Section 15: Magnetic properties of materials
|
|
- Derrick Stephens
- 7 years ago
- Views:
Transcription
1 Physcs 97 Secton 15: Magnetc propertes of materals Defnton of fundamental quanttes When a materal medum s placed n a magnetc feld, the medum s magnetzed. Ths magnetzaton s descrbed by the magnetzaton vector M, the dpole moment per unt volume. Snce the magnetzaton s nduced by the feld, we may assume that M s proportonal to H. That s, M = χ. (1) The proportonalty constant χ s known as the magnetc susceptblty of the medum. Note that the magnetc susceptblty χ bears no physcal relatonshp to the electrc susceptblty, although the same symbol s used for both. Note also that our dscusson assumes that the medum s magnetcally sotropc. ut real crystals are ansotropc, and the susceptblty s represented by a second-rank tensor. In order to avod mathematcal complcatons, however, we shall gnore ansotropc effects n our treatment. Note, that n Eq. (1) we assumed that M s proportonal to, the external feld, and n dong so we gnored such thngs as demagnetzaton feld, whch were ncluded n the electrc case. The neglect of these factors s justfable n the case of paramagnetc and damagnetc materals because M s 5 very small compared to (typcally χ = / M ~ 10 ), unlke the electrc case, n whch χ ~ 1. ut when we deal wth ferromagnetc materals, where M s qute large, ths omsson s no longer tenable, and the above effects must be ncluded. ecause of small value of the magnetc susceptblty we wll not make dstncton between magnetc feld and magnetc nducton. Note also that χ n Eq.(1) can be dependent on the appled magnetc feld. In ths case, we can defne the magnetc susceptblty as follows M χ =. () The magnetzaton can be defned as E M =, (3) where E s the total energy of the system. Defntons () and (3) are more general and can be used n calculatons. Classfcaton of materals All magnetc materals may be grouped nto three magnetc classes, dependng on the magnetc orderng and the sgn, magntude and temperature dependence of the magnetc susceptblty. We wll dscuss propertes of fve classes of materals: damagnetc, paramagnetc, ferromagnetc, antferromagnetc and ferrmagnetc. There s no magnetc order at any temperature n damagnetc and paramagnetc materals, whereas there s a magnetc order at low temperatures n ferromagnetc, antferromagnetc and ferrmagnetc materals. 1
2 Physcs 97 In damagnetc materals the magnetc susceptblty s negatve. Usually ts magntude s of the order of to The negatve value of the susceptblty means that n an appled magnetc feld damagnetc materals acqure the magnetzaton, whch s ponted opposte to the appled feld. In damagnetc materals the susceptblty nearly has a constant value ndependent of temperature. Ionc crystals and nert gas atoms are damagnetc. These substances have atoms or ons wth complete shells, and ther damagnetc behavor s due to the fact that a magnetc feld acts to dstort the orbtal moton. Another class of damagnetc materals s noble metals. All the other classes of materals have postve susceptblty. Wthn these classes the magntude of the susceptblty vares over a very wde range. However, at suffcently hgh temperatures the susceptblty decreases wth ncreasng temperature for all materals n these classes. It was found expermentally that all these materals follow the relatonshp χ = C T ± T C more or less exactly for suffcently hgh T. Here C and T C are postve constants ndependent of temperature and dfferent for each materal. It was found that n some materals T C =0 and ths equaton s obeyed down to the lowest temperatures at whch measurements have been made. Ths class of materals s called paramagnetc. In paramagnetc materals χ s postve - that s, for whch M s parallel to. The susceptblty s however s also very small: 10-4 to The best-known examples of paramagnetc materals are the ons of transton and rare-earth ons. The fact that these ons have ncomplete atomc shells s what s responsble for ther paramagnetc behavor. In all other materals equaton (4) breaks down as temperature decreases. They all have a crtcal temperature below whch the varaton of susceptblty wth temperature s very dfferent from ts varaton above ths temperature. In ferromagnetc materals the crtcal temperature s called the Cure temperature. Above the Cure temperature the susceptblty follow relatonshp (4) wth a negatve sgn. When temperature approaches T C the magnetc susceptblty tends to be nfnte. An nfnte susceptblty means that a fnte magnetzaton can exst even n zero appled feld, whch s the case n permanent magnets. The problem s that the magnetzaton of ferromagnetc materals n zero feld can have a range of dfferent values and consequently cannot be regarded as a property of the materal. However, t s found that f a relatvely small magnetc feld s appled to these materals, the magnetzaton tends to a constant value, whch s called the saturaton magnetzaton M S or spontaneous magnetzaton. elow Cure temperature M S (T) aganst T follows a unversal curve: t tends to a constant value as T=0; as T ncreases, the spontaneous magnetzaton decreases more and more rapdly. At the Cure temperature the magnetzaton dsappeared. Ferrmagnetc materals have non-zero magnetzaton below the Cure temperature whch s smlar to ferromagnetc materals. However, sgnfcant departures from (4) occur over a range of temperatures. Ths behavour s only followed at temperatures large compared wth the Cure temperature. Another dfference between ferrmagnets and ferromagnets s that n ferrmagnetc materals the saturaton magnetzaton aganst temperature behave n a more complcated way. For (4)
3 Physcs 97 example, for some ferrmagnets the magnetzaton can ncrease wth ncreasng temperature and then drops down. Antferromagnetc materals have small postve susceptbltes at all temperatures. At hgh temperatures they follow eq. (4) wth T C usually havng a postve sgn. A crtcal temperature n ths case s called Neel temperature. elow the Neel temperature the susceptblty generally decreases wth decreasng temperature. There s no spontaneous magnetzaton n antferromagnetc materals. Calculaton of atomc susceptbltes In the presence of a unform magnetc feld the Hamltonan of an on (atom) s modfed n the two major ways: (1) In the total knetc energy term the momentum of each electron s replaced by e p p + A, (5) c where A s the vector potental assocated wth the magnetc feld such that = A. (6) We assume that the appled feld s unform so that 1 = A r. (7) () The nteracton energy of the feld wth each electron spn must be added to the Hamltonan: H = µ S, (8) spn where µ s the ohr magneton e µ = = mc / ev G. (9) As the result the total energy of electrons wll have a form 1 e H = + µ m p c r S. (10) We denote by T 0 the knetc energy n the absence of the appled feld,.e. T 0 1 = p. (11) m The cross term s the brackets can be rewrtten takng nto account that ( ) ( ) p r = r p. (1) We note that also r and p are quantum-mechancal operators, here we can work wth these quanttes as wth classcal varables because only non-dagonal components enter ths product (.e. 3
4 Physcs 97 there no terms whch contan, e.g., x components of both r and p whch do not commute). Note: r µ, pν = δ. µν Assumng that the feld s along z drecton, we can rewrte ( ) ( = x + y ) r. (13) Fnally we fnd for the feld-dependent correcton to the total Hamltonan: e H = H T = + + x + y, (14) 0 µ ( L S) 8mc where L s the total orbtal momentum: = ( ) ( ) L r p. (15) The energy correcton due to the appled electrc feld s small compared to electron energes. For 4 example, 1T= µ 1Tesla = ev. Therefore one can compute the changes n the energy levels nduced by the feld wth ordnary perturbaton theory. Equaton (14) s the bass for theores of the magnetc susceptblty of ndvdual atoms, ons, or molecules. Langevn damagnetsm Let us now apply these results to a sold composed of ons or atoms wth all electronc shells flled. Such atoms have zero spn and orbtal angular momentum n ts ground state,.e. 0 S 0 = 0 L 0 = 0. (16) Consequently only last term n eq.(14) contrbutes to the feld-nduced shft n the ground state energy: e e E = 0 H 0 = 0 x + y 0 = 0 r 0 8mc 1 mc ( ), (17) where the last form follows from the sphercal symmetry of the closed-shell on, 0 x 0 = 0 y 0 = 0 z 0 = 0 r (18) It s conventonal to defne a mean square onc radus by r = 1 0 r 0, (19) Z where Z s the total number of electrons n an on. We obtan then for the magnetzaton nduced by the appled magnetc feld, accordng to (3): 4
5 Physcs 97 E e NZ r M = = 6mc whch mples a negatve magnetc susceptblty:, (0) e NZ r χ =, (1) 6mc where N s the number of atoms per unt volume. Damagnetsm s assocated wth the tendency of electrcal charges partally to sheld the nteror of a body from an appled magnetc feld. In electro-magnetsm we are famlar wth Lenz's law: when the magnetc energy flux through an electrcal crcut s changed, an nduced current s set up n such a drecton as to oppose the flux change. Formula (1) can be derved classcally. Consder an electron rotatng about the nucleus n a crcular orbt, and let a magnetc feld be appled perpendcular to the plane of the paper, as shown n Fg. 1. efore ths feld s appled, we have, accordng to Newton's second law, F = mω r () 0 0 where F 0 s the attractve Coulomb force between the nucleus and the electron, and ω 0 s the angular velocty. Fg. 1 Atomc orgn of damagnetsm. The Lorentz force F L opposes the Coulomb force F 0 ; v s the electron velocty. When the feld s appled, an addtonal force starts to act on the electron: the Lorentz force e / c v. For the geometry of Fg.1, the effect s to produce a radally outward force gven by ( ) eω 0 r/c, and Eq. () should therefore be amended to e c ω = ω. (3) F0 0r m r Assumng that s small we can look for a soluton s a form ω = ω0 + ω. (4) 5
6 Physcs 97 Substtutng (4) n the rght-hand part of the Eq.(3) we fnd e ω =, (5) mc whch shows that the rotaton of the electron has been slowed down. Ths reducton n frequency produces a correspondng change n the magnetc moment. Ths can be calculated as follows. The change n the frequency of rotaton s equvalent to the change n the current around the nucleus, whch s I = (charge) x (revolutons per unt tme) = ( Ze) 1 e π mc. The magnetc moment µ of a current loop s gven by the product (current) x (area of the loop)/c, where c appears due to CGS unts. The area of the loop of radus r s πr. We have then 1 e π r e Z r µ = ( Ze) = π mc c 4mc, (6) Here <r > = <x > + <y > s the mean square of the perpendcular dstance of the electron from the feld axs through the nucleus. The mean square dstance of the electrons from the nucleus s <r > = <x > + <y > + <z >. For a sphercally symmetrcal dstrbuton of charge we have <x > = <y > =<z >, so that s <r > n eq.(6) should be replaced by 3/<r >, whch gves dentcal result to eq.(0). Damagnetsm can be found n onc crystals and crystals composed of nert gas atoms, because these substances have atoms or ons wth complete electronc shells. Another class of damagnetc materals s noble metals whch wll be dscussed later. Paramagnetsm of nsulators If atoms n a sold have non-flled electronc shells than we have to take nto account the frst term n the Hamltonan (14). Its contrbuton s then much larger than the contrbuton from the second term so that we can gnore t. We consder the effect of ths term on an on n a ground state whch can be descrbed by quantum numbers L, S, J and J z, where J s the total angular momentum and J z s the projecton of ths momentum nto a quantzaton axs. It can be shown that LSJJ L + S LSJJ = g LSJJ J LSJJ, (7) z z z z where g s the g-factor, whch s gven by J ( J + 1) + S( S + 1) L( L + 1) g = 1+. (8) J ( J + 1) We stress that ths relaton s vald only wthn the (J + 1) dmensonal set of states that make up the degenerate atomc ground state n zero feld;.e., (8) s obeyed only for matrx elements taken between states that are dagonal n J, L, and S. If the splttng between the zero-feld atomc groundstate multplet and the frst excted multplet s large compared wth k T (as s frequently the case), then only the (J + 1) states n the ground-state multplet wll contrbute apprecably to the free 6
7 Physcs 97 energy. In that case (and only n that case) Eq. (8) permts one to nterpret the frst term n the Hamltonan (14) as expressng the nteracton E = (9) of the feld wth a magnetc moment that s proportonal to the total angular momentum of the on, so that = gµ J. (30) The appled magnetc feld lfts degeneracy of the manfold of states and splts t nto J+1 equdstant levels, whch hs known as Zeeman splttng. The energes of these levels are gven by E = gµ J, (31) J z z Where J z s an nteger and has values from J to J. If thermal energy s less or comparable wth the Zeeman splttng, these levels wll be populated dfferently and gve a dfferent contrbuton to the magnetc moment of the on. The magnetzaton of the sold s determned by the average value of the magnetc moment, so that M = N, where N s the concentraton of ons f the sold and s the value of magnetc moment averaged over the oltzmann dstrbuton: J gµ J z kt J ze J z = J J gµ J z kt M = N = Ngµ. (3) J z = J e The summaton can be easly performed due to geometrc progresson and the result for the magnetzaton s M = NgJ µ ( x), (33) where x J gµ J kt = and J ( ) x s the rlloun functon defned by J + 1 J x J ( x) = coth x coth J J J J. (34) Fgure shows the dependence of the magnetzaton for three dfferent ons as a functon of appled magnetc feld. Note that n order to reach the saturaton, very low temperatures and very hgh magnetc felds are requred. At relatvely low felds and not too low temperatures we can expand the coth n (34) assumng that x << 1, so that coth ( x) 1 x +. (35) x 3 We can then fnd for the susceptblty: 7
8 Physcs 97 M NJ ( J + 1) g µ Np µ C χ = = = =, (36) 3k T 3k T T where p s the effectve number of ohr magnetons, defned as [ ( 1) ] 1/ p = g J J +. (37) C s the Cure constant and the form (37) s known as the Cure low. Fg. Magnetc moment versus /T for samples of (I) potassum chromum alum, (II) ferrc ammonum alum, and (III) gadolnum sulfate octahydrate. Over 99.5% magnetc saturaton s acheved at 1.3 K and about 5T. Hund Rules The Hund rules as appled to atoms and ons affrm that electrons wll occupy orbtals n such a way that the ground state s characterzed by the followng: 1. The maxmum value of the total spn S allowed by the excluson prncple;. The maxmum value of the orbtal angular momentum L consstent wth ths value of S; 3. The value of the total angular momentum J s equal to L-S when the shell s less than half full and to L+S when the shell s more than half full. Ths due to the spn-orbt nteracton the constant of whch has opposte sgn dependng on whether s less than half full or more than half full. When the shell s just half full, the applcaton of the frst rule gves L = 0, so that J = S. 8
9 Physcs 97 Table 1. Ground states of ons wth partally flled d- or f-shells, as constructed from Hund's rules Rare-earth ons Experments on rare-earth ons n crystals show that they obey the Cure law, wth an effectve number of magnetons n agreement wth the theory of spn-orbt nteracton. Table confrms ths. In these ons, therefore, the angular momenta L and S are strongly coupled, and the moment of the on can respond freely to the external feld. Table. Effectve Number of Magnetons for Rare-Earth Ions 9
10 Physcs 97 Ths result s not surprsng. In these ons - from La to Lu n the perodc table - the 4f shell s ncompletely flled. The outer 5p shell s completely flled, whle the 5d and 6s shells whch are stll further out are strpped of ther electrons to form the onc crystal. Thus the only ncomplete shell s the 4f shell, and ths s the one n whch the magnetc behavor occurs. Snce electrons n ths shell le deep wthn the on, screened by the outer 5p and 5d shells, they are not apprecably affected by other ons n the crystal. Magnetcally ther behavor s much lke that of a free on. Another reason why the free-on treatment apples to the rare-earth ons s that the spn-orbt nteracton s strong n these substances, because ths nteracton s proportonal to Z, the atomc number of the element concerned, and all the rare-earth ons have large Z's. Typcal values for the spn-orbt and the crystal-feld nteractons n these materals are 10-1 ev and 10 - ev, respectvely. Iron-group ons Table 3 shows that ron-group ons behave magnetcally as f J = S, that s, only the spn moment can contrbute to magnetzaton. We can see ths by means of the followng argument. The magnetc propertes of ths group of elements are due to the electron n the ncomplete 3d shell. Snce electrons n ths outermost shell nteract strongly wth neghborng ons, the orbtal moton s essentally destroyed, or quenched, leavng only the spn moment to contrbute to the magnetzaton. In other words, n these ons, the strength of the crystal feld s much greater than the strength of the spn-orbt nteracton, just the reverse of the stuaton n rare-earth ons. Typcal strengths of the crystal feld and spn-orbt nteractons n the ron group are 1 ev and 10 - ev, respectvely. Table 3. Iron-Group Ions In the case of the transton metal ons from the ron group (partally flled 3d shells) the crystal feld s very much larger than the spn-orbt couplng. Ths perturbaton wll not lft the spn degeneracy, snce t depends only on spatal varables and therefore commutes wth S, but t can completely lft the degeneracy of the orbtal L-multplet, f t s suffcently asymmetrc. The result wll then be a ground-state multplet n whch the mean value of every component of L vanshes (even though L stll has the mean value L(L + 1)). One can nterpret ths classcally as arsng from a precesson of the orbtal angular momentum n the crystal feld, so that although ts magntude s unchanged, all ts components average to zero. 10
11 Physcs 97 Paramagnetsm of conducton electrons Spn paramagnetsm arses from the fact that each conducton electron carres a spn magnetc moment whch tends to algn wth the feld. An electron has spn ½. One mght expect that the conducton would make a Cure-type paramagnetc contrbuton (36) to the magnetzaton of the metal wth J=S =1/: Nµ χ =, (38) k T Ths shows that the susceptblty s nversely proportonal to temperature. Experments show, however, that spn susceptbltes n metals are essentally ndependent of temperature. The observed values are also consderably smaller than those predcted by (38). These facts clearly cast strong doubts on the applcablty of (36) to the conducton electrons. The source of the dffculty les n the fact that Eq. (36) was derved on the bass of localzed electrons obeyng the oltzmann dstrbuton, whereas the conducton electrons are delocalzed and satsfy the Ferm dstrbuton. The proper treatment, takng ths nto account, s llustrated n Fg.4. In the absence of the feld, half the electrons have spns pontng n the postve z-drecton, and the other half n the negatve drecton (Fg. 4a), resultng n a vanshng net magnetzaton. When a feld s appled along the z-drecton, the energy of the spns parallel to s lowered by the amount µ, whle the energy of spns opposte to s rased by the same amount (Fg.4b). The stuaton whch ensues s energetcally unstable, and hence some electrons near the Ferm level begn to transfer from the opposte-spn half to the parallel-spn one, leadng to a net magnetzaton. Note that only relatvely few electrons near the Ferm level are able to flp ther spns and algn wth the feld. The other electrons, lyng deep wthn the Ferm dstrbuton, are prevented from dong so by the excluson prncple. Fg. 4. (a) When = 0, the two halves of the Ferm-Drac dstrbuton are equal, and thus M = 0; (b) When a feld s appled, spns n the antparallel half flp nto the parallel half, resultng n a net parallel magnetzaton. We can now estmate the magnetc susceptblty. The electrons partcpatng n the spn flp occupy an energy nterval of µ (Fg.4). Thus ther concentraton s gven by N eff = ½D(E F )µ, where D(E F ) s the densty of states at the Ferm energy level [the factor ½ s nserted because D(E F ) as defned us earler ncludes both spn drectons, whle n the present crcumstances only one spn drecton s nvolved n the flppng]. Snce each spn flp ncreases the magnetzaton by µ (from -µ to + µ ), t follows that the net magnetzaton s gven by 11
12 Physcs 97 M = N µ = µ D E, (39) eff ( F ) leadng to a paramagnetc susceptblty χ = µ D( E ) (40) F The susceptblty s thus determned by the densty of states at the Ferm level, D(E F ). Accordng to eq. (40), χ s essentally ndependent of temperature. Ths s seen from the fact that temperature has only a small effect on the Ferm-Drac dstrbuton of the electrons, and consequently the dervaton leadng to (40) remans vald. If we apply the results for free electrons for whch D(E F ) = 3N/E F =3N/k TF, eq.(40) then leads to 3Nµ χ =, (41) k T F where T F the Ferm temperature (E F = k T F ). Snce T F s very large, often 30,000 K or hgher, we can see that (41) s smaller than (36) by factor of 10 - n agreement wth experment. In transton metals, the paramagnetc susceptblty s exceptonally large, because D(E F ) s large, by vrtue of the narrow and hgh 3d band. Damagnetsm Conducton electrons also exhbt damagnetsm on account of the cyclotron moton they execute n the presence of the magnetc feld. Each electron loop s equvalent to a dpole moment whose drecton s opposte to that of the appled feld. Classcal treatment shows that the total damagnetc contrbuton of all electrons s zero. Quantum treatment however shows that for free electrons ths causes a damagnetc moment equal to 1/3 of the paramagnetc moment. Therefore the total susceptblty of a free electrons gas s Nµ χ = (4) k T F The net response s therefore paramagnetc. In comparng theoretcal results wth experment, one must also nclude the damagnetc effect of the on cores. Table 4 gves the results for some metals. Table 4. Susceptbltes of Some Monovalent and Dvalent Metals x 10 6 (Room Temperature) 1
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 informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product
More informationMean Molecular Weight
Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationLaws 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 information21 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 informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationFaraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
More informationGoals Rotational quantities as vectors. Math: Cross Product. Angular momentum
Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mt.edu 5.74 Introductory Quantum Mechancs II Sprng 9 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 4-1 4.1. INTERACTION OF LIGHT
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationChapter 12 Inductors and AC Circuits
hapter Inductors and A rcuts awrence B. ees 6. You may make a sngle copy of ths document for personal use wthout wrtten permsson. Hstory oncepts from prevous physcs and math courses that you wll need for
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationSection 2 Introduction to Statistical Mechanics
Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.
More information1 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 informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 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 informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationJet Engine. Figure 1 Jet engine
Jet Engne Prof. Dr. Mustafa Cavcar Anadolu Unversty, School of Cvl Avaton Esksehr, urkey GROSS HRUS INAKE MOMENUM DRAG NE HRUS Fgure 1 Jet engne he thrust for a turboet engne can be derved from Newton
More informationPSYCHOLOGICAL 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 information1. 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 informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationChapter 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 informationSection 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 informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More information+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationCHOLESTEROL 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 informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell
More informationEffects of Extreme-Low Frequency Electromagnetic Fields on the Weight of the Hg at the Superconducting State.
Effects of Etreme-Low Frequency Electromagnetc Felds on the Weght of the at the Superconductng State. Fran De Aquno Maranhao State Unversty, Physcs Department, S.Lus/MA, Brazl. Copyrght 200 by Fran De
More informationRisk-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 informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationChapter 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"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 informationHow 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 informationIntroduction to Statistical Physics (2SP)
Introducton to Statstcal Physcs (2SP) Rchard Sear March 5, 20 Contents What s the entropy (aka the uncertanty)? 2. One macroscopc state s the result of many many mcroscopc states.......... 2.2 States wth
More informationWe 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 informationDamage detection in composite laminates using coin-tap method
Damage detecton n composte lamnates usng con-tap method S.J. Km Korea Aerospace Research Insttute, 45 Eoeun-Dong, Youseong-Gu, 35-333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The con-tap test has the
More informationThe quantum mechanics based on a general kinetic energy
The quantum mechancs based on a general knetc energy Yuchuan We * Internatonal Center of Quantum Mechancs, Three Gorges Unversty, Chna, 4400 Department of adaton Oncology, Wake Forest Unversty, NC, 7157
More informationHow 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 informationViscosity 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 informationCausal, 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 informationBERNSTEIN 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 informationComparison 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 information1 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 informationCHAPTER 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 informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationRing 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- 573 A Possible Detector for the Study of Weak Interactions at Fermi Clash R. Singer Argonne National Laboratory
- 573 A Possble Detector for the Study of Weak nteractons at Ferm Clash R. Snger Argonne Natonal Laboratory The purpose of ths paper s to pont out what weak nteracton phenomena may exst for center-of-mass
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationSPEE 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 informationRotation and Conservation of Angular Momentum
Chapter 4. Rotaton and Conservaton of Angular Momentum Notes: Most of the materal n ths chapter s taken from Young and Freedman, Chaps. 9 and 0. 4. Angular Velocty and Acceleraton We have already brefly
More informationSIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POST-DISCHARGE AIR CORONA REACTOR
XVIII Internatonal Conference on Gas Dscharges and Ther Applcatons (GD 2010) Grefswald - Germany SIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POST-DISCHARGE AIR CORONA REACTOR M. Mezane, J.P. Sarrette,
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationShielding Equations and Buildup Factors Explained
Sheldng Equatons and uldup Factors Explaned Gamma Exposure Fluence Rate Equatons For an explanaton of the fluence rate equatons used n the unshelded and shelded calculatons, vst ths US Health Physcs Socety
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationCHAPTER 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 informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More information1 Battery Technology and Markets, Spring 2010 26 January 2010 Lecture 1: Introduction to Electrochemistry
1 Battery Technology and Markets, Sprng 2010 Lecture 1: Introducton to Electrochemstry 1. Defnton of battery 2. Energy storage devce: voltage and capacty 3. Descrpton of electrochemcal cell and standard
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationL10: 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 informationEnergies of Network Nastsemble
Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada
More informationDEFINING %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 information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationsubstances (among other variables as well). ( ) Thus the change in volume of a mixture can be written as
Mxtures and Solutons Partal Molar Quanttes Partal molar volume he total volume of a mxture of substances s a functon of the amounts of both V V n,n substances (among other varables as well). hus the change
More informationInner core mantle gravitational locking and the super-rotation of the inner core
Geophys. J. Int. (2010) 181, 806 817 do: 10.1111/j.1365-246X.2010.04563.x Inner core mantle gravtatonal lockng and the super-rotaton of the nner core Matheu Dumberry 1 and Jon Mound 2 1 Department of Physcs,
More informationChapter 9. Linear Momentum and Collisions
Chapter 9 Lnear Momentum and Collsons CHAPTER OUTLINE 9.1 Lnear Momentum and Its Conservaton 9.2 Impulse and Momentum 9.3 Collsons n One Dmenson 9.4 Two-Dmensonal Collsons 9.5 The Center of Mass 9.6 Moton
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and
More informationTECHNICAL NOTES 3. Hydraulic Classifiers
TECHNICAL NOTES 3 Hydraulc Classfers 3.1 Classfcaton Based on Dfferental Settlng - The Hydrocyclone 3.1.1 General prncples of the operaton of the hydrocyclone The prncple of operaton of the hydrocyclone
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationActuator forces in CFD: RANS and LES modeling in OpenFOAM
Home Search Collectons Journals About Contact us My IOPscence Actuator forces n CFD: RANS and LES modelng n OpenFOAM Ths content has been downloaded from IOPscence. Please scroll down to see the full text.
More informationFraming and cooperation in public good games : an experiment with an interior solution 1
Framng and cooperaton n publc good games : an experment wth an nteror soluton Marc Wllnger, Anthony Zegelmeyer Bureau d Econome Théorque et Applquée, Unversté Lous Pasteur, 38 boulevard d Anvers, 67000
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationPolitecnico di Torino. Porto Institutional Repository
Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC
More informationProblem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.
Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationInertial Field Energy
Adv. Studes Theor. Phys., Vol. 3, 009, no. 3, 131-140 Inertal Feld Energy C. Johan Masrelez 309 W Lk Sammamsh Pkwy NE Redmond, WA 9805, USA jmasrelez@estfound.org Abstract The phenomenon of Inerta may
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
More informationCity Research Online. Permanent City Research Online URL: http://openaccess.city.ac.uk/1356/
Cooper, E. S., Dssado, L. A. & Fothergll, J. (2005). Applcaton of thermoelectrc agng models to polymerc nsulaton n cable geometry. IEEE Transactons on Delectrcs and Electrcal Insulaton, 12(1), pp. 1-10.
More informationMOLECULAR 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 informationChapter 11 Torque and Angular Momentum
Chapter 11 Torque and Angular Momentum I. Torque II. Angular momentum - Defnton III. Newton s second law n angular form IV. Angular momentum - System of partcles - Rgd body - Conservaton I. Torque - Vector
More informationNPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationA Secure Password-Authenticated Key Agreement Using Smart Cards
A Secure Password-Authentcated Key Agreement Usng Smart Cards Ka Chan 1, Wen-Chung Kuo 2 and Jn-Chou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,
More informationInterlude: Interphase Mass Transfer
Interlude: Interphase Mass Transfer The transport of mass wthn a sngle phase depends drectly on the concentraton gradent of the transportng speces n that phase. Mass may also transport from one phase to
More informationLoudspeaker Voice-Coil Inductance Losses: Circuit Models, Parameter Estimation, and Effect on Frequency Response
44 JOURAL OF THE AUDIO EGIEERIG SOCIETY, VOL. 50, O. 6, 00 JUE Loudspeaker Voce-Col Inductance Losses: Crcut Models, Parameter Estmaton, and Effect on Frequency Response W. Marshall Leach, Jr., Professor
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationLoop 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 informationThe Development of Web Log Mining Based on Improve-K-Means Clustering Analysis
The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More information