Dirac Delta
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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,...more
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Note: This course is being offered this summer by Stanford as an online course for credit. It can be taken individually, or as part of a master’s degree or graduate certificate earned online through the Stanford Center for Professional Development. The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used....more
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Setting Up The Fourier Transform Of A Distribution, Example Of Delta As A Distribution, Distributions Induced By Functions (Includes Many Functions), The Fourier Transform Of A Distribution, The Class Of Tempered Distributions, FT Of A Tempered Distribution, Definition Of The Fourier Transform (By How It Operates On A Test Function), The Inverse Fourier Transform (Proof), Calculations Of Fourier Transforms Using This Definition (Distributions)
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Review Of Basic DFT Definitions, Special Case: Value Of The DFT At 0, Two Special Signals: One Vector, Delta Vector, DFT Of Deltas, Complex Exponentials, DFT As Nxn Matrix Multiplication, Periodicity Of Input/Output Signals In The DFT, Result Of Periodicity: Indexing, Result Of Periodicity: Duality
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Tips For Filling Out Evals, Tomography And Inverting The Radon Transform; Setup, Introducing Coordinates, Delta Along A Line, The Integral Of U Along A Line, Inverting The Radon Transform
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Introduction to the Kawa Development Environment: Evaluation of Expressions
Stanford / Computer Science
Introduction to the Kawa Development Environment: Evaluation of Expressions, Loading Function Definitions From a .Scm File, Mapping Arbitrary Unary Functions Over Lists in Scheme Using the Map Operation, Mapping List Functions (Car, Cdr) Over Lists of Lists, Using Mapping Functions with More than One Input by Passing Multiple Lists into Map, Implementing the Unary Version of Map Using Recursion, Apply, Which Allows You to Specify a...more
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Using the epsilon delta definition to prove a limit.
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Introduction to the Epsilon Delta Definition of a Limit.
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Derivative Of A Distribution, Example: Derivative Of A Unit Step, Example: Derivative Of Sgn(X), Applications To The Fourier Transform (Using The Derivative Theorem), Caveat To Distributions: Multiplying Distributions, Distributions*Functions, Special Case: The Delta Function And Sampling, Convolution In Distributions, Special Case: Convolution When T = Delta, The Scaling Property Of Delta
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Cop Story, Review Of Rapidly Decreasing Functions, Generalized Functions (Distributions) (Delta Function, Etc.), Viewing Delta As A Limit V. Operationally, Definition Of A Distribution, Delta As A Distribution, Discussion Of How To Consider Ordinary Functions In This Space; Pairing Through Integration
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