The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.

The Art of the Probable" addresses the history of scientific ideas, in particular the emergence and development of mathematical probability. But it is neither meant to be a history of the exact sciences per se nor an annex to, say, the Course 6 curriculum in probability and statistics. Rather, our objective is to focus on the formal, thematic, and rhetorical features that imaginative literature shares with texts in the history of probability. These shared issues include (but are not limited to): the attempt to quantify or otherwise explain the presence of chance, risk, and contingency in everyday life; the deduction of causes for phenomena that are knowable only in their effects; and, above all, the question of what it means to think and act rationally in an uncertain world. Our course therefore aims to broaden students’ appreciation for and understanding of how literature interacts with--both reflecting upon and contributing to--the scientific understanding of the world. We are just as centrally committed to encouraging students to regard imaginative literature as a unique contribution to knowledge in its own right, and to see literary works of art as objects that demand and richly repay close critical analysis. It is our hope that the course will serve students well if they elect to pursue further work in Literature or other discipline in SHASS, and also enrich or complement their understanding of probability and statistics in other scientific and engineering subjects they elect to take.

This course covers the fundamentals of astrodynamics, focusing on the two-body orbital initial-value and boundary-value problems with applications to space vehicle navigation and guidance for lunar and planetary missions, including both powered flight and midcourse maneuvers. Other topics include celestial mechanics, Kepler's problem, Lambert's problem, orbit determination, multi-body methods, mission planning, and recursive algorithms for space navigation. Selected applications from the Apollo, Space Shuttle, and Mars exploration programs are also discussed.

Galactic dynamics: potential theory, orbits, collisionless Boltzmann equation, etc. Galaxy interactions. Groups and clusters; dark matter. Intergalactic medium; x-ray clusters. Active galactic nuclei: unified models, black hole accretion, radio and optical jets, etc. Homogeneity and isotropy, redshift, galaxy distance ladder. Newtonian cosmology. Roberston-Walker models and cosmography. Early universe, primordial nucleosynthesis, recombination. Cosmic microwave background radiation. Large-scale structure, galaxy formation.

Size and time scales. Historical astronomy. Astronomical instrumentation. Stars: spectra and classification. Stellar structure equations and survey of stellar evolution. Stellar oscillations. Degenerate and collapsed stars; radio pulsars. Interacting binary systems; accretion disks, x-ray sources. Gravitational lenses; dark matter. Interstellar medium: HII regions, supernova remnants, molecular clouds, dust; radiative transfer; Jeans' mass; star formation. High-energy astrophysics: Compton scattering, bremsstrahlung, synchrotron radiation, cosmic rays. Galactic stellar distributions and populations; Oort constants; Oort limit; and globular clusters.

This undergraduate class is designed to introduce students to the physics that govern the circulation of the ocean and atmosphere. The focus of the course is on the processes that control the climate of the planet.AcknowledgmentsProf. Ferrari wishes to acknowledge that this course was originally designed and taught by Prof. John Marshall.

This course provides an introduction to the physics and chemistry of the atmosphere, including experience with computer codes. It is intended for undergraduates and first year graduate students.

Introduction to the physics of atmospheric radiation and remote sensing including use of computer codes. Radiative transfer equation including emission and scattering, spectroscopy, Mie theory, and numerical solutions. Solution of inverse problems in remote sensing of atmospheric temperature and composition.

Survey of atmospheric and oceanic phenomena including the discussion of observations and theoretical interpretations. Topics covered include: monsoons; El Nino; planetary waves; atmospheric synoptic eddies and fronts; gulf stream rings; hurricanes; surface and internal gravity waves; and tides. In this course, we will look at many important aspects of the circulation of the atmosphere and ocean, from length scales of meters to thousands of km and time scales ranging from seconds to years. We will assume familiarity with concepts covered in course 12.003 (Physics of the Fluid Earth). In the early stages of the present course, we will make somewhat greater use of math than did 12.003, but the math we will use is no more than that encountered in elementary electromagnetic field theory, for example. The focus of the course is on the physics of the phenomena which we will discuss.

The numerical methods, formulation and parameterizations used in models of the circulation of the atmosphere and ocean will be described in detail. Widely used numerical methods will be the focus but we will also review emerging concepts and new methods. The numerics underlying a hierarchy of models will be discussed, ranging from simple GFD models to the high-end GCMs. In the context of ocean GCMs, we will describe parameterization of geostrophic eddies, mixing and the surface and bottom boundary layers. In the atmosphere, we will review parameterizations of convection and large scale condensation, the planetary boundary layer and radiative transfer.

This is the second of a two-semester subject sequence beginning with Atomic and Optical Physics I (8.421) that provides the foundations for contemporary research in selected areas of atomic and optical physics. Topics covered include non-classical states of light–squeezed states; multi-photon processes, Raman scattering; coherence–level crossings, quantum beats, double resonance, superradiance; trapping and cooling-light forces, laser cooling, atom optics, spectroscopy of trapped atoms and ions; atomic interactions–classical collisions, quantum scattering theory, ultracold collisions; and experimental methods.

This Freshman Advising Seminar surveys the many applications of magnets and magnetism. To the Chinese and Greeks of ancient times, the attractive and repulsive forces between magnets must have seemed magical indeed. Through the ages, miraculous curative powers have been attributed to magnets, and magnets have been used by illusionists to produce "magical" effects. Magnets guided ships in the Age of Exploration and generated the electrical industry in the 19th century. Today they store information and entertainment on disks and tapes, and produce sound in speakers, images on TV screens, rotation in motors, and levitation in high-speed trains. Students visit various MIT projects related to magnets (including superconducting electromagnets) and read about and discuss the history, legends, pseudoscience, science, and technology of types of magnets, including applications in medicine. Several short written reports and at least one oral presentation will be required of each participant.

This course illuminates current theories about autism together with challenges faced by people on the autism spectrum. Theories in communicating, interacting socially, managing cognitive and affective overload, and achieving independent lifestyles are covered. In parallel, the course presents state-of-the-art technologies being developed for helping improve both theoretical understanding and practical outcomes. Participants are expected to meet and interact with people on the autism spectrum. Weekly reading, discussion, and a term project are required.

This course provides a challenging introduction to some of the central ideas of theoretical computer science. Beginning in antiquity, the course will progress through finite automata, circuits and decision trees, Turing machines and computability, efficient algorithms and reducibility, the P versus NP problem, NP-completeness, the power of randomness, cryptography and one-way functions, computational learning theory, and quantum computing. It examines the classes of problems that can and cannot be solved by various kinds of machines. It tries to explain the key differences between computational models that affect their power.

Graduate-level introduction to automatic speech recognition. Provides relevant background in acoustic theory of speech production, properties of speech sounds, signal representation, acoustic modeling, pattern classification, search algorithms, stochastic modeling techniques (including hidden Markov modeling), and language modeling. Examines approaches of state-of-the-art speech recognition systems. Introduces students to the rapidly developing field of automatic speech recognition. Its content is divided into three parts. Part I deals with background material in the acoustic theory of speech production, acoustic-phonetics, and signal representation. Part II describes algorithmic aspects of speech recognition systems including pattern classification, search algorithms, stochastic modelling, and language modelling techniques. Part III compares and contrasts the various approaches to speech recognition, and describes advanced techniques used for acoustic-phonetic modelling, robust speech recognition, speaker adaptation, processing paralinguistic information, speech understanding, and multimodal processing.

Offers a foundation in the visual art practice and its critical analysis for beginning architecture students. Emphasis on long-range artistic development and its analogies to architectural thinking and practice. Learn to communicate ideas and experiences through various two-dimensional, three-dimensional, and time-based media, including sculpture, installation, performance, and video. Lectures, visiting artist presentations, field trips, and readings supplement studio practice. Required of and restricted to Course 4 majors. Lab fee.

This course provides students with a basic knowledge of structural analysis and design for buildings, bridges and other structures. The course emphasizes the historical development of structural form and the evolution of structural design knowledge, from Gothic cathedrals to long span suspension bridges. Students will investigate the behavior of structural systems and elements through design exercises, case studies, and load testing of models. Students will design structures using timber, masonry, steel, and concrete and will gain an appreciation of the importance of structural design today, with an emphasis on environmental impact of large scale construction.

Childhood is a source of fascination in most Western cultures. It is both a major inspiration for artistic creation and a political ideal, which aims at protecting future generations. Which role does it play in French society and in other francophone areas? Why is the French national anthem (La Marseillaise) addressed to its 'children'? This course will study the transformation of childhood since the 18th century and the development of sentimentality within the family. We will examine various representations of childhood in literature (e.g. Pagnol, Proust, Sarraute, Laye, Morgiĺvre), movies (e.g. Truffaut), and songs (e.g. Brel, Barbara). Course taught in French.

This course covers the following questions. What are the predominant heat producing elements of the Earth? Where and how much are they? Are they present in the core of the Earth? Detection of antineutrinos generated in the Earth provides: 1) information on the sources of the terrestrial heat, 2) direct test of the Bulk Silicate Earth (BSE) model and 3) testing of non-conventional models of Earth's core. Use of antineutrinos to probe the deep interior of our planet is becoming practical due to recent fundamental advances in the antineutrino detectors.

This course introduces impact craters of the Earth. There are now 170 identified impact craters on the Earth, and this number is growing, ever since the well known discovery of Meteor Crater in 1920s. Currently, multi/inter disciplinary research studies of impact structures are getting conducted in fields like mineralogy, petrology, environmental geology and marine biology. This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the end of the month.