Inverse Theory Course: LTU Kiruna. Day 1
|
|
- Wilfrid Matthews
- 5 years ago
- Views:
Transcription
1 Inverse Theory Course: LTU Kiruna. Day Hugh Pumphrey March 6, 0 Preamble These are the notes for the course Inverse Theory to be taught at LuleåTekniska Universitet, Kiruna in February 00. They are not exhaustive, rather, they are a collection of the formulæ that you will nee to be familiar with, printe up so that you on t have to struggle to copy them off the boar or the projector screen. Inverse Theory: what is it? Inverse theory is a term use for the tools use to attack a class of problems common in various branches of Earth an space science, but which occur in other fiels as well. The main thing that links these problems is that you can not make irect measurements of the thing x you want to measure, but you can measure another thing, y, which is relate to x in a way that you unerstan. The quantities x an y usually consist of more than one number they are vectors. It is common for x to be referre to as the state vector or the moel vector, an for y to be referre to as the measurement vector or the ata vector. 3 Setting up a linear problem We suppose that we can measure y, we want x an we know that they are relate by some function F : y = F (x) F is calle the forwar moel an in general it coul be any kin of function. For a lot of problems we can approximate F by a Taylor series about x = x L : y F (x L ) + K(x x L ) where the matrix K is given by K = y x x=xl These names an symbols are not universal: see appenix A for etails.
2 Just as a tiying job, we choose new variables so that x x x L an y y F (y L ). The problem therefore becomes y = Kx which is a set of simultaneous equations to be solve for x. 4 Solving the linear problem Let y have length m an x have length n. Solving our set of simultaneous equations is straightforwar in principle if m = n as we can immeiately write x = K y Such a problem is calle equi-etermine or well-etermine. But real-worl inverse theory problems often have n > m or n < m. If n > m then we have more unknowns than equations. There will not be a single solution. There will be many solutions an the problem is sai to be uner-etermine. If n < m there will be no solution an the equations are sai to be over-etermine. Note that a set of equations may have n > m but have some equations that contraict each other so that there is no solution instea of infinitely many solutions such a problem is calle mixe-etermine. An a problem with n < m may actually be uner-etermine or mixe-etermine if sufficient of the equations are effectively uplicates of each other. We nee to know how to go about solving all these sorts of systems of equations. 4. The over-etermine problem The over-etermine problem has no exact solution, so we have to look for the next best thing: a value of x which is less ba in some sense than any other value of x. We can efine the error in our solution as e = y Kx, but e is a vector so we nee a single number that is a measure of its length. We choose the sum of the squares of its elements E = e T e = (y Kx) T (y Kx) an look for the solution that makes E as small as possible. This iea occurs over an over again in inverse theory. The thing that we want to minimise is sometimes referre to as a cost function or a penalty function. We can fin the x that minimises E by ifferentiating E with respect to x an setting the result to 0. Differentiating with respect to a vector is a bit tricky the en result in this case is K T Kx = K T y () This is a (usually) equi-etermine set of equation for x which can be solve in the usual manner. They are known as the normal equations. For the case where x has two elements we can raw a contour plot of E (x) this can be quite helpful in unerstaning the nature of the solution that we have obtaine.
3 x x x x Figure : Error surfaces (contours of E = (y Kx) T (y Kx) )for two sets of simultaneous equations. The heavy black lines represent the simultaneous equations an the black ot is the solution. The left-han figure is over-etermine an the solution is a least-squares solution. The right-han figure is equietermine an the solution is exact, so E is zero at the solution. 4. The uner-etermine problem We ll look at this in great etail later in the course. For now, we note that a truly uner-etermine problem has an infinite number of exact solutions. It is sometimes useful to have a formula which will give one of these solutions i.e. any vector x for which y = Kx. To o this, we nee an n m matrix D for which DK = I. Multiply on the right by K T to give DKK T = K T. Now, KK T is a m m matrix which we can probably take the inverse of. We can therefore write D = D(KK T )(KK T ) = K T (KK T ). This solution is sometimes calle the minimum-norm solution. Figure shows a simple example. Dealing with measurement errors. Definition of the covariance matrix Typically, x an y are lists of numbers an can therefore be hanle using the techniques of matrix algebra. Because they are measure quantities (or relate to measure quantities) they have ranom errors in them, so we nee some of the tools for hanling ranom variables. For our purposes a ranom variable is a thing for which you get a ifferent result every time you measure it. Suppose we have a scalar ranom variable v an we measure it N times, 3
4 x x Figure : Error surfaces (contours of E = (y Kx) T (y Kx) )for a set of simultaneous equations with one equation an two unknowns. The black ot is the minimum-norm solution x = K T (KK T ) y calling the jth sample v j. We efine the mean value of v as v = N The sprea of the measurements about the mean is often summarise by the stanar eviation σ σ = N (v j v) N The square of the stanar eviation is calle the variance. If we have two ranom variables: v an u, then we can calculate an aitional quantity: the covariance: cov(v, u) = N (v j v)(u j ū) N N For a vector ranom variable v we efine the mean in a similar way to the scalar case. v = N v j N To express how the iniviual samples v j vary about the mean v we calculate a quantity calle the covariance matrix, efine as: S = N v j N (v j v)(v j v) T 4
5 The iagonal elements of the covariance matrix are the variances of the iniviual elements of v. Each off-iagonal element is the covariances of two ifferent elements of v.. Weighte least-squares Suppose that some of our measurements y are more noisy than others, i.e. they have larger errors. We can escribe these errors by a covariance matrix S, with the iagonal terms being the square error on each element of y. If the errors are correlate, then the off-iagonal terms escribe those correlations. The least-squares approach is now not quite appropriate as it gives the same importance to all elements of y. Instea of minimising E = (y Kx) T (y Kx), we minimise E = (y Kx) T S (y Kx). This weights the elements of y by the inverse of their square errors, so the elements with the largest error get the smallest weight. Any correlations are also correctly accounte for. The normal equations now become K T S Kx = KT S y so that the least-squares solution now becomes ˆx = (K T S K) K T S y Note that the matrix (K T S K) can be shown to be the covariance matrix of ˆx; by explicitly stating the errors on y we get an estimate of how goo our solution is. A Notation Your ata Your moel parameters Matrix in linear forwar moel Covariance matrix of a ranom vector a Matrix relating true moel params to estimate ones Menke Gubbins Rogers Data vector of Data vector of Measurement vector length N length D y of length m Moel vector m of Moel vector m of State vector x of length M length P length n Data kernel G Data kernel A Influence function matrix K [cov a] C(a) or C a S a Moel resolution matrix R Resolution matrix R Averaging matrix A kernel Table : Different names an notations use for the same things in various inverse theory texts.
6 Inverse theory has been evelope in a variety of contexts. You will therefore fin textbooks that escribe essentially the same mathematics but using ifferent notation, ifferent names for things an using ifferent examples. Table shows the ifferent names an symbols that three ifferent textbooks use for the same things. I attempt here to stick to the notation of Rogers. 6
Survey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013
Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationParameter estimation: A new approach to weighting a priori information
Parameter estimation: A new approach to weighting a priori information J.L. Mea Department of Mathematics, Boise State University, Boise, ID 83725-555 E-mail: jmea@boisestate.eu Abstract. We propose a
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More informationFree rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012
Free rotation of a rigi boy 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 1 Introuction In this section, we escribe the motion of a rigi boy that is free to rotate
More informationMA 2232 Lecture 08 - Review of Log and Exponential Functions and Exponential Growth
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationIntroduction to Mechanics Work and Energy
Introuction to Mechanics Work an Energy Lana Sherian De Anza College Mar 15, 2018 Last time non-uniform circular motion an tangential acceleration energy an work Overview energy work a more general efinition
More informationUNDERSTANDING INTEGRATION
UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,
More informationMulti-View Clustering via Canonical Correlation Analysis
Technical Report TTI-TR-2008-5 Multi-View Clustering via Canonical Correlation Analysis Kamalika Chauhuri UC San Diego Sham M. Kakae Toyota Technological Institute at Chicago ABSTRACT Clustering ata in
More informationProblem Sheet 2: Eigenvalues and eigenvectors and their use in solving linear ODEs
Problem Sheet 2: Eigenvalues an eigenvectors an their use in solving linear ODEs If you fin any typos/errors in this problem sheet please email jk28@icacuk The material in this problem sheet is not examinable
More informationDesigning Information Devices and Systems II Fall 2017 Note Theorem: Existence and Uniqueness of Solutions to Differential Equations
EECS 6B Designing Information Devices an Systems II Fall 07 Note 3 Secon Orer Differential Equations Secon orer ifferential equations appear everywhere in the real worl. In this note, we will walk through
More informationImplicit Differentiation. Lecture 16.
Implicit Differentiation. Lecture 16. We are use to working only with functions that are efine explicitly. That is, ones like f(x) = 5x 3 + 7x x 2 + 1 or s(t) = e t5 3, in which the function is escribe
More informationExperiment 2, Physics 2BL
Experiment 2, Physics 2BL Deuction of Mass Distributions. Last Upate: 2009-05-03 Preparation Before this experiment, we recommen you review or familiarize yourself with the following: Chapters 4-6 in Taylor
More informationThe Non-abelian Hodge Correspondence for Non-Compact Curves
1 Section 1 Setup The Non-abelian Hoge Corresponence for Non-Compact Curves Chris Elliott May 8, 2011 1 Setup In this talk I will escribe the non-abelian Hoge theory of a non-compact curve. This was worke
More informationLecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Bair, faculty.uml.eu/cbair University of Massachusetts Lowell 1. Pre-Einstein Relativity - Einstein i not invent the concept of relativity,
More informationARCH 614 Note Set 5 S2012abn. Moments & Supports
RCH 614 Note Set 5 S2012abn Moments & Supports Notation: = perpenicular istance to a force from a point = name for force vectors or magnitue of a force, as is P, Q, R x = force component in the x irection
More informationConductors & Capacitance
Conuctors & Capacitance PICK UP YOUR EXAM;; Average of the three classes is approximately 51. Stanar eviation is 18. It may go up (or own) by a point or two once all graing is finishe. Exam KEY is poste
More informationOptimization of Geometries by Energy Minimization
Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationThe Press-Schechter mass function
The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for
More informationNew Statistical Test for Quality Control in High Dimension Data Set
International Journal of Applie Engineering Research ISSN 973-456 Volume, Number 6 (7) pp. 64-649 New Statistical Test for Quality Control in High Dimension Data Set Shamshuritawati Sharif, Suzilah Ismail
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationComputing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL
Computing Derivatives by J. Douglas Chil, Ph.D. Rollins College Winter Park, FL ii Computing Inefinite Integrals Important notice regaring book materials Texas Instruments makes no warranty, either express
More informationLecture Notes: March C.D. Lin Attosecond X-ray pulses issues:
Lecture Notes: March 2003-- C.D. Lin Attosecon X-ray pulses issues: 1. Generation: Nee short pulses (less than 7 fs) to generate HHG HHG in the frequency omain HHG in the time omain Issues of attosecon
More informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationStable and compact finite difference schemes
Center for Turbulence Research Annual Research Briefs 2006 2 Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long
More informationTutorial on Maximum Likelyhood Estimation: Parametric Density Estimation
Tutorial on Maximum Likelyhoo Estimation: Parametric Density Estimation Suhir B Kylasa 03/13/2014 1 Motivation Suppose one wishes to etermine just how biase an unfair coin is. Call the probability of tossing
More informationMulti-View Clustering via Canonical Correlation Analysis
Keywors: multi-view learning, clustering, canonical correlation analysis Abstract Clustering ata in high-imensions is believe to be a har problem in general. A number of efficient clustering algorithms
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationCapacity Analysis of MIMO Systems with Unknown Channel State Information
Capacity Analysis of MIMO Systems with Unknown Channel State Information Jun Zheng an Bhaskar D. Rao Dept. of Electrical an Computer Engineering University of California at San Diego e-mail: juzheng@ucs.eu,
More informationState-Space Model for a Multi-Machine System
State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal
More informationDiophantine Approximations: Examining the Farey Process and its Method on Producing Best Approximations
Diophantine Approximations: Examining the Farey Process an its Metho on Proucing Best Approximations Kelly Bowen Introuction When a person hears the phrase irrational number, one oes not think of anything
More informationIntegration by Parts
Integration by Parts 6-3-207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an
More informationDifferentiation ( , 9.5)
Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the
More information5.4 Fundamental Theorem of Calculus Calculus. Do you remember the Fundamental Theorem of Algebra? Just thought I'd ask
5.4 FUNDAMENTAL THEOREM OF CALCULUS Do you remember the Funamental Theorem of Algebra? Just thought I' ask The Funamental Theorem of Calculus has two parts. These two parts tie together the concept of
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More informationPhysics 121 for Majors
Physics 121 for Majors Scheule Do Post-Class Quiz #3 Do Pre-Class Quiz #4 HW #2 is ue Wenesay Quiz #1 is ue Saturay, Sept. 16 Lab #1 is set up now, ue Monay Some Department Resources Computers N-212 ESC
More informationA Review of Multiple Try MCMC algorithms for Signal Processing
A Review of Multiple Try MCMC algorithms for Signal Processing Luca Martino Image Processing Lab., Universitat e València (Spain) Universia Carlos III e Mari, Leganes (Spain) Abstract Many applications
More informationMulti-View Clustering via Canonical Correlation Analysis
Kamalika Chauhuri ITA, UC San Diego, 9500 Gilman Drive, La Jolla, CA Sham M. Kakae Karen Livescu Karthik Sriharan Toyota Technological Institute at Chicago, 6045 S. Kenwoo Ave., Chicago, IL kamalika@soe.ucs.eu
More informationA Modification of the Jarque-Bera Test. for Normality
Int. J. Contemp. Math. Sciences, Vol. 8, 01, no. 17, 84-85 HIKARI Lt, www.m-hikari.com http://x.oi.org/10.1988/ijcms.01.9106 A Moification of the Jarque-Bera Test for Normality Moawa El-Fallah Ab El-Salam
More informationSimilarity Measures for Categorical Data A Comparative Study. Technical Report
Similarity Measures for Categorical Data A Comparative Stuy Technical Report Department of Computer Science an Engineering University of Minnesota 4-92 EECS Builing 200 Union Street SE Minneapolis, MN
More informationChapter 6: Integration: partial fractions and improper integrals
Chapter 6: Integration: partial fractions an improper integrals Course S3, 006 07 April 5, 007 These are just summaries of the lecture notes, an few etails are inclue. Most of what we inclue here is to
More informationA Course in Machine Learning
A Course in Machine Learning Hal Daumé III 12 EFFICIENT LEARNING So far, our focus has been on moels of learning an basic algorithms for those moels. We have not place much emphasis on how to learn quickly.
More informationAntiderivatives Introduction
Antierivatives 40. Introuction So far much of the term has been spent fining erivatives or rates of change. But in some circumstances we alreay know the rate of change an we wish to etermine the original
More informationMath 1272 Solutions for Spring 2005 Final Exam. asked to find the limit of the sequence. This is equivalent to evaluating lim. lim.
Math 7 Solutions for Spring 5 Final Exam ) We are gien an infinite sequence for which the general term is a n 3 + 5n n + n an are 3 + 5n aske to fin the limit of the sequence. This is equialent to ealuating
More informationA Second Time Dimension, Hidden in Plain Sight
A Secon Time Dimension, Hien in Plain Sight Brett A Collins. In this paper I postulate the existence of a secon time imension, making five imensions, three space imensions an two time imensions. I will
More informationunder the null hypothesis, the sign test (with continuity correction) rejects H 0 when α n + n 2 2.
Assignment 13 Exercise 8.4 For the hypotheses consiere in Examples 8.12 an 8.13, the sign test is base on the statistic N + = #{i : Z i > 0}. Since 2 n(n + /n 1) N(0, 1) 2 uner the null hypothesis, the
More informationSolving the Schrödinger Equation for the 1 Electron Atom (Hydrogen-Like)
Stockton Univeristy Chemistry Program, School of Natural Sciences an Mathematics 101 Vera King Farris Dr, Galloway, NJ CHEM 340: Physical Chemistry II Solving the Schröinger Equation for the 1 Electron
More informationLecture 6: Generalized multivariate analysis of variance
Lecture 6: Generalize multivariate analysis of variance Measuring association of the entire microbiome with other variables Distance matrices capture some aspects of the ata (e.g. microbiome composition,
More informationPHYS 414 Problem Set 2: Turtles all the way down
PHYS 414 Problem Set 2: Turtles all the way own This problem set explores the common structure of ynamical theories in statistical physics as you pass from one length an time scale to another. Brownian
More informationBohr Model of the Hydrogen Atom
Class 2 page 1 Bohr Moel of the Hyrogen Atom The Bohr Moel of the hyrogen atom assumes that the atom consists of one electron orbiting a positively charge nucleus. Although it oes NOT o a goo job of escribing
More informationFurther Differentiation and Applications
Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle
More informationLecture 6 : Dimensionality Reduction
CPS290: Algorithmic Founations of Data Science February 3, 207 Lecture 6 : Dimensionality Reuction Lecturer: Kamesh Munagala Scribe: Kamesh Munagala In this lecture, we will consier the roblem of maing
More informationPhysics 2112 Unit 5: Electric Potential Energy
Physics 11 Unit 5: Electric Potential Energy Toay s Concept: Electric Potential Energy Unit 5, Slie 1 Stuff you aske about: I on't like this return to mechanics an the potential energy concept, but this
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationIntroduction to Markov Processes
Introuction to Markov Processes Connexions moule m44014 Zzis law Gustav) Meglicki, Jr Office of the VP for Information Technology Iniana University RCS: Section-2.tex,v 1.24 2012/12/21 18:03:08 gustav
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationLecture 10: Logistic growth models #2
Lecture 1: Logistic growth moels #2 Fugo Takasu Dept. Information an Computer Sciences Nara Women s University takasu@ics.nara-wu.ac.jp 6 July 29 1 Analysis of the stochastic process of logistic growth
More informationPure Further Mathematics 1. Revision Notes
Pure Further Mathematics Revision Notes June 20 2 FP JUNE 20 SDB Further Pure Complex Numbers... 3 Definitions an arithmetical operations... 3 Complex conjugate... 3 Properties... 3 Complex number plane,
More informationSlide10 Haykin Chapter 14: Neurodynamics (3rd Ed. Chapter 13)
Slie10 Haykin Chapter 14: Neuroynamics (3r E. Chapter 13) CPSC 636-600 Instructor: Yoonsuck Choe Spring 2012 Neural Networks with Temporal Behavior Inclusion of feeback gives temporal characteristics to
More informationThis module is part of the. Memobust Handbook. on Methodology of Modern Business Statistics
This moule is part of the Memobust Hanbook on Methoology of Moern Business Statistics 26 March 2014 Metho: Balance Sampling for Multi-Way Stratification Contents General section... 3 1. Summary... 3 2.
More information3.2 Shot peening - modeling 3 PROCEEDINGS
3.2 Shot peening - moeling 3 PROCEEDINGS Computer assiste coverage simulation François-Xavier Abaie a, b a FROHN, Germany, fx.abaie@frohn.com. b PEENING ACCESSORIES, Switzerlan, info@peening.ch Keywors:
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationCONTROL CHARTS FOR VARIABLES
UNIT CONTOL CHATS FO VAIABLES Structure.1 Introuction Objectives. Control Chart Technique.3 Control Charts for Variables.4 Control Chart for Mean(-Chart).5 ange Chart (-Chart).6 Stanar Deviation Chart
More informationAnalytische Qualitätssicherung Baden-Württemberg
Analytische Qualitätssicherung Baen-Württemberg Proficiency Test 1/13 TW S1 sweeteners an benzotriazoles in rinking water acesulfam, cyclamate, saccharin, sucralose, 1H-benzotriazole, 4-methyl-1H-benzotriazole,
More informationSome vector algebra and the generalized chain rule Ross Bannister Data Assimilation Research Centre, University of Reading, UK Last updated 10/06/10
Some vector algebra an the generalize chain rule Ross Bannister Data Assimilation Research Centre University of Reaing UK Last upate 10/06/10 1. Introuction an notation As we shall see in these notes the
More informationCalculus Class Notes for the Combined Calculus and Physics Course Semester I
Calculus Class Notes for the Combine Calculus an Physics Course Semester I Kelly Black December 14, 2001 Support provie by the National Science Founation - NSF-DUE-9752485 1 Section 0 2 Contents 1 Average
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationJointly continuous distributions and the multivariate Normal
Jointly continuous istributions an the multivariate Normal Márton alázs an álint Tóth October 3, 04 This little write-up is part of important founations of probability that were left out of the unit Probability
More informationMath 210 Midterm #1 Review
Math 20 Miterm # Review This ocument is intene to be a rough outline of what you are expecte to have learne an retaine from this course to be prepare for the first miterm. : Functions Definition: A function
More informationDEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS
DEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS SHANKAR BHAMIDI 1, JESSE GOODMAN 2, REMCO VAN DER HOFSTAD 3, AND JÚLIA KOMJÁTHY3 Abstract. In this article, we explicitly
More informationVI. Linking and Equating: Getting from A to B Unleashing the full power of Rasch models means identifying, perhaps conceiving an important aspect,
VI. Linking an Equating: Getting from A to B Unleashing the full power of Rasch moels means ientifying, perhaps conceiving an important aspect, efining a useful construct, an calibrating a pool of relevant
More informationSemiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom
PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,
More informationState observers and recursive filters in classical feedback control theory
State observers an recursive filters in classical feeback control theory State-feeback control example: secon-orer system Consier the riven secon-orer system q q q u x q x q x x x x Here u coul represent
More informationarxiv: v1 [hep-lat] 19 Nov 2013
HU-EP-13/69 SFB/CPP-13-98 DESY 13-225 Applicability of Quasi-Monte Carlo for lattice systems arxiv:1311.4726v1 [hep-lat] 19 ov 2013, a,b Tobias Hartung, c Karl Jansen, b Hernan Leovey, Anreas Griewank
More informationMulti-View Clustering via Canonical Correlation Analysis
Kamalika Chauhuri ITA, UC San Diego, 9500 Gilman Drive, La Jolla, CA Sham M. Kakae Karen Livescu Karthik Sriharan Toyota Technological Institute at Chicago, 6045 S. Kenwoo Ave., Chicago, IL kamalika@soe.ucs.eu
More informationTransmission Line Matrix (TLM) network analogues of reversible trapping processes Part B: scaling and consistency
Transmission Line Matrix (TLM network analogues of reversible trapping processes Part B: scaling an consistency Donar e Cogan * ANC Eucation, 308-310.A. De Mel Mawatha, Colombo 3, Sri Lanka * onarecogan@gmail.com
More informationIt's often useful to find all the points in a diagram that have the same voltage. E.g., consider a capacitor again.
17-7 (SJP, Phys 22, Sp ') It's often useful to fin all the points in a iagram that have the same voltage. E.g., consier a capacitor again. V is high here V is in between, here V is low here Everywhere
More informationDerivatives and the Product Rule
Derivatives an the Prouct Rule James K. Peterson Department of Biological Sciences an Department of Mathematical Sciences Clemson University January 28, 2014 Outline Differentiability Simple Derivatives
More informationTHE EFFICIENCIES OF THE SPATIAL MEDIAN AND SPATIAL SIGN COVARIANCE MATRIX FOR ELLIPTICALLY SYMMETRIC DISTRIBUTIONS
THE EFFICIENCIES OF THE SPATIAL MEDIAN AND SPATIAL SIGN COVARIANCE MATRIX FOR ELLIPTICALLY SYMMETRIC DISTRIBUTIONS BY ANDREW F. MAGYAR A issertation submitte to the Grauate School New Brunswick Rutgers,
More informationLogarithmic spurious regressions
Logarithmic spurious regressions Robert M. e Jong Michigan State University February 5, 22 Abstract Spurious regressions, i.e. regressions in which an integrate process is regresse on another integrate
More informationA Random Graph Model for Massive Graphs
A Ranom Graph Moel for Massive Graphs William Aiello AT&T Labs Florham Park, New Jersey aiello@research.att.com Fan Chung University of California, San Diego fan@ucs.eu Linyuan Lu University of Pennsylvania,
More informationTopic 2.3: The Geometry of Derivatives of Vector Functions
BSU Math 275 Notes Topic 2.3: The Geometry of Derivatives of Vector Functions Textbook Sections: 13.2 From the Toolbox (what you nee from previous classes): Be able to compute erivatives scalar-value functions
More information0.1 Differentiation Rules
0.1 Differentiation Rules From our previous work we ve seen tat it can be quite a task to calculate te erivative of an arbitrary function. Just working wit a secon-orer polynomial tings get pretty complicate
More informationDifferentiability, Computing Derivatives, Trig Review. Goals:
Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an
More information