α 2 α 1 β 1 ANTISYMMETRIC WAVEFUNCTIONS: SLATER DETERMINANTS (08/24/14)

Size: px
Start display at page:

Download "α 2 α 1 β 1 ANTISYMMETRIC WAVEFUNCTIONS: SLATER DETERMINANTS (08/24/14)"

Transcription

1 ANTISYMMETRI WAVEFUNTIONS: SLATER DETERMINANTS (08/4/4) Wavefuctos that descrbe more tha oe electro must have two characterstc propertes. Frst, scll electros are detcal partcles, the electros coordates must appear wavefuctos such that the electros are dstgushable. Ths meas that the coordates of electros a atom or molecule must eter to the wavefucto so that the may-electro probablty dstrbuto, Ψ = Ψ*Ψ, every electro s detcal. The secod requremet, ad ths s a more completd rgorous statemet of the Paul excluso prcple, s that the wavefucto for a system of two or more electros must chage sg ay tme we permute the coordates of ay two electros, Ψ(,,,, j, N ) = Ψ(,, j,,, N ). Ths s a property of fermos (amog whch are electros, protos, ad other halftegral sp partcles); systems wth more tha oe detcal fermo, oly probablty dstrbutos correspodg to atsymmetrc wavefuctos are observed. Let us revew the -electro case. If wttempt to costruct a two-electro wavefucto as a product of dvdual electro orbtals, ad, the ether () () or () () alore satsfactory sce we requre that the electros be dstgushable. The combatos () () ± () () do meet the requremet of dstgushablty, but these fuctos just descrbe the spatal dstrbuto of the electros; we must also cosder ther sp. If the two electros have dfferet sp egefuctos, dstgushablty meas that ether α β or α β s satsfactory, but α β ± β α arcceptable as are α α ad β β, of course. As we ve oted, the overall wavefucto for two electros must be atsymmetrc wth respect to terchage of the electros labels. Ths admts four possbltes, as log as both ad are sgly occuped (ormalzato costats cluded): 3 Ψ = ##### Symmetrc "##### $ Atsymmetrc ##" ## $ Ψ = ( ϕ a ()ϕ b () +ϕ b ()ϕ a () ) (α β β α ) ##### Atsymmetrc "##### $ ### Symmetrc "### $ α α ( ϕ a ()ϕ b () ϕ b ()ϕ a () ) ( α β + β α ) β β The left superscrpts o Ψ ad 3 Ψ are the sp multplctes (S + ); the trplet wavefuctos arll egefuctos of Ŝ wth egevalue S(S + ) = ad they are degeerats log as we cosder sp-depedet cotrbutos to the eergy (.e., there s o appled magetc feld ad sp-orbt couplg s eglected). The values of M S (= M S 0 α 0 β

2 m s + m s, the z-compoets of S) for each wavefucto are gve, ad the umber of values M S takes, S +, for a gve S s the sp multplcty for the state. Slater poted out that f we wrte may-electro wavefuctos as (Slater) determats, the atsymmetry requremet s fulflled. Slater determats are costructed usg sporbtals whch the spatal orbtals are combed wth sp fuctos from the outset. We use the otato () ϕ a ()α ad add a bar over the top to dcate spdow, () ϕ a ()β. Slater determats are costructed by arragg sporbtals colums ad electro labels rows ad are ormalzed by dvdg by N, where N s the umber of occuped sporbtals. Ths arragemet s uversally uderstood so the otato for Slater determats ca be made very compact; four Slater determats ca be costructed usg,,, ad :,,, ad What does ths otato mea? To see, let s expad out, step-by-step: φ a () () ϕ = a ()α ϕ b ()α () () ϕ a ()α ϕ b ()α = ϕ a ()α ϕ b ()α ϕ b ()α ϕ a ()α Thus, = ( ϕ a ()ϕ b () ϕ b ()ϕ a () )α α = 3 Ψ( M S = ) Notce that the otato assumes the determat s ormalzed ad that we havdopted the covetos metoed: rug over sporbtals colums ad over electro labels rows. Proceedg the same way for, = () () () () = ϕ a ()β ϕ b ()β ϕ a ()β ϕ b ()β = ϕ a ()β ϕ b ()β ϕ b ()β ϕ a ()β Thus, = ( ϕ a ()ϕ b () ϕ b ()ϕ a () )β β = 3 Ψ( M S = ) ombatos of ad yeld the two wavefuctos wth M S = 0: = = () () () () = ϕ a ()β ϕ b ()α ϕ a ()β ϕ b ()α = ϕ a ()ϕ b () β α ϕ b ()ϕ a () α β () () () () = ϕ a ()α ϕ b ()β ϕ a ()α ϕ b ()β = 3 Ψ( M S = 0) = Ψ( M S = 0) = ϕ a ()ϕ b () α β ϕ b ()ϕ a () β α ( + ) = ( ϕ a ()ϕ b () ϕ b ()ϕ a () ) β α + α β β α α β = ϕ a ()ϕ b () +ϕ b ()ϕ a ()

3 Take partcular ote of the fact that the spatal parts of all three trplet wavefuctos are detcal ad are dfferet from the sglet wavefucto. To summarze, terms of determats the sglet ad trplet wavefuctos are 3 Ψ( M S = ) = ; 3 Ψ( M S = 0) = ( φ a + ) ; 3 Ψ( M S = ) = Ψ = ( φ a ) Determats ca be represeted dagramatcally usg up- ad dow-arrows orbtals a maer famlar to chemsts. However, the dagrams ow take o more precse meags. Whle the M S = ± compoets of the trplet statre represeted as sgle determats, the sglet wavefucto ad the M S = 0 compoet of the trplet state should be wrtte as combatos of the determats: The geeral form of a Slater determat comports wth ths dscusso. Whe expaded, the determat for N electros N sporbtals yelds N terms, geerated by the N possble permutatos of electro labels amog the sporbtals ad dfferg by a multplcatve factor of for terms related by oe parwse permutato. To be explct, wrtte out determatal form we have Ψ(,,,, j, N ) = N Dfferet Sp Orbtals olums #%%%%%%%%%%% $ %%%%%%%%%%% & () () φ p () φ q () φ z () () () φ p () φ q () φ z () " " " " " () () φ p () φ q () φ z () " " " " " ( j) ( j) φ p ( j) φ q ( j) φ z ( j) " " " " " (N ) (N ) φ p (N ) φ q (N ) φ z (N ) Sce electroc wavefuctos for two or more electros should be wrtte as determats, our goal s to determe the symmetry characterstcs of determats, or more specfcally, how the determats form bases for rreducble represetatos. learly, we wat to avod expadg the determat out to exhbt N terms, f possble. To do that, three propertes of determats ca be used (I these expressos, the sporbtals ca carry ether sp, φ() = ϕ() [α or β ]: = electro label): Swap ay two rows (or colums) of a determat, ad the sg chages, Dfferet electros rows 3

4 φ A φ P φ Q = φ A φ Q φ P Therefore, f ay two rows (or colums) are detcal, the determat s zero. Ths guaratees that we ca t volate the Paul prcply usg the same sporbtal twce. olums (or rows) ca be factored, φ A φ P + φ Q φ R = φ A φ P φ R + φ A φ Q φ R Ay costat (cludg ) ca be factored out, φ A c φ p φ R = c φ A φ p φ R The latter two rules wll be useful whe evaluatg the results of symmetry operatos. Let s see how these rules apply to a closed-shell molecule, HO, for whch we wll exam Slater determat costructed from the valece MOs. The four valece MOs for water are depcted herd the trasformato propertes of the MOs are summarzed as follows a b R +a for all symmetry operatos, R. R + for R = E,σ v ad for R =,σ v. R +b for R = E,σ v ad b for R =,σ v. If we combe the orbtal trasformato propertes wth the rules gve above for determats, we ca fd the symmetry of the groud electroc state wavefucto. Each symmetry operato operates o all the the determat ad the rules gve above wll be used to evaluate the rreducble represetatos to whch that groud state determat belogs: a a b b a a b a a ( b )( b )a a ( ) σ a a b b a a b v a a ( b )( b )a a = + a a b b a a a a b b a a σ v a a b b a a = + a a b b a a = + a a b b a a Ths sglet, closed-shell electroc state wavefucto (a Slater determat) belogs the totally symmetrc represetato, A. Sce electros are pared orbtals closedshell molecules, f the doubly-occuped orbtals all belog to oe-dmesoal 4

5 represetatos, the wavefucto wll always belog to the totally symmetrc represetato. Although t s ot as trasparetly true, ths apples to closed-shell molecules wth degeerate orbtals as well. To cosdeust the 3 operato actg upo the groud state determat for NH3, we frst recall how the orbtals trasform: a 3 +a e 3 x e + 3 d x y e y 3 3 e e x y Trasformato of the determats s a bt laborous, but straghtforward ad we ca gore the odegeerate sporbtals: e x e x e y e y 3 ( e + 3 e )( e + 3 e )( 3 e e )( 3 e e ) x y x y x y x y We expad out to four determats obtaed by multplyg through the frst two parethetcal factors,... = 4 e x e x ( 3 e x e y )( 3 e x e y ) e y e y ( 3 e x e y )( 3 e x e y ) 3 4 e y e x ( 3 e x e y )( 3 e x e y ) 3 4 e x e y ( 3 e x e y )( 3 e x e y )... expadg ths out further, we recall that ay determat wth two detcal colums s zero, whch elmates all but oe term for each of these four determats,... = 6 e x e x e y e y e y e y e x e x 3 6 e y e x e x e y 3 6 e x e y e y e x... fally we perform two colum swaps the secod determat ad oe colum swap each of the thrd ad fourth determats, leavg the sg of the secod uchaged ad swtchg the sg of the thrd ad fourth,... = 6 e x e x e y e y e x e x e y e y e x e x e y e y e x e x e y e y So we fally coclude that e x e x e y e y 3 + e x e x e y e y. All the other 3v operatos yeld the same result. The coeffcets work out the ed because the symmetry operatos are orthogoal trasformatos (utary trasformatos, the complex case) - readers are ecouraged to covce themselves that ths s the case. I the smplest ope-shell case, a state s represeted by a sgle determat wth oe upared electro a odegeerate orbtal ad the state symmetry s the sams the symmetry of the half-occuped MO. If two odegeerate orbtals are half occuped, the symmetry of the state s determed by takg the drect product of the two orbtals. 5

6 rreducble represetatos. For the trplet state of methylee (:H), the methylee valece orbtal symmetres are the sams those for water (above) ad the trplet state electroc cofgurato s (a ) (b ) (a ) ( ). The the M S = ± compoets of the trplet state work out qute smply to trasform as, e.g, a a b b a b a = a a b b a a a b b a σ v a = + a a b b a a a b b a σ v a where the closed shells are ot wrtte out. Let s also cofrm that the M S = 0 compoet also behaves as a bass fucto: Let s exame the electroc states of cyclobutadee (), for whch there s a halfoccuped degeerate set of orbtals. The four π orbtals are depcted below; the lowest-eergy cofgurato s (a u ) (e g ). s D4h, but oly the D4 subgroup eed be cosdered becausll the states that ca arse from ths cofgurato are gerade. We focus etrely o the partally occuped e g orbtals, whch trasform as follows, Therre sx determats of terest:,,,,, ad. The frst two clearly belog to a trplet ad trasform as follows: = a a b b a a + a b a ( ) + a ( ) = a + a σ a + a b v + a + a σ a + a b v a e 4 a +eb ; e 4 b ea e 4 a ; e 4 b e x a eb ; e x b ea (x= y) (x= y) ; e 4 b eb ( ) = + e 4 b ( )( ) = + + a ( ) = a + a 6

7 Ths demostrates that ad belog to the A represetato (A g D4h). As the reader ca readly verfy, the combato + also belogs to A (A g ). It s straghtforward to show that has ( g ) symmetry: e 4 b eb ( ) ( ) = + = e b e 4 b ( )( ) ( )( ) = + e b e x b ( eb )( ) ( )( ) = = + e b e (x= y) b ( ) ( ) = e b Fally, ad form bass for a reducble represetato that yelds the A (A g g ) represetatos, e 4 a eb ; e 4 b ( ea )( ) = e 4 a ( )( ) = ; e 4 b ( )( ) = x ( eb )( ) = ; ( ea )( ) = (x= y) (x= y) x ; ( )( ) = The reader ca demostrate that + has A g symmetry ad has g symmetry. (A g ad g projecto operators appled to or wll also geerate thpproprate combatos.) I summary, the (a u ) (e g ) cofgurato gves rse to 3 A g, g, A g, ad g states ad we ve establshed the determatal wavefuctos for each of these states: M S M S g : ( ) 0 3 A g e ( a + ) 0 A g : ( + ) 0 g : e ( a ) 0 7

8 ackgroud: Eerges of Determatal Wavefuctos Several texts quatum chemstry offer rgorous ad complete dervatos for eergy expressos of determatal wavefuctos. I ths documet, we ll provd graphcal method arrvg at the results after provdg a physcal motvato for the method. To accomplsh the latter purpose, let s reexame the determats from whch the sglet ad trplet two-electo wavefuctos were costructed. (A smple example: a helum atom a excted s s cofgurato; = s ad = s): I thbsece of explctly sp-depedet terms the Hamltoa (lk appled magetc feld or sp-orbt couplg), the eerges of these wavefuctos are oly affected by the spatal dstrbuto of the electros specfed by these expressos, so let s exame just the spatal factors: ; Ψ space = ϕ a ()ϕ b () + ϕ b ()ϕ a () 3 Ψ space = ( ϕ ()ϕ () ϕ ()ϕ () a b b a ) Let s evaluate the eerges by takg the expectato values of the Hamltoa (the + sgs apply to the sglet ad the sgs apply to the trplet):,3 E = ϕ ( a ()ϕ b () ±ϕ b ()ϕ a ())H ( ϕ a ()ϕ b () ±ϕ b ()ϕ a ())dτ dτ,3 E = ϕ a ()ϕ b () H ϕ a ()ϕ b ()dτ dτ + ϕ b ()ϕ a () H ϕ b ()ϕ a ()dτ dτ ± ϕ a ()ϕ b () H ϕ b ()ϕ a ()dτ dτ ± ϕ b ()ϕ a () H ϕ b ()ϕ a ()dτ dτ The frst two tegrals have the same valud are evaluated a straghtforward way, e ϕ a ()ϕ b () H ϕ a ()ϕ b () dτ dτ = ϕ a ()ϕ b () ĥ + ĥ + ϕ r a ()ϕ b () dτ dτ = ϕ ()ĥ a ϕ a ()dτ + ϕ ()ĥ b ϕ b ()dτ + e ϕ a ()ϕ b () dτ r dτ = h a + h b + J ab. ( where the ormalzato has bee used: ϕ a dτ = ϕ b dτ = ) The operators ĥ ad ĥ would clude, for the helum s s case, the ketc eergy operators ad electro-uclear oulombc attracto terms for each of the electros geeral they cludll the ketc ad potetal eergy terms that deped oly o each electro s dvdual coordates. h a ad h b are hece referred to as oe electro 8

9 eerges. J ab s called a oulomb tegral ad has a semclasscal terpretato that t ca be vewed as the electro-electro repulso eergy betwee oe electro charge cloud ϕ a ad a secod electro charge cloud ϕ b. The last two tegrals are equal to each other as well, e ± ϕ a ()ϕ b () H ϕ b ()ϕ a () dτ dτ = ± ϕ a ()ϕ b () ĥ + ĥ + ϕ r b ()ϕ a () dτ dτ e = ± ϕ a ()ϕ b () ϕ r b ()ϕ a () dτ dτ = ±e ϕ a ϕ b () ϕ a ϕ b () dτ r dτ = ±K ab E = h a + h b + J ab + K 3 ab E = h a + h b + J ab K ab K ab s called a exchage tegral ad K ab = E 3 E, the sglet-trplet eergy gap. Note that f we had foud the expectato values of or (p. ), the cross-terms that gve the exchage tegrals do t survve due to orthogoalty of the sp fuctos, ad ther eerges are h a + h b + J ab. Exchage tegrals are varably postve sce () ad () wll ted to have same sg whe the tegrad has ts greatest magtude (whe r 0). K ab s largest whe ad exted over the same rego of space. Thtsymmetrc ature of the trplet spatal wavefucto guaratees that the trplet state the electros ad are ever at the same locato ( 3 Ψ = 0 f the two electros have the same coordates),.e., the trplet state les lower eergy because there s less electro-electro repulso. The Rules: Thbovackgroud wll serve to ratoalze the followg rules for evaluatg eerges of determats, whch cota the followg terms: () a oeelectro orbtal eergy, ε, for each electro. ε wll geerally clude two-electro terms volvg e-e repulsos wth thtomc core electros (screeg) whch dstgushes ε from the symbol h used above, () for each parwse e - e repulso, a oulomb term (a Jj cotrbuto), ad (3) a exchage stablzato (a Kj cotrbuto) for each lke-sp e - e teracto. As a example, cosder the fve determats llustrated here. Assocated wth each of these determats are sx oulomb tegral cotrbutos sce there must be sx uque parwse repulsos wth four electros. The two determats wth MS = ± are Proof: Slater, J.. Quatum Theory of Atomc Structure, Vol. I, McGraw-Hll: New York, 960, p

10 assocated wth three exchage stablzatos whle those wth MS = 0 arssocated wth two exchage stablzatos. Ths s algebracally summarzed as E gr = ε a + ε b + J aa + J bb + 4J ab K ab E (3) ex = ε a + ε b + ε c + J aa + J ab + J ac + J bc K ab + K ac Mxg of the two MS = 0 determats, Ψex(A) ad Ψex(), yelds a trplet ad a sglet wavefucto. The trplet eergy must be equal to the eerges for the MS = ± trplet wavefuctos that are represetabls sgle determats. K bc E A ex = E ex = ε a + ε b + ε c + J aa + J ab + J ac + J bc K ab + K ac E A+ ex + E A ex = E A ex + E ex ; but E A+ (3) ex = E ex E () ex = ε a + ε b + ε c + J aa + J ab + J ac + J bc K ab + K ac E () ex = E A ex = E A ex + E (3) ex E ex + K bc ; E () ex E ex States arsg from degeerate orbtals wth two or more electros (3) = +K bc Molecules ad os wth ope shell electroc cofguratos are qute commo trasto metal chemstry. efore proceedg further wth applcatos, however, let s derve some formulas that allow us to work wth characters dervg states for multelectro cofguratos degeerate orbtal sets. Whe -fold degeerate orbtals {ϕ,,ϕ } belogg to rreducble represetato Γ are flled wth, say, two electros or two holes, oe caot smply evaluat drect product to determe the states that derve from such cofguratos. The -dmesoal drect squared represetato (Γ Γ) wll have the parwse products of these orbtals, {ϕ,,ϕ,ϕ ϕ j, j}, as bass fuctos. The characters, χ Γ Γ (R), are just χ Γ (R). Now, f wre costructg permssble sglet/trplet state wavefuctos, the spatal part of the wavefuctos are symmetrc/atsymmetrc wth respect to permutato of the electro labels whle the sp fucto s atsymmetrc/symmetrc: " $$$$$$$ Symmetrc # $$$$$$$ % Ψ = ϕ ()ϕ () ϕ ()ϕ (), < j ϕ ()ϕ j () +ϕ j ()ϕ () " Atsymmetrc $$ # $$ % (α β β α ) 0

11 3 Ψ = ###### Atsymmetrc "###### $ ( ϕ ()ϕ j () ϕ j ()ϕ ()), < j The set, {ϕ,,ϕ }, s thass for a rreducble represetato so the character for each operato wth a class wth respect to ths bass wll be the same, ad depedet of ay choce of orthogoal lear combatos of these orbtals we make. Suppose that we ve sgled out a partcular operato R from each class ad assume that we have chose a lear combato of the orbtals such that the matrx for each R s dagoal: ϕ R r ϕ =,, ; R = The combatos of bass fuctos that dagoalze R wll geerally be dfferet for each operato, but the characters are, as always, the same for every member of a class. Operatg o the spatal parts of the wavefuctos for both the sglets ad the trplets, r r ### Symmetrc "### $ α α ( α β + β α ) β β ; χ(r) = r +"+ r R ϕ ()ϕ () r ϕ () r ϕ () = r ϕ ()ϕ () R ϕ ()ϕ () r ϕ ()ϕ () R ( ϕ ()ϕ j () + ϕ j ()ϕ ()) r ϕ ()ϕ j () + ϕ j ()ϕ () R ( ϕ ()ϕ j () ϕ j ()ϕ ()) r ϕ ()ϕ j () ϕ j ()ϕ () < j < j So the characters for the operatos thass spaed by all the symmetrc sglet ( χ + ) ad atsymmetrc trplet ( χ ) wavefuctos are χ + (R) = r + r ; χ (R) = r = As oted above, the characters for the ormal drect product bass are just χ (R) = r = < j ad sce thass fuctos are chose so our class represetatve operatos have dagoal matrces, the characters for the squares of the operatos are dagoal as well, < j = r + r = r + r = j = < j For all the umbers r,..., r, r = ; geeral, r could b complex umber, r = e α.

12 r 0 0 R ϕ r ϕ =,, ; R = 0 0 ; χ(r ) = r 0 0 r = If we take the sum ad dfferece of the expressos for χ (R) ad χ(r ) ad dvde each by two, we obta formulas for thtsymmetrc ad symmetrc drect products, where the cotext of the dervato gve makes t clear that these two formulas are oly defed whe takg a drect product of a degeerate rreducble represetato wth tself ad they re used to hadle two-electro (or two-hole) cases. It s easy to show that these formulae recover our results for cyclobutadee. The D4 subgroup s aga suffcet, sce the (eg) cofgurato wll geerate oly gerade states: [E E] = A Formulas for three electros (or three holes) a 3-fold- or hgher-degeerate set of orbtals ca be derved usg permutato group theory 3 ad are ad for four electros (or four holes) a 4-fold- or hgher-degeerate set of orbtals: Applcatos to Lgad Feld Theory χ + (R) = χ S =0 (R) = χ (R) + χ(r ) χ (R) = χ S = (R) = χ (R) χ(r ) D 4 E 4 ( 4 ) [E E] = A χ S = (R) = 3 χ 3 (R) χ(r 3 ) χ S = 3 (R) = 6 χ 3 (R) 3χ(R)χ(R ) + χ(r 3 ) χ S =0 (R) = χ 4 (R) 4χ(R)χ(R 3 ) + 3χ (R ) χ S = (R) = 8 χ 4 (R) χ (R)χ(R ) + χ(r 4 ) χ (R ) χ S = (R) = 4 χ 4 (R) 6χ (R)χ(R ) + 8χ(R)χ(R 3 ) 6χ(R 4 ) + 3χ (R ) A uderstadg of bodg trasto-metal complexes, partcularly classcal Werer complexes, demads that wccout for electro-electro repulso o a equal footg wth the quas-depedet electro terms mplct our focus o molecular orbtals ad ther respectve orbtal eerges. I lgad-feld theory, oe seeks to correlate atomc (o) state eerges (.e., Russell-Sauders terms) wth molecular states bult up from molecular orbtal cofguratos. I the lgad-feld approach, the ope-shell wavefuctos arssumed to reta ther d-lke character ad the lgad cotrbuto to See D. I. Ford, J. hem. Ed., 49, (97); Appedx below gves proof of the three electro 3 formulae.

13 the partally-flled orbtals s accouted for through ther effect o orbtal eergy splttg ad by treatg the d-d repulso eerges as adjustable parameters (maly the Racah parameter ) that wll geerally be smaller tha free os because whe the electros are delocalzed oto lgads, repulsos are lesseed by the relef of ther crowdg to relatvely cotracted d orbtals. Thesre matters we wll refer to oly tagetally - our focus wll rema o the symmetry-cotrolled characterstcs of lgad feld theory. Let s tur our atteto to the electroc states of octahedrally-coordated d os, for whch therre three possble cofguratos: (tg), (tg) (eg), ad (eg). The sglyexcted (tg) (eg) cofgurato s hadled easly: therre sx assgmets of the tg electro (three orbtals, sp-up or sp-dow) ad four assgmets of the eg electro, so the state wavefuctos from ths cofgurato are wrtte terms of 4 determats. The symmetres of the states are easly determed by takg the drect product, tg eg =,3 T,3 T where the left superscrpts dcate that both sglets ad trplets ca be formed for each symmetry (wth oe electro each tg ad eg, o Paul Prcple volatos occur). We ll retur to the ssue of fdg wavefuctos for these states below. O E (= 4 ) A x + y + z A E 0 0 (z x y,x y ) T 3 0 (R x, R y, R z ); (x, y, z) T 3 0 (xy, xz, yz) T E T T + T T E E + E E 3 0,3 T,3 T To fd the states arsg from the (tg) ad (eg) cofguratos, the formulas derved the precedg secto arppled, the results of whch are show thugmeted character tablbove for the O group (the g symmetry of the d orbtals s uderstood). If we correlate the d o atomc states wth these molecular states, the dagram show here emerges. Ths s a modfed verso of Fgure 9.4 gve otto s text. Atomc states art left, molecular orbtal cofguratos are show at far rght. The dagram also shows some eergy splttgs that are ot gve the correspodg dagram foud otto s text. I the otto s Fgure 9.4, the cofguratos are labeled as Strog teracto referrg the the stregth of the lgad-metal teracto. Note that eve the strog lgad feld lmt, the electro-electro repulso that causes the state splttg 3 T A E T A A E 3

14 would stll be preset. The fgure gve o the followg paglso cludes eergy splttgs betwee some of the states o both sdes of the dagram the orgs of whch shall be explaed below. Now let s fd wavefuctos for some of these states. For the (tg) cofgurato, Hud s rule predcts that the lowest eergy state wll be the trplet, 3 Tg. The MS = determats are smple to llustrate graphcally ad are show below the correlato dagram: To verfy that these determats do deed belog to the Tg represetato, we frst apply the symmetry operatos to the orbtals... 4

15 3 xy yz ; yz 3 xz ; xz 3 xy ; xy 4z xy ; xz yz ; yz xz 4z 4z...the apply these to the determats, whch do deed behavs Tg bass fuctos: 4z 4z xy xy ; xz xz ; yz yz (x= y) (x= y) (x= y) xy xy ; xz yz ; yz xz 4z xy xz 3 yz xy = xy yz ; xy yz 3 yz xz = yz xz ; xy xz 4z xz yz 3 xy xz χ(3 ) = 0 4z xy yz = xy yz ; xy yz xy xz = xy xz ; xz yz 4z yz xz = xz yz χ(4 ) = xy xz 4z xy xz xy xz = xy xz ; xy yz xy yz = xy yz ; xz yz 4z xz yz = xz yz (x= y) xy yz = xy yz ; xy yz xz yz The determats xyxz, xz yz, ad xy yz are lkewse the wavefuctos wth MS = for the 3 Tg state. The MS = 0 wavefuctos belogg to 3 Tg are combatos of determats: ( xy xz + xy xz ), ( xz yz + xz yz ) ad ( xy yz + xy yz ). The reader may verfy by drect operato that the Tg wavefuctos are the orthogoal combatos of the same determats: ( xy xz xy xz ), ( xz yz xz yz ), 4z χ( 4 ) = (x= y) xy xz = xy xz ; (x= y) yz xz = xz yz χ( ) = ad ( xy yz xy yz ). Fally, we ca costruct a reducble represetato spaed by determats correspodg to two electros each oe of the tg orbtals. Reducto of that represetato shows that these determats form the Ag ad Eg states. 5

16 O E ( 4 ) xy xy, xz xz, yz yz A E The combatos of these determats for these two states, dervato of whch s left as a exercse, ars follows: Ψ( A g ) = xy xy + xz xz + yz yz 3 Ψ a ( E g ) = xy xy xz xz yz yz 6 ; Ψ b ( E g ) = xz xz yz yz. Descet symmetry as a tool for dervg electroc wavefuctos Let s cosder the states we derved for the (tg) (eg) cofgurato:,3 Tg,3 Tg. Perhaps the smplest way to fd determatal wavefuctos for each of these s to proceed by lowerg the symmetry ( ths case, from Oh to D4h) ad explotg the symmetry correlatos that apply to both the orbtals ad the states. Whe Oh symmetry s lowered to D4h symmetry, the correlato of d orbtals goes as llustrated here: The paret (tg) (eg) cofgurato ca yeld four descedet cofguratos D4h: (eg) (ag), (eg) (bg), (bg) (ag), ad (bg) (bg), whch respectvely gve rse to,3 Eg,,3 Eg,,3 g, ad,3 Ag states, as determed by evaluatg drect products usg the D4h orbtals. However, the D4h descedet states must also correlate drectly wth ther Oh paret states,,3 Tg (Oh),3 Eg,,3 Ag (D4h),,3 Tg (Oh),3 Eg,,3 g (D4h). We ca coclude that (bg) (ag) ad (bg) (bg) cofguratos ad ther correspodg determats D4h must respectvely derve from,3 Tg ad,3 Tg Oh. We therefore kow some represetatve wavefuctos for each of these two states, M S M S xy z xy x y 3 T g : xy z + xy z 0 3 T g : xy x y + xy x y 0 xy z xy x y 6

17 M S T g : xy z xy z 0 M S T g : xy x y xy x y 0 Relatve State Eerges Now we wll depart from a purely symmetry-based aalyss ad evaluate the eerges of the trplet states ad several of the sglet states for a octahedral d system. Just as for the qualtatve correlato dagram, the two lmtg cases are thtomc os ad the strog lgad feld lmt. Let s beg wth the strog-feld lmt: = ε t g + J xy,xz K xy,xz = ε t g + (A + ) (3 + ) = ε t g + A 5 = ε t g + ε eg + J xy,z K xy,z = ε t g + ε eg + (A 4 + ) (4 + ) = ε t g + ε eg + A 8 = ε t g + ε eg + J xy,x K y xy,x = ε y t g + ε eg + (A ) () = ε t g + ε eg + A + 4 = ε eg + J z K,x y z = ε,x y eg + (A 4 + ) (4 + ) = ε eg + A 8 = ε t g + J xy,xz + K xy,xz = ε t g + (A + ) + (3 + ) = ε t g + A + + = ε t g + ε eg + J xy,z + K xy,z = ε t g + ε eg + (A 4 + ) + (4 + ) = ε t g + ε eg + A + = ε t g + ε eg + J xy,x + K y xy,x = ε y t g + ε eg + (A ) + () = ε t g + ε eg + A E 3 T g,t g E 3 T g,t g e g E 3 T g,t g e g E 3 A g,e g E T g,t g E T g,t g e g E T g,t g e g I practce, wre terested the relatve eerges of these states, so let s take eergy dffereces to get the excted state eerges relatve to the groud state, E( 3 Tg, tg ): E 3 ( T g,t g ) = ε eg ε t g 3 = Δ o 3 3 E 3 ( T g,t g ) = ε eg ε t g 3 = Δ o 3 3 E 3 T g,t ( g e g ) E 3 ( T g,t g ) = Δ o E ( T g,t g ) E 3 ( T g,t g ) = 6 + o E T g,t ( g e g ) E 3 ( T g,t g ) = Δ o E T g,t ( g e g ) E 3 ( T g,t g ) = Δ o E 3 T g,t g e g E 3 A g,e g At ths pot, we eed to recogze that these expressos arased o thssumpto that each statrses from a sgle (strog feld) cofgurato. As o 0, the eerges of the 3 Tg, 3 Ag, ad the lowest 3 Tg states must be equal scll three states correlate back to the samtomc state ( 3 F). I thbove expressos, however, we ca see that the eergy of the frst two states s 3 relatve to the the lowest 3 Tg stats o 0. The resoluto of ths dffculty les cofgurato teracto (I). Therre two 3 Tg states, for whch we ve wrtte lgad-feld sgle-determat wavefuctos the precedg dscusso, ad they must teract wth each other sce they are of the same symmetry. The wavefuctos wrtte abovre very good approxmatos for large o; the extet of I s small whe the eergy dfferecetwee these lke-symmetry determats from two dfferet cofguratos s large. As o 0, however, these 7

18 wavefuctos mx to yeld the 3 P atomc statd a compoet of the 3 F atomc state. Sce the 3 Tg state s the oly trplet state of that symmetry, t correlates back to the 3 F atomc state wthout ay mxg wth other cofguratos. We therefore choose the eergy of the 3 Tg statt o = 0 as the zero of eergy, so that at o = 0, E( 3 Tg, tg ) = 3 ad E( 3 Tg, tg eg ) =. Now, we kow that f the mxg betwee the two 3 Tg states s accouted for, the lower of the two states must have E = 0 whe o = 0. We ca therefore wrt secular equato that accouts for the I ad t must be of the form, E x x 3 E = 0 E (5)E + 36 x = 0 where x s the matrx elemet due to the teracto of the two 3 Tg states. 4 The lower eergy soluto s E = 0, whch s satsfed f x = 36 ; the hgher eergy root s therefore E = 5. The teracto betwee the two states s due to electro-electro repulso ad s therefore depedet of the lgad-feld splttg,.e., eve for ozero o, the offdagoal etry the secular equato s 6: E ± ( 3 T g ) = o + E E = 0 o + 5 ± Sce the lower of the two 3 Tg states s the groud statd we wsh to express all the other states eerges relatve to the groud state, we must subtract E ( 3 Tg) from each. I partcular, for the trplet states we obta E 3 T g,t g e g E + ( 3 T g ) E ( 3 T g ) E ( 3 T g ) E 3 ( A g,e g ) E ( 3 T g ) = = o The Tg sglet determats also mx va I, ad we kow that as o 0, the eerges of the hgher Tg statd the Tg statoth correlate to the G atomc state wth eergy εd + A We ca therefore wrt secular equato (applcabls o 0) that accouts for the I, o o 5 + = 3 o o ( o ) + 5 o o + 8 o o + 5 A symmetry argumet s used here to deduce the off-dagoal matrx elemet. A drect calculato s gve 4 allhause,. J. Molecular Electroc Structures of Trasto Metal omplexes, McGraw-Hll: New York, 980, p

19 E x x + E = 0 x = f E + = 4 +. As before, x = for all values of o. The geeral I secular equato ad s roots: E ± E ± ( + ) E ( o + ) E = 0 = + o E ( 3 T g ) = 7 + E T g,t g e g o + ± The reader may ow compare these results wth those publshed by Taabd Sugao the epoymously-amed dagram for a d o: o o o + 5 ± E ( 3 T g ) = o + o o o o + 5 Left: a reproducto of the Taabe-Sugao dagram for a d o; Rght: Plots of the state eerges for selected states of a d o as derved ths documet for the parameter choce =

20 Appedx : Relatos Ivolvg oulomb ad Exchage Itegrals p orbtals deftos J 0,0 = J z,z = J x,x = J y, y J, = J, = J, = ( )(J x,x + J x, y ) J J,0 = J,0 = J x, y = J x,z = J y,z, j = ϕ * ()ϕ * j ()( r )ϕ ()ϕ j ()dτ dτ ; K K, = K, = K x, y = J x,x J, j = ϕ * ()ϕ * j ()( r )ϕ ()ϕ j ()dτ dτ x, y K,0 = K,0 = K x, y = K x,z = K y,z d orbtals J 0,0 = J z,z J, = J, = J, = ( )(J xy,xy + J x y ),xy J, = J, = J, = J, = J xy,xz J,0 = J,0 = J xy,z J, = J, = J, = ( )(J xz,xz + J xz, yz ) J,0 = J,0 = J xz,z K, = K xz, yz = J xz,xz J xz, yz K, = K xy,x y = J xy,xy J xy,x y K, = K, = K xy,xz ϕ xz ()ϕ xy ()( r )ϕ yz ()ϕ x y ()dτ dτ K, = K, = K xy,xz + ϕ xz ()ϕ xy ()( r )ϕ yz ()ϕ x y ()dτ dτ K,0 = K,0 = K xy,z K,0 = K,0 = K xz,z Eerges of real d orbtal tegrals terms of Racah parameters J xy,xy = J xz,xz = J yz, yz = J z,z = J x y,x y A J xz, yz = J xy, yz = J xy,xz = J x y = J, yz x y,xz A + J xy,z = J x y,z A 4 + J yz,z = J xz,z A + + J x y,xy A K xy, yz = K xz, yz = K xy,xz = K x y = K, yz x y,xz 3 + K xy,z = K x y,z 4 + K yz,z = K xz,z + K x y,xy ϕ xz ()ϕ xy ()( r )ϕ yz ()ϕ x y ()dτ dτ 3 0

ANOVA Notes Page 1. Analysis of Variance for a One-Way Classification of Data

ANOVA Notes Page 1. Analysis of Variance for a One-Way Classification of Data ANOVA Notes Page Aalss of Varace for a Oe-Wa Classfcato of Data Cosder a sgle factor or treatmet doe at levels (e, there are,, 3, dfferet varatos o the prescrbed treatmet) Wth a gve treatmet level there

More information

APPENDIX III THE ENVELOPE PROPERTY

APPENDIX III THE ENVELOPE PROPERTY Apped III APPENDIX III THE ENVELOPE PROPERTY Optmzato mposes a very strog structure o the problem cosdered Ths s the reaso why eoclasscal ecoomcs whch assumes optmzg behavour has bee the most successful

More information

6.7 Network analysis. 6.7.1 Introduction. References - Network analysis. Topological analysis

6.7 Network analysis. 6.7.1 Introduction. References - Network analysis. Topological analysis 6.7 Network aalyss Le data that explctly store topologcal formato are called etwork data. Besdes spatal operatos, several methods of spatal aalyss are applcable to etwork data. Fgure: Network data Refereces

More information

1. The Time Value of Money

1. The Time Value of Money Corporate Face [00-0345]. The Tme Value of Moey. Compoudg ad Dscoutg Captalzato (compoudg, fdg future values) s a process of movg a value forward tme. It yelds the future value gve the relevat compoudg

More information

IDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki

IDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki IDENIFICAION OF HE DYNAMICS OF HE GOOGLE S RANKING ALGORIHM A. Khak Sedgh, Mehd Roudak Cotrol Dvso, Departmet of Electrcal Egeerg, K.N.oos Uversty of echology P. O. Box: 16315-1355, ehra, Ira sedgh@eetd.ktu.ac.r,

More information

Chapter Eight. f : R R

Chapter Eight. f : R R Chapter Eght f : R R 8. Itroducto We shall ow tur our atteto to the very mportat specal case of fuctos that are real, or scalar, valued. These are sometmes called scalar felds. I the very, but mportat,

More information

Measures of Dispersion, Skew, & Kurtosis (based on Kirk, Ch. 4) {to be used in conjunction with Measures of Dispersion Chart }

Measures of Dispersion, Skew, & Kurtosis (based on Kirk, Ch. 4) {to be used in conjunction with Measures of Dispersion Chart } Percetles Psych 54, 9/8/05 p. /6 Measures of Dsperso, kew, & Kurtoss (based o Krk, Ch. 4) {to be used cojucto wth Measures of Dsperso Chart } percetle (P % ): a score below whch a specfed percetage of

More information

Numerical Methods with MS Excel

Numerical Methods with MS Excel TMME, vol4, o.1, p.84 Numercal Methods wth MS Excel M. El-Gebely & B. Yushau 1 Departmet of Mathematcal Sceces Kg Fahd Uversty of Petroleum & Merals. Dhahra, Saud Araba. Abstract: I ths ote we show how

More information

T = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are :

T = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are : Bullets bods Let s descrbe frst a fxed rate bod wthout amortzg a more geeral way : Let s ote : C the aual fxed rate t s a percetage N the otoal freq ( 2 4 ) the umber of coupo per year R the redempto of

More information

Online Appendix: Measured Aggregate Gains from International Trade

Online Appendix: Measured Aggregate Gains from International Trade Ole Appedx: Measured Aggregate Gas from Iteratoal Trade Arel Burste UCLA ad NBER Javer Cravo Uversty of Mchga March 3, 2014 I ths ole appedx we derve addtoal results dscussed the paper. I the frst secto,

More information

Chapter 3 3-1. Chapter Goals. Summary Measures. Chapter Topics. Measures of Center and Location. Notation Conventions

Chapter 3 3-1. Chapter Goals. Summary Measures. Chapter Topics. Measures of Center and Location. Notation Conventions Chapter 3 3- Chapter Goals Chapter 3 umercal Descrptve Measures After completg ths chapter, you should be able to: Compute ad terpret the mea, meda, ad mode for a set of data Fd the rage, varace, ad stadard

More information

Statistical Pattern Recognition (CE-725) Department of Computer Engineering Sharif University of Technology

Statistical Pattern Recognition (CE-725) Department of Computer Engineering Sharif University of Technology I The Name of God, The Compassoate, The ercful Name: Problems' eys Studet ID#:. Statstcal Patter Recogto (CE-725) Departmet of Computer Egeerg Sharf Uversty of Techology Fal Exam Soluto - Sprg 202 (50

More information

Average Price Ratios

Average Price Ratios Average Prce Ratos Morgstar Methodology Paper August 3, 2005 2005 Morgstar, Ic. All rghts reserved. The formato ths documet s the property of Morgstar, Ic. Reproducto or trascrpto by ay meas, whole or

More information

CHAPTER 2. Time Value of Money 6-1

CHAPTER 2. Time Value of Money 6-1 CHAPTER 2 Tme Value of Moey 6- Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 6-2 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show

More information

Preprocess a planar map S. Given a query point p, report the face of S containing p. Goal: O(n)-size data structure that enables O(log n) query time.

Preprocess a planar map S. Given a query point p, report the face of S containing p. Goal: O(n)-size data structure that enables O(log n) query time. Computatoal Geometry Chapter 6 Pot Locato 1 Problem Defto Preprocess a plaar map S. Gve a query pot p, report the face of S cotag p. S Goal: O()-sze data structure that eables O(log ) query tme. C p E

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL 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 information

The analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0

The analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0 Chapter 2 Autes ad loas A auty s a sequece of paymets wth fxed frequecy. The term auty orgally referred to aual paymets (hece the ame), but t s ow also used for paymets wth ay frequecy. Autes appear may

More information

Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract

Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected

More information

MEASURES OF CENTRAL TENDENCY

MEASURES OF CENTRAL TENDENCY MODULE - 6 Statstcs Measures of Cetral Tedecy 25 MEASURES OF CENTRAL TENDENCY I the prevous lesso, we have leart that the data could be summarsed to some extet by presetg t the form of a frequecy table.

More information

Simple Linear Regression

Simple Linear Regression Smple Lear Regresso Regresso equato a equato that descrbes the average relatoshp betwee a respose (depedet) ad a eplaator (depedet) varable. 6 8 Slope-tercept equato for a le m b (,6) slope. (,) 6 6 8

More information

Chapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization

Chapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization Chapter 3 Mathematcs of Face Secto 4 Preset Value of a Auty; Amortzato Preset Value of a Auty I ths secto, we wll address the problem of determg the amout that should be deposted to a accout ow at a gve

More information

Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R =

Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R = Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS Objectves of the Topc: Beg able to formalse ad solve practcal ad mathematcal problems, whch the subjects of loa amortsato ad maagemet of cumulatve fuds are

More information

Classic Problems at a Glance using the TVM Solver

Classic Problems at a Glance using the TVM Solver C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the

More information

ECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil

ECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil ECONOMIC CHOICE OF OPTIMUM FEEDER CABE CONSIDERING RISK ANAYSIS I Camargo, F Fgueredo, M De Olvera Uversty of Brasla (UB) ad The Brazla Regulatory Agecy (ANEE), Brazl The choce of the approprate cable

More information

CH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID

CH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta CENTE OF GITY ND CENTOID 5. CENTE OF GITY ND CENTE OF MSS FO SYSTEM OF PTICES Ceter of Gravt. The ceter of gravt G s a pot whch locates the resultat

More information

The Digital Signature Scheme MQQ-SIG

The Digital Signature Scheme MQQ-SIG The Dgtal Sgature Scheme MQQ-SIG Itellectual Property Statemet ad Techcal Descrpto Frst publshed: 10 October 2010, Last update: 20 December 2010 Dalo Glgorosk 1 ad Rue Stesmo Ødegård 2 ad Rue Erled Jese

More information

Curve Fitting and Solution of Equation

Curve Fitting and Solution of Equation UNIT V Curve Fttg ad Soluto of Equato 5. CURVE FITTING I ma braches of appled mathematcs ad egeerg sceces we come across epermets ad problems, whch volve two varables. For eample, t s kow that the speed

More information

Common p-belief: The General Case

Common p-belief: The General Case GAMES AND ECONOMIC BEHAVIOR 8, 738 997 ARTICLE NO. GA97053 Commo p-belef: The Geeral Case Atsush Kaj* ad Stephe Morrs Departmet of Ecoomcs, Uersty of Pesylaa Receved February, 995 We develop belef operators

More information

of the relationship between time and the value of money.

of the relationship between time and the value of money. TIME AND THE VALUE OF MONEY Most agrbusess maagers are famlar wth the terms compoudg, dscoutg, auty, ad captalzato. That s, most agrbusess maagers have a tutve uderstadg that each term mples some relatoshp

More information

The simple linear Regression Model

The simple linear Regression Model The smple lear Regresso Model Correlato coeffcet s o-parametrc ad just dcates that two varables are assocated wth oe aother, but t does ot gve a deas of the kd of relatoshp. Regresso models help vestgatg

More information

Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity

Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity Computer Aded Geometrc Desg 19 (2002 365 377 wwwelsevercom/locate/comad Optmal mult-degree reducto of Bézer curves wth costrats of edpots cotuty Guo-Dog Che, Guo-J Wag State Key Laboratory of CAD&CG, Isttute

More information

Banking (Early Repayment of Housing Loans) Order, 5762 2002 1

Banking (Early Repayment of Housing Loans) Order, 5762 2002 1 akg (Early Repaymet of Housg Loas) Order, 5762 2002 y vrtue of the power vested me uder Secto 3 of the akg Ordace 94 (hereafter, the Ordace ), followg cosultato wth the Commttee, ad wth the approval of

More information

n. We know that the sum of squares of p independent standard normal variables has a chi square distribution with p degrees of freedom.

n. We know that the sum of squares of p independent standard normal variables has a chi square distribution with p degrees of freedom. UMEÅ UNIVERSITET Matematsk-statstska sttutoe Multvarat dataaalys för tekologer MSTB0 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multvarat dataaalys för tekologer B, 5 poäg.

More information

An Effectiveness of Integrated Portfolio in Bancassurance

An Effectiveness of Integrated Portfolio in Bancassurance A Effectveess of Itegrated Portfolo Bacassurace Taea Karya Research Ceter for Facal Egeerg Isttute of Ecoomc Research Kyoto versty Sayouu Kyoto 606-850 Japa arya@eryoto-uacp Itroducto As s well ow the

More information

ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN

ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN Colloquum Bometrcum 4 ADAPTATION OF SHAPIRO-WILK TEST TO THE CASE OF KNOWN MEAN Zofa Hausz, Joaa Tarasńska Departmet of Appled Mathematcs ad Computer Scece Uversty of Lfe Sceces Lubl Akademcka 3, -95 Lubl

More information

RUSSIAN ROULETTE AND PARTICLE SPLITTING

RUSSIAN ROULETTE AND PARTICLE SPLITTING RUSSAN ROULETTE AND PARTCLE SPLTTNG M. Ragheb 3/7/203 NTRODUCTON To stuatos are ecoutered partcle trasport smulatos:. a multplyg medum, a partcle such as a eutro a cosmc ray partcle or a photo may geerate

More information

The Gompertz-Makeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev

The Gompertz-Makeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev The Gompertz-Makeham dstrbuto by Fredrk Norström Master s thess Mathematcal Statstcs, Umeå Uversty, 997 Supervsor: Yur Belyaev Abstract Ths work s about the Gompertz-Makeham dstrbuto. The dstrbuto has

More information

Load and Resistance Factor Design (LRFD)

Load and Resistance Factor Design (LRFD) 53:134 Structural Desg II Load ad Resstace Factor Desg (LRFD) Specfcatos ad Buldg Codes: Structural steel desg of buldgs the US s prcpally based o the specfcatos of the Amerca Isttute of Steel Costructo

More information

Reinsurance and the distribution of term insurance claims

Reinsurance and the distribution of term insurance claims Resurace ad the dstrbuto of term surace clams By Rchard Bruyel FIAA, FNZSA Preseted to the NZ Socety of Actuares Coferece Queestow - November 006 1 1 Itroducto Ths paper vestgates the effect of resurace

More information

Sequences and Series

Sequences and Series Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.

More information

10.5 Future Value and Present Value of a General Annuity Due

10.5 Future Value and Present Value of a General Annuity Due Chapter 10 Autes 371 5. Thomas leases a car worth $4,000 at.99% compouded mothly. He agrees to make 36 lease paymets of $330 each at the begg of every moth. What s the buyout prce (resdual value of the

More information

DIRAC s BRA AND KET NOTATION. 1 From inner products to bra-kets 1

DIRAC s BRA AND KET NOTATION. 1 From inner products to bra-kets 1 DIRAC s BRA AND KET NOTATION B. Zwebach October 7, 2013 Cotets 1 From er products to bra-kets 1 2 Operators revsted 5 2.1 Projecto Operators..................................... 6 2.2 Adjot of a lear operator.................................

More information

Overview. Eingebettete Systeme. Model of periodic tasks. Model of periodic tasks. Echtzeitverhalten und Betriebssysteme

Overview. Eingebettete Systeme. Model of periodic tasks. Model of periodic tasks. Echtzeitverhalten und Betriebssysteme Overvew Egebettete Systeme able of some kow preemptve schedulg algorthms for perodc tasks: Echtzetverhalte ud Betrebssysteme 5. Perodsche asks statc prorty dyamc prorty Deadle equals perod Deadle smaller

More information

Questions? Ask Prof. Herz, herz@ucsd.edu. General Classification of adsorption

Questions? Ask Prof. Herz, herz@ucsd.edu. General Classification of adsorption Questos? Ask rof. Herz, herz@ucsd.edu Geeral Classfcato of adsorpto hyscal adsorpto - physsorpto - dsperso forces - Va der Waals forces - weak - oly get hgh fractoal coerage of surface at low temperatures

More information

ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n -SPACE. Yusuf YAYLI 1, Evren ZIPLAR 2. yayli@science.ankara.edu.tr. evrenziplar@yahoo.

ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n -SPACE. Yusuf YAYLI 1, Evren ZIPLAR 2. yayli@science.ankara.edu.tr. evrenziplar@yahoo. ON SLANT HELICES AND ENERAL HELICES IN EUCLIDEAN -SPACE Yusuf YAYLI Evre ZIPLAR Departmet of Mathematcs Faculty of Scece Uversty of Akara Tadoğa Akara Turkey yayl@sceceakaraedutr Departmet of Mathematcs

More information

Relaxation Methods for Iterative Solution to Linear Systems of Equations

Relaxation Methods for Iterative Solution to Linear Systems of Equations Relaxato Methods for Iteratve Soluto to Lear Systems of Equatos Gerald Recktewald Portlad State Uversty Mechacal Egeerg Departmet gerry@me.pdx.edu Prmary Topcs Basc Cocepts Statoary Methods a.k.a. Relaxato

More information

SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN

SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN Wojcech Zelńsk Departmet of Ecoometrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowska 66, -787 Warszawa e-mal: wojtekzelsk@statystykafo Zofa Hausz,

More information

Lecture 7. Norms and Condition Numbers

Lecture 7. Norms and Condition Numbers Lecture 7 Norms ad Codto Numbers To dscuss the errors umerca probems vovg vectors, t s usefu to empo orms. Vector Norm O a vector space V, a orm s a fucto from V to the set of o-egatve reas that obes three

More information

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting MANAGEMENT SCIENCE Vol. 52, No., Jauary 26, pp. 95 ss 25-99 ess 526-55 6 52 95 forms do.287/msc.5.447 26 INFORMS Prcg Amerca-Style Dervatves wth Europea Call Optos Scott B. Laprse BAE Systems, Advaced

More information

STOCHASTIC approximation algorithms have several

STOCHASTIC approximation algorithms have several IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 10, OCTOBER 2014 6609 Trackg a Markov-Modulated Statoary Degree Dstrbuto of a Dyamc Radom Graph Mazyar Hamd, Vkram Krshamurthy, Fellow, IEEE, ad George

More information

Automated Event Registration System in Corporation

Automated Event Registration System in Corporation teratoal Joural of Advaces Computer Scece ad Techology JACST), Vol., No., Pages : 0-0 0) Specal ssue of CACST 0 - Held durg 09-0 May, 0 Malaysa Automated Evet Regstrato System Corporato Zafer Al-Makhadmee

More information

On Savings Accounts in Semimartingale Term Structure Models

On Savings Accounts in Semimartingale Term Structure Models O Savgs Accouts Semmartgale Term Structure Models Frak Döberle Mart Schwezer moeyshelf.com Techsche Uverstät Berl Bockehemer Ladstraße 55 Fachberech Mathematk, MA 7 4 D 6325 Frakfurt am Ma Straße des 17.

More information

Models for Selecting an ERP System with Intuitionistic Trapezoidal Fuzzy Information

Models for Selecting an ERP System with Intuitionistic Trapezoidal Fuzzy Information JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH 00 75 Models for Selectg a ERP System wth Itutostc rapezodal Fuzzy Iformato Guwu We, Ru L Departmet of Ecoomcs ad Maagemet, Chogqg Uversty of Arts ad Sceces, Yogchua,

More information

Compressive Sensing over Strongly Connected Digraph and Its Application in Traffic Monitoring

Compressive Sensing over Strongly Connected Digraph and Its Application in Traffic Monitoring Compressve Sesg over Strogly Coected Dgraph ad Its Applcato Traffc Motorg Xao Q, Yogca Wag, Yuexua Wag, Lwe Xu Isttute for Iterdscplary Iformato Sceces, Tsghua Uversty, Bejg, Cha {qxao3, kyo.c}@gmal.com,

More information

Future Value of an Annuity

Future Value of an Annuity Future Value of a Auty After payg all your blls, you have $200 left each payday (at the ed of each moth) that you wll put to savgs order to save up a dow paymet for a house. If you vest ths moey at 5%

More information

Settlement Prediction by Spatial-temporal Random Process

Settlement Prediction by Spatial-temporal Random Process Safety, Relablty ad Rs of Structures, Ifrastructures ad Egeerg Systems Furuta, Fragopol & Shozua (eds Taylor & Fracs Group, Lodo, ISBN 978---77- Settlemet Predcto by Spatal-temporal Radom Process P. Rugbaapha

More information

Credibility Premium Calculation in Motor Third-Party Liability Insurance

Credibility Premium Calculation in Motor Third-Party Liability Insurance Advaces Mathematcal ad Computatoal Methods Credblty remum Calculato Motor Thrd-arty Lablty Isurace BOHA LIA, JAA KUBAOVÁ epartmet of Mathematcs ad Quattatve Methods Uversty of ardubce Studetská 95, 53

More information

Hypothesis Testing on the Parameters of Exponential, Pareto and Uniform Distributions Based on Extreme Ranked Set Sampling

Hypothesis Testing on the Parameters of Exponential, Pareto and Uniform Distributions Based on Extreme Ranked Set Sampling Mddle-East Joural of Scetfc Research (9): 39-36, ISSN 99-933 IDOSI Publcatos, DOI:.589/dos.mejsr...9.87 Hypothess Testg o the Parameters of Expoetal, Pareto ad Uform Dstrbutos Based o Extreme Raed Set

More information

Performance Attribution. Methodology Overview

Performance Attribution. Methodology Overview erformace Attrbuto Methodology Overvew Faba SUAREZ March 2004 erformace Attrbuto Methodology 1.1 Itroducto erformace Attrbuto s a set of techques that performace aalysts use to expla why a portfolo's performace

More information

On Error Detection with Block Codes

On Error Detection with Block Codes BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 9, No 3 Sofa 2009 O Error Detecto wth Block Codes Rostza Doduekova Chalmers Uversty of Techology ad the Uversty of Gotheburg,

More information

MDM 4U PRACTICE EXAMINATION

MDM 4U PRACTICE EXAMINATION MDM 4U RCTICE EXMINTION Ths s a ractce eam. It does ot cover all the materal ths course ad should ot be the oly revew that you do rearato for your fal eam. Your eam may cota questos that do ot aear o ths

More information

Bayesian Network Representation

Bayesian Network Representation Readgs: K&F 3., 3.2, 3.3, 3.4. Bayesa Network Represetato Lecture 2 Mar 30, 20 CSE 55, Statstcal Methods, Sprg 20 Istructor: Su-I Lee Uversty of Washgto, Seattle Last tme & today Last tme Probablty theory

More information

Methods and Data Analysis

Methods and Data Analysis Fudametal Numercal Methods ad Data Aalyss by George W. Colls, II George W. Colls, II Table of Cotets Lst of Fgures...v Lst of Tables... Preface... Notes to the Iteret Edto...v. Itroducto ad Fudametal Cocepts....

More information

A Study of Unrelated Parallel-Machine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time

A Study of Unrelated Parallel-Machine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time Joural of Na Ka, Vol. 0, No., pp.5-9 (20) 5 A Study of Urelated Parallel-Mache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te Suh-Jeq Yag, Ja-Yuar Guo, Hs-Tao Lee Departet of Idustral

More information

Security Analysis of RAPP: An RFID Authentication Protocol based on Permutation

Security Analysis of RAPP: An RFID Authentication Protocol based on Permutation Securty Aalyss of RAPP: A RFID Authetcato Protocol based o Permutato Wag Shao-hu,,, Ha Zhje,, Lu Sujua,, Che Da-we, {College of Computer, Najg Uversty of Posts ad Telecommucatos, Najg 004, Cha Jagsu Hgh

More information

Speeding up k-means Clustering by Bootstrap Averaging

Speeding up k-means Clustering by Bootstrap Averaging Speedg up -meas Clusterg by Bootstrap Averagg Ia Davdso ad Ashw Satyaarayaa Computer Scece Dept, SUNY Albay, NY, USA,. {davdso, ashw}@cs.albay.edu Abstract K-meas clusterg s oe of the most popular clusterg

More information

Impact of Interference on the GPRS Multislot Link Level Performance

Impact of Interference on the GPRS Multislot Link Level Performance Impact of Iterferece o the GPRS Multslot Lk Level Performace Javer Gozalvez ad Joh Dulop Uversty of Strathclyde - Departmet of Electroc ad Electrcal Egeerg - George St - Glasgow G-XW- Scotlad Ph.: + 8

More information

Probability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES

Probability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES CHAPTR Probablt, Statstcs, ad Relablt or geers ad Scetsts MULTIPL RANDOM VARIABLS Secod dto A. J. Clark School o geerg Departmet o Cvl ad vrometal geerg 6b Probablt ad Statstcs or Cvl geers Departmet o

More information

Proceedings of the 2010 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds.

Proceedings of the 2010 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds. Proceedgs of the 21 Wter Smulato Coferece B. Johasso, S. Ja, J. Motoya-Torres, J. Huga, ad E. Yücesa, eds. EMPIRICAL METHODS OR TWO-ECHELON INVENTORY MANAGEMENT WITH SERVICE LEVEL CONSTRAINTS BASED ON

More information

The Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk

The Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk The Aalyss of Developmet of Isurace Cotract Premums of Geeral Lablty Isurace the Busess Isurace Rsk the Frame of the Czech Isurace Market 1998 011 Scetfc Coferece Jue, 10. - 14. 013 Pavla Kubová Departmet

More information

Fundamentals of Mass Transfer

Fundamentals of Mass Transfer Chapter Fudametals of Mass Trasfer Whe a sgle phase system cotas two or more speces whose cocetratos are ot uform, mass s trasferred to mmze the cocetrato dffereces wth the system. I a mult-phase system

More information

Fractal-Structured Karatsuba`s Algorithm for Binary Field Multiplication: FK

Fractal-Structured Karatsuba`s Algorithm for Binary Field Multiplication: FK Fractal-Structured Karatsuba`s Algorthm for Bary Feld Multplcato: FK *The authors are worg at the Isttute of Mathematcs The Academy of Sceces of DPR Korea. **Address : U Jog dstrct Kwahadog Number Pyogyag

More information

Constrained Cubic Spline Interpolation for Chemical Engineering Applications

Constrained Cubic Spline Interpolation for Chemical Engineering Applications Costraed Cubc Sple Iterpolato or Chemcal Egeerg Applcatos b CJC Kruger Summar Cubc sple terpolato s a useul techque to terpolate betwee kow data pots due to ts stable ad smooth characterstcs. Uortuatel

More information

Session 4: Descriptive statistics and exporting Stata results

Session 4: Descriptive statistics and exporting Stata results Itrduct t Stata Jrd Muñz (UAB) Sess 4: Descrptve statstcs ad exprtg Stata results I ths sess we are gg t wrk wth descrptve statstcs Stata. Frst, we preset a shrt trduct t the very basc statstcal ctets

More information

Generalizations of Pauli channels

Generalizations of Pauli channels Acta Math. Hugar. 24(2009, 65 77. Geeralzatos of Paul chaels Dées Petz ad Hromch Oho 2 Alfréd Réy Isttute of Mathematcs, H-364 Budapest, POB 27, Hugary 2 Graduate School of Mathematcs, Kyushu Uversty,

More information

Using Phase Swapping to Solve Load Phase Balancing by ADSCHNN in LV Distribution Network

Using Phase Swapping to Solve Load Phase Balancing by ADSCHNN in LV Distribution Network Iteratoal Joural of Cotrol ad Automato Vol.7, No.7 (204), pp.-4 http://dx.do.org/0.4257/jca.204.7.7.0 Usg Phase Swappg to Solve Load Phase Balacg by ADSCHNN LV Dstrbuto Network Chu-guo Fe ad Ru Wag College

More information

Report 52 Fixed Maturity EUR Industrial Bond Funds

Report 52 Fixed Maturity EUR Industrial Bond Funds Rep52, Computed & Prted: 17/06/2015 11:53 Report 52 Fxed Maturty EUR Idustral Bod Fuds From Dec 2008 to Dec 2014 31/12/2008 31 December 1999 31/12/2014 Bechmark Noe Defto of the frm ad geeral formato:

More information

arxiv:math/0510414v1 [math.pr] 19 Oct 2005

arxiv:math/0510414v1 [math.pr] 19 Oct 2005 A MODEL FOR THE BUS SYSTEM IN CUERNEVACA MEXICO) JINHO BAIK ALEXEI BORODIN PERCY DEIFT AND TOUFIC SUIDAN arxv:math/05044v [mathpr 9 Oct 2005 Itroducto The bus trasportato system Cuerevaca Mexco has certa

More information

The Time Value of Money

The Time Value of Money The Tme Value of Moey 1 Iversemet Optos Year: 1624 Property Traded: Mahatta Islad Prce : $24.00, FV of $24 @ 6%: FV = $24 (1+0.06) 388 = $158.08 bllo Opto 1 0 1 2 3 4 5 t ($519.37) 0 0 0 0 $1,000 Opto

More information

Conversion of Non-Linear Strength Envelopes into Generalized Hoek-Brown Envelopes

Conversion of Non-Linear Strength Envelopes into Generalized Hoek-Brown Envelopes Covero of No-Lear Stregth Evelope to Geeralzed Hoek-Brow Evelope Itroducto The power curve crtero commoly ued lmt-equlbrum lope tablty aaly to defe a o-lear tregth evelope (relatohp betwee hear tre, τ,

More information

Maintenance Scheduling of Distribution System with Optimal Economy and Reliability

Maintenance Scheduling of Distribution System with Optimal Economy and Reliability Egeerg, 203, 5, 4-8 http://dx.do.org/0.4236/eg.203.59b003 Publshed Ole September 203 (http://www.scrp.org/joural/eg) Mateace Schedulg of Dstrbuto System wth Optmal Ecoomy ad Relablty Syua Hog, Hafeg L,

More information

Projection model for Computer Network Security Evaluation with interval-valued intuitionistic fuzzy information. Qingxiang Li

Projection model for Computer Network Security Evaluation with interval-valued intuitionistic fuzzy information. Qingxiang Li Iteratoal Joural of Scece Vol No7 05 ISSN: 83-4890 Proecto model for Computer Network Securty Evaluato wth terval-valued tutostc fuzzy formato Qgxag L School of Software Egeerg Chogqg Uversty of rts ad

More information

A New Bayesian Network Method for Computing Bottom Event's Structural Importance Degree using Jointree

A New Bayesian Network Method for Computing Bottom Event's Structural Importance Degree using Jointree , pp.277-288 http://dx.do.org/10.14257/juesst.2015.8.1.25 A New Bayesa Network Method for Computg Bottom Evet's Structural Importace Degree usg Jotree Wag Yao ad Su Q School of Aeroautcs, Northwester Polytechcal

More information

Fast, Secure Encryption for Indexing in a Column-Oriented DBMS

Fast, Secure Encryption for Indexing in a Column-Oriented DBMS Fast, Secure Ecrypto for Idexg a Colum-Oreted DBMS Tgja Ge, Sta Zdok Brow Uversty {tge, sbz}@cs.brow.edu Abstract Networked formato systems requre strog securty guaratees because of the ew threats that

More information

FINANCIAL MATHEMATICS 12 MARCH 2014

FINANCIAL MATHEMATICS 12 MARCH 2014 FINNCIL MTHEMTICS 12 MRCH 2014 I ths lesso we: Lesso Descrpto Make use of logarthms to calculate the value of, the tme perod, the equato P1 or P1. Solve problems volvg preset value ad future value autes.

More information

On Cheeger-type inequalities for weighted graphs

On Cheeger-type inequalities for weighted graphs O Cheeger-type equaltes for weghted graphs Shmuel Fredlad Uversty of Illos at Chcago Departmet of Mathematcs 851 S. Morga St., Chcago, Illos 60607-7045 USA Rehard Nabbe Fakultät für Mathematk Uverstät

More information

Aggregation Functions and Personal Utility Functions in General Insurance

Aggregation Functions and Personal Utility Functions in General Insurance Acta Polytechca Huarca Vol. 7, No. 4, 00 Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace Jaa Šprková Departmet of Quattatve Methods ad Iformato Systems, Faculty of Ecoomcs, Matej Bel Uversty Tajovského

More information

Plastic Number: Construction and Applications

Plastic Number: Construction and Applications Scet f c 0 Advaced Advaced Scetfc 0 December,.. 0 Plastc Number: Costructo ad Applcatos Lua Marohć Polytechc of Zagreb, 0000 Zagreb, Croata lua.marohc@tvz.hr Thaa Strmeč Polytechc of Zagreb, 0000 Zagreb,

More information

Network dimensioning for elastic traffic based on flow-level QoS

Network dimensioning for elastic traffic based on flow-level QoS Network dmesog for elastc traffc based o flow-level QoS 1(10) Network dmesog for elastc traffc based o flow-level QoS Pas Lassla ad Jorma Vrtamo Networkg Laboratory Helsk Uversty of Techology Itroducto

More information

ANALYTICAL MODEL FOR TCP FILE TRANSFERS OVER UMTS. Janne Peisa Ericsson Research 02420 Jorvas, Finland. Michael Meyer Ericsson Research, Germany

ANALYTICAL MODEL FOR TCP FILE TRANSFERS OVER UMTS. Janne Peisa Ericsson Research 02420 Jorvas, Finland. Michael Meyer Ericsson Research, Germany ANALYTICAL MODEL FOR TCP FILE TRANSFERS OVER UMTS Jae Pesa Erco Research 4 Jorvas, Flad Mchael Meyer Erco Research, Germay Abstract Ths paper proposes a farly complex model to aalyze the performace of

More information

CSSE463: Image Recognition Day 27

CSSE463: Image Recognition Day 27 CSSE463: Image Recogto Da 27 Ths week Toda: Alcatos of PCA Suda ght: roject las ad relm work due Questos? Prcal Comoets Aalss weght grth c ( )( ) ( )( ( )( ) ) heght sze Gve a set of samles, fd the drecto(s)

More information

RQM: A new rate-based active queue management algorithm

RQM: A new rate-based active queue management algorithm : A ew rate-based actve queue maagemet algorthm Jeff Edmods, Suprakash Datta, Patrck Dymod, Kashf Al Computer Scece ad Egeerg Departmet, York Uversty, Toroto, Caada Abstract I ths paper, we propose a ew

More information

Statistical Decision Theory: Concepts, Methods and Applications. (Special topics in Probabilistic Graphical Models)

Statistical Decision Theory: Concepts, Methods and Applications. (Special topics in Probabilistic Graphical Models) Statstcal Decso Theory: Cocepts, Methods ad Applcatos (Specal topcs Probablstc Graphcal Models) FIRST COMPLETE DRAFT November 30, 003 Supervsor: Professor J. Rosethal STA4000Y Aal Mazumder 9506380 Part

More information

R. Zvan. P.A. Forsyth. paforsyth@yoho.uwaterloo.ca. K. Vetzal. kvetzal@watarts.uwaterloo.ca. University ofwaterloo. Waterloo, ON

R. Zvan. P.A. Forsyth. paforsyth@yoho.uwaterloo.ca. K. Vetzal. kvetzal@watarts.uwaterloo.ca. University ofwaterloo. Waterloo, ON Robust Numercal Methods for PDE Models of sa Optos by R. Zva Departmet of Computer Scece Tel: (59 888-4567 ext. 6 Fax: (59 885-8 rzvayoho.uwaterloo.ca P.. Forsyth Departmet of Computer Scece Tel: (59 888-4567

More information

Taylor & Francis, Ltd. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Experimental Education.

Taylor & Francis, Ltd. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Experimental Education. The Statstcal Iterpretato of Degrees of Freedom Author(s): Wllam J. Mooa Source: The Joural of Expermetal Educato, Vol. 21, No. 3 (Mar., 1953), pp. 259264 Publshed by: Taylor & Fracs, Ltd. Stable URL:

More information

SOLID STATE PHYSICS. Crystal structure. (d) (e) (f)

SOLID STATE PHYSICS. Crystal structure. (d) (e) (f) SOLID STAT PHYSICS y defto, sold state s that partcular aggregato form of matter characterzed by strog teracto forces betwee costtuet partcles (atoms, os, or molecules. As a result, a sold state materal

More information

Capacitated Production Planning and Inventory Control when Demand is Unpredictable for Most Items: The No B/C Strategy

Capacitated Production Planning and Inventory Control when Demand is Unpredictable for Most Items: The No B/C Strategy SCHOOL OF OPERATIONS RESEARCH AND INDUSTRIAL ENGINEERING COLLEGE OF ENGINEERING CORNELL UNIVERSITY ITHACA, NY 4853-380 TECHNICAL REPORT Jue 200 Capactated Producto Plag ad Ivetory Cotrol whe Demad s Upredctable

More information

THE McELIECE CRYPTOSYSTEM WITH ARRAY CODES. MATRİS KODLAR İLE McELIECE ŞİFRELEME SİSTEMİ

THE McELIECE CRYPTOSYSTEM WITH ARRAY CODES. MATRİS KODLAR İLE McELIECE ŞİFRELEME SİSTEMİ SAÜ e Blmler Dergs, 5 Clt, 2 Sayı, THE McELIECE CRYPTOSYSTEM WITH ARRAY CODES Vedat ŞİAP* *Departmet of Mathematcs, aculty of Scece ad Art, Sakarya Uversty, 5487, Serdva, Sakarya-TURKEY vedatsap@gmalcom

More information

Beta. A Statistical Analysis of a Stock s Volatility. Courtney Wahlstrom. Iowa State University, Master of School Mathematics. Creative Component

Beta. A Statistical Analysis of a Stock s Volatility. Courtney Wahlstrom. Iowa State University, Master of School Mathematics. Creative Component Beta A Statstcal Aalyss of a Stock s Volatlty Courtey Wahlstrom Iowa State Uversty, Master of School Mathematcs Creatve Compoet Fall 008 Amy Froelch, Major Professor Heather Bolles, Commttee Member Travs

More information