03-121 Modern Biology
Fall or Spring: 9 units

This is an introductory course that provides the basis for further studies in biochemistry, cell biology, genetics and molecular biology. This course emphasizes the chemical principles underlying biological processes and cell structures as well as the analysis of genetics and heredity from a molecular perspective. This is the introductory biology course for all science and non-science majors. 3 hrs. lecture. Prerequisite: Fundamental knowledge of high school chemistry and biology.

09-105 Introduction to Modern Chemistry
Fall or Spring: 10 Units

This course begins with a very brief survey of some fundamental principles of chemistry and a presentation of chemically interesting applications and sophisticated problems. These will form the basis for introducing various facets of the course that deal ultimately with the relationship between the structure of molecules and their chemical properties and behavior. The subject matter will include principles of atomic structure, chemical bonding, and molecular structures of organic and inorganic compounds including some transition metal complexes. Relevant examples will be drawn from such areas as environmental, materials, and biological chemistry. 3 hrs. lecture, 2 hrs. recitation.

15-100 Introductory/Intermediate Programming
Fall and Spring: 10 units

An introduction to the process of program design and analysis using the Java programming language for students who have no prior programming experience. Topics to be covered include basic data types and their operators, I/O, control structures (selection, loops), classes (including methods and fields), files, arrays, and simple sorting and searching algorithms. If you've taken a programming course before and have used functions, loops, and arrays, you should enroll in 15-111 instead of 15-100.

NOTE: Students who receive a grade of C or less in 15-100 should discuss whether they are adequately prepared for 15-111 with their academic advisor.

15-111 Intermediate/Advanced Programming
Fall and Spring: 10 units

An introduction to the process of program design and analysis using the Java programming language for students with some prior programming experience in any other language. Topics to be covered include an overview of fundamental programming concepts using Java as well as object-based programming techniques, data aggregates, self-referential data structures (e.g., linked lists, stacks, queues, trees, and graphs), and an introduction to the analysis of algorithms that operate on those data structures. Prerequisite: prior experience in any programming language.

NOTE: Students who receive a grade of C or less in 15-111 should discuss whether they are adequately prepared for 15-211 with their academic advisor.

15-200 Advanced Programming/Practicum
Fall and Spring: 9 units

This course assumes prior programming experience in Java (at the level of 15-100) and is designed to expand students' knowledge of computer science and sharpen their programming skills. The course extends object-oriented programming techniques begun in 15-100 and covers data aggregates, data structures, (e.g., linked lists, stacks, queues, trees, and graphs), and an introduction to the analysis of algorithms that operate on those data structures. The course is currently taught in Java, and along with 21-127 serves as a prerequisite for 15-211.

NOTE: Students who receive a C or less in 15-200 should discuss whether they are adequately prepared to take 15-211 with their academic advisor.

Prerequisite: 15-100.

15-211 Fundamental Data Structures and Algorithms
Fall and Spring: 12 units

Fundamental programming concepts are presented together with supporting theoretical bases and practical applications. This course emphasizes the practical application of techniques for writing and analyzing programs: data abstraction, program verification, and performance analysis. These techniques are applied in the design and analysis of fundamental algorithms and data structures. The course is currently taught in Java. Prerequisites: 15-111 and either 15-151 or 21-127

21-120 Differential and Integral Calculus
Fall and Spring: 10 units

Functions, limits, derivatives, curve sketching, Mean Value Theorem, trigonometric functions, related rates, linear and quadratic approximations, maximum-minimum problems. Definite and indefinite integrals; inverse functions, logarithmic, exponential, and hyperbolic functions; applications of integration, integration by substitution and by parts.

3 hours lecture, 2 hours recitation.

21-122 Integration, Differential Equations, and Approximation
Fall and Spring: 10 units

Integration by trigonometric substitution and partial fractions; arclength; improper integrals; Simpson's and Trapezoidal Rules for numerical integration; separable differential equations, first order linear differential equations, homogeneous second order linear differential equations with constant coefficients. Indeterminate forms, Newton's method, Taylor's Theorem including a discussion of the remainder, sequences, series, power series.

3 hours lecture, 2 hours recitation.

Prerequisite: 21-120.

Suggestions of what a student starting calculus should know.

Concepts of Mathematics
Fall or Spring: 9 units

By the end of this course the student should be able to

  1. Construct logically correct proofs using basic proof techniques such as proof by contradiction.
  2. Deploy basic problem solving strategies.
Truth values, connectives, truth tables, contrapositives. Quantifiers. Proof by contradiction. Sets, intersections, unions, differences, the empty set. Integers, divisibility. Proof by induction. Primes, sieve of Eratosthenes, prime factorization. Gcd and lcm, Euclid's algorithm, solving ax + by =c. Congruences, modular arithmetic. Recursion. Linear recurrences. Functions and inverses. Permutations. Binomial coefficients, Catalan number. Inclusion-exclusion. Infinite cardinalities. Binary operations. Groups. Binary relations, equivalence relations. Graphs. Euler characteristic, planar graphs, five color theorem, rationals, reals, polynomials, complex numbers.

3 hours lecture, 2 hours recitation.

No prerequisites.

A prerequisite for 15-211, 21-341, 21-228, 21-355 and other mathematics courses requiring the construction of proofs.

A co-requisite for 18-240.

Recommended as a prerequisite for 21-241.

21-241 Matrix Algebra
Spring and Fall: 9 units

Vectors and matrices, the solution of linear systems of equations, vector spaces and subspaces, orthogonality, determinants, real and complex eigenvalues and eigenvectors, linear transformations. 3 hrs. lec. Prerequisite: 21-127 recommended.

21-256 Multivariate Analysis
Fall or Spring: 9 units

Matrix algebra: vectors, matrices, systems of equations, dot product, cross product, lines and planes. Optimization: partial derivatives, the chain rule, gradient, unconstrained optimization, constrained optimization. Improper integrals. Multiple integration: iterated integrals, probability applications, triple integrals, change of variables. 3 hours lecture, 2 hours recitation. Prerequisite: 21-120.

This course is designed for students in Economics or Business Administration.

21-259 Calculus in Three Dimensions
Fall or Spring: 9 units

Vectors, lines, planes, quadratic surfaces, polar, cylindrical and spherical coordinates, partial derivatives, directional derivatives, gradient, divergence, curl, chain rule, maximum-minimum problems, multiple integrals, parametric surfaces and curves, line integrals, surface integrals, Green-Gauss theorems. 3 hrs. lec. 2 hrs. rec. Prerequisite: 21-120 or 21-121.

21-260 Differential Equations
Fall or Spring: 9 units

Ordinary differential equations: first and second order equations, applications, Laplace transforms; partial differential equations: partial derivatives, separation of variables, Fourier series; systems of ordinary differential equations; applications. 3 hours lecture, 1 hour recitation. Prerequisite: 21-122.

21-270 Introduction to Mathematical Finance
Spring: 9 units

This is a first course for those considering majoring or minoring in Computational Finance. The theme of this course is pricing derivative securities by replication. The simplest case of this idea, static hedging, is used to discuss net present value of a non-random cash flow, internal rate of return, and put-call option parity. Pricing by replication is then considered in a one-period random model. Risk-neutral probability measures, the Fundamental Theorems of Asset Pricing, and an introduction to expected utility maximization and mean-variance analysis are presented in this model. Finally, replication is studied in a multi-period binomial model. Within this model, the replicating strategies for European and American options are determined. 3 hours lecture.

Introduction to Mathematical Finance is a prerequisite for 21-370 Discrete-Time Finance.

21-292 Operations Research I
Spring: 9 units

Operations research offers a scientific approach to decision making, most commonly involving the allocation of scarce resources. This course develops some of the fundamental methods used. Linear programming: the simplex method and its linear algebra foundations, duality, post-optimality and sensitivity analysis; the transportation problem; the critical path method; non-linear programming methods. 3 hours lecture, 1 hour recitation. Prerequisites: 21-122 and 21-241.

21-325 Probability
Fall: 9 units

This course focuses on the understanding of basic concepts in probability theory and illustrates how these concepts can be applied to develop and analyze a variety of models arising in computational biology, finance, engineering and computer science. The firm grounding in the fundamentals is aimed at providing students the flexibility to build and analyze models from diverse applications as well as preparing the interested student for advanced work in these areas. The course will cover core concepts such as probability spaces, random variables, random vectors, multivariate densities, distributions, expectations, sampling and simulation; independence, conditioning, conditional distributions and expectations; limit theorems such as the strong law of large numbers and the central limit theorem; as well as additional topics such as large deviations, random walks and Markov chains, as time permits. 3 hours lecture. Prerequisites: 21-122, 21-259.

21-341 Linear Algebra I
Fall or Spring: 9 units

Vector spaces: subspaces and linear independence, basis and dimension, row equivalence of matrices, general theorems about vector spaces, systems of linear equations, linear manifolds. Linear transformations: addition and multiplication, matrices of linear transformations. Inner products: angle and orthogonality, Gram-Schmidt orthogonalization, orthogonal transformations. Determinants: existence and uniqueness, multiplication theorem. 3 hours lecture. Prerequisite: 21-127.

21-365 Projects in Applied Mathematics
Fall or Spring: 9 units

This course provides students with an opportunity to solve problems posed by area companies. It is also designed to provide experience working as part of a team to solve problems for a client. The background needed might include linear programming, simulation, data analysis, scheduling, numerical techniques, etc. Prerequisite: Permission of the instructor; technical background necessary varies with the project.

21-369 Numerical Methods
Fall or Spring: 9 units

This course provides an introduction to the use of computers to solve scientific problems. Methods for the computational solution of linear algebra systems, nonlinear equations, the interpolation and approximation of functions, differentiation and integration, and ordinary differential equations. Analysis of roundoff and discretization errors and programming techniques. 3 hrs. lec. Prerequisites: 15-100 and 21-259.

21-370 Discrete-Time Finance
Fall: 9 units

This course introduces the Black-Scholes option pricing formula, shows how the binomial model provides a discretization of this formula, and uses this connection to fit the binomial model to data. It then sets the stage for Continuous-Time Finance by discussing in the binomial model the mathematical technology of filtrations, martingales, Markov processes and risk-neutral measures. Additional topics are American options, expected utility maximization, the Fundamental Theorems of Asset Pricing in a multi-period setting, and term structure modeling, including the Heath-Jarrow-Morton model. 3 hours lecture.

The pre-requisite for 21-370 is 21-270, and either 21-256 or 21-259, and the co-requisite is 70-207, 21-325, 36-225, or 36-217.

Students in 21-370 are expected to read and write proofs.

Note that this is a correction of the pre-requisites listed in the 2004-06 catalog.

Discrete-Time Finance is a prerequisite for 21-420 Continuous-Time Finance.

Note that 70-207 Probability and Statistics for Business Applications is an adequate probability co-requisite for this course but is not an adequate pre-requisite for 21-420 Continuous-Time Finance, which requires one of the calculus-based probability courses 21-325, 36-225 or 36-217.

21-372 Partial Differential Equations
Spring: 9 units

This course provides an introduction to partial differential equations and is recommended for majors in mathematics, physical science, or engineering. Boundary value problems on an interval, Fourier series, uniform convergence, the heat, wave, and potential equations on bounded domains, general theory of eigenfunction expansion, the Fourier integral applied to problems on unbounded domains, introduction to numerical methods. 3 hrs. lec. Prerequisites: 21-259 and 21-260.

21-420 Continuous-Time Finance
Spring: 9 units

This course begins with Brownian motion, stochastic integration,and Ito's formula from stochastic calculus. This theory is used to develop the Black-Scholes option pricing formula and the Black-Scholes partial differential equation. Additional topics may include models of credit risk, simulation, and expected utility maximization. 3 hours lecture.

Prerequisites: 21-260 and 21-370 and a calculus based probability course, 21-325, 36-225 or 36-217, is also required as a prerequisite and is usually taken before 21-370.

70-207 is not sufficient preparation in probability for this course.

Note that this is a correction of the pre-requisites listed in the 2004-06 catalog.

21-420 is a prerequisite for 45-816 Studies in Financial Engineering.

33-111 Physics for Science Students I
Fall or Spring: 12 units

Physics I combines the basic principles of mechanics with some quantum physics and relativity to explain nature on both a microscopic and macroscopic scale. The course will build models to describe the universe based on a small number of fundamental physics principles. Som simple computer modelling will be done to develop insight into the solving problems using Newton's laws. Topcs covered will include vectors, momentum, force, gravitation, oscillations, energy, quantum physics, center of mass motion, angular momentum, statistical physics and the laws of thermodynamics. No computer experience is needed. 3 hrs. lecture, 2 hrs. recitation. Pre- or Corequisite: 21-115, 21-116 or 21-121

36-225 Introduction to Probability and Statistics I
Fall: 9 units

This course is the first half of a year-long course which provides an introduction to probability and mathematical statistics for students in mathematics and statistics. The use of probability theory is illustrated with examples drawn from engineering, the sciences, and management. Topics include elementary probability theory, conditional probability and independence, random variables, distribution functions, joint and conditional distributions, laws of large numbers, and the central limit theorem. Not open to students who have received credit for 36-217 or 36-325. Prerequisite: 21-118 or 21-256

36-226 Introduction to Probability and Statistics II
Spring: 9 units

This course is the second half of a year-long course in probability and mathematical statistics. Topics include maximum likelihood estimation, confidence intervales, and hypothesis testing. If time permits there will also be a discussion of linear regression and the analysis of variance. Not open to students who have received credit for 36-326. Prerequisite: 36-225.

36-401 Modern Regression
Fall: 9 units

The material in this course concentrates on methods for the analysis of data. The emphasis is on description, validation, and interpretation. Topics include exploratory data analysis, statistical computing, and regression analysis. Real-world examples will be drawn from engineering and the various physical and social sciences. Students will do projects and write reports. Students who have taken 73-360 need permission of the instructor. Prerequisite: 36-226 or 36-310

36-402 Topic in Data Analysis
Spring: 9 units

The material in this course concentrates on methods for the analysis of data. The emphasis is on description, validation, and interpretation. Topics include the design of experiments, the analysis of categorical data, and if time permits, other topics such as nonlinear regression models, survival analysis, and multivariate methods. Real-world examples will be drawn from engineering and the various physical and social sciences. Students will do projects and write reports. Prerequisite: 36-401.

36-410 Introduction to Probability Models
Spring: 9 units

An introductory-level course in stochastic processes. Topics typically include Poisson processes, Markov chains, birth and death processes, random walks, recurrent events, and renewal theory. Examples are from reliability theory, queuing theory, inventory theory, and various applications in the social and physical sciences. Prerequisite: 36-217 or 36-225

36-461 Statistics Topic
Fall: 9 units

The format and content of the course are flexible and vary, depending on demand. Historical applications of statistical methods to real problems of data analysis may be studied, or special topics in probability and statistics such as decision theory, biostatistics, time series analysis, Bayesian statistics or non-parametric statistics may be covered. Prerequisite: 36-226, 36-310 or 36-326, and either concurrent 36-401 or instructor permission.

45-816 Studies in Financial Engineering
Fall: 6 units

This course focuses on the use of financial engineering and derivative securities in solving practical business problems. Students will work through business cases and give in-class simulated sales pitches to hypothetical clients. The cases highlight the design, valuation and hedging of structured products. In addition, we will look at real options and at derivative pricing with exotic "underlyings" such as energy, weather, and credit. Prerequisite: 21-420

70-122 Introduction to Accounting
Spring and Fall: 9 units

This course provides the knowledge and skills necessary for the student to understand financial statements and financial records and make use of the information for management and investment decisions. Topics include an overview of financial statements and business decisions; the balance sheet, the income statement, and the cash flow statement; sales revenue, receivables, and cash; cost of goods sold and inventory; long-lived assets and depreciation, and amortization; current and long-term liabilities; owners? equity; investments in other corporations; an introduction to financial statement analysis; and international issues dealing with financial statements.

70-391 Finance
Fall: 9 units

The course examines the role of the financial manager in the overall management and control of a firm. Stress is placed on the use of analytical models for improving the decision-making process. Both the short-term management of working capital and the long-term planning of capital structure and investment strategy are covered. Prerequisites: (21-257 or 21-292), 70-122, (36-202, 36-226, 70-207, 36-247, or 36-310).

70-393 FAST I
Fall: 9 units

The Financial Analysis and Securities Trading (FAST) system is an educational technology that teaches applied principles of financial economics using a sophisticated network of personal computers and workstations. Students learn finance using both real time data feeds as well as a simulated trading environment. Prerequisites: 70-391.

70-398 International Finance
Fall or Spring: 9 units

This course covers three main topics: the institutional structure of the various foreign exchange markets, the measurement of risk and return in an international setting and foreign currency risk management. Applications change from year to year to reflect current events. Recent examples are the various international financial crises (Mexico 94-95, South East Asia 97-98, Argentina 2002, etc.), the application of Value at Risk principles to the measuring and managing of the risks associated with foreign investments, and issues related to economic development in China and the U.S. current account deficit.

70-492 Investment Analysis
Fall or Spring: 9 units

Students gain an understanding of financial theories through learning the theory and development of basic computer programs that can be applied in a real world environment. Typical projects include obtaining the efficient frontier of a given set of securities; deciding on the optimal investment strategy for a given set of securities; calculating option prices using Black-Scholes and Binomial option pricing models.

Prerequisite: 70-391.

70-495 Corporate Finance
Fall or Spring: 9 units

This course focuses on how firms make decisions on investments, financing and dividend payout policies, as well as other advanced topics in finance.

Prerequisite: 70-391.

70-497 Options
Fall or Spring: 9 units

In this course students will learn to evaluate contingent claims such as options, futures, swaps and other exotic securities. In addition to covering canonical valuation formulae for standard option and future contracts, students will use numerical simulation methods to evaluate more exotic securities. The course will also cover various aspects of using derivative securities for risk management purposes. Prerequisite: 70-492.

73-100 Principles of Economics
Fall and Spring: 9 units

Literally, an introduction to economic principles, the goal of this course is to give students an understanding as to what constitutes good "economic thinking". This thought process is grounded in the construction and use of economics models. Drawing on issues in both microeconomics and macroeconomics, fundamental principles are shown to transcend particular examples and allow the field to be seen as a coherent, unified whole.

2 hours lecture, 1 hour recitation.

73-150 Microeconomics
Fall and Spring: 9 units

A calculus-based introduction to microeconomics. Topics in partial equilibrium analysis include supply and demand, consumer theory, theory of the firm, profit maximizing behavior, monopoly theory, and perfect competition. The course concludes with an introduction to general equilibrium analysis and the welfare laws.

3 hours lecture, 1 hour recitation.

Prerequisite: 73-100, 21-120

Co-requisites: 21-259 or 21-256.

73-200 Macroeconomics
Fall and Spring: 9 units

Through macroeconomic models built upon microeconomic foundations, insights are developed into economic growth processes and business cycles. Topics include aggregation and measurement, national income, business cycle measurement, economic welfare theorems and social inefficiencies, the effect of government fiscal policy upon employment and productivity, and the relationship between investment, interest rates and economic growth.

Prerequisites: (21-256 or 21-259) and 73-150

73-251 Economic Theory
Fall and Spring: 9 units

This course prepares students for advanced coursework in economics by providing a mathematically intensive overview of economic theory. Students take advantage of their knowledge of multi-dimensional calculus and constrained optimization techniques in order to understand the development and logical consistency of the most commonly employed economic models. Topics include consumer preferences and utility function representations, consumer choice under a budget constraint, substitution and income effects, compensated and uncompensated demands, expected utility theory, risk and insurance, technology and production functions, cost minimization, profit maximizing firms, perfect competition, single-firm markets, game theoretic analysis of markets with few firms, introduction to general equilibrium models and the welfare laws.

73-251 will be replaced by 73-252/253 in Spring 2008.

Prerequisites: (21-256 or 21-259) and 73-100

73-252 Advanced Microeconomic Theory
Fall and Summer, Mini Session: 6 Units

This course provides a mathematically intensive overview of advanced applications of microeconomic theory. Topics include: Marshallian and Hicksean demands, indirect utility functions, substitution/income effects and the Slutsky equation, expected utility theory, risk and insurance, game theory, principle/agent problems, oligopoly, and general equilibrium theory.

3 hours Lecture. 73-252 will be offered in Spring 2008.

Prerequisites: (21-256 or 21-259) and 73-150.

73-253 Advanced Macroeconomic Theory
Fall and Spring, Mini Session: 6 Units

This course provides a mathematically intensive overview of advanced applications of microeconomic theory. Topics may include: Solow and neo-classical growth models, the role of money and its effect on the economy, and the over-lapping generations model.

3 hours Lecture. 73-253 will be offered in Spring 2008.

Prerequisites: (21-256 or 21-259) and 73-252.

73-372 International Money and Finance
Fall or Spring: 9 units

This course is devoted to economic analysis of exchange rate behavior, balance of payments adjustments, the financing of payments imbalances, and related topics in the areas of international monetary, macro, and financial economics. A simple but flexible model of exchange rate determination will be formulated and tested empirically. Considerable emphasis will be given to issues concerning alternative monetary arrangements such as fixed vs. flexible exchange rates, currency unions, and commodity-money standards. Some historical consideration of the pre-1914 gold standard and the 1945- 1971 Bretton Woods system will be included, as well as institutional discussion of the present (and prospective) European Monetary System. Prerequisites: 73-250 and 73-300.

73-420 Monetary Theory and Policy
Offered Intermittently: 9 units

This course is concerned with various topics in monetary and macroeconomics including anticipated inflation, hyperinflation, output effects of monetary policies, alternative techniques of monetary policy implementation, and the interaction of monetary and fiscal policy strategies. Analysis of these issues is conducted by means of simple but explicit dynamic models incorporating rational expectations. In addition, attention is devoted to alternative types of monetary systems _ commodity vs. paper money, for example. This segment of the course includes some consideration of issues relating to a technologically advanced society in which transactions are carried out by means of a "computerized" economy-wide bookkeeping system, rather than by money. Prerequisites: 73-250 and 73-300

76-101 Interpretation and Argument
Fall and Spring: 9 units

Fulfills CCR2 Requirement for H&SS and a Designated Writing Course for other colleges. This course gives students a comprehensive grounding in communication processes. The class focuses on the way in which interpretive arguments function. The goal of the course is to develop skills in both written and verbal communication. In the class, students will develop these skills by reading and understanding the important issues and arguments regarding those issues advanced by a variety of texts, both fiction and non-fiction. They will then be asked to respond to these positions by developing positions of their own, in their writing and in their speaking. The course thus serves as an introduction to the discourse and arguments of the academic community, as well as serving as an introduction to some of the broader issues that the academic community addresses.

90-717 Professional Writing
6 Units

Enrollment limited to 25 students per section

Successful policy and management professionals constantly use writing to inform, persuade and motivate other professionals and, specifically, to gain acceptance of ideas and decisions, many that lead to contracts, grants, and promotions. This course develops a student's ability to write effectively for professional audiences.

The course is rhetorical in its approach: It is taught in the context of problem solving and critical thinking and it analyzes organizational problems based on the interrelationships of audience, purpose, form, social/organizational context and strategic thinking.

90-718 Professional Speaking
6 units

As a managers and analysts of public policy, you will explain new procedures and advocate specific policies to a variety of listeners. The purpose of Professional Speaking 90-718 is to refine your ability to verbally present and visually display specialized knowledge so that its significance for listeners is clear.

Assignments in Professional Speaking focus on organizational skills, argumentation, extemporaneous delivery, and use of computer graphics. Each presentation is an exercise in "thinking aloud." Presenters are asked to identify the outcomes they want to achieve, cite sources of information they use, and design visual aids that reinforce major points of their presentations.

90-729 Organizational Design and Implementation
Fall: 6 units

This course draws on insights and knowledge about organizational behavior with an eye toward using such information for managing in complex organizations. It is intended to provide managers with skills and perspectives that will enable them to work successfully in organizations. Specific topics will include work motivation, leadership, job design, biases in managerial decision making, understanding group processes, building bases of power in organizations, managing conflict, and the relationship between the organization and its environment.

99-101 Computing@Carnegie Mellon
Fall and Spring: Mini Session - 3 units

Computing@Carnegie Mellon is a 3-unit required class that ALL incoming undergraduate students take when they arrive on campus. The course is comprised of mostly Carnegie Mellon specific information and helps students understand what resources are available to them and what responsibilities they have as a user in our computing community. Class is held twice/week for 50 minutes for 1/2 of the semester. There is very little work required outside of class, so students are expected to attend all classes. This course is only offered during the Fall and Spring semesters. There are no text-out opportunities or summer programs. Advanced Placement Computing Courses cannot be credited/substituted for this requirement.