Assignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)
|
|
- Magdalen Hall
- 5 years ago
- Views:
Transcription
1 Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form x qx ( ) = qx (, x,..., x) = aj x xj ( X= ),, j= x we have the max-form qx ( ) = X AX ( A= ( a j ) ). y Moreover, f s a vertble max, ad X = Y ( Y = ), the y qx ( ) = qy ˆ( ) = qy ˆ(, y,..., y) = Y BY for the max B = A. We assume that ( ) A= a j s symmec: aj = aj for all, j =,..., ; as a cosequece, the B s also symmec. The quadratc form qx ( ) s ) postve defte, ) egatve defte, 3) defte f, respectvely: ) for all X 0, we have qx ( ) > 0 ; ) for all X 0, we have qx ( ) < 0; 3) there are X 0, X 0 such that qx ( ) > 0 ad qx ( ) < 0. These three qualtes are mutually exclusve; but a quadratc form may be oe of the above three kds. qx ( ) s ) postve semdefte, 5) egatve semdefte f, respectvely: ) for all X 0, we have qx ( ) 0; 5) for all X 0, we have qx ( ) 0. Our text says oegatve semdefte for postve semdefte. The termology s such that postve semdefte s more geeral tha postve defte ; ad smlarly for egatve semdefte. However, the mportat possblty s that qx ( ) may be postve semdefte, wthout beg postve defte. For stace,
2 qxy (, ) x xy y = + + s postve semdefte sce qxy (, ) = ( x+ y) 0 always; but 0 ad q( ) = q(, ) = ( ) = 0, thus q s ot postve defte. Of course, f defte. q s postve semdefte, the t caot be ether egatve defte, or (I the ma applcato aalyss (calculus), whe the quadratc form formed by the secod partals at a statoary pot s postve defte, oe has a local mmum; whe the quadratc form s egatve defte, oe has a local maxmum; whe t s defte, oe deftely has o local mmum, or a maxmum; ad whe t s semdefte but ot defte, oe does ot kow about local exemum wthout further vestgato. Ths wll be looked at later class.) Wth the quadratc form qx ( ) = X AX, we assocate the symmec blear form ( X, X ) defed by β ( X, X ) X A X β = β ( X, X ) = β ( X, X ) We have the detty qx) ( = β ( XX, ).. β s symmec the sese that [] Let qx ( ) = qx (, yzw,, ) = x y 3z + 5w+ 5xz+ xw+ wz (X Throughout ths questo, q deotes ths partcular quadratc form. x y = ). z w ) Wrte qx ( ) the symmec max form. r s ) Apply the coordate asformato X = Y, Y =, wth the t u asformg max = 0 5/ 7/ /
3 ad wrte qx ( ) the form qxyzw (,,, ) = qrstu ˆ(,,, ) = byy j j (where we wrote y = r, y = s, y = t, y = u) wth sutable coeffcets. Use precse fracto 3 calculato, rather tha decmals; the result should come out precsely dagoal. b j, j= 3) Usg the max approprately, wrte each of the varables rstu,,, as a lear expresso of x, yzw,, ; ad verfy drectly the resultg detty qxyzw (,,, ) = qrs ˆ(,, tu, ). ) Usg the work doe so far, decde what kd q s the defteess classfcato (see above!). rove your asserto: show that q satsfes the defto gve above of the qualty (postve defte,, defte, ). Appealg to some theorem about that qualty s ot eough. 5) Cosder the symmec blear form β assocated wth q. Let X = 0, Y =. 0 3 Calculate β ( X, X)( = q( X)), β ( YY, ) ad β ( X, Y ), ad gve a formula for qax ( + by) the form of a homogeeous quadratc polyomal (quadratc form!) of the varables a ad b. The Gram-Schmdt orthogoalzato process Read secto 7.7, p. 6, the text. Whe you see er product space V, take V to be ay subspace of a stadard space, ad for the er product uv,, read dot product: X, Y = X Y. Note Remark o page 6 especally. Read Example 7.0 o the same page, but omt the ext example 7. (whch s a vector space of a kd that we have ot looked at yet). May also read solved problem 7., p.6. You wll eed to use Gram-Schmdt some problems that follow. 3
4 Recall: A max s called orthogoal f = = I ; equvaletly, f ; ad equvaletly, f the colums,..., of form a orthoormal system: = 0 whe j j; ad = =. If U = ( u,..., u ) ad V = ( v,..., v ) both m are orthoormal bases of a subspace V of some ( partcular whe V =, U = E, the stadard bass, ad V s aother orthoormal bass of ), the = [ U V ], the chage-of-bass max, s a orthogoal max. [] ) rove that f ad Q are orthogoal maces, the so are,, Q. ) Use Gram-Schmdt to fd a orthogoal max the frst colum of / / whch s =. / / Isuctos: Start wth ay bass B of the frst of whose vectors s. You may use some of the stadard vectors e,..., e as the last three vectors of B. Apply Gram- Schmdt to B. Clear deomators termedate results too. Use the symbol to / 3/ 3 3 deote a scaled vector: e.g., =. Scalg should be used also to 3/ 6 6 / 3 6 get smaller umbers: e.g.,. Normalze the system at the ed oly. Wrte 9 3 precse fractos ad square-roots whe ecessary; do ot use decmal approxmato. 3 3) Wth determed ), ad deotg the colums of by,,,, let U = spa(, ). Use ) (that s: do t start from scratch!) ad gve a bass of U, the orthogoal complemet of U.
5 7 ) For U as 3), ad Y =, determe proj ( 0 Y U ) ad proj U ( Y) (Ht: read Theorem 7.8, page 5 ( just before the secto o Gram-Schmdt), ad the Remark followg t. Take ote of the term Fourer coeffcet just before the remark. It wll come back later.) [3] We cosder the followg quadratc forms: q ( X ) = x + 3y z w + xy+ xz+ xw yz yw+ 6zw, q ( X ) = x + y + xy+ xz+ xw+ zw q ( X ) = 3 x 3z 3w yz yw + zw q ( X ) = 3x + 5y + 5z + 5w yz yw zw Treat each of the quadratc forms q = q, q,... as follows. ) Fd a orthogoal asformg max that turs qx ( ) to a puresquare (dagoal) form qy ˆ( ) terms of the ew varables Y ; wrte out what qy ˆ( ) s, ad what the relatoshp s betwee the old ad ew coordate vectors X ad Y. ) Determe whch kd q s the defteess classfcato. For each case whe the q s defte, or semdefte-o-defte, supply examples of vectors X that show the property questo. ( ) 3) Determe max qx X ( ) ad m qx X. ) Suppose that A s a symmec max, s a orthogoal max, ad A s a dagoal. Let B be a polyomal expresso of A : k B = bi 0 +ba ba k. rove that B s symmec, ad B s dagoal. (Hts: f you fd t too hard, verfy the asserto frst for the specal cases B = A, A, I + A, I + 3A+ A,... utl the patter becomes clear.) 5
6 5) Cosder the quadratc form q ( X ) = 5 x + y + 3z + 3w + xy+ xz+ xw+ yz+ yw+ zw. Verfy that ( ) q5 X = X B X for the max B = A + A+ I, where A s the max for whch ( ) q X = X A X, wth q from above. Usg kowledge o q ( X), do the work asked for ), ) ad 3) for q ( X), wth a mmal amout of calculato oly. 5 6) Let A be a symmec max, qx ( ) = X AX the correspodg quadratc form. rove that the fucto QX ( ) defed by QX ( ) = qax ( X) s a quadratc form, ad ay orthogoal max that asforms q to pure-squares form, does the same for Q. Determe the values from the values λ for whch qx ( ) = λ y. = μ for whch qx ( ) = μ y ( X = Y ) = 6
MATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationChapter 9 Jordan Block Matrices
Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.
More informationCHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationLecture 9: Tolerant Testing
Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationMultiple Choice Test. Chapter Adequacy of Models for Regression
Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to
More informationL5 Polynomial / Spline Curves
L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationIdeal multigrades with trigonometric coefficients
Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all
More informationNumerical Analysis Formulae Booklet
Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres... 3. Fte Dfferece Approxmatos... 3 4. Egevalues ad Egevectors of Matrces.... 3 5. Vector ad Matrx
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationUNIT 2 SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
Numercal Computg -I UNIT SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS Structure Page Nos..0 Itroducto 6. Objectves 7. Ital Approxmato to a Root 7. Bsecto Method 8.. Error Aalyss 9.4 Regula Fals Method
More informationLaboratory I.10 It All Adds Up
Laboratory I. It All Adds Up Goals The studet wll work wth Rema sums ad evaluate them usg Derve. The studet wll see applcatos of tegrals as accumulatos of chages. The studet wll revew curve fttg sklls.
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationα1 α2 Simplex and Rectangle Elements Multi-index Notation of polynomials of degree Definition: The set P k will be the set of all functions:
Smplex ad Rectagle Elemets Mult-dex Notato = (,..., ), o-egatve tegers = = β = ( β,..., β ) the + β = ( + β,..., + β ) + x = x x x x = x x β β + D = D = D D x x x β β Defto: The set P of polyomals of degree
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More informationb. There appears to be a positive relationship between X and Y; that is, as X increases, so does Y.
.46. a. The frst varable (X) s the frst umber the par ad s plotted o the horzotal axs, whle the secod varable (Y) s the secod umber the par ad s plotted o the vertcal axs. The scatterplot s show the fgure
More informationLecture 07: Poles and Zeros
Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto
More informationhp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations
HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationMATH 371 Homework assignment 1 August 29, 2013
MATH 371 Homework assgmet 1 August 29, 2013 1. Prove that f a subset S Z has a smallest elemet the t s uque ( other words, f x s a smallest elemet of S ad y s also a smallest elemet of S the x y). We kow
More informationA scalar t is an eigenvalue of A if and only if t satisfies the characteristic equation of A: det (A ti) =0
Chapter 5 a glace: Let e a lear operator whose stadard matrx s wth sze x. he, a ozero vector x s sad to e a egevector of ad f there exsts a scalar sch that (x) x x. he scalar s called a egevale of (or
More information1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.
CS 94- Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy
More informationAlgorithms Theory, Solution for Assignment 2
Juor-Prof. Dr. Robert Elsässer, Marco Muñz, Phllp Hedegger WS 2009/200 Algorthms Theory, Soluto for Assgmet 2 http://lak.formatk.u-freburg.de/lak_teachg/ws09_0/algo090.php Exercse 2. - Fast Fourer Trasform
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationAssignment 7/MATH 247/Winter, 2010 Due: Friday, March 19. Powers of a square matrix
Assgmet 7/MATH 47/Wter, 00 Due: Frday, March 9 Powers o a square matrx Gve a square matrx A, ts powers A or large, or eve arbtrary, teger expoets ca be calculated by dagoalzg A -- that s possble (!) Namely,
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationPTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
More informationQR Factorization and Singular Value Decomposition COS 323
QR Factorzato ad Sgular Value Decomposto COS 33 Why Yet Aother Method? How do we solve least-squares wthout currg codto-squarg effect of ormal equatos (A T A A T b) whe A s sgular, fat, or otherwse poorly-specfed?
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More information2SLS Estimates ECON In this case, begin with the assumption that E[ i
SLS Estmates ECON 3033 Bll Evas Fall 05 Two-Stage Least Squares (SLS Cosder a stadard lear bvarate regresso model y 0 x. I ths case, beg wth the assumto that E[ x] 0 whch meas that OLS estmates of wll
More informationCLASS NOTES. for. PBAF 528: Quantitative Methods II SPRING Instructor: Jean Swanson. Daniel J. Evans School of Public Affairs
CLASS NOTES for PBAF 58: Quattatve Methods II SPRING 005 Istructor: Jea Swaso Dael J. Evas School of Publc Affars Uversty of Washgto Ackowledgemet: The structor wshes to thak Rachel Klet, Assstat Professor,
More informationThe internal structure of natural numbers, one method for the definition of large prime numbers, and a factorization test
Fal verso The teral structure of atural umbers oe method for the defto of large prme umbers ad a factorzato test Emmaul Maousos APM Isttute for the Advacemet of Physcs ad Mathematcs 3 Poulou str. 53 Athes
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationIII-16 G. Brief Review of Grand Orthogonality Theorem and impact on Representations (Γ i ) l i = h n = number of irreducible representations.
III- G. Bref evew of Grad Orthogoalty Theorem ad mpact o epresetatos ( ) GOT: h [ () m ] [ () m ] δδ δmm ll GOT puts great restrcto o form of rreducble represetato also o umber: l h umber of rreducble
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationApplying the condition for equilibrium to this equilibrium, we get (1) n i i =, r G and 5 i
CHEMICAL EQUILIBRIA The Thermodyamc Equlbrum Costat Cosder a reversble reacto of the type 1 A 1 + 2 A 2 + W m A m + m+1 A m+1 + Assgg postve values to the stochometrc coeffcets o the rght had sde ad egatve
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationmeans the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationWe have already referred to a certain reaction, which takes place at high temperature after rich combustion.
ME 41 Day 13 Topcs Chemcal Equlbrum - Theory Chemcal Equlbrum Example #1 Equlbrum Costats Chemcal Equlbrum Example #2 Chemcal Equlbrum of Hot Bured Gas 1. Chemcal Equlbrum We have already referred to a
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationn -dimensional vectors follow naturally from the one
B. Vectors ad sets B. Vectors Ecoomsts study ecoomc pheomea by buldg hghly stylzed models. Uderstadg ad makg use of almost all such models requres a hgh comfort level wth some key mathematcal sklls. I
More informationRandom Variables and Probability Distributions
Radom Varables ad Probablty Dstrbutos * If X : S R s a dscrete radom varable wth rage {x, x, x 3,. } the r = P (X = xr ) = * Let X : S R be a dscrete radom varable wth rage {x, x, x 3,.}.If x r P(X = x
More informationMOLECULAR VIBRATIONS
MOLECULAR VIBRATIONS Here we wsh to vestgate molecular vbratos ad draw a smlarty betwee the theory of molecular vbratos ad Hückel theory. 1. Smple Harmoc Oscllator Recall that the eergy of a oe-dmesoal
More informationENGI 4421 Propagation of Error Page 8-01
ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.
More informationMA 524 Homework 6 Solutions
MA 524 Homework 6 Solutos. Sce S(, s the umber of ways to partto [] to k oempty blocks, ad c(, s the umber of ways to partto to k oempty blocks ad also the arrage each block to a cycle, we must have S(,
More informationLog1 Contest Round 2 Theta Complex Numbers. 4 points each. 5 points each
01 Log1 Cotest Roud Theta Complex Numbers 1 Wrte a b Wrte a b form: 1 5 form: 1 5 4 pots each Wrte a b form: 65 4 4 Evaluate: 65 5 Determe f the followg statemet s always, sometmes, or ever true (you may
More informationF. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties
進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKAL Pure Mathematcs F. Ieualtes. Basc propertes Theorem Let a, b, c be real umbers. () If a b ad b c, the a c. () If a b ad c 0, the ac bc, but f a b ad c 0, the ac bc. Theorem
More informationCIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
More informationMidterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..
More informationCubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem
Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationA New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming
ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research
More informationEECE 301 Signals & Systems
EECE 01 Sgals & Systems Prof. Mark Fowler Note Set #9 Computg D-T Covoluto Readg Assgmet: Secto. of Kame ad Heck 1/ Course Flow Dagram The arrows here show coceptual flow betwee deas. Note the parallel
More informationLattices. Mathematical background
Lattces Mathematcal backgroud Lattces : -dmesoal Eucldea space. That s, { T x } x x = (,, ) :,. T T If x= ( x,, x), y = ( y,, y), the xy, = xy (er product of xad y) x = /2 xx, (Eucldea legth or orm of
More informationd dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin
Learzato of the Swg Equato We wll cover sectos.5.-.6 ad begg of Secto 3.3 these otes. 1. Sgle mache-fte bus case Cosder a sgle mache coected to a fte bus, as show Fg. 1 below. E y1 V=1./_ Fg. 1 The admttace
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationε. Therefore, the estimate
Suggested Aswers, Problem Set 3 ECON 333 Da Hugerma. Ths s ot a very good dea. We kow from the secod FOC problem b) that ( ) SSE / = y x x = ( ) Whch ca be reduced to read y x x = ε x = ( ) The OLS model
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More informationQ-analogue of a Linear Transformation Preserving Log-concavity
Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com
More informationDerivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations
Dervato of -Pot Block Method Formula for Solvg Frst Order Stff Ordary Dfferetal Equatos Kharul Hamd Kharul Auar, Kharl Iskadar Othma, Zara Bb Ibrahm Abstract Dervato of pot block method formula wth costat
More informationMultiple Linear Regression Analysis
LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple
More information7.0 Equality Contraints: Lagrange Multipliers
Systes Optzato 7.0 Equalty Cotrats: Lagrage Multplers Cosder the zato of a o-lear fucto subject to equalty costrats: g f() R ( ) 0 ( ) (7.) where the g ( ) are possbly also olear fuctos, ad < otherwse
More informationThe number of observed cases The number of parameters. ith case of the dichotomous dependent variable. the ith case of the jth parameter
LOGISTIC REGRESSION Notato Model Logstc regresso regresses a dchotomous depedet varable o a set of depedet varables. Several methods are mplemeted for selectg the depedet varables. The followg otato s
More informationSPECIAL CONSIDERATIONS FOR VOLUMETRIC Z-TEST FOR PROPORTIONS
SPECIAL CONSIDERAIONS FOR VOLUMERIC Z-ES FOR PROPORIONS Oe s stctve reacto to the questo of whether two percetages are sgfcatly dfferet from each other s to treat them as f they were proportos whch the
More informationarxiv:math/ v1 [math.gm] 8 Dec 2005
arxv:math/05272v [math.gm] 8 Dec 2005 A GENERALIZATION OF AN INEQUALITY FROM IMO 2005 NIKOLAI NIKOLOV The preset paper was spred by the thrd problem from the IMO 2005. A specal award was gve to Yure Boreko
More informationDr. Shalabh. Indian Institute of Technology Kanpur
Aalyss of Varace ad Desg of Expermets-I MODULE -I LECTURE - SOME RESULTS ON LINEAR ALGEBRA, MATRIX THEORY AND DISTRIBUTIONS Dr. Shalabh Departmet t of Mathematcs t ad Statstcs t t Ida Isttute of Techology
More informationGeneralized Linear Regression with Regularization
Geeralze Lear Regresso wth Regularzato Zoya Bylsk March 3, 05 BASIC REGRESSION PROBLEM Note: I the followg otes I wll make explct what s a vector a what s a scalar usg vec t or otato, to avo cofuso betwee
More informationThe Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
Sc. Math. Japocae Vol. 00, No. 0 0000, 000 000 1 The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally
More information3D Geometry for Computer Graphics. Lesson 2: PCA & SVD
3D Geometry for Computer Graphcs Lesso 2: PCA & SVD Last week - egedecomposto We wat to lear how the matrx A works: A 2 Last week - egedecomposto If we look at arbtrary vectors, t does t tell us much.
More informationInvestigation of Partially Conditional RP Model with Response Error. Ed Stanek
Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a
More information( ) 2 2. Multi-Layer Refraction Problem Rafael Espericueta, Bakersfield College, November, 2006
Mult-Layer Refracto Problem Rafael Espercueta, Bakersfeld College, November, 006 Lght travels at dfferet speeds through dfferet meda, but refracts at layer boudares order to traverse the least-tme path.
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationClass 13,14 June 17, 19, 2015
Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral
More informationExercises for Square-Congruence Modulo n ver 11
Exercses for Square-Cogruece Modulo ver Let ad ab,.. Mark True or False. a. 3S 30 b. 3S 90 c. 3S 3 d. 3S 4 e. 4S f. 5S g. 0S 55 h. 8S 57. 9S 58 j. S 76 k. 6S 304 l. 47S 5347. Fd the equvalece classes duced
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationMultivariate Transformation of Variables and Maximum Likelihood Estimation
Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty
More informationSTA302/1001-Fall 2008 Midterm Test October 21, 2008
STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from
More informationCS5620 Intro to Computer Graphics
CS56 Itro to Computer Graphcs Geometrc Modelg art II Geometrc Modelg II hyscal Sples Curve desg pre-computers Cubc Sples Stadard sple put set of pots { } =, No dervatves specfed as put Iterpolate by cubc
More informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5
THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should
More informationSome Different Perspectives on Linear Least Squares
Soe Dfferet Perspectves o Lear Least Squares A stadard proble statstcs s to easure a respose or depedet varable, y, at fed values of oe or ore depedet varables. Soetes there ests a deterstc odel y f (,,
More informationLecture 12 APPROXIMATION OF FIRST ORDER DERIVATIVES
FDM: Appromato of Frst Order Dervatves Lecture APPROXIMATION OF FIRST ORDER DERIVATIVES. INTRODUCTION Covectve term coservato equatos volve frst order dervatves. The smplest possble approach for dscretzato
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationA Study on Generalized Generalized Quasi hyperbolic Kac Moody algebra QHGGH of rank 10
Global Joural of Mathematcal Sceces: Theory ad Practcal. ISSN 974-3 Volume 9, Number 3 (7), pp. 43-4 Iteratoal Research Publcato House http://www.rphouse.com A Study o Geeralzed Geeralzed Quas (9) hyperbolc
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More information