Non-degenerate Perturbation Theory
|
|
|
- Erik Daniel
- 5 years ago
- Views:
Transcription
1 No-degeerate Perturbato Theory Proble : H E ca't solve exactly. But wth H H H' H" L H E Uperturbed egevalue proble. Ca solve exactly. E Therefore, kow ad. H ' H" called perturbatos Copyrght Mchael D. Fayer, 17
2 Solutos of H E coplete, orthooral set of ket vectors,, 1 wth egevalues ad E, E, E, 1 Kroecker delta 1 Copyrght Mchael D. Fayer, 17
3 Expad wavefucto ad E E E E also have H H H' H" Have seres for H E Substtute these seres to the orgal egevalue equato H E Copyrght Mchael D. Fayer, 17
4 Su of fte uber of ters for all powers of equals. ' H ' E H H E E H H H E E E ' " Coeffcets of the dvdual powers of ust equal. zeroth order - H E frst order - 1 secod order - H H' E E H H' H" E E E Copyrght Mchael D. Fayer, 17
5 Frst order correcto H E E H' Wat to fd E ad. Expad The c H c H c E also substtutg Substtutg ths result. After substtuto ' c E E E H Copyrght Mchael D. Fayer, 17
6 After substtuto ' c E E E H Left ultply by ' c E E E H ' c E E E H uless =, but the E E Therefore, the left sde s. Copyrght Mchael D. Fayer, 17
7 We have E H' The E E E E E H' E uber, kets oralzed, ad trasposg, H' The frst order correcto to the eergy. (Expectato value of H zeroth order state ) Absorbg to E E E H ad E E H' H The frst order correcto to the eergy s the expectato value of H '. Copyrght Mchael D. Fayer, 17
8 Frst order correcto to the wavefucto Aga usg the equato obtaed after substtutg seres expasos ' c E E E H Left ultply by ' c E E E H ' c E E E H c E E H' Equals zero uless =. H ' c E E Coeffcets expaso of ket ters of the zeroth order kets. Copyrght Mchael D. Fayer, 17
9 H ' c E E c E H ' E H ' s the bracket of H ' wth ad. Therefore ' H ( E E) correcto to zeroth order ket The pre o the su ea. zeroth order ket eergy deoator Copyrght Mchael D. Fayer, 17
10 Frst order correctos E E H' H' H ' H H H ( E E) Copyrght Mchael D. Fayer, 17
11 Secod Order Correctos Usg coeffcet Expadg Substtutg ad followg sae type of procedures yelds E ' E E H H H coeffcets have bee absorbed. H H H H Secod order correcto due to frst order pece of H. Secod order correcto due to a addtoal secod order pece of H. H H H H ' ' k k ' k k k k k ( ) k k E E k E E E E E E H Secod order correcto due to frst order pece of H. Secod order correcto due to a addtoal secod order pece of H. Copyrght Mchael D. Fayer, 17
12 Eergy ad Ket Corrected to Frst ad Secod Order E E H' ' E E H H H ' H ( E E) ' ' H kh H H k E E E E E E k k k k ' H E E k k ( k) k Copyrght Mchael D. Fayer, 17
13 Exaple: x 3 ad x 4 perturbato of the Haroc Oscllator E x Vbratoal potetal of olecules ot haroc. Approxately haroc ear potetal u. Expad potetal power seres. Frst addtoal ters potetal after x ter are x 3 ad x 4. Copyrght Mchael D. Fayer, 17
14 p 1 H kx cx qx H p 1 1 kx H aa a a 3 4 quartc force costat cubc force costat haroc oscllator kow solutos 1 E zeroth order egevalues zeroth order egekets 3 4 perturbato H' cx qx c ad q are expaso coeffcets lke. Whe c ad q, H H Copyrght Mchael D. Fayer, 17
15 H H' 3 4 cx qx 3 4 cx qx I Drac represetato 1 x a a k 3 3 x a a Frst cosder cubc ter. Multply out. May ters. 3 3 a, a a, aa a, a. Noe of the ters have the sae uber of rasg ad lowerg operators. 3 x (At secod order wll ot be zero.) Copyrght Mchael D. Fayer, 17
16 4 4 x aa 4k a a 4 has ters wth sae uber of rasg ad lowerg operators. Therefore, 4 x Usg 1/ 1/ a a 1 ad ( 1) 1 1 aaa a a a aa aa aa a aa a aa a a a aaa 1 Oly ters wth the sae uber of rasg ad lowerg operators are o-zero. There are sx ters. Copyrght Mchael D. Fayer, 17
17 Su of the sx ters 4 aa 6( 1 ) Therefore q 3 1 k H Wth k k E q Eergy levels ot equally spaced. Real olecules, levels get closer together q s egatve. Correcto grows wth faster tha zeroth order ter decrease level spacg. Copyrght Mchael D. Fayer, 17
18 Perturbato Theory for Degeerate States H H E 1 1 E 1 ad oralze ad orthogoal 1 ad Degeerate, sae egevalue, E. If wth c c cc cc H E Ay superposto of degeerate egestates s also a egestate wth the sae egevalue. Copyrght Mchael D. Fayer, 17
19 learly depedet states wth sae egevalue syste -fold degeerate Ca for orthooral fro the degerate. Ca for a fte uber of sets of. Nothg uque about ay oe set of degeerate egekets. Copyrght Mchael D. Fayer, 17
20 Wat approxate soluto to H H E ' zeroth order Haltoa perturbato H E zeroth order egeket zeroth order eergy But E s -fold degeerate. Call these egekets belogg to the -fold degeerate E 1 1, orthooral Wth E E E E 1 1 Copyrght Mchael D. Fayer, 17
21 Here s the dffculty perturbed ket zeroth order ket havg egevalue, E 1 But, s a lear cobato of the. c c c 1 1 We do t kow whch partcular lear cobato t s. s the correct zeroth order ket, but we do t kow the c. The correct zero order ket depeds o the ature of the perturbato. p states of the H ato exteral agetc feld p 1, p, p -1 electrc feld p x, p z, p y Copyrght Mchael D. Fayer, 17
22 To solve proble Expad E ad 1 E E E c 1 Soe superposto, but we do t kow the c. Do t kow correct zeroth order fucto. Substtutg the expasos for E ad to H H E ' ad obtag the coeffcets of powers of, gves zeroth order frst order H c E c 1 H E 1 c EH' wat these Copyrght Mchael D. Fayer, 17
23 1 H E 1 c EH' substtute A k k k To solve Need H ' Use proecto operator k k H' H' k k k The proecto operator gves the pece of H ' that s. The the su over all k gves the expaso of H ' ters of the. k Defg H H' ' Hk k k H' k k Kow kow perturbato pece of the Haltoa ad the zeroth order kets. Copyrght Mchael D. Fayer, 17
24 1 H E 1 c EH' H ' Hk k k ch ch k k 1 1 k ths pece becoes Substtutg ths ad Ak k gves Ek E1 Ak k Ec ch k k k 1 k 1 k Result of operatg H o the zeroth order kets. Left ultplyg by Ek E1 Ak k Ec ch k k k 1 k 1 Copyrght Mchael D. Fayer, 17
25 Ek E1 Ak k Ec ch k k k 1 k 1 Correcto to the Eerges Two cases: (the degeerate states) ad >. Left had sde su over k equals zero uless k =. But wth, E E E E The left had sde of the equato =. 1 Therefore, 1 Rght had sde, frst ter o-zero whe =. Bracket = 1, oralzato. Secod ter o-zero whe k =. Bracket = 1, oralzato. The result s 1 Hc Ec We do t kow the c s ad the Es. Copyrght Mchael D. Fayer, 17
26 1 Hc Ec s a syste of of equatos for the c s. H E c H c H c H c H E c H c 1 1 Oe equato for each dex of c. H c H c H E c Besdes trval soluto of c c c oly get soluto f the deterat of the coeffcets vash. H E H H H E H H H H E 1 We kow the H H k k Have th degree equato for the E s. Copyrght Mchael D. Fayer, 17
27 Solve th degree equato get the E s. Now have the correctos to eerges. To fd the correct zeroth order egevectors, oe for each E, substtute E (oe at a te) to syste of equatos. Get syste of equatos for the coeffcets, c s. H E c H c H c H c H E c H c 1 1 Kow the H. H c H c H E c 1 1 * * * 1 1,, 1 cc cc c c There are oly 1 codtos because ca ultply everythg by costat. Use oralzato for th codto. Now we have the correct zeroth order fuctos. Copyrght Mchael D. Fayer, 17
28 The solutos to the th degree equato (expadg deterat) are E E E 1,, Therefore, to frst order, the eerges of the perturbed tally degeerate states are E E E 1 1 Have dfferet E s (uless soe stll degeerate). Wth E E as 1 Copyrght Mchael D. Fayer, 17
29 Correcto to wavefuctos Aga usg equato foud substtutg the expasos to the frst order equato Ek E1 Ak k Ec ch k k k 1 k 1 Left ultply by k Orthogoalty akes other ters zero. Noralzato gves 1 for o-zero brackets. k 1 k k 1 E E A c H Therefore k ch A k k 1 E1 Ek gves 1 gves Noralzato gves A =for. Already have part of wavefucto for Copyrght Mchael D. Fayer, 17
30 Frst order degeerate perturbato theory results E E E 1 k 1 k k E1 Ek ch Correct zeroth order fucto. Coeffcets c k detered fro syste of equatos. Correcto to zeroth order fucto. Copyrght Mchael D. Fayer, 17
Stationary states of atoms and molecules
Statoary states of atos ad olecules I followg wees the geeral aspects of the eergy level structure of atos ad olecules that are essetal for the terpretato ad the aalyss of spectral postos the rotatoal
Solutions to problem set ); (, ) (
Solutos to proble set.. L = ( yp p ); L = ( p p ); y y L, L = yp p, p p = yp p, + p [, p ] y y y = yp + p = L y Here we use for eaple that yp, p = yp p p yp = yp, p = yp : factors that coute ca be treated
( t) ( t) ( t) ρ ψ ψ. (9.1)
Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-1 9. THE DENSTY MATRX Te desty atrx or desty operator s a alterate represetato of te state of a quatu syste for wc we ave prevously used te wavefucto. Altoug
MOLECULAR 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
Chapter 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.
Some 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 (,,
7.0 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
PRACTICAL CONSIDERATIONS IN HUMAN-INDUCED VIBRATION
PRACTICAL CONSIDERATIONS IN HUMAN-INDUCED VIBRATION Bars Erkus, 4 March 007 Itroducto Ths docuet provdes a revew of fudaetal cocepts structural dyacs ad soe applcatos hua-duced vbrato aalyss ad tgato of
A Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions
Appled Matheatcs, 1, 4, 8-88 http://d.do.org/1.4/a.1.448 Publshed Ole Aprl 1 (http://www.scrp.org/joural/a) A Covetoal Approach for the Soluto of the Ffth Order Boudary Value Probles Usg Sth Degree Sple
2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission
/0/0 Topcs Power Flow Part Text: 0-0. Power Trassso Revsted Power Flow Equatos Power Flow Proble Stateet ECEGR 45 Power Systes Power Trassso Power Trassso Recall that for a short trassso le, the power
Coherent Potential Approximation
Coheret Potetal Approxato Noveber 29, 2009 Gree-fucto atrces the TB forals I the tght bdg TB pcture the atrx of a Haltoa H s the for H = { H j}, where H j = δ j ε + γ j. 2 Sgle ad double uderles deote
KURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS. Peter J. Wilcoxen. Impact Research Centre, University of Melbourne.
KURODA S METHOD FOR CONSTRUCTING CONSISTENT INPUT-OUTPUT DATA SETS by Peter J. Wlcoxe Ipact Research Cetre, Uversty of Melboure Aprl 1989 Ths paper descrbes a ethod that ca be used to resolve cossteces
ANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK
Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at http://www.pphm.com 2008 Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION
Some results and conjectures about recurrence relations for certain sequences of binomial sums.
Soe results ad coectures about recurrece relatos for certa sequeces of boal sus Joha Cgler Faultät für Matheat Uverstät We A-9 We Nordbergstraße 5 Joha Cgler@uveacat Abstract I a prevous paper [] I have
Ahmed Elgamal. MDOF Systems & Modal Analysis
DOF Systems & odal Aalyss odal Aalyss (hese otes cover sectos from Ch. 0, Dyamcs of Structures, Al Chopra, Pretce Hall, 995). Refereces Dyamcs of Structures, Al K. Chopra, Pretce Hall, New Jersey, ISBN
Mu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
d 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
Physics 114 Exam 2 Fall Name:
Physcs 114 Exam Fall 015 Name: For gradg purposes (do ot wrte here): Questo 1. 1... 3. 3. Problem Aswer each of the followg questos. Pots for each questo are dcated red. Uless otherwse dcated, the amout
EVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM
EVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM Jose Javer Garca Moreta Ph. D research studet at the UPV/EHU (Uversty of Basque coutry) Departmet of Theoretcal
III-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
Debabrata Dey and Atanu Lahiri
RESEARCH ARTICLE QUALITY COMPETITION AND MARKET SEGMENTATION IN THE SECURITY SOFTWARE MARKET Debabrata Dey ad Atau Lahr Mchael G. Foster School of Busess, Uersty of Washgto, Seattle, Seattle, WA 9895 U.S.A.
Bounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
D. L. Bricker, 2002 Dept of Mechanical & Industrial Engineering The University of Iowa. CPL/XD 12/10/2003 page 1
D. L. Brcker, 2002 Dept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/XD 2/0/2003 page Capactated Plat Locato Proble: Mze FY + C X subject to = = j= where Y = j= X D, j =, j X SY, =,... X 0, =,
CHAPTER 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
( q Modal Analysis. Eigenvectors = Mode Shapes? Eigenproblem (cont) = x x 2 u 2. u 1. x 1 (4.55) vector and M and K are matrices.
4.3 - Modal Aalyss Physcal coordates are ot always the easest to work Egevectors provde a coveet trasformato to modal coordates Modal coordates are lear combato of physcal coordates Say we have physcal
ENGI 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.
Chapter 2 - Free Vibration of Multi-Degree-of-Freedom Systems - II
CEE49b Chapter - Free Vbrato of Mult-Degree-of-Freedom Systems - II We ca obta a approxmate soluto to the fudametal atural frequecy through a approxmate formula developed usg eergy prcples by Lord Raylegh
Overview of the weighting constants and the points where we evaluate the function for The Gaussian quadrature Project two
Overvew of the weghtg costats ad the pots where we evaluate the fucto for The Gaussa quadrature Project two By Ashraf Marzouk ChE 505 Fall 005 Departmet of Mechacal Egeerg Uversty of Teessee Koxvlle, TN
Econometric 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
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
means 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,
C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory
ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?
2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America
SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS Gauss-Sedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca kaw@eg.usf.edu Itroducto Ths worksheet demostrates
Numerical 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
CH E 374 Computational Methods in Engineering Fall 2007
CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 5. The data o the varato of the rato of stagato pressure to statc pressure (r ) wth Mach umber ( M ) for the flow through a duct are as follows:
Basic Concepts in Numerical Analysis November 6, 2017
Basc Cocepts Nuercal Aalyss Noveber 6, 7 Basc Cocepts Nuercal Aalyss Larry Caretto Mecacal Egeerg 5AB Sear Egeerg Aalyss Noveber 6, 7 Outle Revew last class Mdter Exa Noveber 5 covers ateral o deretal
1 Solution to Problem 6.40
1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we
x y exp λ'. x exp λ 2. x exp 1.
egecosmcd Egevalue-egevector of the secod dervatve operator d /d hs leads to Fourer seres (se, cose, Legedre, Bessel, Chebyshev, etc hs s a eample of a systematc way of geeratg a set of mutually orthogoal
THE 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
Lecture 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
Introduction. Free Electron Fermi Gas. Energy Levels in One Dimension
ree Electro er Gas Eergy Levels Oe Deso Effect of eperature o the er-drac Dstrbuto ree Electro Gas hree Desos Heat Capacty of the Electro Gas Electrcal Coductvty ad Oh s Law Moto Magetc elds heral Coductvty
We 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
arxiv:cond-mat/ v2 11 Dec 2000
arxv:cod-mat/0006 v Dec 000 THE NTURE OF THE LON TIME DECY T SECOND ORDER TRNSITION POINT Moshe Schwartz School of Physcs ad stroomy Tel vv Uversty Tel vv, Ramat vv, Israel d S. F. Edwards Cavedsh Laboratory
Decomposition of Hadamard Matrices
Chapter 7 Decomposto of Hadamard Matrces We hae see Chapter that Hadamard s orgal costructo of Hadamard matrces states that the Kroecer product of Hadamard matrces of orders m ad s a Hadamard matrx of
Beam Warming Second-Order Upwind Method
Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet
Lecture 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
PGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
{ }{ ( )} (, ) = ( ) ( ) ( ) 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
Investigation 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
Capacitated Plant Location Problem:
. L. Brcker, 2002 ept of Mechacal & Idustral Egeerg The Uversty of Iowa CPL/ 5/29/2002 page CPL/ 5/29/2002 page 2 Capactated Plat Locato Proble: where Mze F + C subect to = = =, =, S, =,... 0, =, ; =,
9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,
best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best
Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg
Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
arxiv:math/ v1 [math.gm] 8 Dec 2005
![arxiv:math/ v1 [math.gm] 8 Dec 2005](/thumbs/94/118250722.jpg "arxiv: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
The theoretical background of
he theoretcal backgroud of -echologes he theoretcal backgroud of FactSage he followg sldes gve a abrdged overvew of the ajor uderlyg prcples of the calculatoal odules of FactSage. -echologes he bbs Eergy
Module 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
1 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
Analysis of Lagrange Interpolation Formula
P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. www.jset.com ISS 348 7968 Aalyss of Lagrage Iterpolato Formula Vjay Dahya PDepartmet of MathematcsMaharaja Surajmal
LINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
Multiple 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
L5 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
SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi
Faculty of Sceces ad Matheatcs, Uversty of Nš, Serba Avalable at: http://wwwpfacyu/float Float 3:3 (009), 303 309 DOI:098/FIL0903303J SUBCLASS OF ARMONIC UNIVALENT FUNCTIONS ASSOCIATED WIT SALAGEAN DERIVATIVE
UNIT 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
THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the
[ L] υ = (3) [ L] n. Q: What are the units of K in Eq. (3)? (Why is units placed in quotations.) What is the relationship to K in Eq. (1)?
![[ L] υ = (3) [ L] n. Q: What are the units of K in Eq. (3)? (Why is units placed in quotations.) What is the relationship to K in Eq. (1)?](/thumbs/87/96901092.jpg "[ L] υ = (3) [ L] n. Q: What are the units of K in Eq. (3)? (Why is units placed in quotations.) What is the relationship to K in Eq. (1)?")
Chem 78 Spr. M. Wes Bdg Polyomals Bdg Polyomals We ve looked at three cases of lgad bdg so far: The sgle set of depedet stes (ss[]s [ ] [ ] Multple sets of depedet stes (ms[]s, or m[]ss All or oe, or two-state
Polyphase Filters. Section 12.4 Porat
Polyphase Flters Secto.4 Porat .4 Polyphase Flters Polyphase s a way of dog saplg-rate coverso that leads to very effcet pleetatos. But ore tha that, t leads to very geeral vewpots that are useful buldg
Chapter 3. Linear Equations and Matrices
Vector Spaces Physcs 8/6/05 hapter Lear Equatos ad Matrces wde varety of physcal problems volve solvg systems of smultaeous lear equatos These systems of lear equatos ca be ecoomcally descrbed ad effcetly
Standard Deviation for PDG Mass Data
4 Dec 06 Stadard Devato for PDG Mass Data M. J. Gerusa Retred, 47 Clfde Road, Worghall, HP8 9JR, UK. gerusa@aol.co, phoe: +(44) 844 339754 Abstract Ths paper aalyses the data for the asses of eleetary
Algorithms behind the Correlation Setting Window
Algorths behd the Correlato Settg Wdow Itroducto I ths report detaled forato about the correlato settg pop up wdow s gve. See Fgure. Ths wdow s obtaed b clckg o the rado butto labelled Kow dep the a scree
EECE 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
ECE 595, Section 10 Numerical Simulations Lecture 19: FEM for Electronic Transport. Prof. Peter Bermel February 22, 2013
ECE 595, Secto 0 Numercal Smulatos Lecture 9: FEM for Electroc Trasport Prof. Peter Bermel February, 03 Outle Recap from Wedesday Physcs-based devce modelg Electroc trasport theory FEM electroc trasport
A 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
ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K
ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu
Multiple 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
Chapter 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:
Functions 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,
Integral Equation Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, Xin Wang and Karen Veroy
Itroducto to Smulato - Lecture 22 Itegral Equato ethods Jacob Whte Thaks to Deepak Ramaswamy, chal Rewesk, X Wag ad Kare Veroy Outle Itegral Equato ethods Exteror versus teror problems Start wth usg pot
5 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:
Assignment 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,
Lecture Notes 2. The ability to manipulate matrices is critical in economics.
Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets
F. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties
![F. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties](/thumbs/82/86768390.jpg "F. 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
å 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018
Chrs Pech Fal Practce CS09 Dec 5, 08 Practce Fal Examato Solutos. Aswer: 4/5 8/7. There are multle ways to obta ths aswer; here are two: The frst commo method s to sum over all ossbltes for the rak of
Applying 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
A Primer on Summation Notation George H Olson, Ph. D. Doctoral Program in Educational Leadership Appalachian State University Spring 2010
Summato Operator A Prmer o Summato otato George H Olso Ph D Doctoral Program Educatoal Leadershp Appalacha State Uversty Sprg 00 The summato operator ( ) {Greek letter captal sgma} s a structo to sum over
( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix.
Revew + v, + y = v, + v, + y, + y, Cato! v, + y, + v, + y geeral Let A be a atr Let f,g : Ω R ( ) ( ) R y R Ω R h( ) f ( ) g ( ) ( ) ( ) ( ( )) ( ) dh = f dg + g df A, y y A Ay = = r= c= =, : Ω R he Proof
Engineering Vibration 1. Introduction
Egeerg Vbrato. Itroducto he study of the moto of physcal systems resultg from the appled forces s referred to as dyamcs. Oe type of dyamcs of physcal systems s vbrato, whch the system oscllates about certa
Lecture 02: Bounding tail distributions of a random variable
CSCI-B609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
Queueing Networks. γ 3
Queueg Networks Systes odeled by queueg etworks ca roughly be grouped to four categores. Ope etworks Custoers arrve fro outsde the syste are served ad the depart. Exaple: acket swtched data etwork. γ µ
h-analogue of Fibonacci Numbers
h-aalogue of Fboacc Numbers arxv:090.0038v [math-ph 30 Sep 009 H.B. Beaoum Prce Mohammad Uversty, Al-Khobar 395, Saud Araba Abstract I ths paper, we troduce the h-aalogue of Fboacc umbers for o-commutatve
Department of Mathematics UNIVERSITY OF OSLO. FORMULAS FOR STK4040 (version 1, September 12th, 2011) A - Vectors and matrices
Deartet of Matheatcs UNIVERSITY OF OSLO FORMULAS FOR STK4040 (verso Seteber th 0) A - Vectors ad atrces A) For a x atrx A ad a x atrx B we have ( AB) BA A) For osgular square atrces A ad B we have ( )
Lecture 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
Discrete 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
Evaluating Polynomials
Uverst of Nebraska - Lcol DgtalCommos@Uverst of Nebraska - Lcol MAT Exam Expostor Papers Math the Mddle Isttute Partershp 7-7 Evaluatg Polomals Thomas J. Harrgto Uverst of Nebraska-Lcol Follow ths ad addtoal
CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem
CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat
Lecture IV : The Hartree-Fock method
Lecture IV : The Hartree-Fock method I. THE HARTREE METHOD We have see the prevous lecture that the may-body Hamltoa for a electroc system may be wrtte atomc uts as Ĥ = N e N e N I Z I r R I + N e N e
Parallelized methods for solving polynomial equations
IOSR Joural of Matheatcs (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volue 2, Issue 4 Ver. II (Jul. - Aug.206), PP 75-79 www.osrourals.org Paralleled ethods for solvg polyoal equatos Rela Kapçu, Fatr
CIS 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: