A new technique is proposed for NMR image restoration from the influence of main magnetic field non-uniformities. This technique is applicable to direct Fourier NMR imaging. The mathematical basis and details of this technique are fully described. Modification to include image restoration from non-linear field gradient influence is also presented. Computer simulation demonstrates the effectiveness of this technique for both Fourier zeugmatography and spin-warp imaging.
MATH 16 Symmetry. A mathematical treatment of the symmetries of wallpaper patterns. The main goal is to prove that the symmetries of these patterns fall into seventeen distinct types. In addition, students will learn to identify the symmetries of given patterns (with special emphasis on the periodic drawings of M.C. Escher) and to draw such patterns. Three lectures, one section. Recommendations: High school geometry. Engineering students are not permitted to take MATH 16 for credit.
MATH 19 The Mathematics Of Social Choice. Introduction to mathematical methods for dealing with questions arising from social decision making. Topics vary but usually include ranking, determining the strength of, and choosing participants in multicandidate and two-candidate elections, and apportioning votes and rewards to candidates. Recommendations: High school algebra. Engineering students are not permitted to take MATH 19 for credit.
MATH 87 Mathematical Modeling And Computation. A survey of major techniques in the use of mathematics to model physical, biological, economic, and other systems; topics may include derivative-based optimization and sensitivity analysis, linear programming, graph algorithms, probabilistic modeling, Monte-Carlo methods, difference equations, and statistical data fitting. This course includes an introduction to computing using a high-level programming language, and studies the transformation of mathematical objects into computational algorithms. Prerequisites: (1) MATH 34 , 36, or 39, and (2) Math 70 or 72, or permission of instructor. Recommendations: MATH 34, MATH 36 or MATH 39, or consent.
MATH 102 Math-Education: From Numbers to Functions. An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and the pedagogical insights behind the following topics: real numbers, fractions and their multiple representations, introduction to functions: the intuitive and formal definition of function, composition of functions, representations through tables, graphs, dynagraphs, algebraic and verbal expressions, the vertical line criterion, composition of functions, examples of functions coming from arithmetic operations as well as functions commonly used in mathematics and science, functional approach to division with remainder, decimals and decimal representation of rational numbers, divisibility for integers and decomposition into product of powers of primes. Teaching projects with school age students are an integral part of this course. Offered on line with a face-to -face component. Permission of instructor.
MATH 103 Math-Education: Transformations and Equations. An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and the pedagogical insights behind the following topics: transformations of the plane with an emphasis on the comparison with arithmetic operations and the action of transformations on the graphs of functions. Geometric and algebraic interpretations of equations. The use of transformations in the solutions of linear and quadratic equations. Divisibility for integers and polynomials, the euclidean algorithm for the greatest common divisor, divisibility and factorization of polynomials and it solution in the solution of polynomial equations. Teaching projects with school age students are an integral part of the course. This course is offered on line with a face-to -face component. Permission of instructor. MATH 104 Math-Education: Change and Invariance. An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and the pedagogical insights behind the following topics: Helping students with word problems. Functions of several variables. Linear systems of equations and their solutions. Limits of sequences and of functions, limits at infinity. Slope and rate of change for non-linear functions. The derivative function and applications. Teaching projects with school age students are an integral part of the course. This course is offered on line with a face-to -face component. Permission of instructor.
MATH 112 Topics In The History Of Mathematics. The evolution of mathematical concepts and techniques from antiquity to modern times. Recommendations: MATH 34 or 39 or permission of instructor.
MATH 164 The Mathematics of Poverty and Inequality. Mathematical description of wealth distribution (some distribution theory, Lorenz curves), and the quantification of inequality (Hoover index, Gini coefficient, Theil indices, Sen's properties of inequality metrics). Agent-based models of wealth distribution, random walks, Wiener processes, Boltzmann and Fokker-Planck equations, and their application to models of wealth distribution. Wealth condensation and weak solutions. Upward mobility and first-passage times. Methods of mathematical modeling and comparison with empirical observations are emphasized throughout. Prerequisites: MATH 42: Calculus III or equivalent; and MATH 51: Differential Equations or equivalent; or instructor permission. Recommended but not strictly necessary: MATH 135: Real Analysis or equivalent; and MATH 165: Probability or equivalent.
MATH 190 Advanced Special Topics. Content and prerequisites vary from semester to semester. Topics covered in recent years have included mathematical neuroscience, Lie algebras, and nonlinear dynamics and chaos.
Mathematical analysis of sustainability: measurement, flows, networks, rates of change, uncertainty and risk, applying analysis in decision making; using quantitative evidence to support arguments; examples. MATH 033 Mathematics for Sustainability (3) (GQ) This course is one of several offered by the mathematics department with the goal of helping students from non-technical majors partially satisfy their general education quantification requirement. It is designed to provide an introduction to various mathematical modeling techniques, with an emphasis on examples related to environmental and economic sustainability. The course may be used to fulfill three credits of the GQ requirement for some majors, but it does not serve as a prerequisite for any mathematics courses and should be treated as a terminal course. The course provides students with the mathematical background and quantitative reasoning skills necessary to engage as informed citizens in discussions of sustainability related to climate change, resources, pollution, recycling, economic change, and similar matters of public interest. Students apply these skills through writing projects that require quantitative evidence to support an argument. The mathematical content of the course spans six key areas: \"measuring\" (representing information by numbers, problems of measurement, units, estimation skills); \"flowing\" (building and analyzing stock-flow models, calculations using units of energy and power, dynamic equilibria in stock-flow systems, the energy balance of the earth-sun system and the greenhouse effect); \"connecting\" (networks, the bystander effect, feedbacks in stock-flow models); \"changing\" (out-of-equilibrium stock-flow systems, exponential models, stability of equilibria in stock-flow systems, sensitivity of equilibria to changes in a parameter, tipping points in stock-flow models); \"risking\" (probability, expectation, bayesian inference, risk vs uncertainty; \"deciding\" (discounting, uses and limitations of cost-benefit analysis, introduction to game theory and the tragedy of the commons, market-based mechanisms for pollution abatement, ethical considerations).
This course will provide students with the mathematical background and quantitative skills needed to make sound financial decisions. This course introduces personal finance topics including simple interest, simple discount, compound interest, annuities, investments, retirement plans, inflation, depreciation, taxes, credit cards, mortgages, and car leasing. Students will learn how to use linear equations, exponential and logarithmic equations, and arithmetic and geometric sequences to solve real world financial problems. Students will answer questions such as, What is the most they can afford to pay for a car How much do they need to invest in their 401(k) account each month to retire comfortably What credit card is the best option In a society where consumers are presented with a vast array of financial products and providers, students are enabled to evaluate options and make informed, strategic decisions. This course may be used by students from non-technical majors to satisfy 3 credits of their General Education Quantification (GQ) requirement. This course does not serve as a prerequisite for any mathematics courses and should be treated as a terminal course.
This course presents a general view of a number of mathematical topics to a non-technical audience, often relating the mathematical topics to a historical context, and providing students with an opportunity to engage with the mathematics at an introductory level. Although some variation in topics covered may take place among different instructors at different campuses, an example of such a course focuses on a number theory theme throughout the course, beginning with the Greeks' view of integers, the concept of divisors, the calculation of greatest common divisors (which originates with Euclid), the significance of the prime numbers, the i