Midterm examination for a class at MIT covering game theory and its applications to economics. The one-hour-and-twenty-minute open book examination asks open ended theoretical questions. The exam contains questions and solutions.

This OER explores the basic organization of the Pythagorean Solids. It contains both an activity as well as resources for further exploration. It is a product of the OU Academy of the Lynx, developed in conjunction with the Galileo's World Exhibition at the University of Oklahoma.

In this activity, students work with paleoclimate proxy data (d18O, CH4, CO2)from the Byrd and GISP2 ice cores to investigate millennial-scale climate changes during the Last Glacial/Deglacial time periods. Students must prepare a publication quality plot of the data and answer several questions about the similarities and differences between the time-series (north-south phasing, amplitude, symmetry) and use this information to assess the bipolar see-saw mechanism for abrupt climate changes. Students are encouraged to read two journal articles for more information and to synthesize their results with other information from lectures and earlier readings.

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The solutions unit consists of the following: General points for discussion relating to the teaching of the mathematical content in the activities. Step-by-step mathematical solutions to the activities. Annotations to the solutions to assist teachers in their understanding the maths as well as teaching issues relating to the mathematical content represented in the activities. Suggestions of links to alternative activities for the teaching of the mathematical content represented in the activities.

The solutions unit consists of the following: General points for discussion relating to the teaching of the mathematical content in the activities. Step-by-step mathematical solutions to the activities. Annotations to the solutions to assist teachers in their understanding the maths as well as teaching issues relating to the mathematical content represented in the activities. Suggestions of links to alternative activities for the teaching of the mathematical content represented in the activities

At the beginning of the course, each student is assigned a unique blob - or a piece of material of a particular shape with specific material properties (density, bulk modulus, composition, viscosity, volatile content, etc) that is residing within the mantle at a specific environment (depth, pressure, temperature). Then as the semester continues as a topic is covered the student must assess (either quantitatively or qualitatively) what observable would be associated with their blob (for example, gravity anomalies, geoid anomalies, surface expressions, seismic tomography, phase transition topography). The student then develops a portfolio of their blob and its observables to then present at the end of the course with an explanation/interpretation for the source of the blob culiminating at building a geo-story around their anomaly.

Some blobs could be amorphous anomalies whereas other could have physical significance (though best not to tell the students ahead of time so they can make their own discovery as to what the blob is or isn't) such as subducted slabs at the CMB (or 660 km), plumes, lithospheric drip, lithospheric root, or a boring typical piece of the mantle.

(Note: this resource was added to OER Commons as part of a batch upload of over 2,200 records. If you notice an issue with the quality of the metadata, please let us know by using the 'report' button and we will flag it for consideration.)

Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics. It was designed to strengthen the mathematical abilities of graduate students and train them to think on their own.

This course analyzes the functions of a complex variable and the calculus of residues. It also covers subjects such as ordinary differential equations, partial differential equations, Bessel and Legendre functions, and the Sturm-Liouville theory.

This graduate-level course focuses on current research topics in computational complexity theory. Topics include: Nondeterministic, alternating, probabilistic, and parallel computation models; Boolean circuits; Complexity classes and complete sets; The polynomial-time hierarchy; Interactive proof systems; Relativization; Definitions of randomness; Pseudo-randomness and derandomizations;Interactive proof systems and probabilistically checkable proofs.

An open-source textbook covering vector calculus, ordinary and partial differential equations, and Fourier series. The textbook is used in a first-year graduate level course in the Department of Mechanical Engineering at the Colorado School of Mines. It undergoes extensive revisions annually, but is relatively complete.

The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.

This class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.

Learning and Understanding Mathematical Concepts in the Areas of Water Distribution and Water Treatment

In this video, we explore some concepts fundamental to algebra. To streamline the discussion of relationships between physical quantities, we introduce variables, functions, composition, and inverse. By thinking about the concept of an inverse function, we obtain our first glimpse of the imaginary root (i.e. square-root of -1) and the complex plane.

This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms.

This course covers the fundamental notions and results about algebraic varieties over an algebraically closed field. It also analyzes the relations between complex algebraic varieties and complex analytic varieties.

This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.

This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization. The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications.

This is a course on the singular homology of topological spaces. Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, and Poincare duality.

This is the second part of the two-course series on algebraic topology. Topics include basic homotopy theory, obstruction theory, classifying spaces, spectral sequences, characteristic classes, and Steenrod operations.