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L/R 103. Introduction to Calculus. (C) Staff.
Introduction to concepts and methods of calculus for students with little or no previous calculus experience. Polynomial and elementary transcendental functions and their applications, derivatives, extremum problems, curve-sketching, approximations; integrals and the fundamental theorem of calculus.
L/R 104. Calculus, Part I. (C) Staff.
Brief review of High School calculus, applications of integrals, transcendental functions, methods of integration, infinite series, Taylor's theorem. Use of symbolic manipulation and graphics software in calculus.
L/R 114. Calculus, Part II. (C) Staff. Prerequisite(s): Math 104.
Functions of several variables, vector-valued functions, partial derivatives and applications, double and triple integrals, conic sections, polar coordinates, vectors and analytic geometry, first and second order ordinary differential equations. Applications to physical sciences. Use of symbolic manipulation and graphics software in calculus.
L/R 115. Calculus, Part II with Probability and Matrices. (C) Staff. Prerequisite(s): Math 104.
Functions of several variables, partial derivatives, multiple integrals, differential equations; introduction to linear algebra and matrices with applications to linear programming and Markov processes. Elements of probability and statistics. Applications to social and biological sciences. Use of symbolic manipulation and graphics software in calculus.
L/R 116. Honors Calculus. (C) Staff. Prerequisite(s): Math 104.
This is an Honors level version of Math 114 which explores the mathematics more deeply.
L/L 122-123. Community Math Teaching Project. (M) Staff.
This course allows Penn students to teach a series of hands-on activities to students in math classes at University City High School. The semester starts with an introduction to successful approaches for teaching math in urban high schools. The rest of the semester will be devoted to a series of weekly hands-on activities designed to teach fundamental aspects of geometry. The first class meeting of each week, Penn faculty teach Penn students the relevant mathematical background and techniques for a hands-on activity. During the second session of each week, Penn students will teach the hands-on activity to a small group of UCHS students. The Penn students will also have an opportunity to develop their own activity and to implement it with the UCHS students.
L/R 170. Ideas in Mathematics. (C) Natural Science & Mathematics Sector. Class of 2010 and beyond. Staff. May also be counted toward the General Requirement in Natural Science & Mathematics.
Topics from among the following: logic, sets, calculus, probability, history and philosophy of mathematics, game theory, geometry, and their relevance to contemporary science and society.
180. (PPE 180) Analytical Methods in Economics, Law, and Medicine. (M) Staff.
Elementary applications of decision analysis, game theory, probability and statistics to issues in accounting, contracting, finance, law, and medicine, amongst others.
L/L 202. Proving Things: Analysis. (C) Staff. Corequisite(s): Math 104, 114 or 240.
This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying real and complex numbers, sequences, series, continuity, differentiability and integrability. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.
L/L 203. Proving things: Algebra. (C) Staff. Corequisite(s): Math 104, 114 or 240.
This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying arithmetic, algebra, linear algebra, groups, rings and fields. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.
210. Mathematics in the Media. (C) Staff. Prerequisite(s): Math 114, Math 115 or equivalent.
This course counts as a
regular elective for both the Mathematics Major and Minor.
L/R 240. Calculus, Part III. (C) Staff. Prerequisite(s): Calculus II.
Linear algebra: vectors, matrices, systems of linear equations, eigenvalues and eigenvectors. Vector calculus: functions of several variables, vector fields, line and surface integrals, Green's, Stokes' and divergence theorems. Series solutions of ordinary differential equations, Laplace transforms and systems of ordinary differential equations. Use of symbolic manipulation and graphics software.
L/R 241. Calculus, Part IV. (C) Staff. Prerequisite(s): MATH 240.
Sturm-Liouville problems, orthogonal functions, Fourier series, and partial differential equations including solutions of the wave, heat and Laplace equations, Fourier transforms. Introduction to complex analysis. Use of symbolic manipulation and graphics software.
L/R 260. Honors Calculus, Part II. (C) Staff. Prerequisite(s): Calculus II.
This is an honors version of Math 240 which explores the same topics but with greater mathematical rigor.
299. Undergraduate
Research in Mathematics Staff. Prequisite(s): Knowledge
of elementary arithmetic and calculus will be helpful.
Introduction to adeles and their application in Analysis, Geometry and
Number theory.
312. Linear Algebra. (M) Staff. Prerequisite(s): MATH 240. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.
Linear transformations, Gauss Jordan elimination, eigenvalues and eigenvectors, theory and applications. Mathematics majors are advised that MATH 312 cannot be taken to satisfy the major requirements.
313. (CIS 313, MATH513) Computational Linear Algebra. Staff. Prerequisite(s): Math 114 or 115, and some programming experience. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.
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 optimization.
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 decompositions 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.
320. Computer Methods in Mathematical Science I. (A) Staff. Prerequisite(s): MATH 240 or concurrent and ability to program a computer, or permission of instructor.
Students will use symbolic manipulation software and write programs to solve problems in numerical quadrature, equation-solving, linear algebra and differential equations. Theoretical and computational aspects of the methods will be discussed along with error analysis and a critical comparison of methods.
321. Computer Methods in Mathematical Sciences II. (M) Staff. Prerequisite(s): MATH 320.
Continuation of MATH 320.
340. (LGIC210) Discrete Mathematics I. (M) Staff. Prerequisite(s): MATH 114 or Math 115 or permission of the instructor.
Topics will be drawn from some subjects in combinatorial analysis with applications to many other branches of math and science: graphs and networks, generating functions, permutations, posets, asymptotics.
341. (LGIC220) Discrete Mathematics II. Staff. Prerequisite(s): Math 340/Logic 210 or permission of the instructor.
Topics will be drawn from
some subjects useful in the analysis of information and computation:
logic, set theory, theory of computation, number theory, probability,
and basic cryptography.
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350. Number Theory. (M) Staff.
Congruences, Diophantine equations, continued fractions, nonlinear congruences, and quadratic residues.
L/L 360. Advanced Calculus. (C) Staff. Prerequisite(s): MATH 240.
Syllabus for MATH 360-361: a study of the foundations of the differential and integral calculus, including the real numbers and elementary topology, continuous and differentiable functions, uniform convergence of series of functions, and inverse and implicit function theorems. MATH 508-509 is a masters level version of this course.
L/L 361. Advanced Calculus. (C) Staff. Prerequisite(s): MATH 360.
Continuation of MATH 360.
L/L 370. Algebra. (C) Staff. Prerequisite(s): MATH 240. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.
Syllabus for MATH 370-371: an introduction to the basic concepts of modern algebra. Linear algebra, eigenvalues and eigenvectors of matrices, groups, rings and fields. MATH 502-503 is a masters level version of this course.
L/L 371. Algebra. (C) Staff. Prerequisite(s): MATH 370. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.
Continuation of MATH 370.
410. Complex Analysis. (C) Staff. Prerequisite(s): MATH 240 or permission of instructor.
Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.
420. Ordinary Differential Equations. (C) Staff. Prerequisite(s): MATH 240 or permission of instructor.
After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.
425. Partial Differential Equations. (A) Staff. Prerequisite(s): MATH 240 or permission of instructor. Knowledge of PHYS 150-151 will be helpful.
Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.
430. Introduction to Probability. (M) Staff. Prerequisite(s): MATH 240.
Random variables, events, special distributions, expectations, independence, law of large numbers, introduction to the central limit theorem, and applications.
432. Game Theory. (C) Staff.
A mathematical approach to game theory, with an emphasis on examples of actual games. Topics will include mathematical models of games, combinatorial games, two person (zero sum and general sum) games, non-cooperating games and equilibria.
450. (MATH542) Seminar in Computational Mathematics. (M) Staff. Prerequisite(s): Permission of instructor. May, with permission, be repeated for credit.
A seminar devoted to the study of algorithms for solving problems in discrete mathematics.
460 (MATH 500). Geometry-Topology, Differential Geometry. (M) Staff. Prerequisite(s): Math 240/241, Math 360 or 508, or with the permission of the instructor.
Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.
465 (MATH 501). Geometry-Topology, Differential Geometry. (M) Staff. Prerequisite(s): Math 240/241, Math 360 or 508, or with the permission of the instructor.
Review of 2- and 3-dimensional vector calculus, differential geometry of curves and surfaces, Gauss-Bonnet theorem, elementary Riemannian geometry, knot theory, degree theory of maps, transversality.
475. Statistics of Law. (M) Staff. Prerequisite(s): Permission of instructor; no formal mathematical prerequisite, but one year of college calculus would be helpful.
Introduction to probability and statistics with illustrative material drawn from cases. Statistical inference. Basic concepts of information theory. This course may not be taken to satisfy the requirements of the major.
480. (MATH550) Elementary Topics in Advanced Real Analysis. (M) Staff. Prerequisites: A year of analysis at the 300 level or above (for example, Mathematics 360-361, or 508-509); a semester of linear algebra at the 300 level or above (for example, Mathematics 370) or Permission of Instructor.
499. Supervised Study. (C) Staff. Prerequisite(s): Permission of major adviser. Hours and credit to be arranged.
Study under the direction of a faculty member. Intended for a limited number of mathematics majors.
L/L 502. Abstract Algebra. (A) Staff. Prerequisite(s): Math 240. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.
An introduction to groups, rings, fields and other abstract algebraic systems, elementary Galois Theory, and linear algebra -- a more theoretical course than Math 370.
L/L 503. Abstract Algebra. (B) Staff. Prerequisite(s): Math 502 or with the permission of the instructor. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513.
Continuation of Math 502.
504. Graduate Proseminar in Mathematics. (A) Staff.
This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.
505. Graduate Proseminar in Mathematics. (B) Staff.
This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.
L/L 508. Advanced Analysis. (A) Staff. Prerequisite(s): Math 240/241. Math 200/201 also recommended.
Construction of real numbers, the topology of the real line and the foundations of single variable calculus. Notions of convergence for sequences of functions. Basic approximation theorems for continuous functions and rigorous treatment of elementary transcendental functions. The course is intended to teach students how to read and construct rigorous formal proofs. A more theoretical course than Math 360.
L/L 509. Advanced Analysis. (B) Staff. Prerequisite(s): Math 508 or with the permission of the instructor. Linear algebra is also helpful.
Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods.
512. Advanced Linear Algebra. Staff. Prerequisite(s): Math 114 or 115. Math 512 covers Linear Algebra at the advanced level with a theoretical approach. Students can receive credit for at most one of Math 312 and Math 512.
Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Canonical 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.
513. (CIS 313, MATH313) Computational Linear Algebra. Staff.
This course is open only to graduate students from departments other than mathematics. The course as presently taught is merged with Math 313, with additional work required from students enrolled in Math 513. For a description of the topics and pre-requisites, please refer to the description of Math 313.
520. Selections from Algebra. (M) Staff. Corequisite(s): Math 502 or permission of the instructor.
Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.
521. Selections from Algebra. (M) Staff. Corequisite(s): Math 502 or permission of the instructor.
Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.
524. Topics in Modern Applied Algebra. (M) Staff. Prerequisite(s): Math 371 or Math 503.
Topics such as automata, finite state languages, Boolean algebra, computers and logical design will be discussed.
525. Topics in Modern Applied Algebra. (M) Staff. Prerequisite(s): Math 371 or Math 503.
Topics such as automata, finite state languages, Boolean algebra, computers and logical design will be discussed.
530. Mathematics of Finance. (M) Staff. Prerequisite(s): Math 240, Stat 430.
This course presents the basic mathematical tools to model financial markets and to make calculations about financial products, especially financial derivatives. Mathematical topics covered: stochastic processes, partial differential equations and their relationship. No background in finance is assumed.
540. (MATH730) Selections from Classical and Functional Analysis. (M) Staff. Corequisite(s): Math 508 or permission of the instructor.
Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces.
541. Selections from Classical and Functional Analysis. (M) Staff. Corequisite(s): Math 508 or permission of the instructor.
Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces.
542. (MATH450) Calculus of Variations. (M) Staff. Prerequisite(s): Math 241.
Introduction to calculus of variations. The topics will include the variation of a functional, the Euler-Lagrange equations, parametric forms, end points, canonical transformations, the principle of least action and conservation laws, the Hamilton-Jacobi equation, the second variation.
546. (STAT530) Probability Theory. (A) Staff.
547. (STAT531) Stochastic Processes. (M) Staff.
548. Topics in Analysis. (M) Staff. Prerequisite(s): Math 360/361 and Math 370; or Math 508/509 and Math 502.
Topics may vary but typically will include an introduction to topological linear spaces and Banach spaces, and to Hilbert space and the spectral theorem. More advanced topics may include Banach algebras, Fourier analysis, differential equations and nonlinear functional analysis.
549. Topics in Analysis. (M) Staff. Prerequisite(s): Math 548 or with the permission of the instructor.
Continuation of Math 548.
560. Selections from Geometry and Topology. (M) Staff. Corequisite(s): Math 500 or permission of the instructor.
Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory.
561. Selections from Geometry and Topology. (M) Staff. Corequisite(s): Math 500 or permission of the instructor.
Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory.
570. (LGIC310, PHIL006, PHIL506) Introduction to Logic and Computability. (M) Staff. Prerequisite(s): Math 371 or Math 503.
Propositional logic: semantics, formal deductions, resolution method. First order logic: validity, models, formal deductions; Godel's completeness theorem, Lowenheim-Skolem theorem: cut-elimination, Herbrand's theorem, resolution method. Computability: finite automata, Turing machines, Godel's incompleteness theorems. Algorithmically unsolvable problems in mathematics.
SM 571. (LGIC320, MATH671, PHIL412) Introduction to Logic and Computability. (M) Staff. Prerequisite(s): Math 570 or with the permission of the instructor.
Continuation of Math 570.
572. Introduction to Axiomatic set theory. Staff.
Topics will include: the axioms, ordinal and cardinal arithmetic, formal construction of natural numbers and real numbers within set theory, formal treatment of definition by recursion.
574. Mathematical Theory of Computation. (M) Staff. Prerequisite(s): Math 320/321. Spring 09 (prereq. for this version Math240 and an interest in the subject), Prof. Pinsky.
This course will discuss advanced topics in Mathematical Theory of Computation.
575. Mathematical Theory of Computation. (M) Staff. Prerequisite(s): Math 574 or with the permission of the instructor.
Continuation of Math 574.
580. Combinatorial Analysis and Graph Theory. (M) Staff. Prerequisite(s): Permission of the instructor.
Generating functions, enumeration methods, Polya's theorem, combinatorial designs, discrete probability, extremal graphs, graph algorithms and spectral graph theory, combinatorial and computational geometry.
581. Combinatorial Analysis and Graph Theory. (M) Staff. Prerequisite(s): Math 580 or with the permission of the instructor.
Continuation of Math 580.
582. Applied Mathematics and Computation. (M) Staff. Prerequisite(s): Math 240-241. Math 312, Math 360. Knowledge of Math 512 and Math 508 is recommended.
This course offers first-hand experience of coupling mathematics with computing and applications. Topics include: Random walks, randomized algorithms, information theory, coding theory, cryptography, combinatorial optimization, linear programming, permutation networks and parallel computing. Lectures will be supplemented by informal talks by guest speakers from industry about examples and their experience of using mathematics in the real world.
583. Applied Mathematics and Computation. (M) Staff. Prerequisite(s): Math 582 or with the permission of the instructor.
Continuation of Math 582.
584. (BE 584) The Mathematics of Medical Imaging and Measurement. (M) Staff. Prerequisite(s): Math 241, knowledge of linear algebra and basic physics.
In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly. The data which is accessible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT).In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.
585. The Mathematics of Medical Imaging and Measurement. (M) Staff. Prerequisite(s): Math 584 or with the permission of the instructor.
Continuation of Math 584.
590. Advanced Applied Mathematics. (M) Staff. Prerequisite(s): Math 241.
This course offers first-hand experience of coupling mathematics with applications. Topics will vary from year to year. Among them are: Random walks and Markov chains, permutation networks and routing, graph expanders and randomized algorithms, communication and computational complexity, applied number theory and cryptography.
591. Advanced Applied Mathematics. (M) Staff. Prerequisite(s): Math 590 or with the permission of the instructor.
Continuation of Math 590.
594. (PHYS500) Advanced Methods in Applied Mathematics. (M) Staff. Prerequisite(s): Math 241 or Permission of Instructor. Physics 151 would be helpful for undergraduates.
Introduction to mathematics used in physics and engineering, with the goal of developing facility in classical techniques. Vector spaces, linear algebra, computation of eigenvalues and eigenvectors, boundary value problems, spectral theory of second order equations, asymptotic expansions, partial differential equations, differential operators and Green's functions, orthogonal functions, generating functions, contour integration, Fourier and Laplace transforms and an introduction to representation theory of SU(2) and SO(3). The course will draw on examples in continuum mechanics, electrostatics and transport problems.
599. Independent Study. (C)
last updated 05/19/2010