HW10.nb 1. It is basically quadratic around zero and is very steep. One may guess that the harmonic oscillator is a pretty good approximation.


 Cecilia York
 1 years ago
 Views:
Transcription
1 HW0.nb HW #0. Varatonal Method Plot of the potental The potental n the problem s V = 50 He x  L. V = 50 HE x  L 50 H + x L 8x, , <D Ü Graphcs Ü It s bascally quadratc around zero and s very steep. One may guess that the harmonc oscllator s a pretty good approxmaton. Intal guess 8x, 0, 3<D 50 x  50 x If we try to dentfy the frst term wth the harmonc oscllator potental ÅÅÅÅ m w x, we fnd w = 0 because m =. Then one may hope that t would gve the groundstate energy ÅÅÅÅ w = 5. However, ths hope s not qute fulflled. Usng the groundstate wave functon of the harmonc oscllator, y = J ÅÅÅÅÅÅÅÅÅ m w ê4 p N E m w x êh L  m x w ÅÅÅÅÅÅÅÅÅÅÅÅÅ H ÅÅÅÅÅÅ m w ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Lê4 p ê4 ÅÅÅÅÅ
2 HW0.nb y = y ê. 8m Ø, w Ø 0, Ø < 5 x J ÅÅÅÅÅÅÅ 0 Nê4 p K = ÅÅÅÅÅÅÅÅÅ  8x, <D, 8x, , <D ê. 8 Ø, m Ø < m 5 ÅÅÅÅ P = V, 8x, , <D Unque::usym : 5ê s not a symbol or a vald symbol name. The error s 50 H  ê40 + ê0 L Ebar = K + P 5 ÅÅÅÅ + 50 H  ê40 + ê0 L %  39 ê 8 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 39 ê and s bgger than 5%, but s pretty good, accurate wthn 7.%. Therefore, we are motvated to try a Gaussan as a tral functon. Gaussan tral functon y = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ p ê4 D ê Ex êh D L  ÅÅÅÅÅÅÅÅÅ x D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ pê4 è!!! D K = ÅÅÅÅÅÅÅÅÅ  8x, <D, 8x, , <, Assumptons Ø D > 0D ê. 8 Ø, m Ø < m Unque::usym : 5ê s not a symbol or a vald symbol name. ÅÅÅÅÅÅÅÅÅ 4 D P = V, 8x, , <, Assumptons Ø D > 0D Unque::usym : êh4*d^l s not a symbol or a vald symbol name. 50 J  D ÅÅÅÅÅÅ 4 + D N
3 HW0.nb 3 Ebar = K + P ÅÅÅÅÅÅ 50 J  D 4 + D N + ÅÅÅÅÅÅÅÅÅ 4 D 8D, 0 ê <D , 8D Ø <<  39 ê 8 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 39 ê It has mproved, but not yet wthn 5%. Lnear tmes Gaussan One way to mprove t further s the followng. Note that the potental s not party symmetrc, and hence the groundstate wavefuncton s not expected to be an even functon. Because the potental s lower on the rght, we expect the wave functon s skewed towards the rght. Therefore, we can try y = H + k xl E x êh D L  ÅÅÅÅÅÅÅÅÅÅ x D H + k xl It s not normalzed at ths pont, and we calculate ts norm norm = 8x, , <, Assumptons Ø D > 0D Unque::usym : êh4*d^l s not a symbol or a vald symbol name. è!!! ÅÅÅÅ p D H + k D L K = ÅÅÅÅÅÅÅÅÅ  8x, <D, 8x, , <, Assumptons Ø D > 0D ê. 8m Ø, Ø < m Unque::usym : êh4*d^l s not a symbol or a vald symbol name. è!!! p H + 3 k D L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ 8 D P = V, 8x, , <, Assumptons Ø D > 0D Unque::usym : á30à s not a symbol or a vald symbol name. 5 è!!! p D ÅÅÅÅÅÅÅ J  4 D 4 + D ÅÅÅÅÅÅÅ + 4 D 4 k D  4 D k D + k D ÅÅÅÅÅÅ  D 4 k D + D k D ÅÅÅÅÅÅ  D 4 k D 4 + D k D 4 N
4 HW0.nb 4 Ebar = K + P ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ norm j è!!! p H + 3 k j D L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ + 5 è!!! ÅÅÅÅÅÅÅ p D J  4 D 4 + k k 8 D D ÅÅÅÅÅÅÅ + 4 D 4 k D  4 D k D + k D ÅÅÅÅÅÅÅ  D 4 k D + D k D ÅÅÅÅÅÅ  D 4 k D 4 + D k D 4 N y z y z ì H è!!! p D H + k D LL soluton = 8D, 0 ê <, 8k, 0<D , 8D Ø , k Ø <<  39 ê 8 ÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ 39 ê Ths s correct at the % level! The wave functon s therefore y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!!!!!!!!! ÅÅÅÅÅ norm ê. x H xl y PlotA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ è!!!!!!!!!!!! norm ê. 8x, , <E Ü Graphcs Ü As expected, t s more or less a Gaussan, but skewed to the rght. Obvously, there are many ways to mprove Gaussan. I hope you found one successfully. The analytc soluton Ths problem s actually a specal case of the Morse potental V = D HE a u  L D H + a u L
5 HW0.nb 5 Ths potental s a form of the nteratomc potental n datomc molecules proposed by P.M. Morse, Phys. Rev. 34, 57 (99). The varable u s the dstance between two atoms mnus ts equlbrum dstance u = r  r 0. The Schrödnger equaton can be solved analytcally. Expandng t around the mnmum, 8u, 0, 4<D a D u  a 3 D u ÅÅÅÅÅÅÅ a4 D u Harmonc oscllator approxmaton gves w = "########### a D ÅÅÅÅÅÅÅÅÅÅÅÅÅ. The correct energy egenvalues are known to be m E n = whn + ÅÅÅÅ L  a ÅÅÅÅÅÅÅÅÅÅÅÅ m Hn + ÅÅÅÅ L, where the second term s called the anharmoncanharmonc correcton. For large n, the second term wll domnate and the energy appears to become negatve. Clearly, the bound state spectrum does not go forever, and ends at a certan value of n, qute dfferent from the harmonc oscllator. But ths s expcted because the potental energy asymptotes to D for u Ø + and hence states for E > D must be unbound and hence have a contnuous spectrum. The groundstate wave functon s known to have the form d Ea u b a uê y = E E d a u  ÅÅÅÅÅÅÅÅÅÅÅ a b u where b = d , d = è!!!!!!!!!!!! D m ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ. Let us see that t satsfes the Schrödnger equaton. a I ÅÅÅÅÅÅ m SmplfyA 8u, <D + D HEa u  L ym ÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ y è!!! a H4 è!!!!!!! D m + a L  ÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 8 m ê. 8b Ø d  < ê. 9d > è!!!!!!!!!! D m ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ =E a The frst term s ÅÅÅÅ w = a "######## ÅÅÅÅÅÅÅ D m and the zeropont energy wth the harmonc oscllator approxmaton. The second term  a ÅÅÅÅÅÅÅÅÅÅÅÅ s the anharmonc correcton. 8 m The normalzaton s norm = 8u, , <, Assumptons Ø a > 0 && d > 0D Unque::usym : êh4*d^l s not a symbol or a vald symbol name. b d b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ a Actually, ths wave functon could have been guessed f one pays a careful attenton to the asymptotc behavors. For u Ø, the potental asymptotes to a constant D, and hence the wave functon must damp exponentally e k u wth k = è!!!!!!!!!!!!!!! m» E» ë. For u Ø , the potental rses extremely steeply as D e  a u. It suggests that the energy egenvalue becomes quckly rrelevant, and the behavor of the wave functon must be gven purely by the rsng behavor of the potental. By droppng the energy egenvalue and lookng at the Schrödnger equaton,  ÅÅÅÅÅÅÅÅ d y ÅÅÅÅÅÅÅÅÅÅ + D e  a u y = 0, m d u and change the varable to y = e a u, we fnd  ÅÅÅÅÅÅÅÅ m I d y ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅ ÅÅÅÅÅÅÅ d y d y y d y M + D y = 0. The second term n the parentheses s neglble for y Ø. Therefore the wave functon has the behavor y e è!!!!!!!!!!! m D yê a. Combnng the behavor on both ends, the wave functon has precsely the exact form gven above. Ths s the lesson: the onedmensonal potental problem s so smple that there are many ways to study the behavor of the wave functon. On the other hand, the realworld problem nvolves many more degrees of freedom. In many cases, the Hamltonan tself must be guessed.
6 HW0.nb 6 Back to the Morse potental. The case of the homework problem corresponds to D = 50, a =, m =, =. Therefore the groundstate energy s E 0 = ÅÅÅÅ a "######## ÅÅÅÅÅÅÅ D m  a ÅÅÅÅÅÅÅÅÅÅÅÅ 8 m = 5  ÅÅÅÅ 8 = ÅÅÅÅÅÅ SmplfyA y ê. 8b Ø d  < ê. 9d Ø è!!!!!!!!!!!! norm u  ÅÅÅÅÅÅÅÅÅ 9 u ÅÅÅÅÅÅÅÅÅ 567 è!!!!!!!!!! 43 9 u ÅÅÅÅÅÅÅÅÅ è!!!!!!!!!! D m ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a PlotA u  ÅÅÅÅÅÅÅÅÅ 567 è!!!!!!!!!!, 8u, , <, PlotPonts Ø 50E 43 = ê. 8D Ø 50, a Ø, m Ø, Ø <E Ü Graphcs Ü The varatonal lnear tmes Gaussan wave functon was ` x PlotA ` H ` xl, 8x, , <, PlotPonts Ø 50E Ü Graphcs Ü
7 HW0.nb 7 %%D Ü Graphcs Ü Qute close.. Neutrno Oscllaton (a) The untarty matrx s 88, 0, 0<, 80, 3 D, 3 D<, 80, 3 D, 3 D<< êê MatrxForm 0 0 y 0 3 D 3 D j z k 0 3 D 3 D 3 D, 0, 3 D E I d <, 80,, 0<, 3 D E I d, 0, 3 D<< êê MatrxForm 3 D 0 Â d 3 D y 0 0 j z k  Â d 3 D 0 3 D D, D, 0<, D, D, 0<, 80, 0, << êê MatrxForm D D 0 y D D 0 j z k 0 0 U = 88, 0, 0<, 80, 3 D, 3 D<, 80, 3 D, 3 D<<. 3 D, 0, 3 D E I d <, 80,, 0<, 3 D E I d, 0, 3 D<<. D, D, 0<, D, D, 0<, 80, 0, << D 3 D, 3 D D, Â d 3 D<, 3 D D  Â d D 3 D 3 D, D 3 D  Â d D 3 D 3 D, 3 D 3 D<, 8 Â d D 3 D 3 D + D 3 D,  Â d 3 D D 3 D  D 3 D, 3 D 3 D<<
8 HW0.nb 8 Udagger = ê. 8d Ø d<d D 3 D, 3 D D  Â d D 3 D 3 D,  Â d D 3 D 3 D + D 3 D<, 3 D D, D 3 D  Â d D 3 D 3 D,  Â d 3 D D 3 D  D 3 D<, 8 Â d 3 D, 3 D 3 D, 3 D 3 D<< 88, 0, 0<, 80,, 0<, 80, 0, << For the twobytwo case, the untarty matrx s U3 = U ê. 8q Ø 0, q 3 Ø 0< 88, 0, 0<, 80, 3 D, 3 D<, 80, 3 D, 3 D<< % êê MatrxForm 0 0 y 0 3 D 3 D j z k 0 3 D 3 D The man pont n the calculaton s that the Hamltonan can be wrtten as H = UIc p + m c ÅÅÅÅÅÅÅÅÅÅ 3 Å p M U, and hence the tme evoluton operator s e  H tê = U e Hc p+m c 3 ê pl tê U e  m c 3 tê p 0 0 = e  c p tê U 0 e  m c 3 tê p 0 U j k 0 0 e  m 3 c 3 tê p z The ampltude of our nterest s y A = E I c p tê I80,, 0<.U3.DagonalMatrxA9E I m c 3 têh p L, E I m c 3 têh p L, E I m 3 c 3 têh p L =E.  Â c p t ÅÅÅÅÅÅÅÅÅÅÅÅÅ 8<, j  Â c 3 ÅÅÅÅ p 3 D +  Â c3 3 p 3 D y z k Astar = A ê. 8t Ø t< ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Â c p t j Â c3 k p 3 D + Â c3 3 p 3 D y z Psurv = AstarDDD 3 D 4 + CosA c3 t Hm  m 3 L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E p 3 D 3 D + 3 D 4
9 HW0.nb 9 Ths can be rewrtten as P surv = cos 4 q 3 + cos q 3 sn q 3 coshm  m 3 L c 3 t ÅÅÅÅÅÅÅÅÅÅ p + sn4 q 3 = Hcos q 3 + sn q 3 L  cos q 3 sn q 3 + cos q 3 sn q 3 coshm  m 3 L c 3 =  cos q 3 sn q 3 I  coshm  m 3 L c 3 t ÅÅÅÅÅÅÅÅÅÅÅ p M =  4 cos q 3 sn q 3 sn Hm  m 3 L c 3 t ÅÅÅÅÅÅÅÅÅÅÅ 4 p =  sn q 3 sn Hm m 3 L c 3 t ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 p. Ths s nothng but Eq. (5) n the problem. (b) t ÅÅÅÅÅÅÅÅÅÅÅ p For p = GeV, m 3  m =.5ä03 ev ê c, and usng = 6.58 ä0  MeV sec = 6.58 ä06 ev sec and c = 3.00 ä0 0 cm sec , the argument of sn.5ä0 s 3 ev êc c 3 t ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅ 950 t = ÅÅÅÅÅÅÅÅÅÅÅÅÅ 950 c t = 37 L L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = 3.7 c ev ev sec sec c sec 0 0 ä03 ÅÅÅÅÅÅÅ cm km. Takng q 3 = ÅÅÅÅÅÅÅÅ è!!!! and hence sn q 3 =, we fnd P surv =  sn H3.7 ä03 ÅÅÅÅÅÅÅ L km L = cos H3.7 ä03 ÅÅÅÅÅÅÅ L km L. 03 LD, 8L, 0, 0000<E Ü Graphcs Ü The oscllaton occurs on the dstances of thousands of klometers. Ths s what had been observed by the SuperKamokande experment, Phys. Rev. 93, 080 (004), even though the data shows nearly washedout oscllaton due to the soso resoluton n energy and dstance. A very macroscopc quantum phenomenon. (c) For the threestate case, we keep all angles, and we calculate the oscllaton probablty from n m to n e,
10 HW0.nb 0 Amue = E I c p tê I8, 0, 0<.U.DagonalMatrxA9E I m c 3 têh p L, E I m c 3 têh p L, E I m 3 c 3 têh p L =E.Udagger.  Â c p t ÅÅÅÅÅÅÅÅÅÅÅÅÅ 880<, 8<, j Â d Â c3 3 ÅÅÅ p 3 D 3 D 3 D + k  Â c3 p D 3 D 3 D D  Â d D 3 D 3 DL +  Â c3 p 3 D D D 3 D  Â d D 3 D 3 DL y z Amuestar = % ê. 8t Ø t, d Ø d< ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Â c p t j Â d+ Â c3 3 k Â c3 p 3 D 3 D 3 D + p D 3 D 3 D D  Â d D 3 D 3 DL + Â c3 p 3 D D D 3 D  Â d D 3 D 3 DL y z and from n e to n m, Pmue = AmuestarDD D 3 D 3 D D  CosA c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D 3 D D  CosAd  c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p + c3 p E D 3 D 3 D D 3 D 3 D + CosAd  c3 p + c3 p E D 3 D 3 D D 3 D 3 D + D 3 3 D 3 D D 3 D 3 D  CosAd + c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 3 D 3 D D 3 D 3 D  D 3 D 3 D D 3 3 D 3 D + CosAd  c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p + c3 p E D 3 D 3 D D 3 3 D 3 D + 3 D 3 D 3 D  CosA c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D 3 D 3 D + D 4 3 D 3 D 3 D  CosA c3 p E 3 D D 3 D 3 D + CosA c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D D 3 D 3 D + 3 D D 4 3 D 3 D
11 HW0.nb Aemu = E I c p tê I80,, 0<.U.DagonalMatrxA9E I m c 3 têh p L, E I m c 3 têh p L, E I m 3 c 3 têh p L =E.Udagger.  Â c p t ÅÅÅÅÅÅÅÅÅÅÅÅÅ 88<, 80<, j Â d Â c3 3 ÅÅÅ p 3 D 3 D 3 D + k  Â c3 p D 3 D 3 D D  Â d D 3 D 3 DL +  Â c3 p 3 D D D 3 D  Â d D 3 D 3 DL y z Aemustar = % ê. 8t Ø t, d Ø d< ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Â c p t j Â d+ Â c3 3 k Â c3 p 3 D 3 D 3 D + p D 3 D 3 D D  Â d D 3 D 3 DL + Â c3 p 3 D D D 3 D  Â d D 3 D 3 DL y z Pemu = AemustarDD D 3 D 3 D D  CosA c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D 3 D D  CosAd + c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D 3 D D 3 D 3 D + CosAd + c3 p E D 3 D 3 D D 3 D 3 D + D 3 3 D 3 D D 3 D 3 D  CosAd  c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p + c3 p E D 3 3 D 3 D D 3 D 3 D  D 3 D 3 D D 3 3 D 3 D + CosAd + c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D 3 D D 3 3 D 3 D + 3 D 3 D 3 D  CosA c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D 3 D 3 D + D 4 3 D 3 D 3 D  CosA c3 p E 3 D D 3 D 3 D + CosA c3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p E D 3 D D 3 D 3 D + 3 D D 4 3 D 3 D  PemuD 4 3 D SnA c3 t Hm  m L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E 4 p SnA c3 t Hm  m 3 L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E SnA c3 t Hm  m 3 L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E q 4 p 4 p D 3 D q 3 D Indeed, when d 0, two probabltes are dfferent.
12 HW0.nb (d) As we dd n the class, the tmereserval nvarance states that Xa» e  H tê» b\ = Xb è» e  H tê» a è \, and hecepha Ø bl = PHb è Ø a è L. Here, a è s the tmereversed state of a, and b è s the tmereversed state of b. In our case, the tmereversed state has the opposte momentum. (If you worry about the helcty, both the momentum and the spn flp, and remans the same, namely lefthanded.) However, the Hamltonan depends only on the magntude of the momentum and hence the oscllaton probabltes also depend only on the magntude of the momentum. Therefore PHb è Ø a è L = PHb Ø al. Choosng n e and n m of the same momenta to be the ntal and fnal states, we fnd that the tme reversal nvarance would predct PHn m Ø n e L = PHn e Ø n m L. The fact that the tmereversal volaton requres d 0makes sense because the tmereversal opeator nvolves the complex conjugaton, and d s the only complex parameter n the Hamltonan.
Simple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationExperiment 8 Two Types of Pendulum
Experment 8 Two Types of Pendulum Preparaton For ths week's quz revew past experments and read about pendulums and harmonc moton Prncples Any object that swngs back and forth can be consdered a pendulum
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationGibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments)
Gbbs Free Energy and Chemcal Equlbrum (or how to predct chemcal reactons wthout dong experments) OCN 623 Chemcal Oceanography Readng: Frst half of Chapter 3, Snoeynk and Jenkns (1980) Introducton We want
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationA. Te densty matrx and densty operator In general, te manybody wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc syste
G25.2651: Statstcal Mecancs Notes for Lecture 13 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS Te problem of quantum statstcal mecancs s te quantum mecancal treatment of an Npartcle system. Suppose te
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More information5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)
5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages  n "Machnes, Logc and Quantum Physcs"
More informationFaraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
More informationHedging InterestRate Risk with Duration
FIXEDINCOME SECURITIES Chapter 5 Hedgng InterestRate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cashflows Interest rate rsk Hedgng prncples DuratonBased Hedgng Technques Defnton of duraton
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the ageold queston: When the hell
More information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationSection 2 Introduction to Statistical Mechanics
Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.
More informationChapter 31B  Transient Currents and Inductance
Chapter 31B  Transent Currents and Inductance A PowerPont Presentaton by Paul E. Tppens, Professor of Physcs Southern Polytechnc State Unversty 007 Objectves: After completng ths module, you should be
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationThursday, December 10, 2009 Noon  1:50 pm Faraday 143
1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationThe material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (d) superpositions and expectation values
Lecture 7: Expectaton Values The materal n ths lecture covers the followng n Atkns. 11.5 The nformton of a wavefuncton (d) superpostons and expectaton values Lecture onlne Expectaton Values (PDF) Expectaton
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationINVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMAHDR NETWORKS
21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS
More informationChapter 12 Inductors and AC Circuits
hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationMean Molecular Weight
Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 41 4.1. INTERACTION OF LIGHT
More informationRegression Models for a Binary Response Using EXCEL and JMP
SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STATTECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
More informationSemiclassical Statistical Mechanics
Semclasscal Statstcal Mechancs from Statstcal Physcs usng Mathematca James J. Kelly, 9962002 A classcal ensemble s represented by a dstrbuton of ponts n phase space. Two mportant theorems, equpartton
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationFINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals
FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationEXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR
EXAMPLE PROBLEMS SOLVED USING THE SHARP EL733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationTime Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6  The Time Value of Money. The Time Value of Money
Ch. 6  The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21 Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationIntroduction to Statistical Physics (2SP)
Introducton to Statstcal Physcs (2SP) Rchard Sear March 5, 20 Contents What s the entropy (aka the uncertanty)? 2. One macroscopc state s the result of many many mcroscopc states.......... 2.2 States wth
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationChapter 9. Linear Momentum and Collisions
Chapter 9 Lnear Momentum and Collsons CHAPTER OUTLINE 9.1 Lnear Momentum and Its Conservaton 9.2 Impulse and Momentum 9.3 Collsons n One Dmenson 9.4 TwoDmensonal Collsons 9.5 The Center of Mass 9.6 Moton
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More information1 De nitions and Censoring
De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationMulticomponent Distillation
Multcomponent Dstllaton need more than one dstllaton tower, for n components, n1 fractonators are requred Specfcaton Lmtatons The followng are establshed at the begnnng 1. Temperature, pressure, composton,
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationProductForm Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538195174 ORIGINAL ARTICLE ProductForm Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s nonempty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 19982016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationRotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationA frequency decomposition time domain model of broadband frequencydependent absorption: Model II
A frequenc decomposton tme doman model of broadband frequencdependent absorpton: Model II W. Chen Smula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (1 Aprl ) (Proect collaborators: A. Bounam,
More informationESCI 341 Atmospheric Thermodynamics Lesson 9 Entropy
ESCI 341 Atmosherc hermodynamcs Lesson 9 Entroy References: An Introducton to Atmosherc hermodynamcs, sons Physcal Chemstry (4 th edton), Levne hermodynamcs and an Introducton to hermostatstcs, Callen
More informationV1 V. Electronic Spectroscopy. Eigen value: lowest U el is the ground state by electronic dipole selection rule;
V1 V. Electronc Spectroscopy What we have done so far s use B O approxmaton to elmnate φ el (q,q) For example dd datomc, put asde φ el and focused on vbratonal & rotatonal soluton Now must consder electronc
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationSolutions to the exam in SF2862, June 2009
Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodcrevew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r
More information4 Cosmological Perturbation Theory
4 Cosmologcal Perturbaton Theory So far, we have treated the unverse as perfectly homogeneous. To understand the formaton and evoluton of largescale structures, we have to ntroduce nhomogenetes. As long
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationPRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.
PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and mfle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationz(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1
(4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at
More informationsubstances (among other variables as well). ( ) Thus the change in volume of a mixture can be written as
Mxtures and Solutons Partal Molar Quanttes Partal molar volume he total volume of a mxture of substances s a functon of the amounts of both V V n,n substances (among other varables as well). hus the change
More informationInterIng 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 1516 November 2007.
InterIng 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 1516 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationThe quantum mechanics based on a general kinetic energy
The quantum mechancs based on a general knetc energy Yuchuan We * Internatonal Center of Quantum Mechancs, Three Gorges Unversty, Chna, 4400 Department of adaton Oncology, Wake Forest Unversty, NC, 7157
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationWe now turn to our final quantum mechanical topic, molecular spectroscopy.
07 Lecture 34 We now turn to our fnal quantum mechancal topc, molecular spectroscopy. Molecular spectroscopy s ntmately lnked to quantum mechancs. Before the md 190's when quantum mechancs was developed,
More informationFace Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)
Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton
More information