Methods of integration. Infinite series.
Polar coordinates in the plane and complex exponentials. Vector geometry, vector functions and their derivatives. Partial differentiation.
Topics in a Calculus I Course
Maxima and minima. Double integration. Ordinary differential equations: exact, separable, and linear; constant coefficients, undetermined coefficients, variations of parameters. Series solutions. Laplace transforms. Techniques for engineering sciences. Vector fields, gradient fields, divergence, curl. Taylor series in several variables. Conservative fields. Topics include derivative in several variables, Jacobian matrices, extrema and constrained extrema, integration in several variables. The Freshman Seminar Program is designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a small seminar setting.
Freshman Seminars are offered in all campus departments and undergraduate colleges, and topics vary from quarter to quarter. Enrollment is limited to fifteen to twenty students, with preference given to entering freshman. Prerequisites: none. Cross-listed with EDS Students will develop skills in analytical thinking as they solve and present solutions to challenging mathematical problems in preparation for the William Lowell Putnam Mathematics Competition, a national undergraduate mathematics examination held each year.
Students must sit for at least one half of the Putnam exam given the first Saturday in December to receive a passing grade. May be taken for credit up to four times. Independent study or research under direction of a member of the faculty. First course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include groups, subgroups and factor groups, homomorphisms, rings, fields. Second course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra.
Topics include rings especially polynomial rings and ideals, unique factorization, fields; linear algebra from perspective of linear transformations on vector spaces, including inner product spaces, determinants, diagonalization. Third course in a rigorous three-quarter introduction to the methods and basic structures of higher algebra. Topics include linear transformations, including Jordan canonical form and rational canonical form; Galois theory, including the insolvability of the quintic.
Second course in linear algebra from a computational yet geometric point of view. Moore-Penrose generalized inverse and least square problems. Vector and matrix norms. Characteristic and singular values. Canonical forms. Determinants and multilinear algebra. First course in a two-quarter introduction to abstract algebra with some applications. Emphasis on group theory. Topics include definitions and basic properties of groups, properties of isomorphisms, subgroups.
Second course in a two-quarter introduction to abstract algebra with some applications. Emphasis on rings and fields. Topics include definitions and basic properties of rings, fields, and ideals, homomorphisms, irreducibility of polynomials. Elementary number theory with applications. Topics include unique factorization, irrational numbers, residue systems, congruences, primitive roots, reciprocity laws, quadratic forms, arithmetic functions, partitions, Diophantine equations, distribution of primes.
Applications include fast Fourier transform, signal processing, codes, cryptography. Students who have not completed the listed prerequisite s may enroll with consent of instructor. Topics in number theory such as finite fields, continued fractions, Diophantine equations, character sums, zeta and theta functions, prime number theorem, algebraic integers, quadratic and cyclotomic fields, prime ideal theory, class number, quadratic forms, units, Diophantine approximation, p -adic numbers, elliptic curves.
Topics in algebraic and analytic number theory, with an advanced treatment of material listed for MATH B. Prerequisites: Math B or consent of instructor. The course will cover the basic arithmetic properties of the integers, with applications to Diophantine equations and elementary Diophantine approximation theory. Students who have not completed the listed prerequisites may enroll with consent of instructor. Affine and projective spaces, affine and projective varieties. Examples of all the above. Instructor may choose to include some commutative algebra or some computational examples.
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. Required of all departmental majors. An introduction to partial differential equations focusing on equations in two variables.
An introduction to mathematical modeling in the physical and social sciences. Topics vary, but have included mathematical models for epidemics, chemical reactions, political organizations, magnets, economic mobility, and geographical distributions of species. May be taken for credit two times when topics change. Continued study on mathematical modeling in the physical and social sciences, using advanced techniques that will expand upon the topics selected and further the mathematical theory presented in MATH A.
Complex numbers and functions. Analytic functions, harmonic functions, elementary conformal mappings. Complex integration. Power series. Residue theorem. Applications of the residue theorem. Conformal mapping and applications to potential theory, flows, and temperature distributions. Fourier transformations. Laplace transformations, and applications to integral and differential equations.
Cross-listed with EDS A. This course builds on the previous courses where these components of knowledge were addressed exclusively in the context of high-school mathematics. Cross-listed with EDS B. Examine how learning theories can consolidate observations about conceptual development with the individual student as well as the development of knowledge in the history of mathematics. Examine how teaching theories explain the effect of teaching approaches addressed in the previous courses. An introduction to ordinary differential equations from the dynamical systems perspective.
Topics include flows on lines and circles, two-dimensional linear systems and phase portraits, nonlinear planar systems, index theory, limit cycles, bifurcation theory, applications to biology, physics, and electrical engineering. First course in a rigorous three-quarter sequence on real analysis.
Topics include the real number system, basic topology, numerical sequences and series, continuity. Second course in a rigorous three-quarter sequence on real analysis. Topics include differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and special functions. Third course in a rigorous three-quarter sequence on real analysis. Topics include differentiation of functions of several real variables, the implicit and inverse function theorems, the Lebesgue integral, infinite-dimensional normed spaces.
First course in an introductory two-quarter sequence on analysis. Topics include the real number system, numerical sequences and series, limits of functions, continuity. Second course in an introductory two-quarter sequence on analysis. Topics include differentiation, the Riemann integral, sequences and series of functions, uniform convergence, Taylor and Fourier series, special functions.
Rigorous introduction to the theory of Fourier series and Fourier transforms. Topics include basic properties of Fourier series, mean square and pointwise convergence, Hilbert spaces, applications of Fourier series, the Fourier transform on the real line, inversion formula, Plancherel formula, Poisson summation formula, Heisenberg uncertainty principle, applications of the Fourier transform. Students who have not completed listed prerequisite s may enroll with the consent of instructor. A rigorous introduction to systems of ordinary differential equations.
Topics include linear systems, matrix diagonalization and canonical forms, matrix exponentials, nonlinear systems, existence and uniqueness of solutions, linearization, and stability. A rigorous introduction to partial differential equations. Differential geometry of curves and surfaces. Gauss and mean curvatures, geodesics, parallel displacement, Gauss-Bonnet theorem.
Calculus of functions of several variables, inverse function theorem. This course will give students experience in applying theory to real world applications such as internet and wireless communication problems. The course will incorporate talks by experts from industry and students will be helped to carry out independent projects. Topics include graph visualization, labelling, and embeddings, random graphs and randomized algorithms.
May be taken for credit three times. Two- and three-dimensional Euclidean geometry is developed from one set of axioms. Pedagogical issues will emerge from the mathematics and be addressed using current research in teaching and learning geometry. This course is designed for prospective secondary school mathematics teachers. Bezier curves and control lines, de Casteljau construction for subdivision, elevation of degree, control points of Hermite curves, barycentric coordinates, rational curves. Programming knowledge recommended. Spline curves, NURBS, knot insertion, spline interpolation, illumination models, radiosity, and ray tracing.
A hands-on introduction to the use of a variety of open-source mathematical software packages, as applied to a diverse range of topics within pure and applied mathematics. Most of these packages are built on the Python programming language, but no prior experience with mathematical software or computer programming is expected. Extremal combinatorics is the study of how large or small a finite set can be under combinatorial restrictions.
We will give an introduction to graph theory, connectivity, coloring, factors, and matchings, extremal graph theory, Ramsey theory, extremal set theory, and an introduction to probabilistic combinatorics. An introduction to recursion theory, set theory, proof theory, model theory. Turing machines. Undecidability of arithmetic and predicate logic. Proof by induction and definition by recursion.
Cardinal and ordinal numbers. Completeness and compactness theorems for propositional and predicate calculi. A continuation of recursion theory, set theory, proof theory, model theory. Topics will vary from year to year in areas of mathematics and their development. Topics may include the evolution of mathematics from the Babylonian period to the eighteenth century using original sources, a history of the foundations of mathematics and the development of modern mathematics.
Topics to be chosen in areas of applied mathematics and mathematical aspects of computer science. May be taken for credit two times with different topics. Analysis of numerical methods for linear algebraic systems and least squares problems. Orthogonalization methods. Ill conditioned problems. Eigenvalue and singular value computations.
Knowledge of programming recommended. MATH B. Rounding and discretization errors. Calculation of roots of polynomials and nonlinear equations. Approximation of functions. MATH C. Numerical differentiation and integration. Ordinary differential equations and their numerical solution. Basic existence and stability theory. Difference equations. Boundary value problems. MATH A. Introduction to Numerical Optimization: Linear Programming 4. Linear optimization and applications.
Linear programming, the simplex method, duality. Selected topics from integer programming, network flows, transportation problems, inventory problems, and other applications. Three lectures, one recitation. Introduction to Numerical Optimization: Nonlinear Programming 4.
Convergence of sequences in Rn, multivariate Taylor series. Bisection and related methods for nonlinear equations in one variable. Equality-constrained optimization, Kuhn-Tucker theorem. Inequality-constrained optimization. Introduction to convexity: convex sets, convex functions; geometry of hyperplanes; support functions for convex sets; hyperplanes and support vector machines. Linear and quadratic programming: optimality conditions; duality; primal and dual forms of linear support vector machines; active-set methods; interior methods. Convex constrained optimization: optimality conditions; convex programming; Lagrangian relaxation; the method of multipliers; the alternating direction method of multipliers; minimizing combinations of norms.
Conjoined with MATH Floating point arithmetic, direct and iterative solution of linear equations, iterative solution of nonlinear equations, optimization, approximation theory, interpolation, quadrature, numerical methods for initial and boundary value problems in ordinary differential equations. Mathematical background for working with partial differential equations. Survey of finite difference, finite element, and other numerical methods for the solution of elliptic, parabolic, and hyperbolic partial differential equations.
Formerly MATH Graduate students do an extra paper, project, or presentation, per instructor. Mathematical models of physical systems arising in science and engineering, good models and well-posedness, numerical and other approximation techniques, solution algorithms for linear and nonlinear approximation problems, scientific visualizations, scientific software design and engineering, project-oriented.
Graduate students will do an extra paper, project, or presentation per instructor. Probability spaces, random variables, independence, conditional probability, distribution, expectation, variance, joint distributions, central limit theorem. Prior or concurrent enrollment in MATH is highly recommended. Random vectors, multivariate densities, covariance matrix, multivariate normal distribution.
Calculus and analysis
Random walk, Poisson process. Other topics if time permits. Markov chains in discrete and continuous time, random walk, recurrent events. If time permits, topics chosen from stationary normal processes, branching processes, queuing theory. 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. Hypothesis testing. Linear models, regression, and analysis of variance. Goodness of fit tests. Nonparametric statistics. Prior enrollment in MATH is highly recommended. Topics covered may include the following: classical rank test, rank correlations, permutation tests, distribution free testing, efficiency, confidence intervals, nonparametric regression and density estimation, resampling techniques bootstrap, jackknife, etc.
Statistical learning refers to a set of tools for modeling and understanding complex data sets. It uses developments in optimization, computer science, and in particular machine learning. This encompasses many methods such as dimensionality reduction, sparse representations, variable selection, classification, boosting, bagging, support vector machines, and machine learning. Analysis of trends and seasonal effects, autoregressive and moving averages models, forecasting, informal introduction to spectral analysis. Design of sampling surveys: simple, stratified, systematic, cluster, network surveys.
Sources of bias in surveys. Estimators and confidence intervals based on unequal probability sampling. Design and analysis of experiments: block, factorial, crossover, matched-pairs designs. Analysis of variance, re-randomization, and multiple comparisons. 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. Introduction to the theory and applications of combinatorics. Enumeration of combinatorial structures permutations, integer partitions, set partitions. Statistical analysis of data by means of package programs.
Regression, analysis of variance, discriminant analysis, principal components, Monte Carlo simulation, and graphical methods. Emphasis will be on understanding the connections between statistical theory, numerical results, and analysis of real data. 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.
An introduction to the basic concepts and techniques of modern cryptography. Classical cryptanalysis. Probabilistic models of plaintext. Monalphabetic and polyalphabetic substitution. The one-time system. Caesar-Vigenere-Playfair-Hill substitutions. The Enigma. Modern-day developments. The Data Encryption Standard. Public key systems. Security aspects of computer networks. Data protection. Electronic mail. Recommended preparation: programming experience. Renumbered from MATH The object of this course is to study modern public key cryptographic systems and cryptanalysis e. We also explore other applications of these computational techniques e.
A rigorous introduction to algebraic combinatorics. Basic enumeration and generating functions. Enumeration involving group actions: Polya theory. Posets and Sperner property. Partitions and tableaux. An introduction to various quantitative methods and statistical techniques for analyzing data—in particular big data. Quick review of probability continuing to topics of how to process, analyze, and visualize data using statistical language R.
Further topics include basic inference, sampling, hypothesis testing, bootstrap methods, and regression and diagnostics. Offers conceptual explanation of techniques, along with opportunities to examine, implement, and practice them in real and simulated data. An introduction to point set topology: topological spaces, subspace topologies, product topologies, quotient topologies, continuous maps and homeomorphisms, metric spaces, connectedness, compactness, basic separation, and countability axioms.
Students who have not completed prerequisites may enroll with consent of instructor. Examples of all of the above. Instructor may choose further topics such as deck transformations and the Galois correspondence, basic homology, compact surfaces. Students who have not completed the listed prerequisite may enroll with consent of instructor. Topics to be chosen by the instructor from the fields of differential algebraic, geometric, and general topology.
Prerequisites: MATH or consent of instructor. Probabilistic Foundations of Insurance. Short-term risk models. Survival distributions and life tables. Introduction to life insurance. Life Insurance and Annuities. Analysis of premiums and premium reserves. Introduction to multiple life functions and decrement models as time permits. Introduction to the mathematics of financial models.
Basic probabilistic models and associated mathematical machinery will be discussed, with emphasis on discrete time models. Concepts covered will include conditional expectation, martingales, optimal stopping, arbitrage pricing, hedging, European and American options. Students will be responsible for and teach a class section of a lower-division mathematics course. They will also attend a weekly meeting on teaching methods. Does not count toward a minor or major.
Prerequisites: consent of instructor. A variety of topics and current research results in mathematics will be presented by guest lecturers and students under faculty direction. Prerequisites: upper-division status. Subject to the availability of positions, students will work in a local company under the supervision of a faculty member and site supervisor. Units may not be applied toward major graduation requirements.
Prerequisites: completion of ninety units, two upper-division mathematics courses, an overall 2. Department stamp required. Independent reading in advanced mathematics by individual students. Three periods. Prerequisites: permission of department. Honors thesis research for seniors participating in the Honors Program. Research is conducted under the supervision of a mathematics faculty member. Prerequisites: admission to the Honors Program in mathematics, department stamp. Group actions, factor groups, polynomial rings, linear algebra, rational and Jordan canonical forms, unitary and Hermitian matrices, Sylow theorems, finitely generated abelian groups, unique factorization, Galois theory, solvability by radicals, Hilbert Basis Theorem, Hilbert Nullstellensatz, Jacobson radical, semisimple Artinian rings.
Recommended for all students specializing in algebra. Basic topics include categorical algebra, commutative algebra, group representations, homological algebra, nonassociative algebra, ring theory. May be taken for credit six times with consent of adviser as topics vary. Introduction to algebra from a computational perspective. Groups, rings, linear algebra, rational and Jordan forms, unitary and Hermitian matrices, matrix decompositions, perturbation of eigenvalues, group representations, symmetric functions, fast Fourier transform, commutative algebra, Grobner basis, finite fields.
Prerequisites: graduate standing or consent of instructor. Second course in algebra from a computational perspective. Third course in algebra from a computational perspective. Introduction to algebraic geometry. Topics chosen from: varieties and their properties, sheaves and schemes and their properties. May be taken for credit up to three times. Second course in algebraic geometry. Continued exploration of varieties, sheaves and schemes, divisors and linear systems, differentials, cohomology.
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Third course in algebraic geometry. Continued exploration of varieties, sheaves and schemes, divisors and linear systems, differentials, cohomology, curves, and surfaces. First course in graduate-level number theory. Local fields: valuations and metrics on fields; discrete valuation rings and Dedekind domains; completions; ramification theory; main statements of local class field theory.
Second course in graduate-level number theory. Third course in graduate-level number theory. Zeta and L-functions; Dedekind zeta functions; Artin L-functions; the class-number formula and generalizations; density theorems. Topics in algebraic and analytic number theory, such as: L-functions, sieve methods, modular forms, class field theory, p-adic L-functions and Iwasawa theory, elliptic curves and higher dimensional abelian varieties, Galois representations and the Langlands program, p-adic cohomology theories, Berkovich spaces, etc.
May be taken for credit nine times. Prerequisites: graduate standing. Introduction to varied topics in algebraic geometry. Topics will be drawn from current research and may include Hodge theory, higher dimensional geometry, moduli of vector bundles, abelian varieties, deformation theory, intersection theory. Nongraduate students may enroll with consent of instructor. Continued development of a topic in algebraic geometry. May be taken for credit three times with consent of adviser as topics vary. Introduction to varied topics in algebra. In recent years, topics have included number theory, commutative algebra, noncommutative rings, homological algebra, and Lie groups.
Various topics in algebraic geometry. Various topics in number theory. Complex variables with applications. Linear algebra and functional analysis. Fourier analysis of functions and distributions in several variables. In recent years, topics have included applied complex analysis, special functions, and asymptotic methods. May be repeated for credit with consent of adviser as topics vary. Various topics in the mathematics of biological systems. Cauchy theorem and its applications, calculus of residues, expansions of analytic functions, analytic continuation, conformal mapping and Riemann mapping theorem, harmonic functions.
Dirichlet principle, Riemann surfaces. Introduction to varied topics in several complex variables. In recent years, topics have included formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications.
Continued development of a topic in several complex variables. Topics include formal and convergent power series, Weierstrass preparation theorem, Cartan-Ruckert theorem, analytic sets, mapping theorems, domains of holomorphy, proper holomorphic mappings, complex manifolds and modifications. Existence and uniqueness theorems. Cauchy-Kowalewski theorem, first order systems. Hamilton-Jacobi theory, initial value problems for hyperbolic and parabolic systems, boundary value problems for elliptic systems. Introduction to varied topics in differential equations.
In recent years, topics have included Riemannian geometry, Ricci flow, and geometric evolution. Continued development of a topic in differential equations. Topics include Riemannian geometry, Ricci flow, and geometric evolution.
S.O.S. Math - Calculus
Lebesgue integral and Lebesgue measure, Fubini theorems, functions of bounded variations, Stieltjes integral, derivatives and indefinite integrals, the spaces L and C, equi-continuous families, continuous linear functionals general measures and integrations. Metric spaces and contraction mapping theorem; closed graph theorem; uniform boundedness principle; Hahn-Banach theorem; representation of continuous linear functionals; conjugate space, weak topologies; extreme points; Krein-Milman theorem; fixed-point theorems; Riesz convexity theorem; Banach algebras.
Prerequisites: Math A-B-C or consent of instructor. In recent years, topics have included Fourier analysis in Euclidean spaces, groups, and symmetric spaces. Various topics in functional analysis. Convex sets and functions, convex and affine hulls, relative interior, closure, and continuity, recession and existence of optimal solutions, saddle point and min-max theory, subgradients and subdifferentials.
Recommended preparation: course work in linear algebra and real analysis.
Optimality conditions, strong duality and the primal function, conjugate functions, Fenchel duality theorems, dual derivatives and subgradients, subgradient methods, cutting plane methods. Convex optimization problems, linear matrix inequalities, second-order cone programming, semidefinite programming, sum of squares of polynomials, positive polynomials, distance geometry.
Introduction to varied topics in real analysis. In recent years, topics have included Fourier analysis, distribution theory, martingale theory, operator theory. May be taken for credit six times with consent of adviser. Continued development of a topic in real analysis. Topics include Fourier analysis, distribution theory, martingale theory, operator theory.
Various topics in real analysis. Differential manifolds, Sard theorem, tensor bundles, Lie derivatives, DeRham theorem, connections, geodesics, Riemannian metrics, curvature tensor and sectional curvature, completeness, characteristic classes. Differential manifolds immersed in Euclidean space. Lie groups, Lie algebras, exponential map, subgroup subalgebra correspondence, adjoint group, universal enveloping algebra.
Structure theory of semisimple Lie groups, global decompositions, Weyl group. Geometry and analysis on symmetric spaces. Prerequisites: MATH and or consent of instructor. Various topics in Lie groups and Lie algebras, including structure theory, representation theory, and applications. Introduction to varied topics in differential geometry. In recent years, topics have included Morse theory and general relativity.
Continued development of a topic in differential geometry. Topics include Morse theory and general relativity. May be taken for credit three times with consent of adviser. Various topics in differential geometry. Riemannian geometry, harmonic forms. Lie groups and algebras, connections in bundles, homotopy sequence of a bundle, Chern classes. Applications selected from Hamiltonian and continuum mechanics, electromagnetism, thermodynamics, special and general relativity, Yang-Mills fields.
Prerequisites: graduate standing in mathematics, physics, or engineering, or consent of instructor. Propositional calculus and first-order logic. Feasible computability and complexity. Peano arithmetic and the incompleteness theorems, nonstandard models. Introduction to the probabilistic method. Combinatorial applications of the linearity of expectation, second moment method, Markov, Chebyschev, and Azuma inequalities, and the local limit lemma. Introduction to the theory of random graphs. Introduction to probabilistic algorithms.
Game theoretic techniques. Applications of the probabilistic method to algorithm analysis. Markov Chains and Random walks. Applications to approximation algorithms, distributed algorithms, online and parallel algorithms. Advanced topics in the probabilistic combinatorics and probabilistic algorithms. Random graphs. Rohan Gaonkar 11 1 1 bronze badge. What is the content of Young's inequality? James Ronald 2 2 silver badges 10 10 bronze badges. I do not wish to use theorems like the power rule : But sum rules and limit laws are allowed :. Nichlas Uden 3 3 bronze badges. Vladimir Putin 55 6 6 bronze badges.
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