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The physical metallurgy and processing of microalloyed steels and the associated microstructure/processing/property relationship. Usage of microalloyed steels in pipelines including design, forming and welding. Credit cannot be obtained in this course if previous credit has been obtained in MAT E 489. Prerequisite: consent of Instructor.

Terminology, welding processes and materials considerations, mechanisms of welding including the welding arc, molten metal issues, mass and energy balances, heat transfer, basics of procedure development, design of weldments, codes and standards, non-destructive testing, guest lectures from industrial practitioners and specialists. Pre-requisites: Completion of 2 years in any engineering discipline or consent by Instructor.

Classical mechanics and its limitations; basic quantum mechanics; band theory; band diagrams for metals, insulators; Semiconductor and dielectric materials, piezoelectrics and thermoelectrics, and magnetic materials; Intrinsic and doped semiconductors; Optical properties of materials; Light-matter interactions, Prerequisite: PHYS 130, MAT E 202, or by consent of instructor.

Fabrication and application of 1D, 2D, and 3D nanostructured materials. Nanoparticles, carbon nanotubes, graphene, thin films, and nanocomposites. Optical, electrical, and mechanical properties and characterization techniques. Pre-requisite: MAT E 201 or 202.

Survey of nanostructured materials, including processing techniques, properties (mechanical, physical and chemical), characterization, and characterization tools. Introduction to biomedical applications of nanomaterials for diagnosis, therapy and medical implants. Credit may not be obtained in this course if previous credit has been obtained in MAT E 458. Prerequisite: CH E 243 or equivalent, or consent of instructor

Topics of current interest related to process metallurgy, such as welding, process analysis, mathematical modelling and simulation, metal extraction from secondary sources, iron and steel making, physical chemistry of molten systems and production of industrial minerals.

Advanced topics in core fundamentals of materials thermodynamics. Thermodynamic laws, statistical thermodynamics, reaction equilibria, phase diagrams, solutions, changing standard states, electrochemistry, and thermodynamics of surfaces. Prerequisite: MAT E 204 or 301, or consent of Instructor.

Aqueous, molten and solid electrolytes: thermodynamics, structure, transport properties. Applications of conductivity measurements. Electrodes: types, reactions, potential. Electrochemical cells. Applications of EMF measurements. Electrical double layer, electrode kinetics, overpotential. Chlor-alkali industry, electrometallurgy, electrolysis of water, electroplating. Electrochemical energy conversion: primary and secondary batteries, fuel cells. High temperature applications. Prerequisite: Consent of Instructor.

Basic symmetry elements and operations, crystallographic point groups and space groups, application of symmetry in materials analysis. Fundamentals of crystal chemistry, transformations, defects in metals and ionic crystals, interactions between point defects and interfaces. Reciprocal lattice, Brillouin zones, construction of Fermi surfaces, theory of diffraction. Fundamental principles of electron scattering, production and detection of x-rays, diffraction methods, application to crystal structure determination, chemical analysis x-ray spectrometry.

Theoretical strength of solids, Griffith crack theory, mechanisms of brittle and ductile fracture, the ductile to brittle transition, fatigue and creep fracture, environmental effects on fracture. Prerequisites: MAT E 358 or consent of Instructor. Credit cannot be obtained in this course if credit has already been obtained in MAT E 462.

The Kinetics of Materials course delves into the fundamental principles governing the rate and mechanisms of material processes. This course provides students with a comprehensive understanding of the driving forces behind mass transport, diffusion mechanisms, chemical reactions, coarsening, and nucleation theories. It explores the dynamic aspects of materials, focusing on how they change and evolve over time. Through a combination of theoretical discussions and practical applications, students will develop a strong foundation in the kinetics of materials, enabling them to analyze and manipulate material behavior in various engineering and scientific contexts.

Fundamental principles of electron scattering and of the transmission electron microscope, space group theory and application to crystal structure determination, electron diffraction theory, crystallography of precipitation and of defects, TEM imaging theory and application to materials analysis, analytical electron spectrometry. Prerequisite: Graduate student standing or consent of Instructor.

Principles and design of the scanning electron microscope, electron beam-specimen interactions, image formation, x-ray microanalysis in the scanning electron microscope, specimen preparation, application to materials analysis. Prerequisite: Consent of Instructor.

Band theory and solid state properties. Thin film growth at the nanoscale. Semiconductors and dielectric materials, piezoelectrics and thermoelectrics. Semiconductors, doping, p-n junctions, solar cells. Thermoelectric materials and the Seebeck, Thomson, and Peltier Effects. Optical and electrical property measurement.

Weld thermal cycles; fusion zone solidification; phase transformations, heat affected zone phenomena; cracking during welding; ferrous and non-ferrous weldments.

Important ceramic materials and products, processing, typical properties. Structure: binary and ternary compounds, crystalline silicates, glass. Point defects, nonstoichiometry, defect reactions, dislocations. Diffusion, electrochemical transport, examples. Thermal and mechanical properties, thermal shock resistance, electrical conduction. Applications: solid electrolytes, energy conversion systems, refractories, electronics. Prerequisites: Consent of Instructor. Credit cannot be obtained in this course if credit has already been obtained in MAT E 471.

Terminology, welding processes and materials considerations, mechanisms of welding including the welding arc, molten metal issues, mass and energy balances, and heat transfer, basics of procedure development, design of weldments, codes and standards, and non-destructive testing, guest lectures from industrial practitioners and specialists. Completion of a report based on independent research is required. Credit cannot be obtained if previous credit has been obtained for MAT E 481.

Advanced processing and metallurgy of microalloyed steels for pipelines. Steelmaking, casting, microstructural development during thermomechanical processing, pipe fabrication, mechanical and chemical properties and in service performance. Prerequisites: Consent of Instructor.

An advanced treatment of materials engineering topics of current interest to staff and students.

Review of numbers, inequalities, functions, analytic geometry; limits, continuity; derivatives and applications, Taylor polynomials; log, exp, and inverse trig functions. Integration, fundamental theorem of calculus substitution, trapezoidal and Simpson's rules. Prerequisites: Mathematics 30-1 and Mathematics 31. Notes: (1) Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154, or SCI 100. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

Area between curves, techniques of integration. Applications of integration to planar areas and lengths, volumes and masses. First order ordinary differential equations: separable, linear, direction fields, Euler's method, applications. Infinite series, power series, Taylor expansions with remainder terms. Polar coordinates. Rectangular, spherical and cylindrical coordinates in 3-dimensional space. Parametric curves in the plane and space: graphing, arc length, curvature; normal binormal, tangent plane in 3- dimensional space. Volumes and surface areas of rotation. Prerequisite: MATH 100. Notes: (1) Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

Vectors and matrices, solution of linear equations, equations of lines and planes, determinants, matrix algebra, orthogonality and applications (Gram-Schmidt), eigenvalues and eigenvectors and applications, complex numbers. Prerequisite or corequisite: MATH 100. Notes: (1) Credit can be obtained in at most one of MATH 102, 125, or 127. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

A problem-solving approach to discrete mathematics, covering secret codes, public-key codes, error-correcting codes, enumeration, recurrence relations, induction, graph theory, graph algorithms and parallel algorithms. Prerequisite: MATH 30-1, 30-2, or equivalent. Note : Credit can only be obtained in at most one of MATH 111 or MATH 222.

Review of analytic geometry. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. Applications of the derivative. Integration. Fundamental Theorem of Calculus. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100.

Inverse trigonometric functions. Techniques of integration. Improper integrals. Applications of the definite integral. Introduction to differential equations. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

Functions, continuity, and the derivative. Applications of the derivative. Extended limits and L'Hospital's rule. Prerequisites: Mathematics 30-1 and Mathematics 31, or consent of the Department. Notes: (1) This course is designed for students with at least 80 percent in Pure Mathematics 30 or Mathematics 30-1 and Mathematics 31. (2) Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100. (3) Engineering students will receive a weight of 4.0 units for this course.

Integration and the Fundamental Theorem. Techniques and applications of integration. Derivatives and integrals of the exponential, and trigonometric functions. Introduction to infinite series. Introduction to partial derivatives. Prerequisite: MATH 117. (Students with a 100-level calculus course different from MATH 117 may be admitted with consent of the Department.) Notes: (1) Credit can be obtained in at most one MATH 101, 115, 118, 136, 146, 156 or SCI 100. (2) Engineering students will receive a weight of 4.0 units for this course.

Systems of linear equations. Vectors in n-space, vector equations of lines and planes. Matrix algebra, inverses and invertibility. Introduction to linear transformations. Subspaces of n-space. Determinants. Introduction to eigenvalues and eigenvectors. Complex numbers. Dot product, cross product and orthogonality. Applications in a variety of fields. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of MATH 102, 125 or 127.

Systems of linear equations; vectors in Euclidean n-space; span and linear independence in Euclidean n-space; dot and cross product; orthogonality; lines and planes; matrix arithmetic; determinants; introduction to eigenvectors and eigenvalues; introduction to linear transformations; complex numbers; vector space axioms; subspaces and quotients. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of Math 102, 125 or 127.

The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Applications in the context of the life sciences. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100.

Techniques and applications of integration. Improper integrals. Differential equations and mathematical modelling. Partial differentiation. Applications in the context of the life sciences. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Taylor polynomials. Applications in the context of the physical sciences. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100.

Techniques and applications of integration. Improper integrals. Introduction to differential equations. Partial differentiation. Applications in the context of the physical sciences. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Optimization. Applications in the context of business and economics. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100.

Techniques and applications of integration. Improper integrals. Partial differentiation. Multivariate optimization. Probability and calculus. Applications in the context of business and economics. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

Elementary Number Theory, Numeration Systems, Number Systems and Elementary Probability Theory. Math Fair. Prerequisite: Mathematics 30-1 or 30-2, or consent of Department. Notes: (1) This course is restricted to Elementary Education students. (2) This course cannot be used for credit towards a Science degree.

Induction; principles of counting, multinomial coefficients, negative binomial distribution; maximum likelihood estimation, probability axioms; conditional probability, Bayes' rule; independence; probability mass, distribution, and moment generating functions; strong law of large numbers; conditional expectation estimators; gambler's ruin; transience and recurrence; compound processes; applications. Corequisite: One of MATH 101, 118, 136, 146, or 156. Prerequisite: One of MATH 125 or 127. Note: Credit can be obtained in at most two of MATH 181, MATH 281, or STAT 265.

First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 201, 334 or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to Engineering students and Science students in the following programs: Specialization Physics, Specialization Geophysics, Specialization Computing Science, or Specialization Geography (Meteorology). (2) Credit can be obtained in at most one of MATH 201, MATH 334, MATH 336, or MA PH 251. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to Engineering students and Science students in the following programs: Specialization Physics, Specialization Geophysics, Specialization Computing Science, or Specialization Geography (Meteorology). (2) Credit can be obtained in at most one of MATH 201, MATH 334, MATH 336, or MA PH 251. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

Partial differentiation, derivatives of integrals. Multiple integration using rectangular, cylindrical, and spherical coordinates. Vector Field Theory. Prerequisite: MATH 101. Prerequisite or corequisite: MATH 102. Notes: (1) This course may not be taken for credit if credit has already been obtained in MATH 215 or 317. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

Sequences and series, convergence tests, and Taylor series. Curves, tangent vectors, and arc length. Applications of partial differentiation. Polar, cylindrical, and spherical coordinates. Multiple integration. Prerequisite: One of MATH 101, 115, 118, 136, 146 or 156. One of MATH 102, 125 or 127 recommended. Note: This course may not be taken for credit if credit has already been obtained in MATH 209 or 217.

Sets and functions. Induction. Axiomatic introduction of the real numbers. Sequences and series. Continuity and properties of continuous functions. Differentiation. Riemann integral. Corequisite: One of MATH 101, 115, 136, 146, 156 or SCI 100. Note: This course may not be taken for credit if credit has already been obtained in MATH 117. Credit can only be obtained in one of MATH 216 or MATH 314.

Axiomatic development of the real number system. Topology of Rn. Sequences, limits and continuity. Multi-variable calculus: differentiation and integration, including integration in spherical and polar coordinates. The differential and the chain rule. Taylor's Formula, maxima and minima. Introduction to vector field theory. Prerequisites: One of MATH 102, 125 or 127, and either MATH 118 or MATH 216. Notes: (1) MATH 216 may be accepted as corequisite with consent of the Department. (2) Engineering students will receive a weight of 4.0 units for this course.

Vector spaces. Inner product spaces. Examples of n-space and the space of continuous functions. Gram-Schmidt process, QR factorization of a matrix and least squares. Linear transformations, change of basis, similarity and diagonalization. Orthogonal diagonalization, quadratic forms. Applications in a variety of fields. Prerequisites: One of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100, and one of MATH 102, 125 or 127. Note: Credit can be obtained in at most one of MATH 225 or 227.

Groups and their homomorphisms; commutative rings and modules; fields and vector spaces; subgroups and quotient groups, permutation groups; modules, submodules, quotient modules; polynomials rings and their ideals, modules over polynomial rings. Prerequisite: MATH 125. Note: Cannot be taken for credit if credit has been received in MATH 227.

Review of vector space axioms, subspaces and quotients; span; linear independence; Gram-Schmidt process; projections; methods of least squares; linear transformations and their matrix representations with respect to arbitrary bases; change of basis; eigenvectors and eigenvalues; triangularization and diagonalization; canonical forms (Schur, Jordan, spectral theorem). Prerequisite: MATH 127. (Students with MATH 102 or 125 may be admitted with consent of the Department.) Note: Credit can be obtained in at most one of MATH 225 or 227.

Integers. Mathematical induction. Equivalence relations. Commutative rings, including the integers mod n, complex numbers and polynomials. The Chinese remainder theorem. Fields and integral domains. Euclidean domains, principal ideal domains and unique factorization. Quotient rings and homomorphisms. Construction of finite fields. Applications such as public domain encryption, Latin squares and designs, polynomial error detecting codes, and/or addition and multiplication of large integers. Prerequisite: MATH 102, 125 or 127. Note: This course may not be taken for credit if credit has already been obtained in MATH 326.

Basic Euclidean geometry, congruence, parallelism, area, and similarity. Sound axiomatic development with emphasis on problem solving. Constructions and loci, inequalities, maxima and minima, circles, isometries, and additional topics. Prerequisite: Any 100-level MATH course or SCI 100.

Transformation geometry, isometry and homothety, applications in Euclidean geometry; the algebra of transformations, the Classification Theorem, frieze patterns and wall-paper groups. Prerequisite: MATH 241

Accumulation and amount functions, effective, nominal, simple, and compound rates, force of interest and discount, simple and general annuities certain, variable annuities and perpetuities, amortization schedules and sinking funds, bonds and other securities, applications, installment loans, depreciation, depletion, capitalized cost. Prerequisite: One of MATH 101, 115, 118, 136, 146, 156 or SCI 100. Corequisite: MATH 209 or 214.

Divisibility, prime numbers, congruences, quadratic residues, quadratic reciprocity, arithmetic functions and diophantine equations; sums of squares. Prerequisites: MATH 125 or 127.

Reasoning and problem solving in the context of logic, algebra, geometry, and combinatorics. Prerequisite: MATH 160, or consent of Department. Notes: (1) This course is restricted to Elementary Education students. (2) This course cannot be used for credit towards a Science degree.

Review of binomial and negative binomial distributions; continuous random variables; uniform, exponential, and gamma distributions; conditional probability; properties of conditional expectation; stochastic processes; finite-dimensional distributions, Poisson approximation; Poisson measures; counting processes, Markov queues, customer time in queues; steady-state distributions; applications. Corequisite: One of MATH 209, 214, or 217. Credit can be obtained in at most two of MATH 181, STAT 265, or MATH 281. Credit can only be obtained in one of MATH 281 or STAT 371.

Problem solving techniques (pigeonhole principle, invariants, extremal principle, etc.) and survey of problems from various branches of mathematics: calculus, number theory, algebra, combinatorics, probability, geometry, etc. This credit/no-credit course is intended for students interested in mathematics contests and participation in the Putnam Mathematical Competition will be required. Note: This course may be taken for credit up to four times. Prerequisite: consent of the instructor.

Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 300 and 337. (3) Course cannot be taken for credit if credit has been obtained in ECE 341.

Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 300 and 337. (3) Course cannot be taken for credit if credit has been obtained in ECE 341.

Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 300 and 337. (3) Course cannot be taken for credit if credit has been obtained in ECE 341.

Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

Complex numbers. Complex series. Functions of a complex variable. Cauchy's theorem and contour integration. Residue Theorem and its applications. Prerequisite or corequisite: A Calculus IV course.

Construction of real numbers, Heine-Borel and related theorems, differentiation and Riemann integral of functions, topological concepts in metric spaces, sequences, continuous maps, contraction maps, and applications. Prerequisite: MATH 209 or 215 or equivalent. Note: This course may not be taken for credit if credit has already been obtained in MATH 217.

Vector calculus. Line and surface integrals. The divergence, Green's, and Stokes' theorems. Differential forms. Prerequisite: One of MATH 102, 125 or 127, and either MATH 214 or MATH 217. Notes: Credit can be obtained in at most one of MATH 215 and MATH 315. This course may not be taken for credit if credit has already been obtained in MATH 209 or 317.

Implicit function theorem. Proof of the Change of Variables Theorem. Line integrals. Theorems of Green, Gauss and Stokes in their classical form. Differential forms and Stokes' Theorem in their context. Sequences and series of functions. Uniform convergence. Prerequisite: MATH 217.

Graphs, paths and cycles, trees, planarity and duality, coloring problems, digraphs, matching problems, matroid theory. Prerequisite: One of MATH 102, 125 or 127, and any 200-level MATH course. (MATH 216 or MATH 228 recommended.)

Divisibility, prime numbers, congruences, quadratic residues, quadratic reciprocity, arithmetic functions and diophantine equations; sums of squares. Prerequisites: MATH 227 or 228.

Hermitian and unitary matrices, spectral theorem. Jordan canonical form. Cayley-Hamilton Theorem. Bilinear forms, positive-definiteness, Sylvester's Law of inertia, geometric lattices. Numerical methods. Application to discrete system evolution, matrix exponentials and differential equations. Prerequisite: MATH 225. Note: This course may not be taken for credit if credit has already been obtained in MATH 227.

Rings, fields, polynomials, algebras. Homomorphisms, ideals, quotients. Ring extensions, field extensions, construction of finite fields. Integral domains: Euclidean, principal ideal, unique factorization. Chain conditions. Introduction to modules. Modules over a principal ideal domain, finitely generated abelian groups, matrix canonical forms. Prerequisite MATH 227, or both MATH 225 and 228.

Basic group theory: Groups, subgroups, normal subgroups, homomorphisms, quotient groups, coset decomposition, Example: Permutation group and general linear group; basic (commutative) ring theory: Rings, subrings, homomorphisms, ideals, quotient rings, modules over rings, submodules and quotient modules, fraction field; further group theory: Groups operating on a set, Sylow theorems. Prerequisite : One of MATH 226 or MATH 227. Note: Credit can be obtained in at most one of MATH 326 and MATH 327.

Groups, subgroups, homomorphisms. Symmetry groups. Matrix groups. Permutations, symmetric group, Cayley's Theorem. Group actions. Cosets and Lagrange's Theorem. Normal subgroups, quotient groups, isomorphism theorems. Direct and semidirect products. Finite Abelian groups. Prerequisite: MATH 227 or 228. This course may not be taken for credit if credit has already been obtained in MATH 229.

Factorial rings and principal ideal domains; Noetherian rings and modules, Hilbert basis theorem; field extensions, separable and normal extensions; finite Galois theory; solvable groups and equations, construction by ruler and compass, solution by radicals. Prerequisite : MATH 327. Note: Credit can be obtained in a most one of MATH 328 and 329.

First order equations, linear equations of higher order. Power series solution. Laplace transform methods. Introduction to special functions. Introduction to linear systems. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217. Note: Credit can be obtained in at most one of MATH 201, 334 or 336.

First order differential equations. Linear systems of differential equations and linear differential equations of higher order. Stability and qualitative theory of 2-dimensional linear and non-linear systems. Laplace transform methods. Existences and uniqueness theorems. Prerequisites: MATH 225 or 227, and either MATH 209, 217, 314 or both 214 and 216. Note: Credit can be obtained in at most one of MATH 201, 334 and 336.

Boundary value problems of classical Math Physics, orthogonal expansions, classical special functions. Advanced transform techniques. Prerequisites: One of MATH 209, 215, or 217, and one of MATH 201, 334 or 336. Notes: (1) Credit can be obtained in at most one of MATH 300 or 337. (2) Course cannot be taken for credit if credit has been obtained in ECE 341.

Combinatorial geometry and topology, convex sets, sets with constant width, Helly-type problems, extremal problems. Prerequisite: One of MATH 102, 125, 127, 222 or 241.

Projective geometry, Poncelet-Steiner constructions, inverse geometry, Mohr-Mascheroni constructions, Principle of Duality, conic sections. Prerequisite: MATH 241.

Frenet-Seret theory of curves in the plane and in 3-space, examples; local theory of surfaces in 3-space: first and second fundamental forms, Gauss map and Gauss curvature, geodesics and parallel transport, theorema egregium, mean curvature and minimal surfaces. Prerequisites: One of MATH 102, 125 or 127 and one of MATH 209, 215 or 217.

Simple Market Model: one-step binomial model, basic notions and assumptions. Risk-Free Assets: simple interest, zero-coupon bonds, money market account. Risky Assets: dynamic of stock prices, binomial tree model, trinomial tree model. Discrete time market model: stock and money market model, extended models. Portfolio management: risk, two securities, capital asset pricing model. Prerequisite: MATH 253 and one of STAT 265 or MATH 281, or consent of the Department.

Forward and futures contracts: forward and futures prices, hedging with futures. Options: put-call parity, bounds on option prices, time value of options. Option pricing: European and American options in the binomial tree model, Black-Scholes formula. Financial engineering: hedging option positions, hedging business risk. Variable interest rates: maturity-independent yields, general term structure. Stochastic interest rates: arbitrage pricing of bonds, interest rate derivative securities. Prerequisite: MATH 356 or consent of the Department.

Model development, computation, and analysis for problems in the life sciences. Models include differential equations, difference equations and stochastic formulations. Model evaluation and prediction. Applications are chosen from epidemiology, ecology, population biology, physiology and medicine. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217. Note: No previous computing experience is needed.

This course is designed to develop the students' problem-solving abilities along heuristic lines and to illustrate the processes of Applied Mathematics. Students will be encouraged to recognize and formulate problems in mathematical terms, solve the resulting mathematical problems and interpret the solution in real world terms. Typical problems considered include nonlinear programming, optimization problems, diffusion models. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217.

Introduction to optimization. Problem formulation. Linear programming. The simplex method and its variants (revised Simplex method, dual simplex method). Extreme points of polyhedral sets. Theory of linear inequalities (Farkas Lemma). Complementary slackness and duality. Post-optimality analysis. Interior point methods. Applications (elementary games, transportation problems, networks, etc.). Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217.

Approximation of functions by Taylor series, Newton's formulae, Lagrange and Hermite interpolation. Splines. Orthogonal polynomials and least-squares approximation of functions. Direct and iterative methods for solving linear systems. Methods for solving non-linear equations and systems of non-linear equations. Introduction to computer programming. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217. Notes: (1) Credit can be obtained in at most one of MATH 280, 381 or CMPUT 340. (2) Extra classes may be held for students lacking a background in one of the major programming languages such as Fortran, C, C++ or Matlab.

Principles of Monte Carlo methods. Essentials of stochastics. Introduction to financial derivatives pricing. Generating random variables. Simulating stochastic differential equations. Application to financial derivatives pricing and interest rate models. Variance reduction techniques. Prerequisite: STAT 471, or E E 387 and consent of the Department.

Complex number system. Analytic functions. Cauchy's Integral theorem and formula. Applications including the maximum modulus principle, Taylor expansion and Laurent expansion. Harmonic functions. The residue theorem with applications; calculus of residues, argument principle, and Rouche's theorem. Basics of analytic continuation. Additional topics at the instructor's discretion such as: Normal families, The Riemann mapping Theorem, Picard's Theorem. Prerequisite: MATH 314 or 317. Notes: (1) This course is primarily for Honors students in Mathematics or Physics. (2) Offered in alternate years. It may be offered in intervening years if demand is sufficient.

Valuations and their extensions, ramifications; integral dependence, algebraic number fields, ideals and divisors, class number. Prerequisite: MATH 326 or equivalent.

Differentiation of maps in Rn, implicit function and mapping theorems, sequences of functions, Riemann-Stielties integration, additional topics at the discretion of the instructor. Prerequisite: MATH 314. Note: This course may not be taken for credit if credit has already been obtained in MATH 317.

Review of probability tools for discrete financial analysis; Conditional probabilities/expectations. Filtrations, adapted and predictable processes. Martingales, submartingales and supermartingales in discrete-time. Doob decomposition for supermartingales. Predictable representation. Discrete- time financial modes: Arbitrage, complete and incomplete markets. Self-financing property, value and gain processes. Valuation of contingent claims. Binomial model: Model specifications, Perfect hedging. Utility functions and consumption/ investment problems. European and American options in discrete time. Futures and forward contracts in discrete time. Transition to the continuous-time framework. Corequisite: STAT 471 or consent of the Department.

Brief review of set operations and countable sets. Measure theory, integration theory, Lebesgue measure and integrals on R^n, product measure, Tonelli-Fubini theorem. Functions of bounded variation, absolutely continuous functions. Prerequisite: MATH 317 or 414.

Classical Banach spaces. Hahn-Banach, open mapping and closed graphs theorems. Hilbert spaces, orthonormal bases. Elements of spectral theory, spectra of compact operators, spectral theorem for compact self-adjoint operators. Prerequisite: MATH 417. Corequisite: MATH 447.

Permutations and combinations, Binomial Theorem, Principle of Inclusion-Exclusion, recurrence relations, generating functions, orthogonal Latin squares, balanced incomplete block designs, Steiner triple systems, perfect difference sets, Boolean algebra and Finite State Machines. Prerequisites: Either MATH 326 or one of MATH 111 or 228 and a 300-level MATH course (MATH 322 recommended).

Elements of group theory, cosets, Lagrange's theorem, binary group codes, polynomials, finite field theory, error correcting codes. Prerequisites: either (1) MATH 227 or (2) MATH 228 and a 300-level MATH course.

Field extensions. Groups of automorphisms of fields. Galois theory. Finite fields and applications. Solvable groups, the insolvability of the quintic equation. Ruler and compass construction. Prerequisites: MATH 326 (or MATH 228 by consent of the Department) and MATH 328. Note: This course cannot be taken for credit if credit has already been obtained in MATH 427 or 329.

Introduction to commutative algebra, algebraic geometry, and homological algebra. Additional topics at the discretion of the instructor. Prerequisite: MATH 326 or consent of Department.

Group actions, Sylow Theory, solvable and nilpotent groups, Galois Theory. Prerequisite: MATH 328 or consent of the Department.

Elementary existence and uniqueness theorems. Systems of equations, stability, perturbation theory. Introduction to numerical methods. Introduction to phase plane analysis. Prerequisite: One of MATH 201, 334 or 336.

Partial differential equations as physical models. Introduction to basic generalized functions. Theory of linear and quasi-linear first-order equations: general solution, initial value problem, generalized solutions and propagation of singularities, characteristic surfaces, shock formation. Theory of fully nonlinear first order equations: complete solution and the initial value problem. Hamilton-Jacobi equation and its applications. Second order linear equations in n dimensions: classification, canonical form, characteristic surfaces and shock formation, initial and boundary value problem. Prerequisite: MATH 337.

Second order equations in n dimension: classification, canonical form, characteristic surfaces. Laplace equation as a representative of the elliptic equation: the mean value theorem, fundamental solutions and Green functions, the boundary value problems. Wave equation as a representative of hyperbolic equations: initial value problems, the d'Alambert formula, the method of descent, propagation of singularities, Duhamel's principle. Heat equation as a representative of parabolic equations: initial value problems. Introduction to integral transforms: Fourier, Laplace, Hankel transforms. Prerequisite: MATH 337.