differential equations
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This is a basic course on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.
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Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.
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This course will go in-depth in the theory of how war is conducted within the confines of the game Starcraft. There will be lecture on various aspects of the game, from the viewpoint of pure theory to the more computational aspects of how exactly battles are conducted. Calculus and Differential Equations are highly recommended for full understanding of the course. Furthermore, the class will take the theoretical into the practical worl...more
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This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.
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Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, ...more
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Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especial...more
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Topics covered in a first year course in differential equations. Need to understand basic differentiation and integration from Calculus before starting here.
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A special event to mark the publication of Professor Barrow's new book, 'The Book of Universes'. This is a book about universes, a story that revolves around a single unusual and unappreciated fact: that Einstein’s famous theory of relativity describe universes – entire universes. Not many solutions of Einstein’s tantalizing universe equations have ever been found, but those that have are all very remarkable. Some of them describe universe...more
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In order for Social Security to work, people have to believe there's some possibility that the world will last forever, so that each old generation will have a young generation to support it. The overlapping generations model, invented by Allais and Samuelson but here augmented with land, represents such a situation. Financial equilibrium can again be reduced to general equilibrium. At first glance it would seem that the model requires a s...more
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Options introduce an essential nonlineary into portfolio management. They are contracts between buyers and writers, who agree on exercise prices and dates at which the buyer can buy or sell the underlying (such as a stock). Options are priced based on the price and volatility of the underlying asset as well as the duration of the option contract. The Black-Scholes options pricing model is one of the most famous equations in finance and off...more
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This lecture explains what an economic model is, and why it allows for counterfactual reasoning and often yields paradoxical conclusions. Typically, equilibrium is defined as the solution to a system of simultaneous equations. The most important economic model is that of supply and demand in one market, which was understood to some extent by the ancient Greeks and even by Shakespeare. That model accurately fits the experiment from the last...more
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This lecture is all about motion of projectiles (if air drag can be ignored). The objects experience a constant vertical acceleration due to the acceleration of gravity (see also Lecture 12). Professor Lewin reviews the equations for projectile motion, showing that the trajectory is a parabola. He continues with a demonstration that shows how to measure the initial speed of a projectile and how to reach maximum horizontal distance shooting...more
