Vector spaces, linear dependence, basis and dimension, matrices, determinants, solving linear equations, eigenvalues and eigenvectors, quadratic forms, applications such as optimization or linear regression. The same as for Calculus I.
Cannot apply both Calculus courses and Math for Economics courses towards your major. This course is only open to Economics Majors and prospective majors.
Elements of calculus and linear algebra are important to the study of economics. This class is designed to provide the appropriate tools for study in the policy concentration. Examples and motivation are drawn from important topics in economics. Topics covered include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; constrained and unconstrained optimization. A continuation of Mathematics for Economics I. Further topics in vector calculus.
Vector spaces, matrix analysis. Linear and nonlinear programming with applications to game theory. Brief review of multivariate calculus: partial derivatives, chain rule, Riemann integral, change of variables, line integrals. Lagrange multipliers. Inverse and implicit function theorems and their applications.
Introduction to calculus on manifolds: definition and examples of manifolds, tangent vectors and vector fields, differential forms, exterior derivative, line integrals and integration of forms. Gauss' and Stokes' theorems on manifolds. Students should also have some familiarity with introductory physics even at the advanced high school level. An introduction to the dynamical processes that drive the circulation of the atmosphere and ocean, and their interaction.
This is the core of climate science. Lectures will be guided by consideration of observations and experiments, but the goal is to develop an understanding of the unifying principles of planetary fluid dynamics. Topics include the global energy balance, convection and radiation the greenhouse effect , effects of planetary rotation the Coriolis force , structure of the atmospheric circulation the Hadley cell and wind patterns , structure of the oceanic circulation wind-driven currents and the thermohaline circulation , climate and climate variability including El Nino and anthropogenic warming.
Fluid dynamics is the branch of physics that describes motions of fluids as varied as the flow of blood in the human body, the flight of an insect or the motions of weather systems on Earth. The course introduces the key concepts of fluid dynamics: the formalism of continuum mechanics, the conservation of mass, energy and momentum in a fluid, the Euler and Navier-Stokes equations, viscosity and vorticity.
These concepts are applied to study classic problems in fluid dynamics, such as potential flow around a cylinder, the Stokes flow, the propagation of sound and gravity waves and the onset of instability in shear flow. Among the topics to be covered are: the axioms of set theory; Boolean operations on sets; set-theoretic representation of relations, functions and orderings; the natural numbers; theory of transfinite cardinal and ordinal numbers; the axiom of choice and its equivalents; and the foundations of analysis.
If time permits we may also consider some more advanced topics, such as large cardinals or the independence results. Note: This course is intended for math majors and other students with a strong interest in mathematics. An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.
An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences.
In probability: mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal approximation; application to coin-tossing, radioactive decay, etc. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data. The aim of this class is to introduce students to probability theory, with a greater emphasis on rigor, more material, and a faster pace than the Theory of Probability class.
The material will include discrete and continuous probability, and the most fundamental limit theorems law of large numbers and Central Limit Theorem. Students will be made familiar with the classical models, computations on densities, and convergence to universal distributions. They will also be expected to understand the proofs of all the results seen in class, and be able to argue with mathematical rigor.
Techniques for counting and enumeration including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph-theoretic problems. Offered in the fall. Divisibility theory and prime numbers. Linear and quadratic congruences.
The classical number-theoretic functions. Continued fractions. Diophantine equations. Introduction to the mathematics of finance. Topics include: Linear programming with application pricing and quadratic. Interest rates and present value. Basic probability: random walks, central limit theorem, Brownian motion, lognormal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization. Formulation and analysis of mathematical models. Mathematical tool include dimensional analysis, optimization, simulation, probability, and elementary differential equations.
Applications to biology, sports, economics, and other areas of science.
The necessary mathematical and scientific background will be developed as needed. Students will participate in formulating models as well as in analyzing them.
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In numerical analysis one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course gives an introduction to this subject for mathematics majors. Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method. Intended primarily for premedical students with interest and ability in mathematics. Topics of medical importance using mathematics as a tool: control of the heart, optimal principles in the lung, cell membranes, electrophysiology, countercurrent exchange in the kidney, acid-base balance, muscle, cardiac catheterization, computer diagnosis.
Material from the physical sciences and mathematics is introduced as needed and developed within the course. Familiarity with a programming language is recommended. Introduces students to the use of computer simulation as a tool for investigating biological phenomena. The course requirement is to construct three computer models during the semester, to report on results to the class, and to hand in a writeup describing each project. These projects can be done individually, or as part of a team. Topics discussed in class are the circulation of the blood, gas exchange in the lung, electrophysiology of neurons and neural networks, the renal countercurrent mechanism, cross-bridge dynamics in muscle, and the dynamics of epidemic and endemic diseases.
Projects are normally chosen from this list, but may be chosen otherwise by students with other interests. A first course in ordinary differential equations, including analytical solution methods, elementary numerical methods, and modeling. Topics to be covered include: first-order equations including integrating factors; second-order equations including variation of parameters; series solutions; elementary numerical methods including Euler's methods, Runge-Kutta methods, and error analysis; Laplace transforms; systems of linear equations; boundary-value problems. Some optional topics to be chosen at the instructor's discretion include: nonlinear dynamics including phase-plane description; elementary partial differential equations and Fourier series.
Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Non-linear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.
Topics will include dynamics of maps and of first order and second-order differential equations: stability, bifurcations, limit cycles, dissection of systems with fast and slow time scales. Geometric viewpoint, including phase planes, will be stressed. Chaotic behavior will be introduced in the context of one-variable maps the logistic , fractal sets, etc.
Applications will be drawn from physics and biology. There will be homework and projects, and a few computer lab sessions programming experience is not a prerequisite. Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy's theorem and the Cauchy integral formula.
Singularities, residues, and Laurent series. Fractional Linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow etc. This course is an introduction to rigorous analysis on the real line. Topics include: the real number system, sequences and series of numbers, functions of a real variable continuity and differentiability , the Riemann integral, basic topological notions in a metric space, sequences and series of functions including Taylor and Fourier series.
This is an introduction to the rigorous treatment of the foundations of real analysis in one variable. It is based entirely on proofs. Show all. Table of contents 10 chapters Table of contents 10 chapters The Integers Pages Groups Pages Rings Pages Polynomials Pages Vector Spaces and Modules Pages Some Linear Groups Pages Field Theory Pages Finite Fields Pages The Real and Complex Numbers Pages Sets Pages In this image, a vertical jet is deflected into a horizontal sheet by a horizontal impactor.
At the sheet's edge, fluid flows outward along bounding rims that collide to create fluid chains. Photo courtesy A. Hasha and J. Mathematical Problem Solving Putnam Seminar. Topics in Mathematics with Applications in Finance.
Undergraduate Algebra | Department of Mathematics
High-Dimensional Statistics. Street-Fighting Mathematics. Linear Algebra. Introduction to Probability and Statistics. Statistics for Applications. Mathematics of Machine Learning. Combinatorial Analysis. Introduction to Numerical Methods. Applied Category Theory. An undergraduate degree in mathematics provides an excellent basis for graduate work in mathematics or computer science, or for employment in such mathematics-related fields as systems analysis, operations research, or actuarial science.
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