3.1 Basis of single-electron wavefunctions ( orbitals )

Size: px
Start display at page:

Download "3.1 Basis of single-electron wavefunctions ( orbitals )"

Transcription

1 3 Atomc physcs mult-electron atoms 3. Bass of sngle-electron wavefunctons ( orbtals ) Many-partcle quantum mechancs Mult-electron atoms Many-body perturbaton theory (frst-order) Hartree-Fock self-consstent feld Atomc physcs mult-electron atoms Important to note: ths s not a general overvew to atomc physcs. There are many extremely mportant topcs I am not dscussng at all, e.g., spn-orbt effect, theory of angular momentum addton and couplng etc., whch are crucal for understandng quantum mechancs of atoms. However, those are essentally regular quantum mechancs, whch you have seen n other courses, and do not requre a computer. Here, I focus on the aspects of many-body atomc physcs for whch computatonal calculatons are essental. Some excellent sources to pursue these other topcs n more detal are books by Sakura, Johnson, Sobelman 3, and Bethe and Salpeter 4, whch are avalable n the lbrary. 3. Bass of sngle-electron wavefunctons ( orbtals ) Here we brefly revew another method for solvng the radal Schrödnger equaton that s partcularly useful for mult-electron atoms. As we shall see below, the total wavefuncton for a mult-electron atom s formed from combnatons of sngle-partcle wavefunctons (whch we often refer to as orbtals, though that term s used dfferently n many sources). 3.. Algebrac soluton to radal equaton Use some (fnte) set of bass functons {b (r)}, and wrte: P (r) = c b (r). () In general, ths s approxmate, snce the bass set s fnte. As such, a good choce of bass s essental; ths wll be dscussed below. Remember that here P (r) s defned va the radal decomposton for the sngle-partcle wavefunctons and s a soluton to the radal Schrödnger equaton: ψ nlm (r) = P nl(r) Y lm (θ, φ), () r HP = εp. (3) Solvng ths equaton s cast to determnng the expanson coeffcents {c }. Note: n general, the bass states b j are orthogonal or normalsed, and they are not egenstates of the Hamltonan. Usng Drac notaton wth = b, ths gves: H b c = ε b j c j (4) j b j H b c = ε b k b c, (5) J. J. Sakura, Modern Quantum Mechancs (0) [n partcular Chapters 3, 5, 7] W. R. Johnson, Atomc Structure Theory (007) 3 I. I. Sobelman, Atomc Spectra and Radatve Transtons (99) 4 H. A. Bethe and E. E. Salpeter, Quantum Mechancs of One-and Two-Electron Atoms (977)

2 where we multpled on the left by b j (and ntegrated). The result s just a matrx equaton: H j c = ε B j c (6) = Hc = εbc. (7) Ths s a generalsed egenvalue matrx equaton, whch can be solved to yeld a set of egenvalues ε, and egen vectors c. Each egenvector s the set of c expanson coeffcents. Note that f the bass set f orthonormal, the S matrx B j = b b j s just the dentty; however, ths s not true n general. The Fortran routne, DSYGV, wll solve the generalsed egenvalue problem for real, symmetrc matrces. (DSYGV s smlar to DSYEV, but takes n also second matrx, B.) Here, H and B are N b N b square matrces, wth elements: H j = Ĥ j = b (r)ĥb j(r) dr, B j = j = b (r)b j (r) dr, (8) wth N b beng the number of bass states (B-splnes) used the the expanson () not the number of radal grd-ponts used for ntegrals. If the bass was orthonormal, S would just be the dentty matrx; n general t s not. For the general case of H = + V (r), we have r H j = b (r)b j (r) dr + b (r)v (r)b j (r) dr (9) = + b (r)b j(r) dr + b (r)v (r)b j (r) dr (0) (ntegraton by parts). In theory, these ntegrals can be found wth extremely hgh accuracy usng Gaussan quadrature. For our purposes, any reasonable ntegraton scheme wll do just fne. Example 3.: DSYGV Parameters. Documentaton: extern "C" nt dsygv_ ( nt * ITYPE, // = for problems of type Av = ebv char * JOBZ, // = V means calculate egenvectors char * UPLO, // U : upper trangle of matrx s stored, L : lower nt * N, // dmenson of matrx A double * A, // c- style array for matrx A ( ptr to array, ponter to a [ 0]) // On output, A contans matrx of egenvectors nt *LDA, // For us, LDA =N double * B, // c- style array for matrx B [ Av = ebv ] nt *LDB, // For us, LDB =N double * W, // Array of dmenson N - wll hold egenvalues double * WORK, // workspace : array of dmenson LWORK nt * LWORK, // dmenson of workspace : ~ 6* N works well nt * INFO // error code : 0= worked. ); Note: FORTRAN (language LAPACK s wrtten n) uses column-major orderng to access D arrays, wle c and c++ use row-major. Ths means m[][j] n c++ s m[j][] n FORTRAN.. so we often need to transpose the matrx before sendng to LAPACK Our matrx s symmetrc, so ths doesn t matter, except for uplo

3 uplo : U means upper trangle n FORTRAN s stored so lower n c++ [you may just fll entre matrx] For other LAPACK functons, you can often just tell them the matrx s a transpose, so we don t need to waste tme transposng t ourselves Don t forget to declare the dsygv functon wth extern C, and use the -llapack lnker (comple) flag (you may also need the -lblas flag) 3.. B-splne bass functons There are many optons for whch set of bass functons to use n Eq. (). One opton that works very well for atomc physcs s to use B-splnes. b (r) = r/a B Fgure : Every 3rd splne from a set of 30 B-splnes of order k = 7 defned on r [0, 50] a.u. The frst non-zero knot was placed at r 0 = 0 3 a.u., and the knots where dstrbuted unformly over [r 0, R] (though t s typcal to nstead dstrbute them logarthmcally). B-splnes are are a set of N pecewse polynomals of order k, defned over a sub-doman r [0, R]. Each splne functon, b, s non-zero for only a small sub-regon of the doman. Techncal detals (you don t need to know): b (r) n non-zero only for t r < t +n, where {t 0,..., t N } are a seres of N + knots. Frst k knots are placed at r = 0; the fnal k knots are at r = R; the remanng knots are dstrbuted between some non-zero r 0 and R. All splnes go to zero at r = 0, except the 0th splne (ndex 0); the 0th splne (ndex 0) s only non-zero for r < r 0 The kth splne and above [ndex k] are non-zero only for r > r 0 All splnes go to zero at r = R, except for the fnal one (N )th splne 3

4 For any r [0, R], b (r) = Form a complete bass for any polynomal of order k on nterval r [0, R] Typcally, we choose r a.u., R a.u., N = 30 00, and k = Enforce boundary condtons There are several ways to enforce the boundary condtons, ncludng addng extra fcttous nfnte potentals to the Hamltonan at r = 0 and r = R. We wll take a smpler approach, and enforce the boundary condtons by dscardng some of the splnes. Dscard b 0 ths s the only splne non-zero at r = 0, forcng wavefunctons to be P (0) = 0 Dscard fnal splne b N ths s the only splne non-zero at r = R, forcng wavefunctons to be P (R) = 0 ( hard boundary condton) Dscard b We can further take advantage of low-r behavour of the wavefunctons; P (r) r l+. Due to form of the splnes, we can force ths behavour by dscardng b for s and p states. So, wth N total splnes, we actually use only N 3 n the expanson Eq. (). 3. Many-partcle quantum mechancs 3.. Exchange symmetry, product states Quck revew; see Chapter 7 of Sakura 5 for a more complete dscusson (or any other QM textbook). Consder system of two non-nteractng dentcal partcles (they do not nteract wth each other, but may each nteract wth external potental, V ). The total Hamltonan s the sum of those for each partcle (same as n classcal mechancs): H = h = ( ) p m + V (r ). () If ψ a (r) s a soluton to hψ a = ε a ψ a (smlarly for ψ b ), then t s clear that Ψ ab (r, r ) = ψ a (r )ψ b (r ) () s also a soluton to H, wth egenvalue (ε a + ε b ). So s the nterchanged soluton ψ b (r )ψ a (r ) (swapped quantum numbers). In fact, snce two partcles are dentcal, such two solutons are completely ndstngushable. Snce the soluton must of course return to ts orgnal state after two nterchanges of partcles a and b, a sensble general soluton can be wrtten, Ψ ab (r, r ) = [ψ a (r )ψ b (r ) ± ψ b (r )ψ a (r )]. (3) Wthout gong nto the detals, t turns out that for Fermons (/-nteger spn partcles), the total wavefuncton must be ant-symmetrc under the exchange of any two partcles: Ψ ab (r, r ) Fermon = [ψ a (r )ψ b (r ) ψ b (r )ψ a (r )], (4) 5 J. J. Sakura, Modern Quantum Mechancs (0) 4

5 whle for Bosons (nteger spn partcles) the total wavefuncton must be symmetrc under the exchange Ψ ab (r, r ) Boson = [ψ a (r )ψ b (r ) + ψ b (r )ψ a (r )]. (5) Ths s to ensure Fermons obey the Ferm-Drac statstcs, and Bosons obey the Bose-Ensten statstcs. Note that the ant-symmetry of Fermon wavefunctons encodes the Paul excluson prncple (spn-statstcs theorem). The same general form apples for systems of many partcles. So long as the system s nonnteractng, the total wavefuncton can be bult of products of sngle-partcle wavefunctons, called product states (or Fock states). 3.. Slater determnant For many-fermon systems (assumng them to be non-nteractng), we can form wavefunctons usng product states, but we must ensure those wavefunctons are properly ant-symmetrc under exchange. A nce notatonal trck to do ths s to wrte the wavefunctons as determnant, called Slater determnants, of the form: ψ Ψ ab...n (r, r,..., r N ) = a (r ) ψ a (r )... ψ a (r N ) ψ b (r ) ψ b (r )... ψ b (r N ) N!......, (6) ψ n (r ) ψ n (r )... ψ n (r N ) whch s an egenstate of the total Hamltonan H = N h(r ). To avod notaton confuson, note that each r s a dfferent coordnate varable (not a specfc pont on the coordnate grd). It s easest to see that ths works n the case of N = Fermons, but works n general Matrx elements of mult-fermon wavefunctons Let F and G be general one- and two-partcle operators ˆF = ˆf(r ), Ĝ = <j ĝ(r, r j ), (7) where f(r ) acts on the th partcle; g(r, r j ) acts on the par of partcle {, j}. The sum extends over each of the N electrons (or each par of electrons, j; the < j ensures pars are not double-counted). In regular quantum mechancs, ths s all that s needed; there are no three-body operators. It s farly straght-forward, though a lttle cumbersome, to derve the rules for calculatng matrx elements between many-body Slater-determnant wavefunctons. If you do, you wll start to see patterns, whch wll turn nto general rules. I wll not prove the rules here, but just state them; they are easy to verfy for two-partcle wavefunctons. For dagonal matrx elements (expectaton values; ntal state = fnal state): Ψ F Ψ = Ψ G Ψ = <j f ( j g j j g j ). Dagrams representng these are shown n Fg.. Here, the Drac notaton refers to the sngle-partcle wavefunctons: a = ψ a, ab = ψ a (r )ψ b (r ), meanng a h b = ψ a h ψ b dv (9) ab h cd = ψ a(r ) ψ b (r ) h ψ c (r )ψ d (r ) dv dv. (8) 5

6 j j j j Fgure : Dagrams representng nteracton of sngle-partcle operator wth Fermon (left), the drect part of a two-body operator (mddle), and exchange (rght). Dashed lne represents source of nteracton. For non-dagonal matrx elements (transtons), the rules are smlar. Usng a short-hand notaton 6, we have for matrx elements between wavefunctons that dffer by a sngle state: Ψ n F Ψ = n f Ψ n G Ψ = j ( nj g j jn g j ), (0) and for matrx elements between wavefunctons that dffer by two states: Ψ nm j F Ψ = 0 G Ψ = nm g j mn g j. Ψ nm j () 3.3 Mult-electron atoms 3.3. Mean-feld (ndependent partcle) model We can use the above mechansms to study the quantum mechancs of many-electron atoms. H = N ( p Z ) + r <j r r j. () The last term s the electron-electron repulson term, whch ncludes a sum over every par of electrons (the < j ensures pars are not double-counted). Clearly, ths s not a non-nteractng system. So how can we use the above formalsm? Frst, re-wrte the Hamltonan as H = N ( p Z ) r + umf (r ) + δv, (3) where δv = <j r r j u MF (r ). (4) Here, u MF s an (as-of-yet undetermned) potental, whch s taken to be roughly the average electronelectron nteracton term called the mean feld. If t s chosen wdely, then δv should be small, and can therefore be treated perturbatvely. The physcal justfcaton for ths approxmaton s 6 Here, I use a short-hand notaton Ψ n a = a na a Ψ, (a and a are creaton and annhlaton operators) meanng that one partcle n state a s replaced by one n state n. e.g., f Ψ s a 3-partcle wavefuncton wth partcles n states a, b, c, then Ψ n a has partcles n states n, b, c. By state I mean a gven full set of quantum numbers (e.g., n, l, m). 6

7 clear - for a system of large number of nteractng partcles, the ndevdual nteractons between all pars of partcles wll average out, and the average nteracton potental seen by each partcle wll be essentally the same. We therefore frst solve the mean feld Hamltonan wth H H MF H MF = N ( p Z ) r + umf (r ), (5) whch can now be treated as a non-nteractng system. Then, we can deal wth the remanng δv term later, usng perturbaton theory: H = H MF + δv. Ths s called the mean-feld approxmaton, or the ndependent partcle pcture. Ths means we form total (approxmate) many-electron wavefuncton, Ψ, whch satsfes H MF Ψ = EΨ, (6) from Slater determnants made from sngle-electron solutons to the sngle-partcle equaton: hψ = εψ, (7) h = p Z r + umf. (8) In practcle calculatons, we just calculate and store the set of sngle-partcle solutons, ψ, and use the rules from Sec to perform many-body calculatons. Importantly, as can be deduced from these rules, the sngle-partcle energes ε, correspond to the bndng energes (negatve of the onsaton energes) for the ndvdual electrons n the atom Intal choce for mean-feld There are many ways to choose a good startng approxmaton for u MF ; here we wll dscuss a very smple approach. The atomc potental seen by an electron at very small r s essentally unscreened by other electrons. So, n ths regon, we have V (r) Z/r. Smlarly, at very large r there s total screenng by the (Z ) other electrons, so V (r) /r; see Fg. 3. As a very rough frst guess for the mean-feld potental, we can chose a smooth potental that connects these two regons. A smple choce s to use a parametrc potental; one that works reasonably well s the Green potental (Z ) h ( e r/d ) V Gr (r) = r + h (e r/d ), (9) where h and d are parameters, whch may be tuned to gve reasonable results. Note that V nuc (r) + V Gr (r) behaves lke Z/r for small r, and /r for large r. Before modern computers ths was essentally the best one could do. Now, we can do much better. However, we wll use ths Green s potental as a startng pont, and mprove upon t usng perturbaton theory and the Hartree-Fock routne. 3.4 Many-body perturbaton theory (frst-order) 3.4. Sngle-valence systems We wll frstly focus on atoms that have a sngle valence electron above closed-shell core; these are the smplest of the many-electron atoms. All the electrons, besdes one, are n full shells (each l and m occuped for each n) ths s called the core. The sngle valence electron les n an n shell above 7

8 Fgure 3: Total atomc potental seen by electron at very small r s essentally unscreened by other electrons, V (r) Z/r, whle at very large r there s total screenng by the (Z ) other electrons, V (r) /r. the core. Due to the energy scalng /n, the energy requred to excte a electron from the core s much much larger than that requred to excte the valence electron. Therefore, for sngle-valence systems, the core confguraton typcally remans always the same, whle all atomc dynamcs are due to the sngle valence electron. Also, snce the valence electron wavefuncton s mostly located at larger radus, the sngle-partcle core wavefunctons depends only very lttle on the nfluence of the valence electron. From the rules presented n Sec. 3..3, all these greatly smplfy the calculatons. The sngle-partcle energy of the valence electron can be dentfed as (approxmately) the bndng energy of ths electron. Therefore, dfferences n the sngle-partcle energes of dfferent valence states correspond to the transton energes between atoms n those valence states; see Sec Frst-order perturbaton correcton to energes From before, the perturbaton to the total Hamltonan we must consder s: δv = <j r r j u MF (r ), (30) V ee U. (3) (second lne just defnes notaton V ee and U, whch nclude the summatons). quantum mechancs, the frst-order correcton to the total atomc energy s thus Just as n regular δe = Ψ V ee Ψ Ψ U Ψ. (3) To evaluate each of these terms, we must use the rules from Sec The U part s smple, snce these are sngle-partcle operators (depend on sngle electron coordnate only): Ψ U Ψ = u MF, (33) where the sum runs over atomc electrons. Note the sum extends over all electrons, and therefore ncludes summaton over all n, l, m and spn quantum numbers. For sngle-valence atoms, we can break ths nto the core and valence parts: core Ψ U Ψ = c u MF c + v u MF v, (34) c 8

9 where v denotes the state of the valence electron. Notce that the frst part, the sum over the core electrons, s the same for any valence state (t s just the correcton to core energy). Snce all we can observe n experments s the energy for transtons, and for sngle-valence atoms the core essentally remans the same, we can drop ths part (t wll cancel n transtons).e., t does not contrbute to transton energes between valence states, or therefore to the onsaton energy of valence electron. The V ee part s more complcated, snce these are two-partcle operators. From Sec. 3..3, we arrve at [ j r j j r j ] (35) where Ψ V ee Ψ = <j r r r. The frst term s known as the drect contrbuton, and the second s called exchange. Agan, we may separate ths nto terms nvolvng only core states, and those wth core and valence states: core Ψ V ee Ψ = a<c [ ca r ca ca r ac ] core + c [ cv r cv cv r vc ]. (36) Agan, the core part s the same for all valence states, and wll thus cancel n transton and onsaton energes for valence states. Therefore, the frst-order perturbaton theory correcton to the energy for valence state v s Evaluaton of Coulomb ntegrals core [ ] δε v = cv r cv cv r vc v u MF v. (37) c Perturbaton theory requres us to evaluate ntegrals of the form ab r cd, whch correspond to the Coulomb nteracton between electrons. As a remnder, wrtng the ntegral explctly, ths means: ab r cd = ψ a(r )ψ b (r ) r r ψ c(r )ψ d (r ) d 3 r d 3 r. (38) To evaluate ths ntegral, we use the Laplace expanson 7, whch you have lkely come across before n your electrodynamcs course: r r = k=0 r< k r> k+ P k (cos γ), (39) where P s a Legendre polynomal, r < mn( r, r ), r > max( r, r ), and γ s the angle between r and r. For the sphercally symmetrc problems of atom physcs, t becomes much easer to re-wrte ths n terms of the sphercal harmoncs (just usng ther defnton): r r = k=0 4π k + r< k r> k+ q ( ) q Y k, q (θ, φ ) Y k,q (θ, φ ), (40) q= k thus separatng the radal and angular parts n a way that makes calculatng ntegrals wth wavefunctons n the form of Eq. () smple. We get expressons n the form: ab r cd = kq C k,q angular Rk abcd, (4) 7 J. D. Jackson, Classcal Electrodynamcs (00) 9

10 where C angular s the angular ntegral, and Rabcd k s the radal ntegral: Rabcd k P a (r )P b (r ) rk < P r> k+ c (r )P d (r ) dr dr, (4) whch s convenent to wrte as R k abcd = 0 P a (r) y k bd(r) P c (r) dr, (43) where y k bd s called a Hartree screenng functon y k bd(r) = 0 r 0 P b (r ) rk < P r> k+ d (r ) dr (44) P b (r ) (r ) k r k+ P d(r ) dr + r P b (r ) r k (r ) k+ P d(r ) dr. (45) The angular part of the ntegral bols down to ntegrals of sphercal harmoncs; note that t depends only on the angular quantum numbers l and m (and multpolarty k and q), and not the specfc form of the wavefuncton. The angular factor can be calculated geometrcally by dong the analytc ntegrals over angular coordnates. In realty, t s much easer to do these ntegrals algebracally, by makng use of the orthogonalty propertes of the sphercal harmoncs and the quantum theory of angular momentum. Smple n concept, such calculatons are often very complex, and we wll not deal wth them at all here. For more nformaton, see the textbooks by Johnson 8, or (for the partcularly brave) Varshalovch Hartree-Fock self-consstent feld 3.5. Hartree-Fock potental Motvated by the above, we defne a potental, whch we wll call v HF (Hartree-Fock potental), such that a v HF a = [ ] a r a a r a. (46) }{{}}{{} a Drect Exchange Note that ths s the expectaton value of r for the state a.e., the average value of the electronelectron repulson. Therefore, we may use ths potental nstead of u MF when solvng the Schrödnger equaton! I wll justfy ths below, but frst, lets work out explct formulas for v HF. The drect part s easy; we just don t ntegrate over coordnates for ψ a ; vhf drect (r ) = a ψ (r ) r d 3 r. (47) The exchange part s more complex, and cannot be wrtten as a local potental. Instead, we have: vhf exch. ψ a (r ) = ( ) ψ (r )ψ a (r ) d 3 r ψ (r ). (48) r a It s farly easy to check that these formulas agree wth Eq. (46) (v HF = vhf drect for valence states, the summatons smply extend over all core electrons. + v exch. ). Notce that, HF 8 W. R. Johnson, Atomc Structure Theory (007) 9 D. A. Varshalovch, A. N. Moskalev, and V. K. Khersonsk, Quantum Theory of Angular Momentum (988) 0

11 After performng the angular ntegrals, and after summng over magnetc quantum numbers, we can present the effectve drect potental that enters the radal Schrödnger equaton. v drect P v (r) = core n cl c (l c + ) y 0 cc(r) P v (r), (49) where the sum extends over just the prncple and orbtal angular momentum quantum numbers for the electrons n the core, and y 0 s gven by Eq. (44). Only the k = 0 term from the Laplace expanson survves n the drect part. The angular part of the exchange term s a lttle more complcated, and requres some theory we have not yet covered. There s a second way to derve the drect part of the potental, whch further justfes ts use as a potental n the Schrödnger equaton. Consder the electrostatc potental seen by an electron, a, due to the other Z other electrons n an atom. Ths s due to the electrc charge densty of the electron cloud, call t ρ(r). Now, we know that ths s just the same as the electron probablty densty, tmes the electron charge: ρ(r) = e ψ (r). a We can use Gauss law to work out what the electrostatc potental due to ths charge densty s; t wll not be a surprse that we get exactly back the drect part of the potental, Eq. (47) Self-consstent feld method In the prevous secton we derved the Hartree-Fock potental, whch we hope to use n place of the mean-feld potental u MF when solvng the Schrödnger equaton to determne the wavefunctons. However, the Hartree-Fock potental tself depends on the wavefunctons! Therefore, we must start wth an ntal guess for the wavefunctons, and teratvely mprove our approxmaton. Use ntal approxmaton for u MF (e.g., Green potental), solve Schrödnger equaton to generate set of sngle-electron wavefunctons Use these wavefunctons to form v HF, whch s a better potental than the ntal approxmaton Use ths better potental to generate a set of better wavefunctons Now that we have a better set of wavefunctons, can form a better-yet potental, and so-on We contnue ths procedure teratvely, untl the solutons converge. The physcal justfcaton for ths procedure s smple; the nter-electronc potental depends on the dstrbuton of atomc electrons. By contnuously mprovng the model for the electron wavefunctons, we wll mprove our model for the nter-electronc potental, whch wll mprove our resultng wavefunctons and so on. That ths method s an accurate approxmaton can be proven by consderng the frst-order perturbaton theory correcton to the energy, when v HF was used as the potental n the Schrödnger equaton. In fact, due to the very defnton of the Hartree-Fock potental, t s a smple matter to see that ths s exactly zero! Therefore, there are no frst-order correctons to energes (or, ndeed, to the wavefunctons) when the Hartree-Fock potental s used. In order to mprove the accuracy further, we must consder hgher-order perturbaton theory correctons, whch become much more complcated, and s an actve area of current research (honours/phd projects avalable!).

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 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 information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v 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 information

HÜCKEL MOLECULAR ORBITAL THEORY

HÜCKEL 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 information

5.74 Introductory Quantum Mechanics II

5.74 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. 4-1 4.1. INTERACTION OF LIGHT

More information

Recurrence. 1 Definitions and main statements

Recurrence. 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 information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne 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 information

An Alternative Way to Measure Private Equity Performance

An 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 information

PERRON FROBENIUS THEOREM

PERRON 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 information

Calculation of Sampling Weights

Calculation 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 two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Mean Molecular Weight

Mean 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 information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - 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 information

Support Vector Machines

Support 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 information

where the coordinates are related to those in the old frame as follows.

where 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 non-coplanar vectors Scalar product

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 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 annuty-mmedate, and ts present value Study annuty-due, and

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation 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 information

Chapter 7 Symmetry and Spectroscopy Molecular Vibrations p. 1 -

Chapter 7 Symmetry and Spectroscopy Molecular Vibrations p. 1 - Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - 7 Symmetry and Spectroscopy Molecular Vbratons 7 Bases for molecular vbratons We nvestgate a molecule consstng of N atoms, whch has 3N degrees

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear 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 information

Loop Parallelization

Loop Parallelization - - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 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 information

The quantum mechanics based on a general kinetic energy

The 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 information

The OC Curve of Attribute Acceptance Plans

The 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 information

What is Candidate Sampling

What 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 information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit 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 information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Rotation Kinematics, Moment of Inertia, and Torque

Rotation 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 information

How To Understand The Results Of The German Meris Cloud And Water Vapour Product

How To Understand The Results Of The German Meris Cloud And Water Vapour Product Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Laws of Electromagnetism

Laws of Electromagnetism There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of

More information

Introduction to Statistical Physics (2SP)

Introduction 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 information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Research Note APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES * Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE 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 information

NMT 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 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 information

7.5. Present Value of an Annuity. Investigate

7.5. 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 information

1 What is a conservation law?

1 What is a conservation law? MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2015 2016, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,

More information

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS

IDENTIFICATION 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 information

21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque 21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl

More information

How To Calculate The Accountng Perod Of Nequalty

How To Calculate The Accountng Perod Of Nequalty 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 information

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

Inter-Ing 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007.

Inter-Ing 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. Inter-Ing 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

Section 2 Introduction to Statistical Mechanics

Section 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 information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE 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 information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby 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 information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) 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 information

Multiple stage amplifiers

Multiple stage amplifiers Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple

More information

An Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems

An Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems STAN-CS-73-355 I SU-SE-73-013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

Quantization Effects in Digital Filters

Quantization Effects in Digital Filters Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value

More information

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at

More information

Traffic State Estimation in the Traffic Management Center of Berlin

Traffic State Estimation in the Traffic Management Center of Berlin Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

Answer: 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

Answer: 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 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

1. Math 210 Finite Mathematics

1. 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 information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

Conversion between the vector and raster data structures using Fuzzy Geographical Entities

Conversion between the vector and raster data structures using Fuzzy Geographical Entities Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,

More information

Consider a 1-D stationary state diffusion-type equation, which we will call the generalized diffusion equation from now on:

Consider a 1-D stationary state diffusion-type equation, which we will call the generalized diffusion equation from now on: Chapter 1 Boundary value problems Numercal lnear algebra technques can be used for many physcal problems. In ths chapter we wll gve some examples of how these technques can be used to solve certan boundary

More information

Chapter 6 Inductance, Capacitance, and Mutual Inductance

Chapter 6 Inductance, Capacitance, and Mutual Inductance Chapter 6 Inductance Capactance and Mutual Inductance 6. The nductor 6. The capactor 6.3 Seres-parallel combnatons of nductance and capactance 6.4 Mutual nductance 6.5 Closer look at mutual nductance Oerew

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, 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 information

Viscosity of Solutions of Macromolecules

Viscosity of Solutions of Macromolecules Vscosty of Solutons of Macromolecules When a lqud flows, whether through a tube or as the result of pourng from a vessel, layers of lqud slde over each other. The force f requred s drectly proportonal

More information

MOLECULAR PARTITION FUNCTIONS

MOLECULAR PARTITION FUNCTIONS MOLECULR PRTITIO FUCTIOS Introducton In the last chapter, we have been ntroduced to the three man ensembles used n statstcal mechancs and some examples of calculatons of partton functons were also gven.

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

Simple Interest Loans (Section 5.1) :

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 information

Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions

Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda

More information

The Noether Theorems: from Noether to Ševera

The Noether Theorems: from Noether to Ševera 14th Internatonal Summer School n Global Analyss and Mathematcal Physcs Satellte Meetng of the XVI Internatonal Congress on Mathematcal Physcs *** Lectures of Yvette Kosmann-Schwarzbach Centre de Mathématques

More information

Series Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3

Series Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3 Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons

More information

An Interest-Oriented Network Evolution Mechanism for Online Communities

An Interest-Oriented Network Evolution Mechanism for Online Communities An Interest-Orented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 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 information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 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 information

ANALYZING 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 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, 610-519-4390,

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

CHAPTER 8 Potential Energy and Conservation of Energy

CHAPTER 8 Potential Energy and Conservation of Energy CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and non-conservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated

More information

Realistic Image Synthesis

Realistic Image Synthesis Realstc Image Synthess - Combned Samplng and Path Tracng - Phlpp Slusallek Karol Myszkowsk Vncent Pegoraro Overvew: Today Combned Samplng (Multple Importance Samplng) Renderng and Measurng Equaton Random

More information

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom

More information

Numerical Methods 數 值 方 法 概 說. Daniel Lee. Nov. 1, 2006

Numerical Methods 數 值 方 法 概 說. Daniel Lee. Nov. 1, 2006 Numercal Methods 數 值 方 法 概 說 Danel Lee Nov. 1, 2006 Outlnes Lnear system : drect, teratve Nonlnear system : Newton-lke Interpolatons : polys, splnes, trg polys Approxmatons (I) : orthogonal polys Approxmatons

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X 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 information

Section C2: BJT Structure and Operational Modes

Section C2: BJT Structure and Operational Modes Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v

More information

IMPACT ANALYSIS OF A CELLULAR PHONE

IMPACT ANALYSIS OF A CELLULAR PHONE 4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

Mathematics of Finance

Mathematics of Finance 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008 Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

Stochastic epidemic models revisited: Analysis of some continuous performance measures

Stochastic epidemic models revisited: Analysis of some continuous performance measures Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 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 information

Chapter 12 Inductors and AC Circuits

Chapter 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 information

Question 2: What is the variance and standard deviation of a dataset?

Question 2: What is the variance and standard deviation of a dataset? Queston 2: What s the varance and standard devaton of a dataset? The varance of the data uses all of the data to compute a measure of the spread n the data. The varance may be computed for a sample of

More information

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are: polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng

More information

Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Single Layer Perceptrons Kevin Swingler Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts 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 information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

More information