivatives, and basic integration. This is an Honors Course. (Augusta)

MATH 2012: Calculus and Analytical Geometry II (4) math

A continuation of calculus including applications of integration, techniques of integration, improper integrals, sequences, series, and polar coordinates. (Augusta)

MATH 2020: Introduction to Discrete Mathematics (3) math

Introduction to fundamental topics in discrete mathematics. Topics include: sets, functions, elementary number theory, applications to cryptography, basic counting techniques, applications of graphs and relations, and Boolean Algebra. (Augusta)

MATH 3020: Differential Equations (3) math

A study of first-order and linear second-order differential equations with applications. Topics include solution techniques, qualitative behavior, numerical methods, Laplace transformations, and the use of series. (Augusta)

MATH 3210: Math for Business and Economics (3) math

A description of the applications of linear models, simple non-linear models, applied probability, and selected topics from calculus. Additional topics may include a discussion of quadratic models, conditional probability, Bayes’ Theorem, and Markov Chains. (Augusta)

MATH 3250: Introduction to Statistics and Data Analysis (3) math

This course interweaves traditional topics in statistics with elements of data analysis using popular statistical software packages. Topics include descriptive statistics, probability distributions, sampling distributions, statistical inference for means and proportions, categorical analysis, and simple regression, including multiple and non-linear regression. (Augusta)

MATH 3280: Linear Algebra (3) math

A study of vector spaces including finite-dimensional vector spaces, linear transformations, matrices, linear equations and determinants, and eigenvalues. (Augusta)

MATH 3710: Combinatorics (3) math

A first course in enumeration. Topics include permutations and combinations of finite sets and multisets, properties of the binomial coefficients, the inclusion-exclusion formula, recurrences, generating functions, the Fibonacci sequence, and applications of Burnside’s lemma. (Augusta)

MATH 4211: Modern Abstract Algebra I (3) math

A study of abstract algebraic structure. Topics include groups, subgroups, permutation groups, homomorphisms, and quotient groups. (Augusta)

MATH 4310: Modern Geometry (3) math

A modern treatment of geometry primarily from the metric approach, but with some reference to the Euclidean Synthetic approach. Topics include parallelism, similarity, area, constructions, non-Euclidean and finite geometries. (Augusta)

MATH 4320: Theory of Numbers (3) math

A study of the positive integers including divisibility, prime numbers and the theory of congruences. Additional topics may include Fermat’s theorem, the law of quadratic reciprocity, and perfect numbers. (Augusta)

MATH 4350: Numerical Analysis (3) math

A study of non-linear equations, numerical integration and differentiation and numerical solution of initial value problems in ordinary differential equations. (Augusta)

MATH 4420: Introduction to the Theory of Graphs (3) math

A study of graphs, subgraphs, paths, arcs, trees, circuits, digraphs, colorability. (Augusta)

MATH 4510: Complex Variables (3) math

A study of the field of complex numbers, elementary functions of a complex variable, limits, derivatives, analytic functions, mapping by elementary functions, integrals, power series, residues and poles. (Augusta)

CSI 2350: Discrete Structures (3) math

An introduction to the foundations of discrete structures as they apply to computer science, focusing on providing a solid theoretical foundation for further work. Topics include sets, ordered structures, graph and trees, functions, proof techniques, number systems, logic, Boolean algebra, etc. (Baylor)

CSI 3324: Numerical Methods (3) math

Numerical differentiation and integration, linear systems of equations, numerical solutions of ordinary differential equations, curve fitting, and computational techniques. (Baylor)

CSI 4322: Numerical Analysis (3) math

Numerical evaluation of derivatives and integrals, solution of algebraic and differential equations, and approximation theory. (Baylor)

CSI 4328: Numerical Linear Algebra (3) math

Numerical methods for solution of linear equations, eigenvalue problems, and least squares problems, including sparse matrix techniques with applications to partial equations. (Baylor)

MTH 1321: Calculus I (3) math

Differential calculus of a single variable. Introduction to the definite integral and the Fundamental Theorem of Calculus. (Baylor)

MTH 1322: Calculus II (3) math

Integral calculus of a single variable, differential equations, slope fields, and power series. (Baylor)

MTH 2311: Linear Algebra (3) math

Vectors, matrix operations, linear transformations, fundamental properties of vector spaces, systems of linear equations, eigenvalues, and eigenvectors. (Baylor)

MTH 2321: Calculus III (3) math

Differential and integral calculus of several variables, Green's Theorem. (Baylor)

STA 2381: Introductory Statistical Methods (3) math

Parametric statistical methods. Topics range from descriptive statistics through regression and one-way analysis of variance. Applications are typically from biology and medicine. Computer data analysis is required. (Baylor)

MTH 3312: Foundations of Combinatorics and Algebra (3) math

Elementary counting principles, fundamental properties of the integers, the ring of integers modulo n, rings of polynomials, and an introduction to groups, rings and fields. (Baylor)

MTH 3370: Mathematical Methods of Operations Research (3) math

A survey of models and methods used in operations research. Topics include linear programming, dynamic programming, and game theory, with emphasis on the construction of mathematical models for problems arising in a variety of applied areas and an introduction to basic solution techniques. (Baylor)

STA 3381: Probability and Statistics (3) math

Introduction to the fundamentals of probability, random variables, discrete and continuous probability distributions, expectations, sampling distributions, topics of statistical inference such as confidence intervals, tests of hypotheses, and regression. (Baylor)

MTH 4312: Cryptology (3) math

Introduction to cryptology, the study of select codes and ciphers. Included is a historical context, a survey of modern crypto systems, and an exposition of the role of mathematical topics such as number theory and elliptic curves in the subject. Mathematical software will be available. (Baylor)

MATH 224: Differential Calculus (2) math

This is a 2-credit course in differential calculus covering limits, continuity, and differentiation. (Binghamton)

MATH 225: Integral Calculus (2) math

This is a 2-credit course in integral calculus covering optimization and integration. (Binghamton)

MATH 226: Integration Tech & Application (2) math

This is a 2-credit course covering the calculus of transcendental & inverse functions, L’Hospital’s Rule, integral techniques, improper integrals, calculus of parametric curves, and polar coordinates. (Binghamton)

MATH 227: Infinite Series (2) math

This is a 2-credit course covering sequences, series, power series, and Taylor series. (Binghamton)

MATH 304: Linear Algebra (4) math

Vector spaces, linear transformations, determinants, characteristic values, inner products. (Binghamton)

MATH 314: Discrete Mathematics (4) math

Logic, sets, relations, functions, induction, recursion, counting methods, graphs, trees. Some abstract algebra. (Binghamton)

MATH 323: Calculus III (4) math

Calculus of functions of several variables. Every semester. (Binghamton)

MATH 327: Probability with Stat Methods (4) math

Development of probabilistic concepts in discrete and absolutely continuous cases. Classical combinatorial methods, independence, random variables, distributions, moments, transformations, conditioning, confidence intervals, estimation. Open to Watson School students only. Does not serve as a prerequisite for MATH 448 or for any actuarial science courses. (Binghamton)

MATH 330: Number Systems (4) math

Careful discussion of the real numbers, the rational numbers and the integers, including a thorough study of induction and recursion. Countable and uncountable sets. The methodology of mathematics: basic logic, the use of quantifiers, equivalence relations, sets and functions. Methods of proof in mathematics. Training in how to discover and write proofs. (Binghamton)

MATH 371: Ordinary Diff. Equations (4) math

Ordinary differential equations from quantitative and qualitative point of view including existence and uniqueness theory, first and second order equations and higher order equations, systems of first ­order equations, Laplace transforms, series solutions methods. MATH 371 contains the topics of MATH 324 and includes additional topics of the theory of existence and uniqueness, and systems of linear equations. The topics are studied from a more advanced mathematical viewpoint than in MATH 324.. (Binghamton)

MATH 381: Graph Theory (4) math

Directed and undirected graphs, trees, connectivity, Eulerian and Hamiltonian graphs, planar graphs, coloring of graphs, graph parameters, optimization and graph algorithms. Spring only. (Binghamton)

MATH 386: Combinatorics (4) math

Topics from among counting techniques, generating function and recurrence relations, pigeonhole principle, Ramsey's Theorem, Latin squares, combinatorial designs. Fall only. (Binghamton)

MATH 407: Intro to the Theory of Numbers (4) math

Classical number theory. Divisibility, prime numbers, quadratic reciprocity, Diophantine equations. (Binghamton)

MATH 448: Mathematical Statistics (4) math

Estimation, confidence intervals and hypothesis testing. Introduction to linear models, categorical data and nonparametric statistics. Students who obtain B- or better in this course can apply for the VEE Mathematical Statistics credits from Society of Actuaries. (Binghamton)

CSCI2243: Logic and Computation (3) math

A course in the mathematical foundations of Computer Science, illustrated throughout with applications such as sets and functions, propositional and predicate logic, induction and recursion, basic number theory, and mathematical models of computation such as formal languages, finite state machines, and Turing machines. (Boston)

CSCI2244: Randomness and Computation (3) math

This course presents the mathematical and computational tools needed to solve problems that involve randomness. For example, an understanding of random variables allows us to efficiently generate the enormous prime numbers needed for information security, and to quantify the expected performance of a machine learning algorithm beyond a small data sample. An understanding of covariance allows high quality compression of audio and video. Topics include combinatorics and counting, random experiments and probability, random variables and distributions, computational modeling of randomness, Bayes' rule, laws of large numbers, vectors and matrices, covariance and principal axes, and Markov chains. (Boston)

MATH1102: Calculus I for Math and Science Majors (4) math

MATH 1102 is a first course in the calculus of one variable intended for Chemistry, Computer Science, Geology/Geophysics, Mathematics, and Physics majors. It is open to others who are qualified and desire a more rigorous calculus course than MATH 1100. Topics covered include the algebraic and analytic properties of the real number system, functions, limits, derivatives, and an introduction to integration. (Boston)

MATH1103: Calculus II for Math and Science Majors (4) math

MATH 1103 is a continuation of MATH 1102. Topics covered in the course include several algebraic techniques of integration, many applications of integration, and infinite sequences and series. (Boston)

MATH2202: Multivariable Calculus (4) math

Topics include vectors in two and three dimensions, analytic geometry of three dimensions, parametric curves, partial derivatives, the gradient, optimization in several variables, multiple integration with change of variables across different coordinate systems, line integrals, and Green's Theorem. (Boston)

MATH2210: Linear Algebra (3) math

This course is an introduction to the techniques of linear algebra in Euclidean space. Topics covered include matrices, determinants, systems of linear equations, vectors in n-dimensional space, complex numbers, and eigenvalues. The course is required of mathematics majors and minors, but is also suitable for students in the social sciences, natural sciences, and management. (Boston)

CS 131: Combinatoric Structures (4) math

Representation, analysis, techniques, and principles for manipulation of basic combinatoric structures used in computer science. Rigorous reasoning is emphasized. (BU)

CS 235: Algebraic Algorithms (4) math

Basic concepts and algorithms for manipulation of algebraic objects, such as residues, matrices, polynomials; and applications to various CS areas, such as cryptography and fault-tolerance. Emphasis on rigorous reasoning and analysis. (BU)

CS 237: Probability in Computing (4) math

Introduction to basic probabilistic concepts and methods used in computer science. Develops an understanding of the crucial role played by randomness in computing, both as a powerful tool and as a challenge to confront and analyze. Emphasis on rigorous reasoning, analysis, and algorithmic thinking. (BU)

CS 531: Advanced Optimization Algorithms (4) math

Optimization algorithms, highlighting the fruitful interactions between discrete and continuous. Intended audience is advanced master students and doctoral students. Topics include gradient descent algorithms, online optimization, linear and semidefinite programming, duality, network optimization, submodular optimization, approximation algorithms via continuous relaxations. (BU)

CS 583: Audio Computation (4) math

Introduction to algorithms, data structures, and applications in computer manipulation of audio signals. Topics include the physical properties of sound and of musical instruments, representation and synthesis of musical and environmental sounds, analysis of audio signals using the Fourier Transform, and topics of current interest in research, including the use of deep learning for analysis of audio signals. (BU)

MA 123: Calculus I (4) math

Limits; derivatives; differentiation of algebraic and transcendental functions. Applications to maxima, minima, and convexity of functions. The definite integral; the fundamental theorem of integral calculus. Carries MCS divisional credit in CAS. (BU)

MA 124: Calculus II (4) math

Logarithmic, exponential, and trigonometric functions. Sequences and series; Taylor's series with the remainder. Methods of integration. Calculus I and II together constitute an introduction to calculus of a function of a single real variable. Carries MCS divisional credit in CAS. (BU)

MA 213: Basic Statistics and Probability (4) math

Elementary treatment of probability densities, means, variances, correlation, independence, the central limit theorem, confidence intervals, and p-values. Students will be able to answer questions such as how can a pollster use a sample to predict the uncertainty of an election? Carries MCS divisional credit in CAS. (BU)

MA 214: Applied Statistics (4) math

Inference about proportions, goodness of fit, student's t-distribution, tests for normality; two-sample comparisons, regression and correlation, tests for linearity and outliers, residual analysis, contingency tables, analysis of variance. Carries MCS divisional credit in CAS. (BU)

MA 225: Multivariate Calculus (4) math

Vectors, lines, planes. Multiple integration, cylindrical and spherical coordinates. Partial derivatives, directional derivatives, scalar and vector fields, the gradient, potentials, approximation, multivariate minimization, Stokes's and related theorems. (Cannot be taken for credit in addition to MA 230.) (BU)

MA 242: Linear Algebra (4) math

Matrix algebra, solution of linear systems, determinants, Gaussian elimination, fundamental theory, row-echelon form. Vector spaces, bases, norms. Computer methods. Eigenvalues and eigenvectors, canonical decomposition. Applications. (BU)

MA 293: Discrete Mathematics (4) math

Propositional logic, set theory. Elementary probability theory. Number theory. Combinatorics with applications. (BU)

MA 294: Applied Abstract Algebra (4) math

Abstract algebra and its applications to combinatorics. A first exposure to groups, rings, and fields via significant combinatorial applications. (BU)

MA 531: Mathematical Logic (4) math

The investigation of logical reasoning with mathematical methods. The syntax and semantics of sentential logic and quantificational logic. The unifying Godel Completeness Theorem, and models of theories. A look at the Godel Incompleteness Theorem and its ramifications. (BU)

MA 532: Foundations of Mathematics (4) math

Axiomatic set theory as a foundation for mathematics and as a field of mathematics: Axiom of Choice, the Continuum Hypothesis, and consistency results. Also offered as CAS PH 461. (BU)

MA 541: Modern Algebra I (4) math

Basic properties of groups, Sylow theorems, basic properties of rings and ideals, Euclidean rings, polynomial rings. (BU)

MA 542: Modern Algebra II (4) math

Vector spaces and modules, Galois theory, linear transformations and matrices, canonical forms, bilinear and quadratic forms. (BU)

MA 555: Numerical Analysis I (4) math

Numerical solutions of equations, iterative methods, analysis of sequences. Theory of interpolation and functional approximation, divided differences. Numerical differentiation and integration. Polynomial theory. Ordinary differential equations. (BU)

MA 556: Numerical Analysis II (4) math

Numerical linear algebra; norms, elimination methods, error analysis, conditioning, eigenvalues, iterative methods, least squares and nonlinear functional minimization. Partial differentiation equation boundary value and initial value problems. Finite element methods. Legendre and Chebyshev polynomials. Treatment in greater depth of selected topics from MA 555. (BU)

MA 569: Optimization Methods of Operations Research (4) math

Optimization of linear functions: linear programming, simplex method; transportation, assignment, and network problems. Optimization of non-linear functions: unconstrained optima, constrained optima and Lagrange multipliers, Kuhn-Tucker conditions, calculus of variations, and Euler's equation. (BU)

MA 570: Stochastic Methods of Operations Research (4) math

Poisson processes, Markov chains, queuing theory. Matrix differential equations, differential-difference equations, probability-generating functions, single- and multiple-channel queues, steady-state and transient distributions. (BU)

MA 575: Linear Models (0) math

Post-introductory course on linear models. Topics to be covered include simple and multiple linear regression, regression with polynomials or factors, analysis of variance, weighted and generalized least squares, transformations, regression diagnostics, variable selection, and extensions of linear models. (BU)

MA 581: Probability (4) math

Basic probability, conditional probability, independence. Discrete and continuous random variables, mean and variance, functions of random variables, moment generating function. Jointly distributed random variables, conditional distributions, independent random variables. Methods of transformations, law of large numbers, central limit theorem. Cannot be taken for credit in addition to MA 381. (BU)

MA 582: Mathematical Statistics (4) math

Point estimation including unbiasedness, efficiency, consistency, sufficiency, minimum variance unbiased estimator, Rao-Blackwell theorem, and Rao-Cramer inequality. Maximum likelihood and method of moment estimations; interval estimation; tests of hypothesis, uniformly most powerful tests, uniformly most powerful unbiased tests, likelihood ratio test, and chi-square test. (BU)

MA 583: Introduction to Stochastic Processes (4) math

Basic concepts and techniques of stochastic process as they are most often used to construct models for a variety of problems of practical interest. Topics include Markov chains, Poisson process, birth and death processes, queuing theory, renewal processes, and reliability. (BU)

MA 589: Computational Statistics (4) math

Topics from computational statistics that are relevant to modern statistical applications: random number generation, sampling, Monte Carlo methods, computational inference, MCMC methods, graphical models, data partitioning, and bootstrapping. Emphasis on developing solid conceptual understanding of the methods through applications. (BU)

COSI 29a: Discrete Structures (4) math

Covers topics in discrete mathematics with applications within computer science. Some of the topics to be covered include graphs and matrices; principles of logic and induction; number theory; counting, summation, and recurrence relations; discrete probability. Usually offered every year. (Brandeis)

MATH 8a: Introduction to Probability and Statistics (4) math

Discrete probability spaces, random variables, expectation, variance, approximation by the normal curve, sample mean and variance, and confidence intervals. Does not require calculus; only high school algebra and graphing of functions. Usually offered every semester. (Brandeis)

MATH 10a: Techniques of Calculus (a) (4) math

Introduction to differential (and some integral) calculus of one variable, with emphasis on techniques and applications. Usually offered every semester in multiple sections. (Brandeis)

MATH 15a: Linear Algebra (4) math

Matrices, determinants, linear equations, vector spaces, eigenvalues, quadratic forms, linear programming. Emphasis on techniques and applications. Usually offered every semester. (Brandeis)

MATH 22a: Honors Linear Algebra (4) math

MATH 22a covers linear algebra. The material is similar to MATH 15a but with some additional content, a more theoretical emphasis, and more attention to proofs. Usually offered every fall. (Brandeis)

MATH 122a: Numerical Methods and Big Data (4) math

Introduces fundamental techniques of numerical linear algebra widely used for data science and scientific computing. The purpose of this course is to introduce methods that are useful in applications and research. Usually offered every year. (Brandeis)

MATH 124a: Optimization (4) math

Explores the theory of mathematical optimization and its fundamental algorithms, emphasizing problems arising in machine learning, economics, and operations research. Topics include linear and integer programming, convex analysis, and duality. Usually offered every second year. (Brandeis)

MATH 125a: Mathematics for Machine Learning (4) math

Serves as a first course in machine learning and general data science, with a focus on the mathematics underlying the various modern machine learning algorithms. The course covers the fundamental concepts of statistical distribution, information theory, statistical learning, optimization and matrix factorizations, as well as classic algorithms such as tree methods, kernel methods and various neural network models. A few important real world examples of current interest will be considered such as computer vision, natural language processing, search engine, recommendation systems, finance, and biology. Usually offered every second year. (Brandeis)

CSCI 0530: Coding the Matrix: an Introduction to Linear Algebra for Computer Science (1) math

An introduction to vectors and matrices and their role in computer science. The course material consists of three components: (1) concepts, theorems, and proofs, (2) procedures and programs, and (3) applications and working with data. The course revolves around weekly lab sessions in each of which students apply the concepts to a real task with real data. Lab topics include transformations in 2-d graphics, error-correcting codes, image compression using wavelets, synthesizing a new perspective in a photo, face recognition, news story categorization, cancer diagnosis using machine learning, matching airplanes to destinations, Google's PageRank method. Other topics addressed in the course include linear programming, zero-sum games, rudimentary cryptographic methods, linear regression, and discrete linear dynamical systems such as a spreading computer virus. (Brown)

CSCI 1440: Algorithmic Game Theory (1) math

This course examines topics in game theory from a computer scientist's perspective. Through the lens of computation, this course will focus on the design and analysis of systems involving self-interested agents, investigating how strategic behavior should influence algorithm design, which game-theoretic solution concepts are practical to implement, and the ramifications of conflicts of interest between system designers and participating agents. Topics include: auctions and mechanism design, equilibria, and learning. (Brown)

CSCI 1450: Advanced Introduction to Probability for Computing and Data Science (1) math

Probability and statistics have become indispensable tools in computer science. Probabilistic methods and statistical reasoning play major roles in machine learning, cryptography, network security, communication protocols, web search engines, robotics, program verification, and more. This course introduces the basic concepts of probability and statistics, focusing on topics that are most useful in computer science applications. Topics include: modeling and solution in sample space, random variables, simple random processes and their probability distributions, Markov processes, limit theorems, and basic elements of Bayesian and frequentist statistical inference. (Brown)

CSCI 1510: Introduction to Cryptography and Computer Security (1) math

This course studies the tools for guaranteeing safe communication and computation in an adversarial setting. We develop notions of security and give provably secure constructions for such cryptographic objects as cryptosystems, signature schemes and pseudorandom generators. We also review the principles for secure system design. (Brown)

MATH 0090: Single Variable Calculus, Part I () math

An intensive course in the calculus of one variable including limits; differentiation; maxima and inima, and the chain rule for polynomials, rational functions, trigonometric functions, and exponential functions. Introduction of integration with applications to area and volumes of revolution. Mathematics 0090 and 0100 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences. (Brown)

MATH 0100: Single Variable Calculus, Part II (1) math

A continuation of the material of MATH 90 including further development of techniques of integration. Other topics covered are infinite series, power series, Taylor's formula, polar coordinates, parametric equations, introduction to differential equations, and numerical methods. MATH 90 and 100 or the equivalent are recommended for all students intending to concentrate in mathematics or the sciences. MATH 100 may not be taken in addition to MATH 170 or MATH 190. (Brown)

MATH 0170: Single Variable Calculus, Part II (Accelerated) (1) math

This course, which covers roughly the same material and has the same prerequisites as MATH 100, covers integration techniques, sequences and series, parametric and polar curves, and differential equations of first and second order. Topics will generally include more depth and detail than in MATH 100. MATH 170 may not be taken in addition to MATH 100 or MATH 190. (Brown)

MATH 0180: Multivariable Calculus (1) math

Three-dimensional analytic geometry. Differential and integral calculus of functions of two or three variables: partial derivatives, multiple integrals, Green's Theorem, Stokes' theorem, and the divergence theorem. (Brown)

MATH 0190: Single Variable Calculus, Part II (Physics/Engineering) (1) math

This course, which covers roughly the same material and has the same prerequisites as MATH 100, is intended for students with a special interest in physics or engineering. The main topics are: calculus of vectors and paths in two and three dimensions; differential equations of the first and second order; and infinite series, including power series. (Brown)

MATH 0200: Multivariable Calculus (Physics/Engineering) (1) math

This course, which covers roughly the same material as MATH 180, is intended for students with a special interest in physics or engineering. The main topics are: geometry of three-dimensional space; partial derivatives; Lagrange multipliers; double, surface, and triple integrals; vector analysis; Stokes' theorem and the divergence theorem, with applications to electrostatics and fluid flow. (Brown)

APMA 0350: Applied Ordinary Differential Equations (1) math

This course provides a comprehensive introduction to ordinary differential equations and their applications. During the course, we will see how applied mathematicians use ordinary differential equations to solve practical applications, from understanding the underlying problem, creating a differential-equations model, solving the model using analytical, numerical, or qualitative methods, and interpreting the findings in terms of the original problem. We will also learn about the underlying rigorous theoretical foundations of differential equations. (Brown)

MATH 0350: Multivariable Calculus With Theory (1) math

This course provides a rigorous treatment of multivariable calculus. Topics covered include vector analysis, partial differentiation, multiple integration, line integrals, Green's theorem, Stokes' theorem, and the divergence theorem. MATH 0350 covers the same material as MATH 0180, but with more emphasis on theory and on understanding proofs. (Brown)

APMA 0360: Applied Partial Differential Equations I (1) math

This course provides an introduction to partial differential equations and their applications. We will learn how to use partial differential equations to solve problems that arise in practical applications, formulating questions about a real-world problem, creating a partial differential equation model that can help answer these questions, solving the resulting system using analytical, numerical, and qualitative methods, and interpreting the results in terms of the original application. To help us support and justify our approaches and solutions, we will also learn about theoretical foundations of partial differential equations. (Brown)

MATH 0520: Linear Algebra (1) math

A first course in linear algebra designed to develop students' problem solving skills, mathematical writing skills, and gain facility with the applications and theory of linear algebra. (Brown)

MATH 0540: Linear Algebra With Theory (1) math

This course provides a rigorous introduction to the theory of linear algebra. Topics covered include: matrices, linear equations, determinants, and eigenvalues; vector spaces and linear transformations; inner products; Hermitian, orthogonal, and unitary matrices; and Jordan normal form. (Brown)

APMA 1160: An Introduction to Numerical Optimization (01) math

This course provides a thorough introduction to numerical methods and algorithms for solving non-linear continuous optimization problems. A particular attention will be given to the mathematical underpinnings to understand the theoretical properties of the optimization problems and the algorithms designed to solve them. Topics will include: line search methods, trust-region methods, nonlinear conjugate gradient methods, an introduction to constrained optimization (Karush-Kuhn-Tucker conditions, mini-maximization, saddle-points of Lagrangians). Some applications in signal and image processing will be explored. (Brown)

APMA 1170: Introduction to Computational Linear Algebra (1) math

Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods. (Brown)

APMA 1180: Introduction to Numerical Solution of Differential Equations (1) math

Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of multi-step and multi-stage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Students will be required to use Matlab (or other computer languages) to implement the mathematical algorithms under consideration: experience with a programming language is therefore strongly recommended. (Brown)

MATH 1210: Probability (1) math

Basic probability theory including random variables, distribution functions, independence, expectation, variance, and conditional expectation. Classical examples of probability density and mass functions (binomial, geometric, normal, exponential) and their applications. Stochastic processes including discrete and continuous time Poisson processes, Markov chains, and Brownian motion. (Brown)

MATH 1530: Abstract Algebra (1) math

A proof-based course that introduces the principles and concepts of modern abstract algebra. Topics will include groups, rings, and fields, with applications to number theory, the theory of equations, and geometry. Previous proof-writing experience is not required. MATH 1530 is required of all students concentrating in mathematics. (Brown)

MATH 1610: Probability (1) math

Basic probability theory including random variables, distribution functions, independence, expectation, variance, and conditional expectation. Classical examples of probability density and mass functions (binomial, geometric, normal, exponential) and their applications. Stochastic processes including discrete and continuous time Poisson processes, Markov chains, and Brownian motion. (Brown)

APMA 1650: Statistical Inference I (1) math

APMA 1650 is an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing. (Brown)

APMA 1655: Honors Statistical Inference I (1) math

Students may opt to enroll in APMA 1655 for more in depth coverage of APMA 1650. Enrollment in 1655 will include an optional recitation section and required additional individual work. Applied Math concentrators are encouraged to take 1655. (Brown)

APMA 1690: Computational Probability and Statistics (1) math

Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, dimensionality reduction. (Brown)

APMA 1740: Recent Applications of Probability and Statistics (1) math

This course develops the mathematical foundations of modern applications of statistics to the computational, cognitive, engineering, and neural sciences. The course is rigorous, but the emphasis is on application. Topics include: Gibbs ensembles and their relation to maximum entropy, large deviations, exponential models, and information theory; statistical estimation and classification; graphical models, dynamic programming, MCMC, parameter estimation, and the EM algorithm. (Brown)

CS 6 a: Introduction to Discrete Mathematics (3) math

First term: a survey emphasizing graph theory, algorithms, and applications of algebraic structures. Graphs: paths, trees, circuits, breadth-first and depth-first searches, colorings, matchings. Enumeration techniques; formal power series; combinatorial interpretations. Topics from coding and cryptography, including Hamming codes and RSA. (Caltech)

CS 13: Mathematical Foundations of Computer Science (9) math

This course introduces key mathematical concepts used in computer science, and in particular it prepares students for proof-based CS courses such as CS 21 and CS 38. Mathematical topics are illustrated via applications in Computer Science. CS 1 is a co-requisite as there will be a small number of programming assignments. The course covers basic set theory, induction and inductive structures (e.g., lists and trees), asymptotic analysis, and elementary combinatorics, number theory, and graph theory. Applications include number representation, basic cryptography, basic algorithms on trees, numbers, and polynomials, social network graphs, compression, and simple error-correcting codes. (Caltech)

CS 112: Bayesian Statistics (36) math

This course provides an introduction to Bayesian Statistics and its applications to data analysis in various fields. (Caltech)

CS 120: Quantum Cryptography (9) math

This course is an introduction to quantum cryptography: how to use quantum effects, such as quantum entanglement and uncertainty, to implement cryptographic tasks with levels of security that are impossible to achieve classically. The course covers the fundamental ideas of quantum information that form the basis for quantum cryptography, such as entanglement and quantifying quantum knowledge. We will introduce the security definition for quantum key distribution and see protocols and proofs of security for this task. We will also discuss the basics of device-independent quantum cryptography as well as other cryptographic tasks and protocols, such as bit commitment or position-based cryptography. Not offered 2023-24. (Caltech)

CS 126 ab: Information Theory (9) math

Shannon's mathematical theory of communication, 1948-present. Entropy, relative entropy, and mutual information for discrete and continuous random variables. Shannon's source and channel coding theorems. Mathematical models for information sources and communication channels, including memoryless, Markov, ergodic, and Gaussian. Calculation of capacity and rate-distortion functions. Universal source codes. Side information in source coding and communications. Network information theory, including multiuser data compression, multiple access channels, broadcast channels, and multiterminal networks. Discussion of philosophical and practical implications of the theory. This course, when combined with EE 112, CS 127, CS 161, and CS 167, should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression. (Caltech)

CS 127: Error-Correcting Codes (36) math

This course develops from first principles the theory and practical implementation of the most important techniques for combating errors in digital transmission and storage systems. Topics include highly symmetric linear codes, such as Hamming, Reed-Muller, and Polar codes; algebraic block codes, such as Reed-Solomon and BCH codes, including a self-contained introduction to the theory of finite fields; and low-density parity-check codes. Students will become acquainted with encoding and decoding algorithms, design principles and performance evaluation of codes. (Caltech)

CS 136: Information Theory and Applications (9) math

This class introduces information measures such as entropy, information divergence, mutual information, information density, and establishes the fundamental importance of those measures in data compression, statistical inference, and error control. The course does not require a prior exposure to information theory; it is complementary to EE 126a. Not offered 2023-24. (Caltech)

CS 157: Statistical Inference (9) math

Statistical Inference is a branch of mathematical engineering that studies ways of extracting reliable information from limited data for learning, prediction, and decision making in the presence of uncertainty. This is an introductory course on statistical inference. The main goals are: develop statistical thinking and intuitive feel for the subject; introduce the most fundamental ideas, concepts, and methods of statistical inference; and explain how and why they work, and when they don't. Topics covered include summarizing data, fundamentals of survey sampling, statistical functionals, jackknife, bootstrap, methods of moments and maximum likelihood, hypothesis testing, p-values, the Wald, Student's t-, permutation, and likelihood ratio tests, multiple testing, scatterplots, simple linear regression, ordinary least squares, interval estimation, prediction, graphical residual analysis. (Caltech)

CS 160: Fundamentals of Information Transmission and Storage (9) math

Basics of information theory: entropy, mutual information, source and channel coding theorems. Basics of coding theory: error-correcting codes for information transmission and storage, block codes, algebraic codes, sparse graph codes. Basics of digital communications: sampling, quantization, digital modulation, matched filters, equalization. (Caltech)

CS 177 ab: Discrete Differential Geometry: Theory and Applications (9) math

Working knowledge of multivariate calculus and linear algebra as well as fluency in some implementation language is expected. Subject matter covered: differential geometry of curves and surfaces, classical exterior calculus, discrete exterior calculus, sampling and reconstruction of differential forms, low dimensional algebraic and computational topology, Morse theory, Noether's theorem, Helmholtz-Hodge decomposition, structure preserving time integration, connections and their curvatures on complex line bundles. Applications include elastica and rods, surface parameterization, conformal surface deformations, computation of geodesics, tangent vector field design, connections, discrete thin shells, fluids, electromagnetism, and elasticity. Part b not offered 2023-24. (Caltech)

CS 178: Numerical Algorithms and their Implementation (9) math

This course gives students the understanding necessary to choose and implement basic numerical algorithms as needed in everyday programming practice. Concepts include: sources of numerical error, stability, convergence, ill-conditioning, and efficiency. Algorithms covered include solution of linear systems (direct and iterative methods), orthogonalization, SVD, interpolation and approximation, numerical integration, solution of ODEs and PDEs, transform methods (Fourier, Wavelet), and low rank approximation such as multipole expansions. Not offered 2023-24. (Caltech)

Ma 1 abc: Calculus of One and Several Variables and Linear Algebra (9) math

Special section of Ma 1 a, 12 units (5-0-7). Review of calculus. Complex numbers, Taylor polynomials, infinite series. Comprehensive presentation of linear algebra. Derivatives of vector functions, multiple integrals, line and path integrals, theorems of Green and Stokes. Ma 1 b, c is divided into two tracks: analytic and practical. Students will be given information helping them to choose a track at the end of the fall term. (Caltech)

Ma 2: Differential Equations (9) math

The course is aimed at providing an introduction to the theory of ordinary differential equations, with a particular emphasis on equations with well known applications ranging from physics to population dynamics. The material covered includes some existence and uniqueness results, first order linear equations and systems, exact equations, linear equations with constant coefficients, series solutions, regular singular equations, Laplace transform, and methods for the study of nonlinear equations (equilibria, stability, predator-prey equations, periodic solutions and limiting cycles). (Caltech)

Ma 3: Introduction to Probability and Statistics (9) math

This course is an introduction to the main ideas of probability and statistics. The first half is devoted to the fundamental concepts of probability theory, including basic combinatorics, random variables, independence, conditional probability, and the central limit theorem. The second half is devoted to statistical reasoning, including methods for the collection, organization, analysis, and interpretation of data. Topics covered will include parameter estimation, hypothesis testing, confidence intervals, Bayesian inference, and linear regression. The course will emphasize the application of statistics to engineering and the sciences. (Caltech)

Ma 121 a: Combinatorial Analysis (9) math

First half of: A survey of modern combinatorial mathematics, starting with an introduction to graph theory and extremal problems. Flows in networks with combinatorial applications. Counting, recursion, and generating functions. Theory of partitions. (0, 1)-matrices. Partially ordered sets. Latin squares, finite geometries, combinatorial designs, and codes. Algebraic graph theory, graph embedding, and coloring. (Caltech)

CS 202: Mathematics of Computer Science (6) math

This course introduces some of the formal tools of computer science, using a variety of applications as a vehicle. You’ll learn how to encode data so that when you scratch the back of a DVD, it still plays just fine; how to distribute “shares” of your floor’s PIN so that any five of you can withdraw money from the floor bank account (but no four of you can); how to play chess; and more. Topics that we’ll explore along the way include: logic and proofs, number theory, elementary complexity theory and recurrence relations, basic probability, counting techniques, and graphs. (Carleton)

CS 341: History of Computing in England Program: Cryptography (6) math

Modern cryptographic systems allow parties to communicate in a secure way, even if they don’t trust the channels over which they are communicating (or maybe even each other). Cryptography is at the heart of a huge range of applications: online banking and shopping, password-protected computer accounts, and secure wireless networks, to name just a few. In this course, we will introduce and explore some fundamental cryptographic primitives. Topics will include public-key encryption, digital signatures, code;-breaking techniques (like those used at Bletchley Park during WWII to break the Enigma machine (Carleton)

MATH 111: Introduction to Calculus (6) math

An introduction to the differential and integral calculus. Derivatives, antiderivatives, the definite integral, applications, and the fundamental theorem of calculus. (Carleton)

MATH 236: Mathematical Structures (6) math

Basic concepts and techniques used throughout mathematics. Topics include logic, mathematical induction and other methods of proof, problem solving, sets, cardinality, equivalence relations, functions and relations, and the axiom of choice. Other topics may include: algebraic structures, graph theory, and basic combinatorics. (Carleton)

15-151: Mathematical Foundations for Computer Science (12) math

*CS majors only* This course is offered to incoming Computer Science freshmen and focuses on the fundamental concepts in Mathematics that are of particular interest to Computer Science such as logic, sets,induction, functions, and combinatorics. These topics are used as a context in which students learn to formalize arguments using the methods of mathematical proof. This course uses experimentation and collaboration as ways to gain better understanding of the material. Open to CS freshmen only. NOTE: students must achieve a C or better (CMU)

16-211: Foundational Mathematics of Robotics (12) math

This course will cover core mathematics concepts used in many advanced robotics courses at the RI. Perhaps unlike prior courses in math, the focus of this class will be to ground concepts in robotics algorithms or applications. For example: How to move and manipulate objects in 3D space (coordinate transforms, rotations). How to move an articulated robots end-effector in Cartesian space (Jacobians, gradient optimization). How to have a robot learn to recognize a vision input (neural networks, back propagation). How to plan navigate a robot optimally (dynamic programming, A* Search). (CMU)

15-259: Probability and Computing (12) math

Probability theory is indispensable in computer science today. In areas such as artificial intelligence and computer science theory, probabilistic reasoning and randomization are central. Within networks and systems, probability is used to model uncertainty and queuing latency. This course gives an introduction to probability as it is used in computer science theory and practice, drawing on applications and current research developments as motivation. The course has 3 parts: Part I is an introduction to probability, including discrete and continuous random variables, heavy tails, simulation, Laplace transforms, z-transforms, and applications of generating functions. Part II is an in-depth coverage of concentration inequalities, like the Chernoff bound and SLLN bounds, as well as their use in randomized algorithms. Part III covers Markov chains (both discrete-time and continuous-time) and stochastic processes and their application to queuing systems performance modeling. This is a fast-paced class which will cover more material than the other probability options and will cover it in greater depth. (CMU)

15-260: Statistics and Computing (3) math

Statistics is essential for a wide range of fields including machine learning, artificial intelligence, bioinformatics, and finance. This mini course presents the fundamental concepts and methods in statistics in six lectures. The course covers key topics in statistical estimation, inference, and prediction. This course is only open to students enrolled in 15-259. Enrollment for 15-260, mini 4, starts around mid semester. (CMU)

15-317: Constructive Logic (9) math

This multidisciplinary junior-level course is designed to provide a thorough introduction to modern constructive logic, its roots in philosophy, its numerous applications in computer science, and its mathematical properties. Some of the topics to be covered are intuitionistic logic, inductive definitions, functional programming, type theory, realizability, connections between classical and constructive logic, decidable classes. (CMU)

15-326: Computational Microeconomics (9) math

Use of computational techniques to operationalize basic concepts from economics. Expressive marketplaces: combinatorial auctions and exchanges, winner determination problem. Game theory: normal and extensive-form games, equilibrium notions, computing equilibria. Mechanism design: auction theory, automated mechanism design. (CMU)

15-327: Monte Carlo Methods and Applications (9) math

The Monte Carlo method uses random sampling to solve computational problems that would otherwise be intractable, and enables computers to model complex systems in nature that are otherwise too difficult to simulate. This course provides a first introduction to Monte Carlo methods from complementary theoretical and applied points of view, and will include implementation of practical algorithms. Topics include random number generation, sampling, Markov chains, Monte Carlo integration, stochastic processes, and applications in computational science. Students need a basic background in probability, multivariable calculus, and some coding experience in any language. (CMU)

15-354: Computational Discrete Mathematics (12) math

This course is about the computational aspects of some of the standard concepts of discrete mathematics (relations, functions, logic, graphs, algebra, automata), with emphasis on efficient algorithms. We begin with a brief introduction to computability and computational complexity. Other topics include: iteration, orbits and fixed points, order and equivalence relations, propositional logic and satisfiability testing, finite fields and shift register sequences, finite state machines, and cellular automata. Computational support for some of the material is available in the form of a Mathematica package. (CMU)

15-355: Modern Computer Algebra (9) math

The goal of this course is to investigate the relationship between algebra and computation. The course is designed to expose students to algorithms used for symbolic computation, as well as to the concepts from modern algebra which are applied to the development of these algorithms. This course provides a hands-on introduction to many of the most important ideas used in symbolic mathematical computation, which involves solving system of polynomial equations (via Groebner bases), analytic integration, and solving linear difference equations. Throughout the course the computer algebra system Mathematica will be used for computation. (CMU)

15-356: Introduction to Cryptography (12) math

This course is aimed as an introduction to modern cryptography. This course will be a mix of applied and theoretical cryptography. We will cover popular primitives such as: pseudorandom functions, encryption, signatures, zero-knowledge proofs, multi-party computation, and Blockchains. In addition, we will cover the necessary number-theoretic background. We will cover formal definitions of security, as well as constructions based on well established assumptions like factoring. Please see the course webpage for a detailed list of topics. (CMU)

15-359: Probability and Computing (12) math

Probability theory has become indispensable in computer science. In areas such as artificial intelligence and computer science theory, probabilistic methods and ideas based on randomization are central. In other areas such as networks and systems, probability is becoming an increasingly useful framework for handling uncertainty and modeling the patterns of data that occur in complex systems. This course gives an introduction to probability as it is used in computer science theory and practice, drawing on applications and current research developments as motivation and context. Topics include combinatorial probability and random graphs, heavy tail distributions, concentration inequalities, various randomized algorithms, sampling random variables and computer simulation, and Markov chains and their many applications, from Web search engines to models of network protocols. The course will assume familiarity with 3-D calculus and linear algebra. (CMU)

21-120: Differential and Integral Calculus (10) math

Functions, limits, derivatives, logarithmic, exponential, and trigonometric functions, inverse functions; L'Hospital's Rule, curve sketching, Mean Value Theorem, related rates, linear and approximations, maximum-minimum problems, inverse functions, definite and indefinite integrals; integration by substitution and by parts. Applications of integration, as time permits. (Three 50 minute lectures, two 50 minute recitations) (CMU)

21-122: Integration and Approximation (10) math

Integration by trigonometric substitution and partial fractions; arclength; improper integrals; Simpson's and Trapezoidal Rules for numerical integration; separable differential equations, Newton's method, Euler's method, Taylor's Theorem, including a discussion of the remainder, sequences, series, power series. Parametric curves, polar coordinates, vectors, dot product. (Three 50 minute lectures, two 50 minute recitations) (CMU)

36-202: Methods for Statistics & Data Science (9) math

This course builds on the principles and methods of statistical reasoning developed in 36-200 (or its equivalents). The course covers simple and multiple regression, basic analysis of variance methods, logistic regression, and introduction to data mining including classification and clustering. Students will also learn the principles of overfitting, training vs testing, ensemble methods, variable selection, and bootstrapping. Course objectives include applying the basic principles and methods that underlie statistical practice and empirical research to real data sets and interdisciplinary problems. Learning the Data Analysis Pipeline is strongly emphasized through structured coding and data analysis projects. In addition to three lectures a week, students attend a computer lab once a week for "hands-on" practice of the material covered in lecture. There is no programming language pre-requisite. Students will learn the basics of R Markdown and related analytics tools. (CMU)

36-218: Probability Theory for Computer Scientists (9) math

Probability theory is the mathematical foundation for the study of both statistics and of random systems. This course is an intensive introduction to probability,from the foundations and mechanics to its application in statistical methods and modeling of random processes. Special topics and many examples are drawn from areas and problems that are of interest to computer scientists and that should prepare computer science students for the probabilistic and statistical ideas they encounter in downstream courses and research. A grade of C or better is required in order to use this course as a pre-requisite for 36-226, 36-326, and 36-410. If you hold a Statistics primary/additional major or minor you will be required to complete 36-226. For those who do not have a major or minor in Statistics, and receive at least a B in 36-218, you will be eligible to move directly onto 36-401. (CMU)

36-225: Introduction to Probability Theory (9) math

This course is the first half of a year-long course which provides an introduction to probability and mathematical statistics for students in the data sciences. Topics include elementary probability theory, conditional probability and independence, random variables, distribution functions, joint and conditional distributions, law of large numbers, and the central limit theorem. (CMU)

36-226: Introduction to Statistical Inference (9) math

This course is the second half of a year long course in probability and mathematical statistics. Topics include maximum likelihood estimation, confidence intervals, hypothesis testing, and properties of estimators, such as unbiasedness and consistency. If time permits there will also be a discussion of linear regression and the analysis of variance. (CMU)

21-241: Matrices and Linear Transformations (11) math

A first course in linear algebra intended for scientists, engineers, mathematicians and computer scientists. Students will be required to write some straightforward proofs. Topics to be covered: complex numbers, real and complex vectors and matrices, rowspace and columnspace of a matrix, rank and nullity, solving linear systems by row reduction of a matrix, inverse matrices and determinants, change of basis, linear transformations, inner product of vectors, orthonormal bases and the Gram-Schmidt process, eigenvectors and eigenvalues, diagonalization of a matrix, symmetric and orthogonal matrices. (CMU)

21-242: Matrix Theory (11) math

A component of the honors program, 21-242 is a more demanding version of 21-241 (Matrix Algebra and Linear Transformations), of greater scope, with increased emphasis placed on rigorous proofs. Topics to be covered: complex numbers, real and complex vectors and matrices, rowspace and columnspace of a matrix, rank and nullity, solving linear systems by row reduction of a matrix, inverse matrices and determinants, change of basis, linear transformations, inner product of vectors, orthonormal bases and the Gram-Schmidt process, eigenvectors and eigenvalues, diagonalization of a matrix, symmetric and orthogonal matrices, hermitian and unitary matrices, quadratic forms. (CMU)

21-259: Calculus in Three Dimensions (10) math

Vectors, lines, planes, quadratic surfaces, polar, cylindrical and spherical coordinates, partial derivatives, directional derivatives, gradient, divergence, curl, chain rule, maximum-minimum problems, multiple integrals, parametric surfaces and curves, line integrals, surface integrals, Green-Gauss theorems. (Three 50 minute lectures, two 50 minute recitations) (CMU)

21-266: Vector Calculus for Computer Scientists (10) math

This course is an introduction to vector calculus making use of techniques from linear algebra. Topics covered include scalar-valued and vector-valued functions, conic sections and quadric surfaces, new coordinate systems, partial derivatives, tangent planes, the Jacobian matrix, the chain rule, gradient, divergence, curl, the Hessian matrix, linear and quadratic approximation, local and global extrema, Lagrange multipliers, multiple integration, parametrised curves, line integrals, conservative vector fields, parametrised surfaces, surface integrals, Green's theorem, Stokes's theorem and Gauss's theorem. (Three 50 minute lectures, one 50 minute recitation) (CMU)

21-268: Multidimensional Calculus (11) math

A serious introduction to multidimensional calculus that makes use of matrices and linear transformations. Results will be stated carefully and rigorously. Students will be expected to write some proofs; however, some of the deeper results will be presented without proofs. Topics to be covered include functions of several variables, limits, and continuity, partial derivatives, differentiability, chain rule, inverse and implicit functions, higher derivatives, Taylor's theorem, optimization, multiple integrals and change of variables, line integrals, surface integrals, divergence theorem and Stokes's theorem. (Three 50 minute lectures, one 50 minute recitation) (CMU)

21-269: Vector Analysis (10) math

A component of the honors program, 21-269 is a more demanding version of 21-268 of greater scope, with greater emphasis placed on rigorous proofs. Topics to be covered typically include: the real field, sups, infs, and completeness; geometry and topology of metric spaces; limits, continuity, and derivatives of maps between normed spaces; inverse and implicit function theorems, higher derivatives, Taylor's theorem, extremal calculus, and Lagrange multipliers. Integration. Iterated integration and change of variables. (Three 50 minute lectures, one 50 minute recitation) (CMU)

36-315: Statistical Graphics and Visualization (9) math

Graphical displays of quantitative information take on many forms, and they help us understand data and statistical methods by (hopefully) clearly communicating arguments, results, and ideas. This course introduces students to the most common forms of graphical displays and their uses and misuses. Ideally, graphs are designed according to three key elements: The data structure, the graph's audience, and the designer's intended message. Students will learn how to create well-designed graphs and understand them from a statistical perspective. Furthermore, the course will consider complex data structures that are becoming increasingly common in data visualizations (temporal, spatial, and text data); we will discuss common ways to process these data that make them easy to visualize. As time permits, we may also consider more advanced graphical methods (e.g., interactive graphics and computer-generated animations). In addition to two weekly lectures, there will be weekly computer labs and homework assignments where students use R to visualize and analyze real datasets. Along the way, students also make monthly Piazza posts discussing the strengths and weaknesses of a graph they found online, thereby critiquing real graphical designs found in the wild. The course culminates in a group final project, where students make public-facing data visualizations and analyses for a real dataset. All assignments will be in R; although this is not a programming class, using programming-based statistical software like R is essential to create modern-day graphics, and this class will give you practice using this kind of software. Throughout, communication skills (usually written or visual, but sometimes spoken) will play an important role. Indeed, if it's true that "a picture speaks a thousand words," then ideally the one thousand words you are communicating with your graphics are statistically correct, clear, and compelling. (CMU)

21-325: Probability (9) math

This course focuses on the understanding of basic concepts in probability theory and illustrates how these concepts can be applied to develop and analyze a variety of models arising in computational biology, finance, engineering and computer science. The firm grounding in the fundamentals is aimed at providing students the flexibility to build and analyze models from diverse applications as well as preparing the interested student for advanced work in these areas. The course will cover core concepts such as probability spaces, random variables, random vectors, multivariate densities, distributions, expectations, sampling and simulation; independence, conditioning, conditional distributions and expectations; limit theorems such as the strong law of large numbers and the central limit theorem; as well as additional topics such as large deviations, random walks and Markov chains, as time permits. (Three 50 minute lectures) (CMU)

36-401: Modern Regression (9) math

This course is an introduction to the real world of statistics and data analysis using linear regression modeling. We will explore real data sets, examine various models for the data, assess the validity of their assumptions, and determine which conclusions we can make (if any). We will use the R programming language to implement our analyses and produce graphs and tables of results. Data analysis is a bit of an art; there may be several valid approaches. We will strongly emphasize the importance of critical thinking about the data and the question of interest. Our overall goal is to use data and a basic set of modeling tools to answer substantive questions, and to present the results in a scientific report. (CMU)

36-402: Advanced Methods for Data Analysis (9) math

This course introduces modern methods of data analysis, building on the theory and application of linear models from 36-401. Topics include nonlinear regression, nonparametric smoothing, density estimation, generalized linear and generalized additive models, simulation and predictive model-checking, cross-validation, bootstrap uncertainty estimation, multivariate methods including factor analysis and mixture models, and graphical models and causal inference. Students will analyze real-world data from a range of fields, coding small programs and writing reports. (CMU)

80-413: Category Theory (9) math

Category theory is a formal framework devoted to studying the structural relationships between mathematical objects. Developed in the mid-20th century to attack geometrical problems, subsequent progress has revealed deep connections to algebra and logic, as well as to mathematical physics and computer science. The course emphasizes two perspectives. On one hand, we develop the basic theory of categories, regarded as mathematical structures in their own right. At the same time, we will consider the application of these results to concrete examples from logic and algebra. Some familiarity with abstract algebra or logic required. (CMU)

ECSE 246: Signals and Systems (4) math

Mathematical representation, characterization, and analysis of continuous-time signals and systems. Development of elementary mathematical models of continuous-time dynamic systems. Time domain and frequency domain analysis of linear time-invariant systems. Fourier series, Fourier transforms, and Laplace transforms. Sampling theorem. Filter design. Introduction to feedback control systems and feedback controller design. (Case)

CSDS 302: Discrete Mathematics (3) math

A general introduction to basic mathematical terminology and the techniques of abstract mathematics in the context of discrete mathematics. Topics introduced are mathematical reasoning, Boolean connectives, deduction, mathematical induction, sets, functions and relations, algorithms, graphs, combinatorial reasoning. Offered as CSDS 302, ECSE 302 and MATH 304. (Case)

CSDS 313: Introduction to Data Analysis (3) math

This course provides a conceptual and hands-on introduction to reasoning with data. Introduction of basic statistical concepts; models vs. observations, common distributions, parameters vs. statistics, statistical inference, hypothesis testing, multiple hypotheses, confidence intervals. Use of computational approaches to address statistical problems; data representation, empirical assessment of statistical significance, assessment of the association between variables, dimensionality reduction, model building, evaluation, and validation. Data visualization and accessibility/interpretability of patterns in data and predictive models. Computational thinking and critical approaches in data science; common mistakes and issues in data analysis, causality vs. correlation, confounders, statistical artifacts, Simpson's paradox, base rate fallacy, stage migration, survivorship bias, censoring, misleading visualization. Offered as CSDS 313 and CSDS 413. (Case)

ECSE 313: Signal Processing (3) math

Fourier series and transforms. Analog and digital filters. Fast-Fourier transforms, sampling, and modulation for discrete time signals and systems. Consideration of stochastic signals and linear processing of stochastic signals using correlation functions and spectral analysis. The course will incorporate the use of Grand Challenges in the areas of Energy Systems, Control Systems, and Data Analytics in order to provide a framework for problems to study in the development and application of the concepts and tools studied in the course. Various aspects of important engineering skills relating to leadership, teaming, emotional intelligence, and effective communication are integrated into the course. (Case)

ECSE 346: Engineering Optimization (3) math

Optimization techniques including linear programming and extensions; transportation and assignment problems; network flow optimization; quadratic, integer, and separable programming; geometric programming; and dynamic programming. Nonlinear optimization topics: optimality criteria, gradient and other practical unconstrained and constrained methods. Computer applications using engineering and business case studies. The course will incorporate the use of Grand Challenges in the areas of Energy Systems, Control Systems, and Data Analytics in order to provide a framework for problems to study in the development and application of the concepts and tools studied in the course. Various aspects of important engineering skills relating to leadership, teaming, emotional intelligence, and effective communication are integrated into the course. (Case)

CSDS 386: Quantum Computing, Information, and Devices (3) math

An introduction to the math, physics, engineering, and computer science underlying the rapidly emerging fields of quantum computing, quantum information, and quantum devices. (Case)

CSDS 394: Introduction to Information Theory (3) math

This course is intended as an introduction to information and coding theory with emphasis on the mathematical aspects. It is suitable for advanced undergraduate and graduate students in mathematics, applied mathematics, statistics, physics, computer science and electrical engineering. Course content: Information measures-entropy, relative entropy, mutual information, and their properties. Typical sets and sequences, asymptotic equipartition property, data compression. Channel coding and capacity: channel coding theorem. Differential entropy, Gaussian channel, Shannon-Nyquist theorem. Information theory inequalities (400 level). Additional topics, which may include compressed sensing and elements of quantum information theory. (Case)

CSDS 413: Introduction to Data Analysis (3) math

This course provides a conceptual and hands-on introduction to reasoning with data. Introduction of basic statistical concepts; models vs. observations, common distributions, parameters vs. statistics, statistical inference, hypothesis testing, multiple hypotheses, confidence intervals. Use of computational approaches to address statistical problems; data representation, empirical assessment of statistical significance, assessment of the association between variables, dimensionality reduction, model building, evaluation, and validation. Data visualization and accessibility/interpretability of patterns in data and predictive models. Computational thinking and critical approaches in data science; common mistakes and issues in data analysis, causality vs. correlation, confounders, statistical artifacts, Simpson's paradox, base rate fallacy, stage migration, survivorship bias, censoring, misleading visualization. (Case)

ECSE 416: Convex Optimization for Engineering (3) math

This course will focus on the development of a working knowledge and skills to recognize, formulate, and solve convex optimization problems that are so prevalent in engineering. Applications in control systems; parameter and state estimation; signal processing; communications and networks; circuit design; data modeling and analysis; data mining including clustering and classification; and combinatorial and global optimization will be highlighted. New reliable and efficient methods, particular those based on interior-point methods and other special methods to solve convex optimization problems will be emphasized. Implementation issues will also be underscored. (Case)

CSDS 455: Applied Graph Theory (3) math

This course serves as an introduction to many of the important aspects of graph theory. Topics include connectivity, flows, matchings, planar graphs, and graph coloring with additional topics selected from extremal graphs, random graphs, bounded treewidth graphs, social networks and small world graphs. The class will explore the underlying mathematical theory with a specific focus on the development and analysis of graph algorithms. (Case)

CSDS 486: Quantum Computing, Information, and Devices (3) math

An introduction to the math, physics, engineering, and computer science underlying the rapidly emerging fields of quantum computing, quantum information, and quantum devices. The course is taught by a group of faculty from physics, engineering, computer science, and math, and is geared towards students with diverse backgrounds and interests in these fields. Students will select a concentration in one of these four areas, and the coursework, while still covering all topics, will be adjusted to focus on the selected area in the most detail. Note that the listed prerequisites depend on choice of concentration. Topics will include: 1. (Mathematics) Introduction to linear algebra, convex geometry, fundamental theory of quantum information. 2. (Physics) Introduction to the quantum mechanics of two-level systems (qubits). Survey of physics and materials for qubit technologies. 3. (Computer Science) Basic quantum gates and circuits, introduction to the theory of algorithms, survey of quantum algorithms. 4. (Engineering) Quantum architectures, mapping algorithms onto circuits. The course consists of lectures, homework, and group projects. Group projects will aim to synthesize the diverse backgrounds of the students and instructors to capture the interdisciplinary nature of the field. Students taking the course for graduate credit will complete an additional literature research project and presentation, in addition to enhanced problem sets. (Case)

CSDS 494: Introduction to Information Theory (3) math

This course is intended as an introduction to information and coding theory with emphasis on the mathematical aspects. It is suitable for advanced undergraduate and graduate students in mathematics, applied mathematics, statistics, physics, computer science and electrical engineering. Course content: Information measures-entropy, relative entropy, mutual information, and their properties. Typical sets and sequences, asymptotic equipartition property, data compression. Channel coding and capacity: channel coding theorem. Differential entropy, Gaussian channel, Shannon-Nyquist theorem. Information theory inequalities (400 level). Additional topics, which may include compressed sensing and elements of quantum information theory. (Case)

MATH 121: Calculus for Science and Engineering I (4) math

Functions, analytic geometry of lines and polynomials, limits, derivatives of algebraic and trigonometric functions. Definite integral, antiderivatives, fundamental theorem of calculus, change of variables. (Case)

MATH 122: Calculus for Science and Engineering II (4) math

Continuation of MATH 121. Exponentials and logarithms, growth and decay, inverse trigonometric functions, related rates, basic techniques of integration, area and volume, polar coordinates, parametric equations. Taylor polynomials and Taylor's theorem. (Case)

MATH 124: Calculus II (4) math

Review of differentiation. Techniques of integration, and applications of the definite integral. Parametric equations and polar coordinates. Taylor's theorem. Sequences, series, power series. Complex arithmetic. Introduction to multivariable calculus. (Case)

MATH 201: Introduction to Linear Algebra for Applications (3) math

Matrix operations, systems of linear equations, vector spaces, subspaces, bases and linear independence, eigenvalues and eigenvectors, diagonalization of matrices, linear transformations, determinants. Less theoretical than MATH 307. Appropriate for majors in science, engineering, economics. (Case)

MATH 223: Calculus for Science and Engineering III (3) math

Introduction to vector algebra; lines and planes. Functions of several variables: partial derivatives, gradients, chain rule, directional derivative, maxima/minima. Multiple integrals, cylindrical and spherical coordinates. Derivatives of vector valued functions, velocity and acceleration. Vector fields, line integrals, Green's theorem. (Case)

MATH 224: Elementary Differential Equations (3) math

A first course in ordinary differential equations. First order equations and applications, linear equations with constant coefficients, linear systems, Laplace transforms, numerical methods of solution. (Case)

MATH 227: Calculus III (3) math

Vector algebra and geometry. Linear maps and matrices. Calculus of vector valued functions. Derivatives of functions of several variables. Multiple integrals. Vector fields and line integrals. (Case)

MATH 228: Differential Equations (3) math

Elementary ordinary differential equations: first order equations; linear systems; applications; numerical methods of solution. (Case)

STAT 243: Statistical Theory with Application I (3) math

Introduction to fundamental concepts of statistics through examples including design of an observational study, industrial simulation. Theoretical development motivated by sample survey methodology. Randomness, distribution functions, conditional probabilities. Derivation of common discrete distributions. Expectation operator. Statistics as random variables, point and interval estimation. Maximum likelihood estimators. Properties of estimators. (Case)

STAT 244: Statistical Theory with Application II (3) math

Extension of inferences to continuous-valued random variables. Common continuous-valued distributions. Expectation operator. Maximum likelihood estimators for the continuous case. Simple linear, multiple and polynomial regression. Properties of regression estimators when errors are Gaussian. Regression diagnostics. Class or student projects gathering real data or generating simulated data, fitting models and analyzing residuals from fit. (Case)

MATH 307: Linear Algebra (3) math

A course in linear algebra that studies the fundamentals of vector spaces, inner product spaces, and linear transformations on an axiomatic basis. Topics include: solutions of linear systems, matrix algebra over the real and complex numbers, linear independence, bases and dimension, eigenvalues and eigenvectors, singular value decomposition, and determinants. Other topics may include least squares, general inner product and normed spaces, orthogonal projections, finite dimensional spectral theorem. This course is required of all students majoring in mathematics and applied mathematics. More theoretical than MATH 201. (Case)

STAT 312: Basic Statistics for Engineering and Science (3) math

For advanced undergraduate students in engineering, physical sciences, life sciences. Comprehensive introduction to probability models and statistical methods of analyzing data with the object of formulating statistical models and choosing appropriate methods for inference from experimental and observational data and for testing the model's validity. Balanced approach with equal emphasis on probability, fundamental concepts of statistics, point and interval estimation, hypothesis testing, analysis of variance, design of experiments, and regression modeling. (Case)

STAT 312R: Basic Statistics for Engineering and Science Using R Programming (3) math

For advanced undergraduate students in engineering, physical sciences, life sciences. Comprehensive introduction to probability models and statistical methods of analyzing data with the object of formulating statistical models and choosing appropriate methods for inference from experimental and observational data and for testing the model's validity. Balanced approach with equal emphasis on probability, fundamental concepts of statistics, point and interval estimation, hypothesis testing, analysis of variance, design of experiments, and regression modeling. (Case)

STAT 313: Statistics for Experimenters (3) math

For advanced undergraduates in engineering, physical sciences, life sciences. Comprehensive introduction to modeling data and statistical methods of analyzing data. General objective is to train students in formulating statistical models, in choosing appropriate methods for inference from experimental and observational data and to test the validity of these models. Focus on practicalities of inference from experimental data. Inference for curve and surface fitting to real data sets. Designs for experiments and simulations. Student generation of experimental data and application of statistical methods for analysis. Critique of model; use of regression diagnostics to analyze errors. (Case)

STAT 325: Data Analysis and Linear Models (3) math

Basic exploratory data analysis for univariate response with single or multiple covariates. Graphical methods and data summarization, model-fitting using S-plus computing language. Linear and multiple regression. Emphasis on model selection criteria, on diagnostics to assess goodness of fit and interpretation. Techniques include transformation, smoothing, median polish, robust/resistant methods. Case studies and analysis of individual data sets. Notes of caution and some methods for handling bad data. Knowledge of regression is helpful. (Case)

MATH 327: Convexity and Optimization (3) math

Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role. Among the topics discussed are basic properties of convex sets (extreme points, facial structure of polytopes), separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems. Offered as MATH 327, MATH 427, and OPRE 427. (Case)

STAT 332: Statistics for Signal Processing (3) math

For advanced undergraduate students or beginning graduate students in engineering, physical sciences, life sciences. Introduction to probability models and statistical methods. Emphasis on probability as relative frequencies. Derivation of conditional probabilities and memoryless channels. Joint distribution of random variables, transformations, autocorrelation, series of irregular observations, stationarity. Random harmonic signals with noise, random phase and/or random amplitude. Gaussian and Poisson signals. Modulation and averaging properties. Transmission through linear filters. Power spectra, bandwidth, white and colored noise. ARMA processes and forecasting. Optimal linear systems, signal-to-noise ratio, Wiener filter. Completion of additional assignments required from graduate students registered in this course. (Case)

STAT 333: Uncertainty in Engineering and Science (3) math

Phenomena of uncertainty appear in engineering and science for various reasons and can be modeled in different ways. The course integrates the mainstream ideas in statistical data analysis with models of uncertain phenomena stemming from three distinct viewpoints: algorithmic/computational complexity; classical probability theory; and chaotic behavior of nonlinear systems. Descriptive statistics, estimation procedures and hypothesis testing (including design of experiments). Random number generators and their testing. Monte Carlo Methods. Mathematica notebooks and simulations will be used. Graduate students are required to do an extra project. (Case)

MATH 380: Introduction to Probability (3) math

Combinatorial analysis. Permutations and combinations. Axioms of probability. Sample space and events. Equally likely outcomes. Conditional probability. Bayes' formula. Independent events and trials. Discrete random variables, probability mass functions. Expected value, variance. Bernoulli, binomial, Poisson, geometric, negative binomial random variables. Continuous random variables, density functions. Expected value and variance. Uniform, normal, exponential, Gamma random variables. The De Moivre-Laplace limit theorem. Joint probability mass functions and densities. Independent random variables and the distribution of their sums. Covariance. Conditional expectations and distributions (discrete case). Moment generating functions. Law of large numbers. Central limit theorem. Additional topics (time permitting): the Poisson process, finite state space Markov chains, entropy. (Case)

MATH 408: Introduction to Cryptology (3) math

Introduction to the mathematical theory of secure communication. Topics include: classical cryptographic systems; one-way and trapdoor functions; RSA, DSA, and other public key systems; Primality and Factorization algorithms; birthday problem and other attack methods; elliptic curve cryptosystems; introduction to complexity theory; other topics as time permits. (Case)

OPRE 207: Statistics for Business and Management Science I (3) math

Organizing and summarizing data. Mean, variance, moments. Elementary probability, conditional probability. Commonly encountered distributions including binomial. Poisson, uniform, exponential, normal distributions. Central limit theorem. Sample quantities, empirical distributions. Reference distributions (chi-square, z-, t-, F-distributions). Point and interval estimation: hypothesis tests. (Case)

MA117: Elementary Probability and Statistics (1) math

An introduction to the ideas of probability, including counting techniques, random variables and distributions. Elementary parametric statistical tests with examples drawn from the social sciences and life sciences. (Colorado)

MA120: Applied Linear Algebra (1) math

The study of systems of linear equations and matrix algebra with an emphasis on applications. Topics include the use of matrices to represent linear systems, independence and bases, invertibility, and eigenvalues. The use of computer algebra systems is emphasized. Applications will be drawn from economics, statistics, computer science, biology, and other fields. (Colorado)

MA126: Calculus 1 (1) math

Introduction to calculus for functions of one variable. Focus is on the definition, methods, and applications of derivatives. Integrals are briefly introduced. Students normally begin the calculus sequence with this course if they have solid precalculus preparation and have not previously studied calculus. Students who need a thorough review of precalculus should take MA125 instead; students who have previously studied calculus should consider MA129 instead. (Colorado)

MA129: Calculus 2 (1) math

Development of the definite integral, techniques of integration, and applications of the definite integral. Modeling with differential equations. Taylor polynomials and non-Cartesian coordinate systems in two dimensions. Students who have successfully completed a first course in calculus that focused on derivatives should consider this as an appropriate next course. (Colorado)

MA201: Foundations of Discrete Mathematics (1) math

An introduction to combinatorics, graph theory, and combinatorial geometry. The topics are fundamental for the study of many areas of mathematics as well as for the study of computer science, with applications to cryptography, linear programming, coding theory, and the theory of computing. (Colorado)

MA217: Introduction to Probability and Statistics (1) math

A calculus-based introduction to probability theory and statistical inference. Topics include probability, random variables, discrete and continuous distributions, sampling distributions, confidence intervals, hypothesis testing, and linear regression. This course also provides basic introduction to statistical programming language R. (Colorado)

MA251: Number Theory (1) math

A careful study of major topics in elementary number theory, including divisibility, factorization, prime numbers, perfect numbers, congruences, Diophantine equations and primitive roots. (Colorado)

COMS W3202: Finite Mathematics (3) math

(Columbia)

COMS W3203: Discrete Mathematics (4) math

Logic and formal proofs, sequences and summation, mathematical induction, binomial coefficients, elements of finite probability, recurrence relations, equivalence relations and partial orderings, and topics in graph theory (including isomorphism, traversability, planarity, and colorings) (Columbia)

COMS W3210: Scientific Computation (3) math

Introduction to computation on digital computers. Design and analysis of numerical algorithms. Numerical solution of equations, integration, recurrences, chaos, differential equations. Introduction to Monte Carlo methods. Properties of floating point arithmetic. Applications to weather prediction, computational finance, computational science, and computational engineering. (Columbia)

COMS W3251: Computational Linear Algebra (4) math

(Columbia)

COMS W4203: Graph Theory (3) math

General introduction to graph theory. Isomorphism testing, algebraic specification, symmetries, spanning trees, traversability, planarity, drawings on higher-order surfaces, colorings, extremal graphs, random graphs, graphical measurement, directed graphs, Burnside-Polya counting, voltage graph theory (Columbia)

COMS W4205: Combinatorial Theory (3) math

Sequences and recursions, calculus of finite differences and sums, elementary number theory, permutation group structures, binomial coefficients, Stilling numbers, harmonic numbers, generating functions (Columbia)

MATH BC2006: Combinatorics (3) math

(Columbia)

APMA E2000: Multv. Calc. For Engi & App Sci (4) math

Differential and integral calculus of multiple variables. Topics include partial differentiation; optimization of functions of several variables; line, area, volume, and surface integrals; vector functions and vector calculus; theorems of Green, Gauss, and Stokes; applications to selected problems in engineering and applied science (Columbia)

APMA E2101: Intro to Applied Mathematics (3) math

A unified, single-semester introduction to differential equations and linear algebra with emphases on (1) elementary analytical and numerical technique and (2) discovering the analogs on the continuous and discrete sides of the mathematics of linear operators: superposition, diagonalization, fundamental solutions. Concepts are illustrated with applications using the language of engineering, the natural sciences, and the social sciences. Students execute scripts in Mathematica and MATLAB (or the like) to illustrate and visualize course concepts (programming not required) (Columbia)

APMA E3101: Applied Math I: Linear Algebra (3) math

Matrix algebra, elementary matrices, inverses, rank, determinants. Computational aspects of solving systems of linear equations: existence-uniqueness of solutions, Gaussian elimination, scaling, ill-conditioned systems, iterative techniques. Vector spaces, bases, dimension. Eigenvalue problems, diagonalization, inner products, unitary matrices (Columbia)

STAT GU4001: Introduction to Probability and Statistics (3) math

A calculus-based tour of the fundamentals of probability theory and statistical inference. Probability models, random variables, useful distributions, conditioning, expectations, law of large numbers, central limit theorem, point and confidence interval estimation. (Columbia)

MATH GU4041: Intro Modern Algebra I (3) math

Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups (Columbia)

MATH GU4051: Topology (3) math

Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces (Columbia)

MATH GU4061: Intro Modern Analysis I (3) math

Real numbers, metric spaces, elements of general topology, sequences and series, continuity, differentiation, integration, uniform convergence, Ascoli-Arzela theorem, Stone-Weierstrass theorem (Columbia)

MATH UN1101: Calculus I (3) math

Functions, limits, derivatives, introduction to integrals, or an understanding of pre-calculus will be assumed. (Columbia)

MATH UN1102: Calculus II (3) math

Methods of integration, applications of the integral, Taylors theorem, infinite series. (Columbia)

MATH UN1201: Calculus III (3) math

Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramers rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC) (Columbia)

STAT UN1201: Calc-Based Intro to Statistics (3) math

Designed for students who desire a strong grounding in statistical concepts with a greater degree of mathematical rigor than in STAT W1111. Random variables, probability distributions, pdf, cdf, mean, variance, correlation, conditional distribution, conditional mean and conditional variance, law of iterated expectations, normal, chi-square, F and t distributions, law of large numbers, central limit theorem, parameter estimation, unbiasedness, consistency, efficiency, hypothesis testing, p-value, confidence intervals, maximum likelihood estimation. (Columbia)

MATH UN1202: Calculus IV (3) math

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC) (Columbia)

MATH UN1205: Accelerated Multivariable Calc (4) math

Vectors in dimensions 2 and 3, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, optimization, Lagrange multipliers, double and triple integrals, line and surface integrals, vector calculus. This course is an accelerated version of MATH UN1201 - MATH UN1202. (Columbia)

MATH UN1207: Honors Mathematics A (4) math

Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC) (Columbia)

MATH UN1208: Honors Mathematics B (4) math

Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC) (Columbia)

MATH UN2010: Linear Algebra (3) math

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (Columbia)

MATH UN2015: Linear Algebra and Probability (3) math

Linear algebra with a focus on probability and statistics. The course covers the standard linear algebra topics: systems of linear equations, matrices, determinants, vector spaces, bases, dimension, eigenvalues and eigenvectors, the Spectral Theorem and singular value decompositions. It also teaches applications of linear algebra to probability, statistics and dynamical systems giving a background sufficient for higher level courses in probability and statistics. The topics covered in the probability theory part include conditional probability, discrete and continuous random variables, probability distributions and the limit theorems, as well as Markov chains, curve fitting, regression, and pattern analysis. The course contains applications to life sciences, chemistry, and environmental life sciences. No a prior i background in the life sciences is assumed. This course is best suited for students who wish to focus on applications and practical approaches to problem solving. It is recommended to students majoring in engineering, technology, life sciences, social sciences, and economics. Math majors, joint majors, and math concentrators must take MATH UN2010 Linear Algebra, which focuses on linear algebra concepts and foundations that are needed for upper-level math courses. MATH UN2015 (Linear Algebra and Probability) does NOT replace MATH UN2010 (Linear Algebra) as prerequisite requirements of math courses. Students may not receive full credit for both courses MATH UN2010 and MATH UN2015 (Columbia)

MATH UN2500: Analysis and Optimization (3) math

Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (Columbia)

MATH UN3007: Complex Variables (3) math

Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping. (Columbia)

MATH UN3020: Number Theory and Cryptography (3) math

Congruences. Primitive roots. Quadratic residues. Contemporary applications (Columbia)

MATH UN3386: Differential Geometry (3) math

Local and global differential geometry of submanifolds of Euclidean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem. (Columbia)

MATH UN3951: Undergraduate Seminars I (3) math

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow (Columbia)

MATH UN3952: Undergraduate Seminars II (3) math

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow (Columbia)

CS 1380: Data Science for All (4) math

For description, see STSCI 1380. (Cornell)

CS 2800: Discrete Structures (4) math

Covers the mathematics that underlies most of computer science. (Cornell)

CS 2802: Discrete Structures - Honors (3) math

Covers the mathematics that underlies most of computer science. This course is an honors version of CS 2800. (Cornell)

CS 2850: Networks (3) math

For description, see ECON 2040. (Cornell)

CS 3220: Computational Mathematics for Computer Science (3) math

Introduction to computational mathematics covering topics in (numerical) linear algebra, statistics, and optimization. Topics included are those of particular relevance to upper-division computer science courses in machine learning, numerical analysis, graphics, vision, robotics, and more. An emphasis is placed both on understanding core mathematical concepts and introducing associated computational methodologies. (Cornell)

CS 4210: Numerical Analysis and Differential Equations (4) math

For description, see MATH 4250. (Cornell)

CS 4220: Numerical Analysis: Linear and Nonlinear Problems (4) math

Introduction to the fundamentals of numerical linear algebra: direct and iterative methods for linear systems, eigenvalue problems, singular value decomposition. (Cornell)

CS 4850: Mathematical Foundations for the Information Age (4) math

Covers the mathematical foundations for access to information. Topics include high dimensional space, random graphs, singular value decomposition, Markov processes, learning theory, and algorithms for massive data. (Cornell)

CS 4852: Networks II: Market Design (3) math

For description, see INFO 4220. (Cornell)

CS 4860: Applied Logic (3) math

For description, see MATH 4860. (Cornell)

MATH 1110: Calculus I (4) math

Topics include functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, and the area under a curve. (Cornell)

MATH 1120: Calculus II (4) math

Focuses on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations and their applications. Also covers infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations. (Cornell)

MATH 1910: Calculus for Engineers (4) math

Essentially a second course in calculus. Topics include techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, tests of convergence, and power series. (Cornell)

MATH 1920: Multivariable Calculus for Engineers (4) math

Introduction to multivariable calculus. Topics include partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green’s theorem, Stokes’ theorem, and the divergence theorem. (Cornell)

MATH 2210: Linear Algebra (4) math

Topics include vector algebra, linear transformations, matrices, determinants, orthogonality, eigenvalues, and eigenvectors. Applications are made to linear differential or difference equations. The lectures introduce students to formal proofs. Students are required to produce some proofs in their homework and on exams. (Cornell)

MATH 2930: Differential Equations for Engineers (4) math

Introduction to ordinary and partial differential equations. Topics include: first-order equations (separable, linear, homogeneous, exact); mathematical modeling (e.g., population growth, terminal velocity); qualitative methods (slope fields, phase plots, equilibria, and stability); numerical methods; second-order equations (method of undetermined coefficients, application to oscillations and resonance, boundary-value problems and eigenvalues); and Fourier series. A substantial part of this course involves partial differential equations, such as the heat equation, the wave equation, and Laplace’s equation. (Cornell)

MATH 2940: Linear Algebra for Engineers (4) math

Linear algebra and its applications. Topics include: matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces; applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems. (Cornell)

MATH 4710: Basic Probability (4) math

Introduction to probability theory, which prepares the student to take MATH 4720. The course begins with basics: combinatorial probability, mean and variance, independence, conditional probability, and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and the central limit theorem are stated and their implications for statistics are discussed. (Cornell)

COSC 30: Discrete Mathematics in Computer Science (1) math

This course develops the mathematical foundations of computer science that are not calculus-based. It covers basic set theory, logic, mathematical proof techniques, and a selection of discrete mathematics topics such as combinatorics (counting), discrete probability, number theory, and graph theory. The mathematics is frequently motivated using computer science applications. (Dartmouth)

COSC 70: Foundations of Applied Computer Science (1) math

This course introduces core computational and mathematical techniques for data analysis and physical modeling, foundational to applications including computational biology, computer vision, graphics, machine learning, and robotics. The approaches covered include modeling and optimizing both linear and nonlinear systems, representing and computing with uncertainty, analyzing multi-dimensional data, and sampling from complex domains. The techniques are both grounded in mathematical principles and practically applied to problems from a broad range of areas. (Dartmouth)

COSC 71: Numerical Methods in Computation (1) math

A study and analysis of important numerical and computational methods for solving engineering and scientific problems. The course will include methods for solving linear and nonlinear equations, doing polynomial interpolation, evaluating integrals, solving ordinary differential equations, and determining eigenvalues and eigenvectors of matrices. The student will be required to write and run computer programs. (Dartmouth)

COSC 84: Mathematical Optimization and Modeling (1) math

Planning, scheduling, and design problems in large organizations, economic or engineering systems can often be modeled mathematically using variables satisfying linear equations and inequalities. This course explores these models: the types of problems that can be handled, their formulation, solution, and interpretation. It introduces the theory underlying linear programming, a natural extension of linear algebra that captures these types of models, and also studies the process of modeling concrete problems, the algorithms to solve these models, and the solution and analysis of these problems using a modeling language. It also discusses the relation of linear programming to the more complex frameworks of nonlinear programming and integer programming. These paradigms broaden linear programming to respectively allow for nonlinear equations and inequalities, or for variables to be constrained to be integers. (Dartmouth)

COSC 49.08: Information Theory in Computer Science (1) math

This course introduces students to information theory, a mathematical formalism for quantifying and reasoning about communication. While traditionally a part of electrical engineering, it has found several powerful applications in the theory of algorithms and complexity and adjacent fields such as combinatorics and game theory. The first third of the course will teach students the basics of information theory (Shannon entropy, mutual information, Kullback-Liebler divergence). The rest of the course will sample topics from error correcting codes, communication complexity, data structures, and optimization, in each case highlighting applications of information theory. (Dartmouth)

CPS 338: Computational Mathematics (1) math

Numerical analysis as implemented on computers. Polynomial and rational approximations, numerical differentiation and integration, systems of linear equations, matrix inversion, eigenvalues, first and second order differential equations. (F&M)

MAT 109: Calculus I (1) math

Introduction to the basic concepts of calculus and their applications. Functions, derivatives and limits; exponential, logarithmic and trigonometric functions; the definite integral and the Fundamental Theorem of Calculus. (F&M)

MAT 110: Calculus II (1) math

Techniques of integration, applications of integration, separable first-order differential equations, convergence tests for infinite series, Taylor polynomials and Taylor series. (F&M)

MAT 216: Probability and Statistics I (1) math

Introduction to single variable probability and statistics. Random variables. Binomial, geometric, Poisson, exponential and gamma distributions, among others. Counting techniques. Estimation and hypothesis tests on a single parameter. (F&M)

MAT 229: Linear Algebra and Differential Equations (1) math

Systems of linear equations and matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, nth order linear differential equations, systems of first order differential equations. (F&M)

CPS/MAT 237: Computational Discrete Mathematics (1) math

Basic set theory, combinatorics (the theory of counting), finite difference equations, and graph theory with related algorithms. (F&M)

CS 2050: Introduction to Discrete Mathematics for Computer Science (3) math

Proof methods, strategy, correctness of algorithms over discrete structures. Induction and recursion. Complexity and order of growth. Number theoretic principles and algorithms. Counting and computability. Credit not allowed for both CS 2050 and CS 2051. (Georgia Tech)

CS 2051: Honors - Induction to Discrete Mathematics for Computer Science (3) math

Proof methods, strategy, correctness of algorithms over discrete structures. Induction and recursion. Complexity and order of growth. Number theoretic principles and algorithms. Counting and computability. Credit not allowed for both CS 2051 and CS 2050. (Georgia Tech)

MATH 1551: Differential Calculus (2) math

Differential calculus including applications and the underlying theory of limits for functions and sequences. Credit not awarded for both MATH 1551 and MATH 1501, MATH 1503, or MATH 1550. (Georgia Tech)

MATH 1552: Integral Calculus (4) math

Integral calculus: Definite and indefinite integrals, techniques of integration, improper integrals, infinite series, applications. Credit not awarded for both MATH 1552 and MATH 1502, MATH 1504, MATH 1512 or MATH 1555. (Georgia Tech)

MATH 1554: Linear Algebra (4) math

Linear algebra eigenvalues, eigenvectors, applications to linear systems, least squares, diagnolization, quadratic forms. (Georgia Tech)

MATH 2550: Introduction to Multivariable Calculus (2) math

Vectors in three dimensions, curves in space, functions of several variables, partial derivatives, optimization, integration of functions of several variables. Vector Calculus not covered. Credit will not be awarded for both MATH 2550 and MATH 2605 or MATH 2401 or MATH 2551 or MATH 1555. (Georgia Tech)

MATH 2552: Differential Equations (4) math

Methods for obtaining numerical and analytic solutions of elementary differential equations. Applications are also discussed with an emphasis on modeling. Credit not awarded for both MATH 2552 and MATH 2403 or MATH 2413 or MATH 2562. (Georgia Tech)

MATH 2562: Honors Differential Equations (4) math

The topics covered parallel those of MATH 2552 with a somewhat more intensive and rigorous treatment. (Georgia Tech)

MATH 3012: Applied Combinatorics (3) math

Elementary combinatorial techniques used in discrete problem solving: counting methods, solving linear recurrences, graph and network models, related algorithms, and combinatorial designs. (Georgia Tech)

MATH 3215: Introduction to Probability and Statistics (3) math

This course is a problem-oriented introduction to the basic concepts of probability and statistics, providing a foundation for applications and further study. (Georgia Tech)

MATH 3406: A Second Course in Linear Algebra (3) math

This course will cover important topics in linear algebra not usually discussed in a first-semester course, featuring a mixture of theory and applications. (Georgia Tech)

MATH 3670: Probability and Statistics with Applications (3) math

Introduction to probability, probability distributions, point estimation, confidence intervals, hypothesis testing, linear regression and analysis of variance. Students cannot receive credit for both MATH 3670 and MATH 3770 or ISYE 3770 or CEE 3770. (Georgia Tech)

MATH 4012: Algebraic Structures in Coding Theory (3) math

Introduction to linear error correcting codes with an emphasis on the algebraic tools required, including matrices vector spaces, groups, polynomial rings, and finite fields. (Georgia Tech)

MATH 4022: Introduction to Graph Theory (3) math

The fundamentals of graph theory: trees, connectivity, Euler torus, Hamilton cycles, matchings, colorings, and Ramsey theory. (Georgia Tech)

MATH 4032: Combinatorial Analysis (3) math

Combinatorial problem-solving techniques including the use of generating functions, recurrence relations, Polya theory, combinatorial designs, Ramsey theory, matroids, and asymptotic analysis. (Georgia Tech)

MATH 4107: Introduction to Abstract Algebra I (3) math

This course develops in the theme of 'Arithmetic congruence and abstract algebraic structures'. Strong emphasis on theory and proofs. (Georgia Tech)

MATH 4150: Introduction to Number Theory (3) math

Primes and unique factorization, congruences, Chinese remainder theorem, Diophantine equations, Diophantine approximations, quadratic reciprocity. Applications such as fast multiplication, factorization, and encryption. (Georgia Tech)

MATH 4255: Monte Carlo Methods (3) math

Probability distributions, limit laws, and applications through the computer. (Georgia Tech)

MATH 4280: Introduction to Information Theory (3) math

The measurement and quantification of information. These ideas are applied to the probabilistic analysis of the transmission of information over a channel along which random distortion of the message occurs. (Georgia Tech)

MATH 4305: Topics in Linear Algebra (3) math

Finite dimensional vector spaces, inner product spaces, least squares, linear transformations, the spectral theorem for normal transformations. Applications to convex sets, positive matrices, difference equations. (Georgia Tech)

MATH 4580: Linear Programming (3) math

A study of linear programming problems, including the simplex method, duality, and sensitivity analysis with applications to matrix games, integer programming, and networks. (Georgia Tech)

MATH 4640: Numerical Analysis I (3) math

Introduction to numerical algorithms for some basic problems in computational mathematics. Discussion of both implementation issues and error analysis. (Georgia Tech)

MATH 4782: Quantum Information and Quantum Computing (3) math

Introduction to quantum computing and quantum information theory, formalism of quantum mechanics, quantum gates, algorithms, measurements, coding, and information. Physical realizations and experiments. Crosslisted with PHYS 4782. (Georgia Tech)

ISYE 2027: Probability with Applications (3) math

Topics include conditional probability, density and distribution functions from engineering, expectation, conditional expectation, laws of large numbers, central limit theorem, and introduction to Poisson Processes. (Georgia Tech)

ISYE 2028: Basic Statistical Methods (3) math

Point and interval estimation of systems parameters, statistical decision making about differences in system parameters, analysis and modeling of relationships between variables. (Georgia Tech)

COMPSCI 20: Discrete Mathematics for Computer Science (4) math

Widely applicable mathematical tools for computer science, including topics from logic, set theory, combinatorics, number theory, probability theory, and graph theory. Practice in reasoning formally and proving theorems. (Harvard)

MATH MA: Introduction to Functions and Calculus I (4) math

The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students. (Harvard)

MATH 1A: Introduction to Calculus (4) math

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines. (Harvard)

MATH 1B: Integration, Series and Differential Equations (4) math

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it. (Harvard)

MATH 21B: Linear Algebra and Differential Equations (4) math

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series. (Harvard)

MATH 22A: Vector Calculus and Linear Algebra I (4) math

MATH 22 covers multivariable calculus and linear algebra for students interested in mathematical sciences. It covers the same topics as Mathematics 21, but does so with more rigor. Students are taught techniques of proof and mathematical reasoning. The workload and content is comparable with the Mathematics 21 sequence. But unlike the latter, the linear algebra and calculus are more interlinked. The content of Math 22a is mostly aligned with Math 21b (linear algebra), and the content of Math 22b is mostly aligned with Math 21a (multivariable calculus). (Harvard)

APMTH 22A: Solving and Optimizing (4) math

This course covers a combination of linear algebra and multivariate calculus with an eye towards solving systems of equations and optimization problems. Students will learn how to prove some key results, and will also implement these ideas with code.Linear algebra: matrices, vector spaces, bases and dimension, inner products, least squares problems, eigenvalues, eigenvectors, singular values, singular vectors.Multivariate calculus: partial differentiation, gradient and Hessian, critical points, Lagrange Multipliers. (Harvard)

MATH 23A: Linear Algebra and Real Analysis I (4) math

Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Single-variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl. Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results. Notes: Students are expected to watch videos of the lectures from Fall 2015 before attending class. Weekly two-hour classes will consist of a one-hour seminar in which students present key definitions and proofs and a one-hour activity-based session in which students work in small groups to solve problems. Students are expected to continue in either MATH 23B (recommended for students who are thinking of concentrating in mathematics, the physical sciences, or engineering) or MATH 23C (recommended for students who are not sure of their concentration, or who are thinking about a concentration in the social sciences, economics, computer science, life sciences or data science). Either alternative will provide a solid foundation for a concentration in mathematics or any field that uses mathematics. (Harvard)

MATH 23C: Mathematics for Computation, Statistics, and Data Science (4) math

Proof strategies and logic. Sets, countability, sigma fields, and axiomatic foundations of probability. Summation of series and evaluation of multiple integrals, with emphasis on calculation of expectation and variance. Abstract vector spaces and inner product spaces, with applications to analysis of large datasets. Key functions and theorems of mathematical statistics. A brief introduction to classical vector calculus as used in electromagnetic theory. Students will learn to use some of the statistical and graphical display tools in the R scription language. (Harvard)

MATH 25A: Theoretical Linear Algebra and Real Analysis I (4) math

A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness. (Harvard)

MATH 55A: Studies in Algebra and Group Theory (4) math

A rigorous introduction to abstract algebra, including group theory and linear algebra. This course covers the equivalent of Mathematics 25a and Mathematics 122, and prepares students for Mathematics 123 and other advanced courses in number theory and algebra. (A course in analysis such as Mathematics 25b or 55b is recommended for Spring semester.) (Harvard)

MATH 101: Sets, Groups and Real Analysis (4) math

This course provides an introduction to conceptual and axiomatic mathematics, the writing of proofs, and mathematical culture, with sets, groups and real analysis as the main topics. (Harvard)

APMTH 107: Graph Theory and Combinatorics (4) math

Topics in combinatorial mathematics that find frequent application in computer science, engineering, and general applied mathematics. Specific topics taken from graph theory, enumeration techniques, optimization theory, combinatorial algorithms, and discrete probability. (Harvard)

STAT 110: Introduction to Probability (4) math

A comprehensive introduction to probability. Basics: sample spaces and events, conditional probability, and Bayes' Theorem. Univariate distributions: density functions, expectation and variance, Normal, t, Binomial, Negative Binomial, Poisson, Beta, and Gamma distributions. Multivariate distributions: joint and conditional distributions, independence, transformations, and Multivariate Normal. Limit laws: law of large numbers, central limit theorem. Markov chains: transition probabilities, stationary distributions, convergence. (Harvard)

APMTH 115: Mathematical Modeling (4) math

Abstracting the essential components and mechanisms from a natural system to produce a mathematical model, which can be analyzed with a variety of formal mathematical methods, is perhaps the most important, but least understood, task in applied mathematics. This course approaches a number of problems without the prejudice of trying to apply a particular method of solution. Topics drawn from biology, economics, engineering, physical and social sciences. (Harvard)

MATH 152: Discrete Mathematics (4) math

An introduction to finite groups, finite fields, quaternions, finite geometry, finite topology, combinatorics, and graph theory. A recurring theme of the course is the symmetry group of the regular icosahedron. Taught in a seminar format: students will gain experience in presenting proofs at the blackboard. (Harvard)

MATH 154: Probability Theory (4) math

An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes. (Harvard)

STAT 185: Introduction to Unsupervised Learning (4) math

An introductory course in unsupervised learning with an emphasis on dimensionality reduction and clustering. Topics include principal component analysis, nonnegative matrix factorization, and spectral clustering. In this course we will study these techniques and others with a focus on high-dimensional geometry and insights provided by linear algebra. Numerous data example will be included throughout the course. (Harvard)

MATH MB: Introduction to Functions and Calculus II (4) math

Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for MATH 1B. (Harvard)

MIT 21M.387: Fundamentals of Music Processing (4) math

Analyzes recorded music in digital audio form using advanced signal processing and optimization techniques to understand higher-level musical meaning. Covers fundamental tools like windowing, feature extraction, discrete and short-time Fourier transforms, chromagrams, and onset detection. Addresses analysis methods including dynamic time warping, dynamic programming, self-similarity matrices, and matrix factorization. Explores a variety of applications, such as event classification, audio alignment, chord recognition, structural analysis, tempo and beat tracking, content-based audio retrieval, and audio decomposition. Students taking graduate version complete different assignments. (Harvard)

6.1200: Mathematics for Computer Science (12) math

Elementary discrete mathematics for science and engineering, with a focus on mathematical tools and proof techniques useful in computer science. Topics include logical notation, sets, relations, elementary graph theory, state machines and invariants, induction and proofs by contradiction, recurrences, asymptotic notation, elementary analysis of algorithms, elementary number theory and cryptography, permutations and combinations, counting tools, and discrete probability. (MIT)

6.120A: Discrete Mathematics and Proof for Computer Science (6) math

Subset of elementary discrete mathematics for science and engineering useful in computer science. Topics may include logical notation, sets, done relations, elementary graph theory, state machines and invariants, induction and proofs by contradiction, recurrences, asymptotic notation, elementary analysis of algorithms, elementary number theory and cryptography, permutations and combinations, counting tools. (MIT)

6.3000: Signal Processing (12) math

Fundamentals of signal processing, focusing on the use of Fourier methods to analyze and process signals such as sounds and images. Topics include Fourier series, Fourier transforms, the Discrete Fourier Transform, sampling, convolution, deconvolution, filtering, noise reduction, and compression. Applications draw broadly from areas of contemporary interest with emphasis on both analysis and design. (MIT)

6.3010: Signals, Systems and Inference (12) math

Covers signals, systems and inference in communication, control and signal processing. Topics include input-output and state-space models of linear systems driven by deterministic and random signals; time- and transform-domain representations in discrete and continuous time; and group delay. State feedback and observers. Probabilistic models; stochastic processes, correlation functions, power spectra, spectral factorization. Least-mean square error estimation; Wiener filtering. Hypothesis testing; detection; matched filters. (MIT)

6.3700: Introduction to Probability (12) math

An introduction to probability theory, the modeling and analysis of probabilistic systems, and elements of statistical inference. Probabilistic models, conditional probability. Discrete and continuous random variables. Expectation and conditional expectation, and further topics about random variables. Limit Theorems. Bayesian estimation and hypothesis testing. Elements of classical statistical inference. Bernoulli and Poisson processes. Markov chains. Students taking graduate version complete additional assignments. (MIT)

6.5610: Applied Cryptography and Security (12) math

Emphasis on applied cryptography. May include: basic notion of systems security, cryptographic hash functions, symmetric cryptography (one-time pad, block ciphers, stream ciphers, message authentication codes), hash functions, key-exchange, public-key cryptography (encryption, digital signatures), elliptic curve cryptography, secret-sharing, fully homomorphic encryption, zero-knowledge proofs, and electronic voting. Assignments include a final group project. Topics may vary year to year. (MIT)

6.5620: Cryptography and Cryptanalysis (12) math

A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives of public-key encryption, digital signatures, pseudo-random number generation, and basic protocols and their computational complexity requirements. (MIT)

6.5630: Advanced Topics in Cryptography (12) math

In-depth exploration of recent results in cryptography. (MIT)

6.7201: Optimization Methods (12) math

Introduces the principal algorithms for linear, network, discrete, robust, nonlinear, and dynamic optimization. Emphasizes methodology and the underlying mathematical structures. Topics include the simplex method, network flow methods, branch and bound and cutting plane methods for discrete optimization, optimality conditions for nonlinear optimization, interior point methods for convex optimization, Newton's method, heuristic methods, and dynamic programming and optimal control methods. Expectations and evaluation criteria differ for students taking graduate version; consult syllabus or instructor for specific details. (MIT)

CS 1800: Discrete Structures (4) math

Introduces the mathematical structures and methods that form the foundation of computer science. Studies structures such as sets, tuples, sequences, lists, trees, and graphs. Discusses functions, relations, ordering, and equivalence relations. Examines inductive and recursive definitions of structures and functions. Discusses principles of proof such as truth tables, inductive proof, and basic logic. Also covers the counting techniques and arguments needed to estimate the size of sets, the growth of functions, and the space-time complexity of algorithms. (Northeastern)

CS 1802: Seminar for CS 1800 (1) math

Accompanies CS 1800. Illustrates topics from the lecture course through discussions, quizzes, and homework assignments. (Northeastern)

CS 2810: Mathematics of Data Models (4) math

Studies the methods and ideas in linear algebra, multivariable calculus, and statistics that are most relevant for the practicing computer scientist doing machine learning, modeling, or hypothesis testing with data. Covers least squares regression, finding eigenvalues to predict a linear system's behavior, performing gradient descent to fit a model to data, and performing t-tests and chi-square tests to determine whether differences between populations are significant. Includes applications to popular machine-learning methods, including Bayesian models and neural networks. (Northeastern)

CY 4770: Cryptography (4) math

Studies the design of cryptographic schemes that enable secure communication and computation. Emphasizes cryptography as a mathematically rigorous discipline with precise definitions, theorems, and proofs and highlights deep connections to information theory, computational complexity, and number theory. Topics include pseudorandomness; symmetric-key cryptosystems and block ciphers such as AES; hash functions; public-key cryptosystems, including ones based on factoring and discrete logarithms; signature schemes; secure multiparty computation and applications such as auctions and voting; and zero-knowledge proofs. (Northeastern)

MATH 1341: Calculus 1 for Science and Engineering (4) math

Covers definition, calculation, and major uses of the derivative, as well as an introduction to integration. Topics include limits; the derivative as a limit; rules for differentiation; and formulas for the derivatives of algebraic, trigonometric, and exponential/logarithmic functions. Also discusses applications of derivatives to motion, density, optimization, linear approximations, and related rates. Topics on integration include the definition of the integral as a limit of sums, antidifferentiation, the fundamental theorem of calculus, and integration by substitution. (Northeastern)

MATH 1342: Calculus 2 for Science and Engineering (4) math

Covers further techniques and applications of integration, infinite series, and introduction to vectors. Topics include integration by parts; numerical integration; improper integrals; separable differential equations; and areas, volumes, and work as integrals. Also discusses convergence of sequences and series of numbers, power series representations and approximations, 3D coordinates, parameterizations, vectors and dot products, tangent and normal vectors, velocity, and acceleration in space. Requires prior completion of MATH 1341 or permission of head mathematics advisor. (Northeastern)

MATH 1365: Introduction to Mathematical Reasoning (4) math

Covers the basics of mathematical reasoning and problem solving to prepare incoming math majors for more challenging mathematical courses at Northeastern. Focuses on learning to write logically sound mathematical arguments and to analyze such arguments appearing in mathematical books and courses. Includes fundamental mathematical concepts such as sets, relations, and functions. (Northeastern)

MATH 2321: Calculus 3 for Science and Engineering (4) math

Extends the techniques of calculus to functions of several variables; introduces vector fields and vector calculus in two and three dimensions. Topics include lines and planes, 3D graphing, partial derivatives, the gradient, tangent planes and local linearization, optimization, multiple integrals, line and surface integrals, the divergence theorem, and theorems of Green and Stokes with applications to science and engineering and several computer lab projects. Requires prior completion of MATH 1342 or MATH 1252. (Northeastern)

MATH 2331: Linear Algebra (4) math

Uses the Gauss-Jordan elimination algorithm to analyze and find bases for subspaces such as the image and kernel of a linear transformation... (Northeastern)

MATH 2341: Differential Equations and Linear Algebra for Engineering (4) math

Studies ordinary differential equations, their applications, and techniques for solving them including numerical methods... (Northeastern)

MATH 3081: Probability and Statistics (4) math

Focuses on probability theory. Topics include sample space; conditional probability and independence; discrete and continuous probability distributions for one and for several random variables; expectation; variance; special distributions including binomial, Poisson, and normal distributions; law of large numbers; and central limit theorem. Also introduces basic statistical theory including estimation of parameters, confidence intervals, and hypothesis testing. (Northeastern)

MATH 3175: Group Theory (4) math

Presents basic concepts and techniques of the group theory: symmetry groups, axiomatic definition of groups, important classes of groups (abelian groups, cyclic groups, additive and multiplicative groups of residues, and permutation groups), Cayley table, subgroups, group homomorphism, cosets, the Lagrange theorem, normal subgroups, quotient groups, and direct products. Studies structural properties of groups. Possible applications include geometry, number theory, crystallography, physics, and combinatorics (Northeastern)

MATH 3527: Number Theory 1 (4) math

Introduces number theory. Topics include linear diophantine equations, congruences, design of magic squares, Fermat's little theorem, Euler's formula, Euler's phi function, computing powers and roots in modular arithmetic, the RSA encryption system, primitive roots and indices, and the law of quadratic reciprocity. As time permits, may cover diophantine approximation and Pell's equation, elliptic curves, points on elliptic curves, and Fermat's last theorem. (Northeastern)

COMP_SCI 212-0: Mathematical Foundations of Comp Science (1) math

Basic concepts of finite and structural mathematics. Sets, axiomatic systems, the propositional and predicate calculi, and graph theory. Application to computer science: sequential machines, formal grammars, and software design. (Northwestern)

COMP_SCI 307-0: Introduction to Cryptography (1) math

This course covers the basic knowledge in understanding and using cryptography. The main focus is on definitions, theoretical foundations, and rigorous proofs of security, with some programming practice. Topics include symmetric and public-key encryption, message integrity, hash functions, block-cipher design and analysis, number theory, and digital signatures. (Northwestern)

COMP_SCI 341-0: Mechanism Design (1) math

Applying algorithms and microeconomics to derive a theory of the design of mechanisms that produce desired outcomes despite counteractive inputs by outside agents. Key application areas: auctions, markets, networking protocols. (Northwestern)

STAT 210-0: Introduction to Probability and Statistics (1) math

A mathematical introduction to probability theory and statistical methods, including properties of probability distributions, sampling distributions, estimation, confidence intervals, and hypothesis testing. STAT 210-0 is primarily intended for economics majors. (Northwestern)

MATH 218-1: Single-Variable Calculus with Precalculus (1) math

Functions and graphs. Limits. Continuity. Differentiation. Linearization. (Northwestern)

MATH 218-2: Single-Variable Calculus with Precalculus (1) math

Extreme value theorem, mean value theorem, and curve-sketching. Related rates. Optimization. Transcendental and inverse functions. (Northwestern)

MATH 218-3: Single-Variable Calculus with Precalculus (1) math

Definite integrals, antiderivatives, and the fundamental theorem of calculus. Areas and volumes. Techniques of integration, numerical integration, and improper integrals. First-order linear and separable ordinary differential equations. (Northwestern)

MATH 220-1: Single-Variable Differential Calculus (1) math

Limits. Differentiation. Linear approximation and related rates. Extreme value theorem, mean value theorem, and curve-sketching. Optimization. (Northwestern)

MATH 220-2: Single-Variable Integral Calculus (1) math

Definite integrals, antiderivatives, and the fundamental theorem of calculus. Transcendental and inverse functions. Areas and volumes. Techniques of integration, numerical integration, and improper integrals. First-order linear and separable ordinary differential equations. (Northwestern)

MATH 228-1: Multivariable Differential Calculus for Engineering (1) math

Vectors, vector functions, partial derivatives, Taylor polynomials, and optimization. Emphasis on engineering applications. (Northwestern)

MATH 230-1: Multivariable Differential Calculus (1) math

Vectors, vector functions, partial derivatives, and optimization. Not open to students in the McCormick School of Engineering. (Northwestern)

MATH 240-0: Linear Algebra (1) math

Elementary linear algebra: systems of linear equations, matrix algebra, subspaces, determinants, eigenvalues, eigenvectors, and orthogonality. (Northwestern)

MATH 310-1: Probability and Stochastic Processes (1) math

Axioms of probability. Conditional probability and independence. Random variables. Joint distributions. Expectation. Limit theorems: the weak law of large numbers and the central limit theorem. (Northwestern)

MTH1111: Modeling and Simulation of the Physical World (2) math

This course provides an introduction to mathematical modeling and computer simulation of physical systems. Working with a broad range of examples, students practice the steps involved in modeling and analyzing a physical system, learn the role of models in explaining and predicting the behavior of the physical world, and develop skills with the programming and computational tools necessary for simulation. Students work in a studio environment on increasingly open-ended projects, and learn how to present their results, with an emphasis on visual and oral communication. (Olin)

MTH2110: Discrete Math (4) math

Discrete Mathematics is a course that will introduce students to advanced counting and partitioning techniques as well as widely applicable discrete structures such as graphs and trees. This class will emphasize creative problem solving, mathematical writing, and collaboratively carrying out small-group projects. (Olin)

MTH2130: Probability and Statistics (variable) math

An introduction to probability and statistics, with applications to science, engineering, and social science. Topics include discrete and continuous probability distributions; moments; conditional probability; Bayes' Rule; point and interval estimation; hypothesis testing. (Olin)

MTH2131: Data Science (2) math

Data Science is a powerful toolkit for using data to answer questions and guide decision making. It involves skills and knowledge from statistics, software engineering, machine learning, and data engineering. In this class, students work on data science projects that involve collecting data or finding data sources, exploratory data analysis and interactive visualization, statistical analysis, predictive analytics, model selection and validation. Course work involves readings and case studies on ethical practice in data science. This course may be used to satisfy the Probability and Statistics requirement. (Olin)

MTH2133: Computational Bayesian Statistics (2) math

Bayesian statistics provide a powerful toolkit for modeling random processes and making predictions. The ideas behind these tools are simple, but expressing them mathematically can make them hard to learn and apply. This class takes a computational approach, which allows students with programming experience to use that knowledge as leverage. Students will work through a series of exercises in the book, Think Bayes, and help develop new material. (Olin)

MTH2135: Neurotechnology, Brains and Machines (2) math

Neurotechnology falls in the intersection of engineering, data science, and neuroscience. This area involves work in how humans can use machines to understand how we think and how to make machines that can think. Advances in neurotechnology will likely lead to new treatments for brain disorders, repair and augmentation of our sensory and motor systems, and shifts in computation strategies. In this course, students will learn about cutting-edge technologies used to understand and emulate the brain, develop statistical data analysis skills to conduct and understand neurotechnology research, and discuss the cultural and ethical implications of these advances. Course work will involve analysis of data from neuroscience, reading and synthesizing articles from research journals, and project work. (Olin)

MTH2136: Astronomy and Statistics: AstroStats (2) math

It's not science unless you quantify your errors. Learn statistics and error analysis by studying our dynamic solar system. The first half of the class will provide you with a toolbox of standard statistical methods. You will learn these methods by studying data from planets, moons, and asteroids. The second half consists of student-designed projects. Your project will investigate an element of our solar system, and will include rigorous error analysis. This course will use data from NASA and ESA missions. (Olin)

ENGX2000: Quantitative Engineering Analysis 1 (13) mathengr

Quantitative Engineering Analysis 1 is the first in a series of interdisciplinary math, science and engineering courses. The application of quantitative analysis of mathematical models and/or data can enable, improve, and speed up the engineering design process. Using quantitative analysis to answer engineering questions, you will be able to make the choices necessary to successfully complete an engineering design. Whether you are selecting the best part from a catalog, choosing an appropriate material, sizing a component, determining the effect of certain influences on your design, or optimizing your design within a parameter space, you often need to obtain (through experiment or calculation) and interpret quantitative information to inform your decisions. There are many different approaches to getting and interpreting the data you need: you may conduct an experiment, do a rough estimation, perform a detailed calculation based on mathematical models, or create a computer simulation. If you want to engineer effectively, you must be able to choose and use appropriate quantitative tools for a given situation. In this class, you will be introduced to various approaches to perform quantitative engineering analysis through real-world examples. You will learn how to select between different tools and different approaches within the context of an engineering challenge, how to use many different tools for quantitative analysis, and how to acquire new tools on your own in the future. This course fulfills the linear algebra requirement. (Olin)

ENGX2006: Quantitative Engineering Analysis 2 (13) mathengr

Quantitative Engineering Analysis 2 is the 2nd course in a 3 course interdisciplinary sequence. The application of quantitative analysis of mathematical models and/or data can enable, improve, and speed up the engineering design process. Using quantitative analysis to answer engineering questions, you will be able to make the choices necessary to successfully complete an engineering design. Whether you are selecting the best part from a catalog, choosing an appropriate material, sizing a component, determining the effect of certain influences on your design, or optimizing your design within a parameter space, you often need to obtain (through experiment or calculation) and interpret quantitative information to inform your decisions. There are many different approaches to getting and interpreting the data you need: you may conduct an experiment, do a rough estimation, perform a detailed calculation based on mathematical models, or create a computer simulation. If you want to engineer effectively, you must be able to choose and use appropriate quantitative tools for a given situation. In this class, you will be introduced to various approaches to perform quantitative engineering analysis through real-world examples. You will learn how to select between different tools and different approaches within the context of an engineering challenge, how to use many different tools for quantitative analysis, and how to acquire new tools on your own in the future. This course fulfills the multivariable calculus requirement. (Olin)

ENGX2011: Quantitative Engineering Analysis 3 (33) mathengr

Quantitative Engineering Analysis 3 is the third course in the 12-credit QEA sequence required for some degree programs. The course will revisit, reinforce, and build upon the contextualized math, science, and engineering tools and skills developed during QEA 1 and 2. Conceptual material in QEA 3 will draw from topics including ordinary differential equations, Fourier transforms, and equations of motion. QEA 3 will endeavor to place this foundational material in the broader engineering context, drawing connections to relevant examples and applications in engineering and beyond. The course will teach students how to select the appropriate set of tools and techniques for a given situation, ask critical questions about the consequences of their work, and develop the skills needed to acquire new knowledge beyond the course material. This course fulfills the ordinary differential equations requirement, and when coupled with Quantitative Engineering Analysis 2. (Olin)

MATH 058: Introduction to Statistics with lab (1) math

An introduction to the methodology and tools vital to the researcher in both the sciences and social sciences. Introduction to probability; binomial, normal, t and Chi-squared distributions; testing hypotheses; confidence intervals; analysis of variance; and regression and correlation analysis. Concepts will be applied to current data using statistical computer software. (Pomona)

MATH 060: Linear Algebra (1) math

Emphasizes vector spaces and linear transformations. Linear independence and bases, null spaces and ranks of linear transformations, the algebra of linear transformations, the representation of linear transformations by matrices. Additional topics may include Gaussian elimination, inner product spaces; determinants, eigenvalues; and applications of linear algebra. (Pomona)

COS 240: Reasoning About Computation (1) math

An introduction to mathematical topics relevant to computer science. Combinatorics, probability and graph theory will be covered in the context of computer science applications. The course will present a computer science approach to thinking and modeling. Students will be introduced to fundamental concepts in theoretical computer science, such as NP-completeness and cryptography that arise from the world view of efficient computation. R. Raz, M. Braverman (Princeton)

COS 302: Mathematics for Numerical Computing and Machine Learning (1) math

This course provides a comprehensive and practical background for students interested in continuous mathematics for computer science. The goal is to prepare students for higher-level subjects in artificial intelligence, machine learning, computer vision, natural language processing, graphics, and other topics that require numerical computation. This course is intended students who wish to pursue these more advanced topics, but who have not taken (or do not feel comfortable) with university-level multivariable calculus (e.g., MAT 201/203) and probability (e.g., ORF 245 or ORF 309). R. Adams (Princeton)

COS 323: Computing and Optimization for the Physical and Social Sciences (1) math

(Princeton)

COS 342: Introduction to Graph Theory. Same as MAT 375 (1) math

(Princeton)

COS 433: Cryptography (1) math

An introduction to modern cryptography with an emphasis on fundamental ideas. The course will survey both the basic information and complexity-theoretic concepts as well as their (often surprising and counter-intuitive) applications. (Princeton)

COS 488: Introduction to Analytic Combinatorics (1) math

Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines. This course combines motivation for the study of the field with an introduction to underlying techniques, by covering as applications the analysis of numerous fundamental algorithms from computer science. The second half of the course introduces Analytic Combinatorics, starting from basic principles. (Princeton)

MAT 103: Calculus I (1) math

First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. (Princeton)

MAT 104: Calculus II (1) math

Continuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers. (Princeton)

MAT 201: Multivariable Calculus (1) math

Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields, and Stokes's theorem. (Princeton)

MAT 202: Linear Algebra with Applications (1) math

Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems. (Princeton)

MAT 203: Advanced Vector Calculus (1) math

Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 216/218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. (Princeton)

MAT 204: Advanced Linear Algebra with Applications (1) math

Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. (Princeton)

MAT 217: Honors Linear Algebra (1) math

A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. (Princeton)

MAT 218: Multivariable Analysis and Linear Algebra II (1) math

Continuation of Multivariable Analysis and Linear Algebra I (MAT 216) from the fall. A rigorous course in analysis with an emphasis on proof rather than applications. Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini. (Princeton)

CS 31400: Numerical Methods (3) math

Iterative methods for solving nonlinear equations; direct and iterative methods for solving linear systems; approximations of functions, derivatives, and integrals; error analysis. (Purdue)

CS 35500: Introduction To Cryptography (3) math

An introduction to cryptography basics: Classic historical ciphers including Caesar, Vigenere, and Vernam ciphers; modern ciphers including DES, AES, Pohlig-Hellman, and RSA; signatures and digests; key exchange; simple protocols; block and stream ciphers; network-centric protocols. (Purdue)

CS 50025: Foundations Of Decision Making (1) math

This course provides an overview of data science methods used for data-driven discovery, extraction of knowledge, and informed decision making. The course covers fundamental computational methods and statistical techniques used to correctly reason about uncertainty, conduct hypothesis tests, infer causal relationships, and apply and evaluate predictive models. The course highlights how sampling biases can impact fairness in decision making. Throughout, students get hands-on experience on how to make correct and explainable inferences from data. Experience in Python Progamming, Probability, Statistics and Linear Algebra is required. Typically offered Fall Spring Summer. (Purdue)

CS 51500: Numerical Linear Algebra (3) math

Direct and iterative solvers of dense and sparse linear systems of equations, numerical schemes for handling symmetric algebraic eigenvalue problems, and the singular-value decomposition and its applications in linear least squares problems. (Purdue)

CS 51501: Parallelism In Numerical Linear Algebra (3) math

This course examines both theoretical and practical aspects of numerical algorithm design and implementation on parallel computing platforms. In particular, it provides an understanding of the tradeoff between arithmetic complexity and management of hierarchical memory structures, roundoff characteristics if different from the sequential schemes, and performance evaluation and enhancement. Applications are derived from a variety of computational science and engineering areas. (Purdue)

CS 52000: Computational Methods In Optimization (3) math

A treatment of numerical algorithms and software for optimization problems with a secondary emphasis on linear and nonlinear systems of equations: unconstrained and constrained optimization; line search methods; trust region methods; Quasi-Newton methods; linear programming; calculating derivatives; quadratic programming; global optimization, including simulated annealing. (Purdue)

MA 16100: Plane Analytic Geometry And Calculus I (5) math

Introduction to differential and integral calculus of one variable, with applications. Some schools or departments may allow only 4 credit hours toward graduation for this course. Designed for students who have not had at least a one-semester calculus course in high school, with a grade of “A” or “B”. Not open to students with credit in MA 16500. Demonstrated competence in college algebra and trigonometry. (Purdue)

MA 16200: Plane Analytic Geometry And Calculus II (5) math

Continuation of MA 16100. Vectors in two and three dimensions, techniques of integration, infinite series, conic sections, polar coordinates, surfaces in three dimensions. Some schools or departments may allow only 4 credit hours toward graduation for this course. (Purdue)

MA 16500: Analytic Geometry And Calculus I (4) math

Introduction to differential and integral calculus of one variable, with applications. Conic sections. Designed for students who have had at least a one-semester calculus course in high school, with a grade of “A” or “B”, but are not qualified to enter MA 16200 or MA 16600 or the advanced placement courses MA 17300 or the honors calculus course MA 18100. Demonstrated competence in college algebra and trigonometry. (Purdue)

MA 16600: Analytic Geometry And Calculus II (4) math

Continuation of MA 16500. Vectors in two and three dimensions. Techniques of integration, infinite series, polar coordinates, surfaces in three dimensions. Not open to students with credit in MA 16200. (Purdue)

STAT 24200: Introduction To Data Science (3) math

This course provides a broad introduction to the field of data science. The course focuses on using computational methods and statistical techniques to analyze massive amounts of data and to extract knowledge. It provides an overview of foundational computational and statistical tools for data acquisition and cleaning, data management and big data systems. The course surveys the complete data science process from data to knowledge and gives students hands-on experience with tools and methods. Basic knowledge of Python required. (Purdue)

MA 26100: Multivariate Calculus (4) math

Planes, lines, and curves in three dimensions. Differential calculus of several variables; multiple integrals. Introduction to vector calculus. Not open to students with credit in MA 17400 or 27100. (Purdue)

MA 26500: Linear Algebra (3) math

Introduction to linear algebra. Systems of linear equations, matrix algebra, vector spaces, determinants, eigenvalues and eigenvectors, diagonalization of matrices, applications. Not open to students with credit in MA 26200, 27200, 35000 or 35100. (Purdue)

MA 26600: Ordinary Differential Equations (3) math

First order equations, second and n’th order linear equations, series solutions, solution by Laplace transform, systems of linear equations. It is preferable but not required to take MA 26500 either first or concurrently. Not open to students with credit in MA 26200, 27200, 36000, 36100, or 36600. (Purdue)

MA 27101: Honors Multivariate Calculus (5) math

This course is the Honors version of MA 26100, Multivariate Calculus; it will also include a review of infinite series. The course is intended for first-year students who have credit for Calculus I and II. There will be a significant emphasis on conceptual explanation, but not on formal proof. Permission of department is required. (Purdue)

MA 34100: Foundations Of Analysis (3) math

An introductory course in rigorous analysis, covering real numbers, sequences, series, continuous functions, differentiation, and Riemann integration. MA 30100 is helpful but not required. (Purdue)

STAT 35000: Introduction To Statistics (3) math

This course provides a data-oriented introduction to applied statistics, covering exploratory data analysis, experimental design, probability distributions, simulation, sampling distributions, and the Central Limit Theorem. Students will learn the fundamentals of statistical inference, including confidence intervals and hypothesis tests for population means, paired and independent comparisons of means, analysis of variance, and regression. The course emphasizes hands-on experience with statistical software and is primarily intended for students majoring in the mathematical sciences. (Purdue)

MA 35100: Elementary Linear Algebra (3) math

Systems of linear equations, finite dimensional vector spaces, matrices, determinants, eigenvalues and eigenvector applications to analytical geometry. (Purdue)

MA 35301: Linear Algebra II (3) math

Theoretical background for methods and results that appear in MA 35100. Inner products, orthogonality, and applications including least squares. (Purdue)

STAT 35500: Statistics For Data Science (3) math

An introduction to methodologies for data analysis and simulation. Populations and sampling. Distributions and summaries of distributions. Algorithms for sampling and resampling. Foundational statistical concepts including confidence intervals, hypothesis testing, correlation. Introduction to classification and regression. Essential use is made of statistical software throughout. (Purdue)

MA 36200: Topics In Vector Calculus (3) math

Multivariate calculus; partial differentiation; implicit function theorems and transformations; line and surface integrals; vector fields; theorems of Gauss, Green, and Stokes. Credit granted for only one of MA 36200 and MA 51000. (Purdue)

MA 36600: Ordinary Differential Equations (4) math

An introduction to ordinary differential equations with emphasis on problem solving and applications. The one-hour computer lab will give students an opportunity for hands-on experience with both the theory and applications of the subject. (Purdue)

MA 38500: Introduction To Logic (3) math

Propositional calculus and predicate calculus with applications to mathematical proofs, valid arguments, switching theory, and formal languages. (Purdue)

MA 41600: Probability (3) math

An introduction to mathematical probability suitable as a preparation for actuarial science, statistical theory, and mathematical modeling. General probability rules, conditional probability and Bayes theorem, discrete and continuous random variables, moments and moment generating functions, joint and conditional distributions, standard discrete and continuous distributions and their properties, law of large numbers and central limit theorem. (Purdue)

STAT 41700: Statistical Theory (3) math

An introduction to the mathematical theory of statistical inference, emphasizing inference for standard parametric families of distributions. Properties of estimators. Maximum likelihood estimation. Sufficient statistics. Hypothesis tests and confidence intervals. Distribution theory for common statistics based on normal distributions, including linear regression. Bayesian Statistics include posterior inference, posterior mean, maximum a-posteriori estimator, credible intervals, and Bayesian hypothesis testing. (Purdue)

STAT 42000: Introduction To Time Series (3) math

An introduction to time series analysis suitable for students of actuarial science, engineering, and the sciences. Model building and forecasting with ARMA and ARIMA models. Basic financial volatility models (ARCH and GARCH). Resampling methods for confidence intervals. Basics of spectral analysis, including spectral density estimation and periodograms. (Purdue)

MA 42100: Linear Programming And Optimization Techniques (3) math

Solution of linear programming problems by the simplex method, duality theory, transportation problems, assignment problems, network analysis, dynamic programming. (Purdue)

MA 43200: Elementary Stochastic Processes (3) math

An introduction to some classes of stochastic processes that arise in probabilistic models of time-dependent random processes. The main stochastic processes studied will be discrete time Markov chains and Poisson processes. Other possible topics covered may include continuous time Markov chains, renewal processes, queueing networks, and martingales. (Purdue)

MA 44000: Honors Real Analysis I (3) math

Real analysis in one and n-dimensional Euclidean spaces. Topics include the completeness property of real numbers, topology of Euclidean spaces, Heine-Borel theorem, convergence of sequences and series in Euclidean spaces, limit superior and limit inferior, Bolzano-Weierstrass theorem, continuity, uniform continuity, limits and uniform convergence of functions, Riemann or Riemann-Stieltjes integrals. (Purdue)

MA 45300: Elements Of Algebra I (3) math

Fundamental properties of integers, polynomials, groups, rings, and fields, with emphasis on problem solving and applications. Not open to students with credit in MA 45000. (Purdue)

STAT 50600: Statistical Programming And Data Management (3) math

Use of the SAS software system for managing statistical data. How to write programs to access, explore, prepare, and analyze data. Using the DATA step and procedures to access, transform, and summarize data. Introduction to the SAS macro language. Prepares students for the base SAS certification exam. (Purdue)

STAT 51100: Statistical Methods (3) math

Descriptive statistics; elementary probability; sampling distributions; inference, testing hypotheses, and estimation; normal, binomial, Poisson, hypergeometric distributions; one-way analysis of variance; contingency tables; regression. For statistics majors and minors, credit should be allowed in no more than one of STAT 30100, STAT 30301, STAT 35000, STAT 35500, STAT 50100, and in no more than one of STAT 50300 and STAT 51100. (Purdue)

STAT 51200: Applied Regression Analysis (3) math

Inference in simple and multiple linear regression, residual analysis, transformations, polynomial regression, model building with real data, nonlinear regression. One-way and two-way analysis of variance, multiple comparisons, fixed and random factors, analysis of covariance. Use of existing statistical computer programs. (Purdue)

STAT 51300: Statistical Quality Control (3) math

A strong background in control charts including adaptations, acceptance sampling for attributes and variables data, standard acceptance plans, sequential analysis, statistics of combinations, moments and probability distributions, applications. (Purdue)

STAT 51400: Design Of Experiments (3) math

Fundamentals, completely randomized design; randomized complete blocks; latin square; multi-classification; factorial; nested factorial; incomplete block and fractional replications for 2n, 3n, 2m x 3n; confounding; lattice designs; general mixed factorials; split plot; analysis of variance in regression models; optimum design. Use of existing statistical programs. (Purdue)

STAT 52200: Sampling And Survey Techniques (3) math

This course covers basic sampling design and analysis techniques. Sampling designs include: simple random, stratified, clustered, multi-staged, and systematic samples. Methods of estimation appropriate to design features and efficiency and costs related to sample design are covered. (Purdue)

STAT 52500: Intermediate Statistical Methodology (3) math

Statistical methods for analyzing data based on general/generalized linear models, including linear regression, analysis of variance (ANOVA), analysis of covariance (ANCOVA), random and mixed effects models, and logistic/loglinear regression models. Application of these methods to real-world problems using SAS statistical software. (Purdue)

COMP 323: Introduction to Mathematical Cryptography (3) math

The course introduces students to modern cryptographic techniques, focusing mainly on mathematical tools. The course covers topics such as Diffie-Hellman key exchange, the ElGamal public key crypto system, integer factorization and RSA, and elliptic curves and lattices in cryptography (Rice)

COMP 409: Advanced Logic in Computer Science (3) math

Logic has been called 'the calculus of computer science'. The argument is that logic plays a fundamental role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of Computer Science as disparate as artificial intelligence (automated reasoning), architecture (logic gates), software engineering (specification and verification), programming languages (semantics, logic programming), databases (relational algebra and SQL), algorithms (complexity and expressiveness), and theory of computation (general notions of computability). (Rice)

COMP 423: Intro to Math Cryptography (3) math

The course introduces students to modern cryptographic techniques, focusing mainly on mathematical tools. The course covers topics such as Diffie-Hellman key exchange, the ElGamal public key crypto system, integer factorization and RSA, and elliptic curves and lattices in cryptography. (Rice)

COMP 448: Concrete Mathematics (3) math

Concrete mathematics is a blend of continuous and discrete mathematics. Major topics include sums, recurrences, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability and asymptotic methods. (Rice)

MATH 101: Single Variable Calculus I (3) math

Limits, continuity, differentiation, integration, and the Fundamental Theorem of Calculus. (Rice)

MATH 102: Single Variable Calculus II (3) math

Continuation of MATH 101. Includes further techniques of integration, as well as infinite sequences and series, Taylor polynomials and Taylor series, parametric equations, arc length, polar coordinates, complex numbers, and Fourier polynomials. (Rice)

MATH 105: Ap/Oth Credit in Calculus I (3) math

Provides transfer credit based on student performance on approved examinations in calculus (Rice)

MATH 106: Ap/Oth Credit in Calculus II (3) math

Provides transfer credit based on student performance on approved examinations in calculus, such as the BC Calculus Advanced Placement exam or the International Baccalaureate higher-level calculus exams. (Rice)

MATH 111: Calculus: Differentiation (3) math

Study of calculus, forming with MATH 112 a version of MATH 101/102 that does not cover infinite series. MATH 111 covers functions, limits, continuity, and derivatives and their applications. (Rice)

MATH 112: Calculus: Integration and Its Applications (3) math

Continuation of the study of calculus from MATH 111. Integration, the Fundamental Theorem of Calculus, techniques of integration and applications. (Rice)

MATH 212: Multivariable Calculus (3) math

Calculus of multiple variables. Vectors, partial derivatives and gradients, double and triple integrals, vector fields, line and surface integrals, Green's theorem, Stokes's theorem, and Gauss's theorem. (Rice)

MATH 232: Honors Multivariable Calculus (3) math

Calculus of several variables (partial derivatives and gradients, double and triple integrals, vector fields, line and surface integrals, Green's theorem, Stokes's theorem, Gauss's theorem). Content is similar to that of MATH 212, but MATH 232 will use linear algebra to extend results to R^n. There will also be an emphasis on rigorous mathematical arguments. (Rice)

CMOR 302: Matrix Analysis (3) math

Equilibria and the solution of linear systems and linear least squares problems. Eigenvalue problem and its application to solve dynamical systems. Singular value decomposition and its application. (Rice)

CMOR 303: Matrix Analysis Data Science (3) math

Solution of linear systems and linear least squares problems. Eigenvalue problem and singular value decomposition. Introduction to gradient based methods. Applications to data science. (Rice)

STAT 310: Probability and Statistics (3) math

Probability and the central concepts and methods of statistics including probability, random variables, distributions of random variables, expectation, sampling distributions, estimation, confidence intervals, and hypothesis testing (Rice)

STAT 311: Honors Probability and Mathematical Statistics (3) math

Probability and the central concepts and methods of statistics including probability, random variables, distributions of random variables, expectation, sampling distributions, estimation, confidence intervals, and hypothesis testing. Advanced topics (not covered in STAT 310 or STAT 315) include the modeling stochastic phenomena and asymptotic statistical theory. Intended for students wishing to understand more rigorous statistical theory and for those contemplating a BS degree in Statistics or graduate school in statistical science. (Rice)

STAT 312: Prob & Stat for Engineers (3) math

Probability and the central concepts and methods of statistics including probability, distributions of random variables, expectation, sampling distributions, estimation, confidence intervals, and hypothesis testing. Examples are predominantly from civil and environmental engineering. (Rice)

STAT 315: Statistics for Data Science (4) math

An introduction to mathematical statistics and computation for applications to data science. Topics include probability, random variables expectation, sampling distributions, estimation, confidence intervals, hypothesis testing and regression. A weekly lab will cover the statistical package, R, and data projects. (Rice)

MATH 354: Honors Linear Algebra (3) math

Vector spaces, linear transformations and matrices, theory of systems of linear equations, determinants, eigenvalues and diagonalizability, inner product spaces; and optional material chosen from: dual vector spaces, spectral theorem for self-adjoint operators, Jordan canonical form. Content is similar to that of MATH 355, but with more emphasis on theory. The course will include instruction on how to construct mathematical proofs. This course is appropriate for potential Mathematics majors and others interested in learning how to construct rigorous mathematical arguments. (Rice)

MATH 355: Linear Algebra (3) math

Systems of linear equations, matrices, vector spaces, linear transformations, determinants, inner products, eigenvalues and eigenvectors, and the Spectral Theorem for real symmetric matrices (Rice)

CSSE 479: Cryptography (4) math

Introduction to basic ideas of modern cryptography with emphasis on mathematical background and practical implementation. Topics include: the history of cryptography and cryptanalysis, public and private key cryptography, digital signatures, and limitations of modern cryptography. Touches upon some of the societal issues of cryptography. (Rose-Hulman)

MA 111: Calculus I (5) math

Calculus and analytic geometry in the plane. Algebraic and transcendental functions. Limits and continuity. Differentiation, geometric and physical interpretations of the derivative, Newton’s method. Introduction to integration and the Fundamental Theorem of Calculus. A student cannot earn credit for both MA 105 and MA 111. (Rose-Hulman)

MA 112: Calculus II (5) math

Techniques of integration, numerical integration, applications of integration. L’Hopital’s rule and improper integrals. Separable first order differential equations, applications of separable first order differential equations. Series of constants, power series, Taylor polynomials, Taylor and McLaurin series. (Rose-Hulman)

MA 113: Calculus III (5) math

Vectors and parametric equations in three dimensions. Functions of several variables, partial derivatives, maxima and minima of functions of several variables, multiple integrals, and other coordinate systems. Applications of partial derivatives and multiple integrals. (Rose-Hulman)

MA 221: Matrix Algebra & Differential Equations I (4) math

First order scalar differential equations including basic solution techniques and numerical methods. Second order linear, constant coefficient differential equations, including both the homogeneous and non-homogeneous cases. Basic matrix algebra with emphasis on understanding systems of linear equations from algebraic and geometric viewpoints, and eigenvalues and eigenvectors. Introduction to complex arithmetic. Applications to problems in science and engineering. (Rose-Hulman)

MA 276: Introduction to Proofs (4) math

Introduction to writing mathematical proofs. Logic: direct proof, contradiction, contrapositive, counterexamples. Induction. Recursion. Sets: relations (order, equivalence), functions. Properties of infinite sets. Basic number theory. Important preparation for further courses in theoretical mathematics. (Rose-Hulman)

MA 374: Combinatorics (4) math

A first course in combinatorics. Basic counting principles, permutations, combinations. Combinatorial proof. The pigeonhole principle. The principle of inclusion/exclusion. Generating functions. Recurrence relations. Additional topics in combinatorics, which may include permutation groups and Burnside's Lemma, Polya enumeration, multivariate generating functions, combinatorial designs, Ramsey theory, order relations, or other topics at the discretion of the instructor. (Rose-Hulman)

MA 381: Introduction to Probability with Applications to Statistics (4) math

Introduction to probability theory; axioms of probability, sample spaces, and probability laws (including conditional probabilities). Univariate random variables (discrete and continuous) and their expectations including these distributions: binomial, Poisson, geometric, uniform, exponential, and normal. Introduction to moment generating functions. Introduction to jointly distributed random variables. Univariate and joint transformations of random variables. The distribution of linear combinations of random variables and an introduction to the Central Limit Theorem. Applications of probability to statistics. (Rose-Hulman)

CS 109: Introduction to Probability for Computer Scientists (35) math

Topics include: counting and combinatorics, random variables, conditional probability, independence, distributions, expectation, point estimation, and limit theorems. Applications of probability in computer science including machine learning and the use of probability in the analysis of algorithms. (Stanford)

CS 109ACE: Problem-solving Lab for CS 109 (1) math

Additional problem solving practice for the introductory CS course CS 109. Sections are designed to allow students to acquire a deeper understanding of CS and its applications, work collaboratively, and develop a mastery of the material. Enrollment limited to 30 students, permission of instructor required. Concurrent enrollment in CS 109 required. (Stanford)

CS 205L: Continuous Mathematical Methods with an Emphasis on Machine Learning (3) math

A survey of numerical approaches to the continuous mathematics used throughout computer science with an emphasis on machine and deep learning... (Stanford)

CS 250: Algebraic Error Correcting Codes (EE 387) (3) math

Introduction to the theory of error correcting codes, emphasizing algebraic constructions, and diverse applications throughout computer science and engineering. Topics include basic bounds on error correcting codes; Reed-Solomon and Reed-Muller codes; list-decoding, list-recovery and locality. Applications may include communication, storage, complexity theory, pseudorandomness, cryptography, streaming algorithms, group testing, and compressed sensing. (Stanford)

CS 269O: Introduction to Optimization Theory (MS&E 213) (3) math

Introduction of core algorithmic techniques and proof strategies that underlie the best known provable guarantees for minimizing high dimensional convex functions. Focus on broad canonical optimization problems and survey results for efficiently solving them, ultimately providing the theoretical foundation for further study in optimization. (Stanford)

CS 351: Open Problems in Coding Theory (3) math

Coding theory is the study of how to encode data to protect it from noise. Coding theory touches CS, EE, math, and many other areas, and there are exciting open problems at all of these frontiers. In this class, we will explore these open problems by reading recent research papers and thinking about some open problems together. Required work will involve reading and presenting research papers, as well as working in small groups at these open problems and presenting progress. (Solving an open problem is not required!) Topics will depend on student interest and may include locality, coded computation, index coding, interactive communication, and group testing. (Stanford)

MATH 19: Calculus (3) math

Introduction to differential calculus of functions of one variable. Review of elementary functions (including exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative. (Stanford)

MATH 20: Calculus (3) math

The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initial-value problems, exponential and logistic models, direction fields, and parametric curves. (Stanford)

MATH 21: Calculus (4) math

This course addresses a variety of topics centered around the theme of 'calculus with infinite processes', largely the content of BC-level AP Calculus that isn't in the AB-level syllabus. (Stanford)

MATH 51: Linear Algebra, Multivariable Calculus, and Modern Applications (5) math

This course provides unified coverage of linear algebra and multivariable differential calculus, and the free course e-text connects the material to many fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion includes orthogonality, linear independence, matrix algebra, and eigenvalues with applications such as least squares, linear regression, and Markov chains (relevant to population dynamics, molecular chemistry, and PageRank); the singular value decomposition (essential in image compression, topic modeling, and data-intensive work in many fields) is introduced in the final chapter of the e-text. The multivariable calculus portion includes unconstrained optimization via gradients and Hessians (used for energy minimization), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underlie many machine learning algorithms, such as backpropagation), and Newton's method (an ingredient in GPS and robotics). The course emphasizes computations alongside an intuitive understanding of key ideas. The widespread use of computers makes it important for users of math to understand concepts: novel users of quantitative tools in the future will be those who understand ideas and how they fit with examples and applications. (Stanford)

MATH 52: Integral Calculus of Several Variables (5) math

Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. (Stanford)

MATH 53: Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications (5) math

Ordinary differential equations and initial value problems, linear systems of such equations with an emphasis on second-order constant-coefficient equations, stability analysis for non-linear systems (including phase portraits and the role of eigenvalues), and numerical methods. Partial differential equations and boundary-value problems, Fourier series and initial conditions, and Fourier transform for non-periodic phenomena. Throughout the development we harness insights from linear algebra, and software widgets are used to explore course topics on a computer (no coding background is needed). The free e-text provides motivation from applications across a wide array of fields (biology, chemistry, computer science, economics, engineering, and physics) described in a manner not requiring any area-specific expertise, and it has an appendix on Laplace transforms with many worked examples as a complement to the Fourier transform in the main text. (Stanford)

MATH 104: Applied Matrix Theory (4) math

Linear algebra for applications in science and engineering. The course introduces the key mathematical ideas in matrix theory, which are used in modern methods of data analysis, scientific computing, optimization, and nearly all quantitative fields of science and engineering. While the choice of topics is motivated by their use in various disciplines, the course will emphasize the theoretical and conceptual underpinnings of this subject. Topics include orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares methods, and algorithms for solving systems of linear equations; applications include clustering, principal component analysis and dimensionality reduction, regression. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and ENGR 108 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of ENGR 108 is on a few linear algebra concepts, and many applications. (Stanford)

MATH 107: Graph Theory (4) math

An introductory course in graph theory establishing fundamental concepts and results in variety of topics. Topics include: basic notions, connectivity, cycles, matchings, planar graphs, graph coloring, matrix-tree theorem, conditions for hamiltonicity, Kuratowski's theorem, Ramsey and Turan-type theorem. (Stanford)

MATH 108: Introduction to Combinatorics and Its Applications (4) math

Topics: graphs, trees (Cayley's Theorem, application to phylogony), eigenvalues, basic enumeration (permutations, Stirling and Bell numbers), recurrences, generating functions, basic asymptotics. (Stanford)

MATH 109: Groups and Symmetry (4) math

Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. Honors math majors and students who intend to do graduate work in mathematics should take 120. WIM. (Stanford)

MATH 110: Number Theory for Cryptography (4) math

Number theory and its applications to modern cryptography. Topics include: congruences, primality testing and factorization, public key cryptography, and elliptic curves, emphasizing algorithms. Includes an introduction to proof-writing. This course develops math background useful in CS 255. WIM. (Stanford)

MATH 113: Linear Algebra and Matrix Theory (4) math

Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; dual space and dual basis; eigenvectors and eigenvalues; diagonalization. Includes an introduction to proof-writing. ( MATH 104 offers a more application-oriented treatment.) (Stanford)

CPSC 049: The Probabilistic Method (1) math

In mathematics and theoretical computer science, we often consider classes of objects (say graphs, circuits or matrices) and we'd like to know if there are objects that have certain nice properties. One way to show these nice objects exist is to look at a random object, and show it has the nice property with nonzero probability. If this is true, there must be some object with this nice property. This is the Probabilistic Method in a nutshell. It has become an essential tool for understanding structure of lots and lots of things in theoretical computer science and combinatorics, even in problems and applications which involve no randomness at all. This class will start from the ground up, first introducing discrete probability theory, then covering the probabilistic method in detail: how it works, extensions, and most of all lots of applications. We'll also spend a few weeks discussing NP-Completeness and randomized algorithms. (Swarthmore)

CSCE 442: Scientific Programming (3) math

Introduction to numerical algorithms fundamental to scientific and engineering applications of computers; elementary discussion of error; algorithms, efficiency; polynomial approximations, quadrature and systems of algebraic and differential equations. (Texas A&M)

MATH 140: Mathematics for Business and Social Sciences (3) math

Application of common algebraic functions, including polynomial, exponential, logarithmic and rational, to problems in business, economics and the social sciences; includes mathematics of finance, including simple and compound interest and annuities; systems of linear equations; matrices; linear programming; and probability, including expected value. (Texas A&M)

MATH 142: Business Calculus (3) math

Limits and continuity; techniques and applications of derivatives including curve sketching and optimization; techniques and applications of integrals; emphasis on applications in business, economics, and social sciences. (Texas A&M)

MATH 147: Calculus I for Biological Sciences (4) math

Introduction to differential calculus in a context that emphasizes applications in the biological sciences. (Texas A&M)

MATH 148: Calculus II for Biological Sciences (4) math

Introduction to integral calculus in a context that emphasizes applications in the biological sciences; ordinary differential equations and analytical geometry. (Texas A&M)

MATH 151: Engineering Mathematics I (4) math

Rectangular coordinates, vectors, analytic geometry, functions, limits, derivatives of functions, applications, integration, computer algebra. (Texas A&M)

MATH 152: Engineering Mathematics II (4) math

Differentiation and integration techniques and their applications (area, volumes, work), improper integrals, approximate integration, analytic geometry, vectors, infinite series, power series, Taylor series, computer algebra. (Texas A&M)

MATH 168: Finite Mathematics (3) math

Linear equations and applications; systems of linear equations, matrix algebra and applications, linear programming, probability and applications, statistics. (Texas A&M)

MATH 171: Calculus I (4) math

Vectors, functions, limits, derivatives, Mean Value Theorem, applications of derivatives, integrals, Fundamental Theorem of Calculus. Designed to be more demanding than MATH 151. (Texas A&M)

MATH 172: Calculus II (4) math

Techniques of integration, applications of integrals, improper integrals, sequences, infinite series, vector algebra and solid analytic geometry. Designed to be more demanding than MATH 152. (Texas A&M)

STAT 211: Principles of Statistics I (3) math

Introduction to probability and probability distributions; sampling and descriptive measures; inference and hypothesis testing; linear regression, analysis of variance. (Texas A&M)

STAT 212: Principles of Statistics II (3) math

Design of experiments, model building, multiple regression, nonparametric techniques and contingency tables. (Texas A&M)

MATH 251: Engineering Mathematics III (3) math

Vector algebra, calculus of functions of several variables, partial derivatives, directional derivatives, gradient, multiple integration, line and surface integrals, Green's and Stokes' theorems. (Texas A&M)

STAT 301: Introduction to Biometry (3) math

Intended for students in animal sciences. Introduces fundamental concepts of biometry including measures of location and variation, probability, tests of significance, regression, correlation and analysis of variance which are used in advanced courses and are being widely applied to animal-oriented industry. (Texas A&M)

STAT 302: Statistical Methods (3) math

Intended for undergraduates in the biological sciences. Introduction to concepts of random sampling and statistical inference; estimation and testing hypotheses of means and variances; analysis of variance; regression analysis; chi-square tests. (Texas A&M)

STAT 303: Statistical Methods (3) math

Intended for undergraduates in the social sciences. Introduction to concepts of random sampling and statistical inference, estimation and testing hypotheses of means and variances, analysis of variance, regression analysis, chi-square tests. (Texas A&M)

MATH 304: Linear Algebra (3) math

Introductory course in linear algebra covering abstract ideas of vector space and linear transformation as well as models and applications of these concepts, such as systems of linear equations, matrices and determinants; MATH 323 designed to be a more demanding version of this course. (Texas A&M)

MATH 308: Differential Equations (3) math

Ordinary differential equations, solutions in series, solutions using Laplace transforms, systems of differential equations. (Texas A&M)

PHIL 240: Introduction to Logic (3) math

Introduction to formal methods of deductive and inductive logic including, but not limited to, truth-tables, formal deduction and probability theory. (Texas A&M)

CS 61: Discrete Mathematics (4) math

Sets, relations and functions, logic and methods of proof, combinatorics, graphs and digraphs. (Tufts)

MATH 32: Calculus I (4) math

Differential and integral calculus: limits and continuity, the derivative and techniques of differentiation, extremal problems, related rates, the definite integral, Fundamental Theorem of Calculus, derivatives and integrals of trigonometric functions, logarithmic and exponential functions. (Tufts)

MATH 34: Calculus II (4) math

Applications of the integral, techniques of integration, separable differential equations, improper integrals. Sequences, series, convergence tests, Taylor series. Polar coordinates, complex numbers. (Tufts)

MATH 39: Honors Calculus I-II (8) math

Intended for students who have had at least the AB syllabus of advanced placement mathematics in secondary school. Stresses the theoretical aspects of the subject, including proofs of basic results. Topics include: convergence of sequences and series; continuous functions, Intermediate Value and Extreme Value Theorems; definition of the derivative, formal differentiation, finding extrema, curve-sketching, Mean Value Theorems; basic theory of the Riemann integral, Fundamental Theorem of Calculus and formal integration, improper integrals; Taylor series, power series and analytic functions. (Tufts)

MATH 42: Calculus III (4) math

Vectors in two and three dimensions, applications of the derivative of vector-valued functions of a single variable. Functions of several variables, continuity, partial derivatives, the gradient, directional derivatives. Multiple integrals and their applications. Line integrals, Green's theorem, divergence theorem, Stokes’ theorem. (Tufts)

MATH 44: Honors Calculus III (4) math

Analysis of real- and vector-valued functions of one or more variables using tools from linear and multilinear algebra; stress is on theoretical aspects of the subject, including proofs of basic results. (Tufts)

MATH 61: Discrete Mathematics (4) math

Sets, relations and functions, logic and methods of proof, combinatorics, graphs and digraphs. (Tufts)

MATH 65: Bridge to Higher Mathematics (4) math

Introduction to rigorous reasoning, applicable across all areas of mathematics, to prepare for proof-based courses at the 100 level. Writing proofs with precise reasoning and clear exposition. Topics may include induction, functions and relations, combinatorics, modular arithmetic, graph theory, and convergence of sequences and series of real numbers. (Tufts)

MATH 70: Linear Algebra (4) math

Introduction to the theory of vector spaces and linear transformations over the real or complex numbers, including linear independence, dimension, matrix multiplication, similarity and change of basis, inner products, eigenvalues and eigenvectors, and some applications. (Tufts)

MATH 166: Statistics (4) math

Statistical estimation, sampling distributions of estimators, hypothesis testing, regression, analysis of variance, and nonparametric methods. (Tufts)

MA205: Calculus II (4) math

This course provides a foundation for the continued study of mathematics and for the subsequent study of the physical sciences, social sciences, and engineering. MA205 covers topics in multivariable differential and integral calculus, vectors and geometry of Euclidean space, vector functions, and vector calculus. Throughout the course mathematical models motivate the study of topics such as optimization, accumulation, change in several variables, and other topics from the natural, social, and decision sciences. An understanding of course material is enhanced through the use of computer algebra systems. (West Point)

MA255: Adv Multivariable Calculus (4.5) math

This is the second course of a two-semester advanced mathematics sequence for selected cadets who have validated single variable calculus and demonstrated strength in the mathematical sciences. It is designed to provide a foundation for the continued study of mathematics, sciences, and engineering. This course consists of an advanced coverage of topics in multivariable calculus. Topics may include a study of infinite sequences and series, vectors and the geometry of space, vector functions, partial derivatives, multiple integrals, and vector calculus. An understanding of course material is enhanced through the use of a computer algebra system. (West Point)

MA371: Linear Algebra (3) math

This course emphasizes both the computational and theoretical aspects of linear algebra one encounters in many subjects ranging from economics to engineering. The course covers solutions of linear systems of equations and the algebra of matrices. The foundational aspects of vector spaces and linear transformations to include linear dependence and independence, subspaces, bases and dimension, inner products, and orthonormalization are developed. This is rounded out with a detailed investigation of eigenvalues and eigenvectors as they relate to diagonalization, quadratic equations, and systems of differential equations. The Invertible Matrix Theorem is explored as the conceptual/theoretical thread of the course. A computer algebra system is used to explore concepts and compute solutions to problems. Applications of the course material are included in the form of special problems to illustrate its wide scope. (West Point)

MA372: Introduction to Discrete Math (3) math

The purpose of this course is to introduce topics in Discrete Mathematics, providing a foundation for further study and application. The topics covered are useful to both the applied mathematician and the computer scientist. They include propositional logic, elements of set theory, combinatorics, relations, functions, partitions, methods of proof, induction and recursion, digraphs, trees, finite state machines, and algebraic systems. Specific applications to computer science are presented. (West Point)

MA376: Applied Statistics (3) math

This course builds on the foundations presented in the core probability and statistics course to provide a broad introduction to some of the most common models and techniques in applied statistics. The mathematical basis for each of the models and techniques is presented with particular emphasis on the development of the required test statistics and their distributions. Topics covered include hypothesis testing, analysis of variance, categorical data analysis, regression analysis, and nonparametric methods. (West Point)

MA383: Foundations of Math (3) math

This course introduces the student to the methods and language of upper division mathematics. It presents formal set theory, and introduces the student to the methods of formulating and writing mathematical proofs. Finally, it provides the student a rigorous introduction to the theory of relations, functions, and infinite sets. (West Point)

MA385: Chaos and Fractals (3) math

This course introduces topics in fractal geometry and chaotic dynamical systems, providing a foundation for applications and further study. The topics from fractal geometry include the military applications of image analysis and data storage. The chaotic dynamical systems studied in the course are one-, two-, and three-dimensional, nonlinear, discrete and continuous dynamical systems. Topics include the logistics equation, the Henon attractor, the Lorenz equations, bifurcation theory, Julia sets, and the Mandelbrot set. These topics have applications in many fields of science, and examples from biology, meteorology, engineering, and the social sciences are studied. The course integrates concepts introduced in the core mathematics courses. (West Point)

MA386: Intro to Numerical Analysis (3) math

This course develops an understanding of the methods for solving mathematical problems using a digital computer. Algorithms leading to solution of mathematical problems will be examined for consistency, stability, and convergence. After a brief review of calculus theory, a study of error analysis and computer arithmetic will provide the framework for the study of the following topics: solutions of equations of one variable, solutions of linear and nonlinear systems of equations, the use of polynomials to approximate discrete data, curve fitting, numerical integration and differentiation, and the approximation of continuous functions. Special problems will incorporate computer graphics and the use of mathematical software libraries to produce numerical solutions of applied problems. (West Point)

MA388: Sabermetrics (3) math

This course builds on the statistical foundation of the core mathematics sequence by exploring the application of statistical concepts to sports analytics. Students develop skills and apply statistical techniques appropriate for baseball and other sports including: regression, forecasting, and stochastic processes. Guest lectures and a course trip section to discuss Sabermetrics at the baseball Hall of Fame in Cooperstown, NY are part of this course (when available). Software packages (Mathematica, Excel) are used as decision support tools to investigate application problems and augment understanding of course material. (West Point)

MA391: Mathematical Modeling (3) math

This course is designed to give cadets the opportunity to develop skills in model construction and model analysis while addressing interesting scenarios with practical applications from a wide variety of fields. The course addresses the complex process of translating real-world events into mathematical language, solving the resulting mathematical model (iterating as necessary), and interpreting the results in terms of real-world issues. Topics may include model development from data, optimization, dynamic models, and deterministic and stochastic model development. Interdisciplinary projects based on actual modeling scenarios are used to integrate the various topics into a coherent theme. (West Point)

MA394: Fundamentals/Network Science (3) math

MA394 exposes cadets to the basic concepts of networks and gives them an opportunity to apply techniques learned in the course to real-world problems. Students will develop skills and problem-solving strategies for modeling complex networks associated with physical, informational, and social phenomena. Software packages are used as decision support tools to investigate application problems and augment understanding of the course material. (West Point)

MA461: Graph Theory and Networks (3) math

This course introduces the student to the techniques, algorithms, and structures used in graph theory and network flows in order to solve real world discrete optimization problems. Basic definitions relating to graphs and digraphs, together with a large number of examples and applications are provided. Cadets learn to implement new graph theory techniques in their area of study. Emphasis is on modeling, algorithms, and optimization. (West Point)

MA462: Combinatorics (3) math

This course introduces the basic techniques and modes of combinatorial problem-solving important to the field of computer science and mathematical sciences such as operations research. Applications of combinatorics are also related to fields such as genetics, organic chemistry, electrical engineering and political science. Combinatorial enumeration and logical structure are stressed. Applications and examples provide the structure of progression through topics which include counting methods, generating functions, recurrence relations, and enumeration techniques. (West Point)

MA464: Applied Algebra W/ Cryptology (3) math

We study the underlying algebra of computer science structures as well as sets, set functions, Boolean algebra, finite state machines, groups, and modular arithmetic. We introduce and study mathematical aspects of cryptology with an emphasis on cryptanalysis of encryption ciphers. We study early paper-and-pencil systems through current computer algorithms for encryption. We employ algebraic principles in both design and analysis of encryption systems, be it matrix, linear feedback shift register sequence, or linear congruential random number generator sequence efforts. Further, we investigate the mathematics of breaking machine ciphers and of designing modern public-key crypto systems. (West Point)

MA466: Abstract Algebra (3) math

This is an introductory course in modern algebra for cadets who plan to do graduate work in mathematics or theoretical work in the physical sciences or engineering. The emphasis of the course is on group theory, considering such topics as cyclic and abelian groups, normal sub-groups and factor groups, series of groups, and solvable groups. Selected applications are interspersed with the material on group theory. The course concludes with an introduction to rings and fields. One special problem is provided to allow the student to do independent research in an area of the student's interest. (West Point)

MA476: Mathematical Statistics (3) math

This course builds on the foundation presented in the core probability and statistics course to provide a mathematical presentation of the important topics in mathematical statistics. The course begins with a review of probability concepts from the core course, adding additional topics such as transformations of random variables and moment generating functions. To provide the mathematical basis for much of statistical practice, certain limit theorems and sampling distributions are proven. The central focus of the course is distribution theory, to include the theory of estimation and the theory of hypothesis testing. (West Point)

MA477: Theory & Appl of Data Science (3) math

This course builds on the foundations presented in the core probability and statistics course and the applied statistics course to develop a broad base of Advanced Data Science to some of the most common techniques in the field. The mathematical basis for each method is presented with focus on both the statistical theory and application. Topics covered may include classification and regression trees, regularization methods, splines and localized regression, and model validation. (West Point)

CS 70: Discrete Mathematics and Probability Theory (4) math

Logic, infinity, and induction; applications include undecidability and stable marriage problem. Modular arithmetic and GCDs; applications include primality testing and cryptography. Polynomials; examples include error correcting codes and interpolation. Probability including sample spaces, independence, random variables, law of large numbers; examples include load balancing, existence arguments, Bayesian inference. (Berkeley)

CS 171: Cryptography (4) math

Cryptography or cryptology is the science of designing algorithms and protocols for enabling parties to communicate and compute securely in an untrusted environment (e.g. secure communication, digital signature, etc.) Over the last four decades, cryptography has transformed from an ad hoc collection of mysterious tricks into a rigorous science based on firm complexity-theoretic foundations. This modern complexity-theoretic approach to cryptography will be the focus. E.g., in the context of encryption we will begin by giving a precise mathematical definition for what it means to be a secure encryption scheme and then give a construction (realizing this security notion) assuming various computational hardness assumptions (e.g. factoring). (Berkeley)

CS 174: Combinatorics and Discrete Probability (4) math

Permutations, combinations, principle of inclusion and exclusion, generating functions, Ramsey theory. Expectation and variance, Chebychev's inequality, Chernov bounds. Birthday paradox, coupon collector's problem, Markov chains and entropy computations, universal hashing, random number generation, random graphs and probabilistic existence bounds. (Berkeley)

MATH 1A: Calculus (4) math

This course is intended for STEM majors. An introduction to differential and integral calculus of functions of one variable, with applications and an introduction to transcendental functions. (Berkeley)

MATH 1B: Calculus (4) math

Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations. (Berkeley)

MATH 53: Multivariable Calculus (4) math

Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes. (Berkeley)

MATH 54: Linear Algebra and Differential Equations (4) math

Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series. (Berkeley)

MATH 56: Linear Algebra (4) math

This is a first course in Linear Algebra. Core topics include: algebra and geometry of vectors and matrices; systems of linear equations and Gaussian elimination; eigenvalues and eigenvectors; Gram-Schmidt and least squares; symmetric matrices and quadratic forms; singular value decomposition and other factorizations. Time permitting, additional topics may include: Markov chains and Perron-Frobenius, dimensionality reduction, or linear programming. This course differs from Math 54 in that it does not cover Differential Equations, but focuses on Linear Algebra motivated by first applications in Data Science and Statistics. (Berkeley)

CSE 20: Discrete Mathematics (4) math

This course will introduce the ways logic is used in computer science: for reasoning, as a language for specifications, and as operations in computation. Concepts include sets, relations, functions, equivalence relations, partial orders, number systems, and proof methods (especially induction and recursion). Propositional and predicate logic will be introduced and applied to various computer science domains such as circuit design, databases, cryptography, and program correctness. (UCSD)

CSE 21: Mathematics for Algorithms and Systems (4) math

This course will cover mathematical concepts used to model and analyze algorithms and computer systems. Topics include counting techniques, data representations, analysis of algorithms, recurrence relations, graphs and trees, and basic probability and its applications. (UCSD)

CSE 103: A Practical Introduction to Probability and Statistics (4) math

Distributions over the real line. Independence, expectation, conditional expectation, mean, variance. Hypothesis testing. Learning classifiers. Distributions over R^n, covariance matrix. Binomial, Poisson distributions. Chernoff bound. Entropy. Compression. Arithmetic coding. Maximal likelihood estimation. Bayesian estimation. CSE 103 is not duplicate credit for ECE 109, ECON 120A, or MATH 183. (UCSD)

CSE 106: Discrete and Continuous Optimization (4) math

One frequently deals with problems in engineering, data science, business, economics, and other disciplines for which algorithmic solutions that optimize a given quantity under constraints are desired. This course is an introduction to the models, theory, methods, and applications of discrete and continuous optimization. (UCSD)

CSE 107: Introduction to Modern Cryptography (4) math

Topics include private and public-key cryptography, block ciphers, data encryption, authentication, key distribution and certification, pseudorandom number generators, design and analysis of protocols, zero-knowledge proofs, and advanced protocols. Emphasizes rigorous mathematical approach including formal definitions of security goals and proofs of protocol security. (UCSD)

MATH 15A: Introduction to Discrete Mathematics (4) math

Basic discrete mathematical structure: sets, relations, functions, sequences, equivalence relations, partial orders, and number systems. Methods of reasoning and proofs: propositional logic, predicate logic, induction, recursion, and pigeonhole principle. Infinite sets and diagonalization. Basic counting techniques; permutation and combinations. Applications will be given to digital logic design, elementary number theory, design of programs, and proofs of program correctness. Students who have completed MATH 109 may not receive credit for MATH 15A. Credit not offered for both MATH 15A and CSE 20. (UCSD)

MATH 18: Linear Algebra (4) math

Matrix algebra, Gaussian elimination, determinants. Linear and affine subspaces, bases of Euclidean spaces. Eigenvalues and eigenvectors, quadratic forms, orthogonal matrices, diagonalization of symmetric matrices. Applications. Computing symbolic and graphical solutions using MATLAB. Students may not receive credit for both MATH 18 and 31AH. (UCSD)

MATH 20A: Calculus for Science and Engineering (4) math

Foundations of differential and integral calculus of one variable. Functions, graphs, continuity, limits, derivative, tangent line. Applications with algebraic, exponential, logarithmic, and trigonometric functions. Introduction to the integral. (Two credits given if taken after MATH 1A/10A and no credit given if taken after MATH 1B/10B or MATH 1C/10C. Formerly numbered MATH 2A.) (UCSD)

MATH 20B: Calculus for Science and Engineering (4) math

Integral calculus of one variable and its applications, with exponential, logarithmic, hyperbolic, and trigonometric functions. Methods of integration. Infinite series. Polar coordinates in the plane and complex exponentials. (Two units of credits given if taken after MATH 1B/10B or MATH 1C/10C.) (UCSD)

MATH 20C: Calculus and Analytic Geometry for Science and Engineering (4) math

Vector geometry, vector functions and their derivatives. Partial differentiation. Maxima and minima. Double integration. (Two units of credit given if taken after MATH 10C. Credit not offered for both MATH 20C and 31BH. Formerly numbered MATH 21C.) (UCSD)

MATH 31AH: Honors Linear Algebra (4) math

First quarter of three-quarter honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Topics include real/complex number systems, vector spaces, linear transformations, bases and dimension, change of basis, eigenvalues, eigenvectors, diagonalization. (UCSD)

MATH 31BH: Honors Multivariable Calculus (4) math

Second quarter of three-quarter honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Topics include derivative in several variables, Jacobian matrices, extrema and constrained extrema, integration in several variables. (Credit not offered for both MATH 31BH and 20C.) (UCSD)

MATH 31CH: Honors Vector Calculus (4) math

Third quarter of honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Topics include change of variables formula, integration of differential forms, exterior derivative, generalized Stoke’s theorem, conservative vector fields, potentials. (UCSD)

MATH 109: Mathematical Reasoning (4) math

This course uses a variety of topics in mathematics to introduce the students to rigorous mathematical proof, emphasizing quantifiers, induction, negation, proof by contradiction, naive set theory, equivalence relations and epsilon-delta proofs. (UCSD)

MATH 154: Discrete Mathematics and Graph Theory (4) math

Basic concepts in graph theory, including trees, walks, paths, and connectivity, cycles, matching theory, vertex and edge-coloring, planar graphs, flows and combinatorial algorithms, covering Hall’s theorems, the max-flow min-cut theorem, Euler’s formula, and the travelling salesman problem. (UCSD)

MATH 181A: Introduction to Mathematical Statistics I (4) math

Multivariate distribution, functions of random variables, distributions related to normal. Parameter estimation, method of moments, maximum likelihood. Estimator accuracy and confidence intervals. Hypothesis testing, type I and type II errors, power, one-sample t-test. (UCSD)

MATH 183: Statistical Methods (4) math

Introduction to probability. Discrete and continuous random variables–binomial, Poisson and Gaussian distributions. Central limit theorem. Data analysis and inferential statistics: graphical techniques, confidence intervals, hypothesis tests, curve fitting. (UCSD)

MATH 184: Enumerative Combinatorics (4) math

Introduction to the theory and applications of combinatorics. Enumeration of combinatorial structures (permutations, integer partitions, set partitions). Bijections, inclusion-exclusion, ordinary and exponential generating functions. (UCSD)

MATH 186: Probability and Statistics for Bioinformatics (4) math

This course will cover discrete and random variables, data analysis and inferential statistics, likelihood estimators and scoring matrices with applications to biological problems. Introduction to Binomial, Poisson, and Gaussian distributions, central limit theorem, applications to sequence and functional analysis of genomes and genetic epidemiology. (UCSD)

MAE 8: MATLAB Programming for Engineering Analysis (4) math

Computer programming in MATLAB with elementary numerical analysis of engineering problems. Arithmetic and logical operations, arrays, graphical presentation of computations, symbolic mathematics, solutions of equations, and introduction to data structures. (UCSD)

CMPSC 111: Introduction to Computational Science (4) math

Introduction to the numerical algorithms that form the foundations of data science, machine learning, and computational science and engineering. Matrix computation, linear equation systems, eigenvalue and singular value decompositions, numerical optimization. The informed use of mathematical software environments and libraries, such as python/numpy/scipy. (UCSB)

CMPSC 178: Introduction to Cryptography (4) math

An introduction to the basic concepts and techniques of cryptography and cryptanalysis. Topics include: The Shannon Theory, classical systems, the enigma machine, the data encryption standard, public key systems, digital signatures, file security. (UCSB)

MATH 3A: Calculus with Applications, First Course (4) math

Differential Calculus including analytic geometry, functions and limits, derivatives, techniques and applications of differentiation; introduction to integration; logarithmic and trigonometric functions. (UCSB)

MATH 3B: Calculus with Applications, Second Course (4) math

Integral calculus including definite and indefinite integrals, techniques of integration; introduction to sequences and series; with applications in mathematics and physics. (UCSB)

MATH 4A: Linear Algebra with Applications (4) math

Systems of linear equations, matrix algebra, determinants, vector spaces and subspaces, basis and dimension, linear transformations, eigenvalues and eigenvectors, diagonalization, and orthogonality. (UCSB)

MATH 4B: Differential Equations (4) math

First and second order differential equations, separation of variables, linear differential equations, systems of first order equations, nonlinear differential equations and stability. (UCSB)

MATH 6A: Vector Calculus with Applications, First Course (4) math

Calculus of functions of several variables, vector valued functions of one variable, derivative and integrals of vector functions, double and triple integrals, properties and applications of integrals, change of variables. (UCSB)

PSTAT 120A: Probability and Statistics (4) math

Concepts of probability; random variables; combinatorial probability; discrete and continuous distributions; joint distributions, expected values; moment generating functions; law of large numbers and central limit theorems. (UCSB)

PSTAT 120B: Probability and Statistics (4) math

Distribution of sample mean and sample variance; t, chi-squared and F distributions; summarizing data by statistics and graphs; estimation theory for single samples: sufficiency, efficiency, consistency, method of moments, maximum likelihood; hypothesis testing: likelihood ratio test; confidence intervals. (UCSB)

CS 173: Discrete Structures (3) math

Discrete mathematical structures frequently encountered in the study of Computer Science. Sets, propositions, Boolean algebra, induction, recursion, relations, functions, and graphs. (Illinois)

CS 357: Numerical Methods I (3) math

Fundamentals of numerical methods for students in science and engineering; floating-point computation, systems of linear equations, approximation of functions and integrals, the single nonlinear equation, and the numerical solution of ordinary differential equations; various applications in science and engineering; programming exercises and use of high quality mathematical library routines. (Illinois)

CS 361: Probability & Statistics for Computer Science (3) math

Introduction to probability theory and statistics with applications to computer science. Topics include: visualizing datasets, summarizing data, basic descriptive statistics, conditional probability, independence, Bayes theorem, random variables, joint and conditional distributions, expectation, variance and covariance, central limit theorem, Markov inequality, Chebyshev inequality, law of large numbers, Markov chains, simulation, the PageRank algorithm, populations and sampling, sample mean, standard error, maximum likelihood estimation, Bayes estimation, hypothesis testing, confidence intervals, linear regression, principal component analysis, classification, and decision trees. (Illinois)

CS 407: Cryptography (3) math

Same as ECE 407. (Illinois)

CS 450: Numerical Analysis (3) math

Linear system solvers, optimization techniques, interpolation and approximation of functions, solving systems of nonlinear equations, eigenvalue problems, least squares, and quadrature; numerical handling of ordinary and partial differential equations. (Illinois)

CS 481: Advanced Topics in Stochastic Processes & Applications (3) math

Modeling and analysis of stochastic processes. Transient and steady-state behavior of continuous-time Markov chains; renewal processes; models of queuing systems (birth-and-death models, embedded-Markov-chain models, queuing networks); reliability models; inventory models. Familiarity with discrete-time Markov chains, Poisson processes, and birth-and-death processes is assumed. Same as CS 481. (Illinois)

CS 482: Simulation (3) math

Use of discrete-event simulation in modeling and analysis of complex systems. Data structures and event-list management; verification and validation of simulation models; input modeling, including selection of probability distributions and random variate generation; statistical analysis of output data. Same as CS 482. (Illinois)

STAT 107: Data Science Discovery (4) math

Data Science Discovery is the intersection of statistics, computation, and real-world relevance. As a project-driven course, students perform hands-on-analysis of real-world datasets to analyze and discover the impact of the data. Throughout each experience, students reflect on the social issues surrounding data analysis such as privacy and design. Same as CS 107 and IS 107. This course satisfies the General Education Criteria for: Quantitative Reasoning I (Illinois)

STAT 200: Statistical Analysis (3) math

Survey of statistical concepts, data analysis, designed and observational studies and statistical models. Statistical computing using a statistical package such as R or a spreadsheet. Topics to be covered include data summary and visualization, study design, elementary probability, categorical data, comparative experiments, multiple linear regression, analysis of variance, statistical inferences and model diagnostics. May be taken as a first statistics course for quantitatively oriented students, or as a second course to follow a basic concepts course. (Illinois)

STAT 212: Biostatistics (3) math

Application of statistical reasoning and statistical methodology to biology. Topics include descriptive statistics, graphical methods, experimental design, probability, statistical inference and regression. In addition, techniques of statistical computing are covered. (Illinois)

MATH 221: Calculus I (4) math

First course in calculus and analytic geometry for students with some calculus background; basic techniques of differentiation and integration with applications including curve sketching; antidifferentation, the Riemann integral, fundamental theorem, exponential and trigonometric functions. (Illinois)

MATH 231: Calculus II (3) math

Second course in calculus and analytic geometry: techniques of integration, conic sections, polar coordinates, and infinite series. (Illinois)

MATH 241: Calculus III (4) math

Third course in calculus and analytic geometry including vector analysis: Euclidean space, partial differentiation, multiple integrals, line integrals and surface integrals, the integral theorems of vector calculus. (Illinois)

MATH 257: Linear Algebra with Computational Applications (3) math

Introductory course incorporating linear algebra concepts with computational tools, with real world applications to science, engineering and data science. Topics include linear equations, matrix operations, vector spaces, linear transformations, eigenvalues, eigenvectors, inner products and norms, orthogonality, linear regression, equilibrium, linear dynamical systems and the singular value decomposition. (Illinois)

MATH 285: Intro Differential Equations (3) math

Techniques and applications of ordinary differential equations, including Fourier series and boundary value problems, and an introduction to partial differential equations. Intended for engineering majors and others who require a working knowledge of differential equations. (Illinois)

MATH 347: Fundamental Mathematics (3) math

Fundamental ideas used in many areas of mathematics. Topics will include: techniques of proof, mathematical induction, binomial coefficients, rational and irrational numbers, the least upper bound axiom for real numbers, and a rigorous treatment of convergence of sequences and series. This will be supplemented by the instructor from topics available in the various texts. Students will regularly write proofs emphasizing precise reasoning and clear exposition. Credit is not given for both MATH 347 and MATH 348. (Illinois)

STAT 400: Statistics and Probability I (4) math

Introduction to mathematical statistics that develops probability as needed; includes the calculus of probability, random variables, expectation, distribution functions, central limit theorem, point estimation, confidence intervals, and hypothesis testing. Offers a basic one-term introduction to statistics and also prepares students for STAT 410 and STAT 425. Same as MATH 463. (Illinois)

STAT 410: Statistics and Probability II (3) math

Continuation of STAT 400. Includes moment-generating functions, transformations of random variables, normal sampling theory, sufficiency, best estimators, maximum likelihood estimators, confidence intervals, most powerful tests, unbiased tests, and chi-square tests. Same as MATH 464. (Illinois)

MATH 412: Graph Theory (3) math

Examines basic concepts and applications of graph theory, where graph refers to a set of vertices and edges that join some pairs of vertices; topics include subgraphs, connectivity, trees, cycles, vertex and edge coloring, planar graphs and their colorings. Draws applications from computer science, operations research, chemistry, the social sciences, and other branches of mathematics, but emphasis is placed on theoretical aspects of graphs. (Illinois)

MATH 413: Intro to Combinatorics (3) math

Permutations and combinations, generating functions, recurrence relations, inclusion and exclusion, Polya's theory of counting, and block designs. Same as CS 413. (Illinois)

MATH 414: Mathematical Logic (3) math

Introduction to the formalization of mathematics and the study of axiomatic systems; expressive power of logical formulas; detailed treatment of propositional logical and predicate logic; compactness theorem and Godel completeness theorem, with applications to specific mathematical theories; algorithmic aspects of logical formulas. Proofs are emphasized in this course, which can serve as an introduction to abstract mathematics and rigorous proof; some ability to do mathematical reasoning required. (Illinois)

MATH 415: Applied Linear Algebra (3) math

Introductory course emphasizing techniques of linear algebra with applications to engineering; topics include matrix operations, determinants, linear equations, vector spaces, linear transformations, eigenvalues, and eigenvectors, inner products and norms, orthogonality, equilibrium, and linear dynamical systems. (Illinois)

MATH 416: Abstract Linear Algebra (3) math

Rigorous proof-oriented course in linear algebra. Topics include determinants, vector spaces over fields, linear transformations, inner product spaces, eigenvectors and eigenvalues, Hermitian matrices, Jordan Normal Form. (Illinois)

MATH 417: Intro to Abstract Algebra (3) math

Fundamental theorem of arithmetic, congruences. Permutations. Groups and subgroups, homomorphisms. Group actions with applications. Polynomials. Rings, subrings, and ideals. Integral domains and fields. Roots of polynomials. Maximal ideals, construction of fields. (Illinois)

MATH 424: Honors Real Analysis (3) math

A rigorous treatment of basic real analysis via metric spaces recommended for those who intend to pursue programs heavily dependent upon graduate level Mathematics. Metric space topics include continuity, compactness, completeness, connectedness and uniform convergence. Analysis topics include the theory of differentiation, Riemann-Darboux integration, sequences and series of functions, and interchange of limiting operations. As part of the honors sequence, this course will be rigorous and abstract. (Illinois)

STAT 425: Statistical Modeling I (34) math

This is the foundation for advanced statistical modeling with a focus on multiple strategies for analyzing data. The course explores linear regression, least squares estimates, F-tests, analysis of residuals, regression diagnostics, transformations, model building, generalized and weighted least squares, PCA, A/B testing, randomization tests, ANOVA, random effects, mixed effects, and longitudinal data. Statistical computing is an integral part of the course. (Illinois)

STAT 426: Statistical Modeling II (34) math

This is a continuation in the study of advanced statistical modeling techniques with a focus on categorical data. The course explores logistic regression, generalized linear models, goodness-of-fit, link functions, count regression, log-linear models, probability models for contingency tables, and ordinal response models. Statistical computing is an integral part of the course. (Illinois)

MATH 427: Honors Abstract Algebra (3) math

Group theory, counting formulae, factorization, modules with applications to Abelian groups and linear operators. As part of the honors sequence, this course will be rigorous and abstract. (Illinois)

STAT 428: Statistical Computing (34) math

Examines statistical packages, numerical analysis for linear and nonlinear models, graphics, and random number generation and Monte Carlo methods. Same as CSE 428. (Illinois)

STAT 431: Applied Bayesian Analysis (34) math

Introduction to the concepts and methodology of Bayesian statistics, for students with fundamental knowledge of mathematical statistics. Topics include Bayes' rule, prior and posterior distributions, conjugacy, Bayesian point estimates and intervals, Bayesian hypothesis testing, noninformative priors, practical Markov chain Monte Carlo, hierarchical models and model graphs, and more advanced topics as time permits. Implementations in R and specialized simulation software. (Illinois)

STAT 432: Basics of Statistical Learning (34) math

Topics in supervised and unsupervised learning are covered, including logistic regression, support vector machines, classification trees and nonparametric regression. Model building and feature selection are discussed for these techniques, with a focus on regularization methods, such as lasso and ridge regression, as well as methods for model selection and assessment using cross validation. Cluster analysis and principal components analysis are introduced as examples of unsupervised learning. (Illinois)

STAT 434: Survival Analysis (34) math

Introduction to the analysis of time-to-event outcomes. Topics center around three main procedures: the Kaplan-Meier estimator, the log-rank test, and Cox regression. Emphasis on big-picture concepts, basic methodological understanding, and practical implementation in R. (Illinois)

MATH 441: Differential Equations (3) math

Basic course in ordinary differential equations; topics include existence and uniqueness of solutions and the general theory of linear differential equations; treatment is more rigorous than that given in MATH 285. (Illinois)

MATH 444: Elementary Real Analysis (3) math

Careful treatment of the theoretical aspects of the calculus of functions of a real variable intended for those who do not plan to take graduate courses in Mathematics. Topics include the real number system, limits, continuity, derivatives, and the Riemann integral. (Illinois)

MATH 446: Applied Complex Variables (3) math

For students who desire a working knowledge of complex variables; covers the standard topics and gives an introduction to integration by residues, the argument principle, conformal maps, and potential fields. (Illinois)

MATH 447: Real Variables (3) math

Careful development of elementary real analysis for those who intend to take graduate courses in Mathematics. Topics include completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration. (Illinois)

STAT 448: Advanced Data Analysis (4) math

Several of the most widely used techniques of data analysis are discussed with an emphasis on statistical computing. Topics include linear regression, analysis of variance, generalized linear models, and analysis of categorical data. In addition, an introduction to data mining is provided considering classification, model building, decision trees, and cluster analysis. Same as CSE 448. (Illinois)

MATH 461: Probability Theory (3) math

Introduction to mathematical probability; includes the calculus of probability, combinatorial analysis, random variables, expectation, distribution functions, moment-generating functions, and central limit theorem. (Illinois)

MATH 484: Nonlinear Programming (3) math

Iterative and analytical solutions of constrained and unconstrained problems of optimization; gradient and conjugate gradient solution methods; Newton's method, Lagrange multipliers, duality and the Kuhn-Tucker theorem; and quadratic, convex, and geometric programming. (Illinois)

MATH 231: Elements of Discrete Mathematics I (4) math

Sets, mathematical logic, induction, sequences, and functions. Sequence. (UO)

MATH 232: Elements of Discrete Mathematics II (4) math

Relations, theory of graphs and trees with applications, permutations and combinations. (UO)

MATH 246: Calculus for the Biological Sciences I (4) math

For students in biological science and related fields. Emphasizes modeling and applications to biology. Differential calculus and applications. Sequence. Students cannot receive credit for more than one of MATH 241, MATH 246, MATH 251. (UO)

MATH 247: Calculus for the Biological Sciences II (4) math

For students in biological science and related fields. Emphasizes modeling and applications to biology. Integral calculus and applications. Students cannot receive credit for more than one of MATH 242, MATH 247, MATH 252. (UO)

MATH 251: Calculus I (4) math

Standard sequence for students of physical and social sciences and of mathematics. Differential calculus and applications. Students cannot receive credit for more than one of MATH 241, MATH 246, MATH 251. Sequence with MATH 252 and MATH 253. (UO)

MATH 252: Calculus II (4) math

Standard sequence for students of physical and social sciences and of mathematics. Integral calculus. Sequence. Students cannot receive credit for more than one of MATH 242, MATH 247, MATH 252. (UO)

MATH 253: Calculus III (4) math

Standard sequence for students of physical and social sciences and of mathematics. Introduction to improper integrals, infinite sequences and series, Taylor series, and differential equations. Sequence. (UO)

MATH 261: Calculus with Theory I (4) math

Covers both applications of calculus and its theoretical background. Axiomatic treatment of the real numbers, limits, and the least upper bound property. (UO)

MATH 262: Calculus with Theory II (4) math

Covers both applications of calculus and its theoretical background. Differential and integral calculus. (UO)

MATH 263: Calculus with Theory III (4) math

Covers both applications of calculus and its theoretical background. Sequences and series, Taylor's theorem. (UO)

MATH 316: Fundamentals of Analysis I (4) math

Rigorous treatment of topics introduced in calculus such as limits, sequences, series, the Cauchy condition, and continuity. Development of mathematical proof in these contexts. Sequence with MATH 317. (UO)

MATH 341: Elementary Linear Algebra (4) math

Vector and matrix algebra; n-dimensional vector spaces; systems of linear equations; linear independence and dimension; linear transformations; rank and nullity; determinants; eigenvalues; inner product spaces; theory of a single linear transformation. Sequence. (UO)

MATH 342: Elementary Linear Algebra (4) math

Vector and matrix algebra; n-dimensional vector spaces; systems of linear equations; linear independence and dimension; linear transformations; rank and nullity; determinants; eigenvalues; inner product spaces; theory of a single linear transformation. (UO)

MATH 343: Statistical Models and Methods (4) math

Review of theory and applications of mathematical statistics including estimation and hypothesis testing. Students cannot get credit for both MATH 343 and DSCI 345M/MATH 345M. (UO)

MATH 345M: Probability and Statistics for Data Science (4) math

Introduction to probability and statistics, with an emphasis upon topics relevant for data science. Multilisted with DSCI 345M. Students cannot get credit for both MATH 343 and DSCI 345M/MATH 345M. (UO)

MATH 347: Fundamentals of Number Theory I (4) math

A study of congruences, the Chinese remainder theorem, the theory of prime numbers and divisors, Diophantine equations, and quadratic reciprocity. Development of mathematical proof in these contexts. Sequence with MATH 348. (UO)

MATH 351: Elementary Numerical Analysis I (4) math

Basic techniques of numerical analysis and their use on computers. Topics include root approximation, linear systems, interpolation, integration, and differential equations. Sequence. (UO)

MATH 352: Elementary Numerical Analysis II (4) math

Basic techniques of numerical analysis and their use on computers. Topics include root approximation, linear systems, interpolation, integration, and differential equations. (UO)

MATH 391: Fundamentals of Abstract Algebra I (4) math

Introduction to algebraic structures including groups, rings, fields, and polynomial rings. Sequence. (UO)

MATH 425: Statistical Methods I (4) math

Statistical methods for upper-division and graduate students anticipating research in nonmathematical disciplines. Presentation of data, sampling distributions, tests of significance, confidence intervals, linear regression, analysis of variance, correlation, statistical software. (UO)

MATH 461: Introduction to Mathematical Methods of Statistics I (4) math

Discrete and continuous probability models; useful distributions; applications of moment-generating functions; sample theory with applications to tests of hypotheses, point and confidence interval estimates. Sequence. (UO)

MATH 462: Introduction to Mathematical Methods of Statistics II (4) math

Discrete and continuous probability models; useful distributions; applications of moment-generating functions; sample theory with applications to tests of hypotheses, point and confidence interval estimates. (UO)

CIS 2610: Discrete Probability, Stochastic Processes, and Statistical Inference (1) math

The purpose of this course is to provide a 1 CU educational experience which tightly integrates the theory and applications of discrete probability, discrete stochastic processes, and discrete statistical inference in the study of computer science. The intended audience for this class is both those students who are CS majors as well as those intending to be CS majors. Specifically, it will be assumed that the students will know: Set Theory, Mathematical Induction, Number Theory, Functions, Equivalence Relations, Partial-Order Relations, Combinatorics, and Graph Theory at the level currently covered in CIS 1600. This course could be taken immediately following CIS 1600. Computation and Programming will play an essential role in this course. The students will be expected to use the Maple programming environment in homework exercises which will include: numerical and symbolic computations, simulations, and graphical displays. (Penn)

CIS 3333: Mathematics of Machine Learning (1) math

Machine learning is the study of algorithms (e.g. gradient descent) that learn functions (e.g. deep networks) from experience (e.g. data). Behind this simple statement is a lot of mathematical scaffolding: statistics for handling data, optimization for understanding learning algorithms, and linear algebra to create expressive models. This course provides the background to be able to understand mathematical concepts commonly used in machine learning. Topics include continuous probability, parametric distributions, and concentration inequalities from statistics; inner product spaces, functional analysis and Hilbert spaces from linear algebra; and multivariate calculus, Taylor’s theorem, and convexity from optimization. (Penn)

CIS 4670: Scientific Computing (1) math

This course will focus on numerical algorithms and scientific computing techniques that are practical and efficient for a number of canonical science and engineering applications. Built on top of classical theories in multi-variable calculus and linear algebra (as a prerequisite), the lectures in this course will strongly focus on explaining numerical methods for applying these mathematical theories to practical engineering problems. (Penn)

CIS 5150: Fundamentals of Linear Algebra and Optimization (1) math

This course provides firm foundations in linear algebra and basic optimization techniques. Emphasis is placed on teaching methods and tools that are widely used in various areas of computer science. Both theoretical and algorithmic aspects will be discussed. (Penn)

CIS 5560: Cryptography (1) math

This course is an introduction to cryptography, both theory and applications, intended for advanced undergraduates and graduate students. Topics covered include symmetric cryptography, message authentication, public-key cryptography, digital signatures, cryptanalysis, cryptographic security, and secure channels, as well as a selection of more advanced topics such as zero-knowledge proofs, secure multiparty computation, privacy-enhancing technologies, or lattice-based cryptography. (Penn)

CIS 5670: Scientific Computing (1) math

This course will focus on numerical algorithms and scientific computing techniques that are practical and efficient for a number of canonical science and engineering applications. Built on top of classical theories in multi-variable calculus and linear algebra (as a prerequisite), the lectures in this course will strongly focus on explaining numerical methods for applying these mathematical theories to practical engineering problems. Students will be expected to implement solutions and software tools using MATLAB/C++, practice state-of-the-art parallel computing paradigms, and learn scientific visualization techniques using modern software packages. (Penn)

MATH 1400: Calculus, Part I (1) math

Brief review of High School calculus, applications of integrals, transcendental functions, methods of integration, infinite series, Taylor's theorem, and first order ordinary differential equations. Use of symbolic manipulation and graphics software in calculus. (Penn)

MATH 1410: Calculus, Part II (1) math

Functions of several variables, vector-valued functions, partial derivatives and applications, double and triple integrals, conic sections, polar coordinates, vectors and vector calculus, first order ordinary differential equations. Applications to physical sciences. Use of symbolic manipulation and graphics software in calculus. (Penn)

MATH 1610: Honors Calculus (1) math

Students who are interested in math or science might also want to consider a more challenging Honors version of Calculus II and III, Math 1610 and Math 2600 (the analogues of Math 1410 and Math 2400, respectively). These courses will cover essentially the same material as 1610 and 2400, but more in depth and involve discussion of the underlying theory as well as computations. (Penn)

MATH 2400: Calculus, Part III (1) math

Linear algebra: vectors, matrices, systems of linear equations, vector spaces, subspaces, spans, bases, and dimension, eigenvalues, and eigenvectors, matrix exponentials. Ordinary differential equations: higher-order homogeneous and inhomogeneous ODEs and linear systems of ODEs, phase plane analysis, non-linear systems. Fall or Spring. (Penn)

MATH 2600: Honors Calculus, Part II (1) math

This is an honors version of Math 2400 which explores the same topics but with greater mathematical rigor. Not Offered Every Year (Penn)

MATH 3120: Linear Algebra (1) math

Linear transformations, Gauss Jordan elimination, eigenvalues and eigenvectors, theory and applications. Mathematics majors are advised that MATH 3120 cannot be taken to satisfy the major requirements. Not Offered Every Year (Penn)

MATH 3130: Computational Linear Algebra (1) math

Many important problems in a wide range of disciplines within computer science and throughout science are solved using techniques from linear algebra. This course will introduce students to some of the most widely used algorithms and illustrate how they are actually used. Some specific topics: the solution of systems of linear equations by Gaussian elimination, dimension of a linear space, inner product, cross product, change of basis, affine and rigid motions, eigenvalues and eigenvectors, diagonalization of both symmetric and non-symmetric matrices, quadratic polynomials, and least squares optimazation. Applications will include the use of matrix computations to computer graphics, use of the discrete Fourier transform and related techniques in digital signal processing, the analysis of systems of linear differential equations, and singular value deompositions with application to a principal component analysis. The ideas and tools provided by this course will be useful to students who intend to tackle higher level courses in digital signal processing, computer vision, robotics, and computer graphics. (Penn)

MATH 3140: Advanced Linear Algebra (1) math

Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and Unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra. (Penn)

STAT 4300: Probability (1) math

Discrete and continuous sample spaces and probability; random variables, distributions, independence; expectation and generating functions; Markov chains and recurrence theory. (Penn)

ESE 3010: Engineering Probability (1) math

This course introduces students to the mathematical foundations of the theory of probability and its rich applications. The course begins with an exploration of combinatorial probabilities in the classical setting of games of chance, proceeds to the development of an axiomatic, fully mathematical theory of probability, and concludes with the discovery of the remarkable limit laws and the eminence grise of the classical theory, the central limit theorem. The topics covered include: discrete and continuous probability spaces , distributions, mass functions, densities; conditional probability; independence; the Bernoulli schema: the binomial, Poisson, and waiting time distributions; uniform, exponential, normal, and related densities; expectation, variance, moments; conditional expectation; generating functions, characteristic functions; inequalities, tail bounds, and limit laws. But a bald listing of topics does not do justice to the subject: the material is presented in its lush and glorious historical context, the mathematical theory buttressed and made vivid by rich and beautiful applications drawn from the world around us. The student will see surprises in election-day counting of ballots, a historical wager the sun will rise tomorrow, the folly of gambling, the sad news about lethal genes, the curiously persistent illusion of the hot hand in sports, the unreasonable efficacy of polls and its implications to medical testing, and a host of other beguiling settings. (Penn)

CSCI 170: Discrete Methods in Computer Science (4) math

Sets, functions, series. Big-O notation and algorithm analysis. Propositional and first-order logic. Counting and discrete probability. Graphs and basic graph algorithms. Basic number theory. (USC)

MATH 125g: Calculus I (4) math

Limits; continuity, derivatives and applications; antiderivatives; the fundamental theorem of calculus; exponential and logarithmic functions. (USC)

MATH 126g: Calculus II (4) math

A continuation of MATH 125g: trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates. (USC)

MATH 129: Calculus II for Engineers and Scientists (4) math

Trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates. Engineering and physics applications. (USC)

MATH 225: Linear Algebra and Linear Differential Equations (4) math

Matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues, systems of linear differential equations. (USC)

MATH 226g: Calculus III (4) math

Vectors, vector valued functions; differential and integral calculus of functions of several variables; Green’s theorem, Divergence theorem, Stoke’s theorem. (USC)

MATH 229: Calculus III for Engineers and Scientists (4) math

A continuation of MATH 129; vectors, vector valued functions; differential and integral calculus of functions of several variables; Green’s theorem. Engineering and physics applications. (USC)

MATH 235: Linear Algebra and Applications (4) math

Matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues, linear differential equations, singular value decomposition, image compression, graphs, networks and linear programming. (USC)

MATH 407: Probability Theory (4) math

Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws. (USC)

EE 141L: Applied Linear Algebra for Engineering (4) math

Introduction to linear algebra with engineering applications. Weekly laboratory exercises using MATLAB. (USC)

CMPU 250: Modeling, Simulation and Analysis (1) math

Principles of computation in the sciences, driven by current applications in biology, physics, chemistry, natural and social sciences, and computer science. Topics include: Discrete and continuous stochastic models, random number generation, elementary statistics, numerical analysis and algorithms, discrete event simulation, and point and interval parameter estimation. Students pursue projects that involve modeling phenomena in two to three different fields and simulate the model in order to understand mechanisms and/or explore new hypotheses or conditions. (Vassar)

CSE 240: Logic and Discrete Mathematics (3) math

Introduces elements of logic and discrete mathematics that allow reasoning about computational structures and processes. Generally, the areas of discrete structures, proof techniques, probability and computational models are covered. Topics typically include propositional and predicate logic; sets, relations, functions and graphs; direct and indirect proof methods, induction and recursion; finite state machines and regular languages. (Washington U.)

CSE 442T: Introduction to Cryptography (3) math

This course is an introduction to modern cryptography, with an emphasis on its theoretical foundations. Topics will include one-way functions, pseudorandom generators, public key encryption, digital signatures, and zero-knowledge proofs. (Washington U.)

Math 131: Calculus I (3) math

Derivatives of algebraic, trigonometric and transcendental functions, techniques of differentiation, Mean Value Theorem, applications of the derivative. The definite integral and Fundamental Theorem of Calculus. Areas. Simpler integration techniques. (Washington U.)

Math 132: Calculus II (3) math

Continuation of Math 131. A brief review of the definite integral and Fundamental Theorem of Calculus. Techniques of integration, applications of the integral, sequences and series, Taylor polynomials and series, and some material on differential equations. (Washington U.)

Math 233: Calculus III (3) math

Multivariable calculus. Topics include differential and integral calculus of functions of two or three variables: vectors and curves in space, partial derivatives, multiple integrals, line integrals, vector calculus at least through Green's Theorem. (Washington U.)

Math 309: Matrix Algebra (3) math

An introductory course in linear algebra that focuses on Euclidean n-space, matrices and related computations. Topics include: systems of linear equations, row reduction, matrix operations, determinants, linear independence, dimension, rank, change of basis, diagonalization, eigenvalues, eigenvectors, orthogonality, symmetric matrices, least square approximation, quadratic forms. Introduction to abstract vector spaces. (Washington U.)

Math 310: Foundations for Higher Mathematics (3) math

Introduction to the rigorous techniques used in more advanced mathematics. Topics include postpositional logic, use of quantifiers, set theory, methods of proof and disproof (counterexamples), foundations of mathematics. Use of these tools in the construction of number systems and in other areas such as elementary number theory, combinatorial arguments and elementary proofs in analysis. (Washington U.)

Math 310W: Foundations for Higher Mathematics with Writing (3) math

This course introduces the rigorous techniques used in more advanced mathematics. Topics include basic logic, set theory, methods of proof and counterexamples, foundations of mathematics, construction of number systems, counting methods, combinatorial arguments and elementary analysis. At least three papers will be required, with at least one revision. (Washington U.)

Math 3200: Elementary to Intermediate Statistics and Data Analysis (3) math

An introduction to probability and statistics. Major topics include elementary probability, special distributions, experimental design, exploratory data analysis, estimation of mean and proportion, hypothesis testing and confidence, regression, and analysis of variance. Emphasis is placed on development of statistical reasoning, basic analytic skills, and critical thinking in empirical research studies. The use of the statistical software R is integrated into lectures and weekly assignments. Required for students pursuing a major or minor in mathematics or wishing to take 400-level or above statistics courses. (Washington U.)

Math 3211: Statistics for Data Science I (3) math

(Washington U.)

Math 4211: Statistics for Data Science II (3) math

This builds on the foundation from the first course (SDS I) and further develops the theory of statistical hypotheses testing. It also covers advanced computer intensive statistical methods, such as the Bootstrap, that will make extensive use of R. The emphasis of the course is to expose students to modern statistical modeling tools beyond linear models that allow for flexible and tractable interaction among response variables and covariates/feature sets. Statistical modeling and analysis of real datasets is a key component of the course. (Washington U.)

Math 439: Linear Statistical Models (3) math

Theory and practice of linear regression, analysis of variance (ANOVA) and their extensions, including testing, estimation, confidence interval procedures, modeling, regression diagnostics and plots, polynomial regression, colinearity and confounding, model selection, geometry of least squares, etc. The theory will be approached mainly from the frequentist perspective, and use of the computer (mostly R) to analyze data will be emphasized. (Washington U.)

MATH 225: Combinatorics and Graph Theory (1) math

Combinatorics is the art of counting possibilities: for instance, how many different ways are there to distribute 20 apples to 10 kids? Graph theory is the study of connected networks of objects. Both have important applications to many areas of mathematics and computer science. The course will be taught emphasizing creative problem-solving as well as methods of proof, such as proof by contradiction and induction. Topics include: selections and arrangements, generating functions, recurrence relations, graph coloring, Hamiltonian and Eulerian circuits, and trees. (Wellesley)

MATH 228: Discrete Mathematics (1) math

This course is a survey of discrete mathematical processes. Students will be introduced to the process of writing formal mathematical proofs, including mathematical induction. Topics may include set theory, logic, number theory, finite fields, permutations, elementary combinatorics, or graph theory. (Wesleyan)

MATH 261: Introduction to Abstract Algebra (1) math

This course is an introduction to abstract principles based on the special properties of the integers, rational, real and complex numbers. The course will cover general algebraic structures as well as their quotients and homomorphisms, with emphasis on fundamental results about groups and rings. (Wesleyan)

CSCI 441: Information Theory and Applications (1) math

What is information? And how do we communicate information effectively? This course will introduce students to the fundamental ideas of Information Theory including entropy, communication channels, mutual information, and Kolmogorov complexity. These ideas have surprising connections to a fields as diverse as physics (statistical mechanics, thermodynamics), mathematics (ergodic theory and number theory), statistics and machine learning (Fisher information, Occam's razor), and electrical engineering (communication theory). (Williams)

MATH 200: Discrete Mathematics (1) math

In contrast to calculus, which is the study of continuous processes, this course examines the structure and properties of finite sets. Topics to be covered include mathematical logic, elementary number theory, mathematical induction, set theory, functions, relations, elementary combinatorics and probability, and graphs. Emphasis will be given on the methods and styles of mathematical proofs, in order to prepare the students for more advanced math courses. (Williams)

MATH 328: Combinatorics (1) math

Combinatorics is a branch of mathematics that focuses on enumerating, examining, and investigating the existence of discrete mathematical structures with certain properties. This course provides an introduction to the fundamental structures and techniques in combinatorics including enumerative methods, generating functions, partition theory, the principle of inclusion and exclusion, and partially ordered sets. (Williams)

MATH 334: Graph Theory (1) math

A graph is a collection of vertices, joined together by edges. In this course, we will study the sorts of structures that can be encoded in graphs, along with the properties of those graphs. We'll learn about such classes of graphs as multi-partite, planar, and perfect graphs, and will see applications to such optimization problems as minimum colorings of graphs, maximum matchings in graphs, and network flows. (Williams)

MATH 341: Probability (1) math

The historical roots of probability lie in the study of games of chance. Modern probability, however, is a mathematical discipline that has wide applications in a myriad of other mathematical and physical sciences. Drawing on classical gaming examples for motivation, this course will present axiomatic and mathematical aspects of probability. Included will be discussions of random variables (both discrete and continuous), distribution and expectation, independence, laws of large numbers, and the well-known Central Limit Theorem. Many interesting and important applications will also be presented, including some from classical Poisson processes, random walks and Markov Chains. (Williams)

CPSC 202: Mathematical Tools for Computer Science (1) math

Introduction to formal methods for reasoning and to mathematical techniques basic to computer science. Topics include propositional logic, discrete mathematics, and linear algebra. Emphasis on applications to computer science: recurrences, sorting, graph traversal, Gaussian elimination. (Yale)

CPSC 416: Lattices and Post-Quantum Cryptography (1) math

This course explores the role of lattices in modern cryptography. In the last decades, novel computational problems, whose hardness is related to lattices, have been instrumental in cryptography by offering: (a) a basis for 'post-quantum' cryptography, (b) cryptographic constructions based on worst-case hard problems, (c) numerous celebrated cryptographic protocols unattainable from other cryptographic assumptions. (Yale)

CPSC 417: Advanced Topics in Cryptography: Cryptography and Computation (1) math

Traditional cryptography is mostly concerned with studying the foundations of securing communication via, for example, encryption and message authentication codes. This class studies the applications of cryptography in securing computation. (Yale)

MATH 120: Calculus of Functions of Several Variables (1) math

Analytic geometry in three dimensions, using vectors. Real-valued functions of two and three variables, partial derivatives, gradient and directional derivatives, level curves and surfaces, maxima and minima. Parametrized curves in space, motion in space, line integrals; applications. Multiple integrals, with applications. Divergence and curl. The theorems of Green, Stokes, and Gauss. (Yale)

MATH 225: Linear Algebra (1) math

An introduction to the theory of vector spaces, matrix theory and linear transformations, determinants, eigenvalues, inner product spaces, spectral theorem. The course focuses on conceptual understanding and serves as an introduction to writing mathematical proofs. For an approach focused on applications rather than proofs, consider MATH 222. Students with a strong mathematical background or interest are encouraged to consider MATH 226. (Yale)

MATH 226: Linear Algebra (Intensive) (1) math

A fast-paced introduction to the theory of vector spaces, matrix theory and linear transformations, determinants, eigenvalues, inner product spaces, spectral theorem. Topics are covered at a deeper level than in MATH 225, and additional topics may be covered, for example canonical forms or the classical groups. The course focuses on conceptual understanding. Familiarity with writing mathematical proofs is recommended. For a less intensive course, consider MATH 225. For an approach focused on applications, consider MATH 222. (Yale)

MATH 244: Discrete Mathematics (1) math

Basic concepts and results in discrete mathematics: graphs, trees, connectivity, Ramsey theorem, enumeration, binomial coefficients, Stirling numbers. Properties of finite set systems. (Yale)

MATH 475: Senior Essay (1) math

Interested students may write a senior essay under the guidance of a faculty member, and give an oral report to the department. Students wishing to write a senior essay should consult the director of undergraduate studies at least one semester in advance of the semester in which they plan to write the essay. (Yale)