Chapter 6 1-D Continuous Groups

Size: px
Start display at page:

Download "Chapter 6 1-D Continuous Groups"

Transcription

1 Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores: 1. the group of rotations in a plane SO 2, 2. the group of translations in 1-D T 1. These groups are 1-D (i.e. the group eleents only depend on one continuous paraeter) and they are necessarily abelian. A Lie group is an infinite group whose eleents can be paraetrized soothly and analytically. The definition of these properties requires introducing algebraic and geoetric structures beyond group ultiplication in pure group theory (which has been our only concern so far for discrete groups). The precise forulation of the general theory of Lie groups requires considerable care; it involves notions of topology and differential geoetry. All known continuous syetry groups of interest in physics, however, are groups of atrices for which the additional algebraic and geoetric structures are already failiar. These groups are usually referred to as linear Lie groups or classical Lie groups. We shall introduce the ost iportant features and techniques of the theory of linear Lie groups by studying the iportant syetry groups of space and tie, starting fro this chapter. Experience with these physically useful exaples should provide a good foundation for studying the general theory of Lie groups. [ Chevalley, Gilore, Miller, Pontrjagin, Weyl ] Suary by Sections 1. Definition of SO 2 & its general properties. 2. Generator of the group as deterined by the group structure near e.. 3. IRs of SO(2) derived using eigenvectors of the generator. Role of global properties. 4. Invariant integration easure on the group anifold. Orthonorality and copleteness properties of the rep functions. Their relation to Fourier analysis.. 5. Multi- valued reps & their relation to the topology of the group anifold. 6. T Generators are identified with easurable physical quantities. All ideas introduced in this chapter will find significant generalizations in later chapters. 6.1 The Rotation Group SO(2) Consider a syste syetric under rotations in a plane, around a fixed point O.

2 2 6._1DContinuousGroups.nb Let e 1 and e 2 be the Cartesian orthonoral basis vectors [ see Fig. 6.1 ]. A rotation through angle Φ by R Φ gives (6.1-1) R Φ e 1 e 1 cosφ e 2 sinφ or equivalently, R Φ e 2 e 1 sinφ e 2 cosφ (6.1-2) R Φ e i e j R Φ j i with the atrix R Φ j i given by cosφ sinφ (6.1-3) Φ sinφ cosφ Fig. 6.1 Rotation in a plane. Let x e i x i Then x transfors under rotation R Φ according to: Writing x x ' R Φ x R Φ e i x i e j R Φ j i xi x ' e i x ' i we obtain (6.1-4) x ' j R Φ j i xi or in atrix for ' Φ Geoetrically, it is obvious that the length of vectors reains invariant under rotations, i.e. or x 2 x i x i x ' 2 x i ' x ' i T ' T ' T T Φ Φ Thus (6.1-5) T Φ Φ Φ where T denotes the transpose of, and is the unit atrix. Real atrices satisfying the condition (6.1-5) are called orthogonal atrices and denoted by O n, where n is their diension. In the present case, n 2.

3 6._1DContinuousGroups.nb Eq. (6.1-5) iplies that det T Φ Φ det so that det T Φ det Φ 1 [ det det det ] det Φ 2 1 [ det T det ] det Φ ±1 Now, the identity operator is given by E R 0 or in atrix for 0 with det 0 det 1 Operators that evolve continuously fro E ust therefore satisfy (6.1-6) det Φ 1 Φ They are called proper rotations. Matrices satisfying (6.1-6) are said to be special. In the present case, they are special orthogonal atrices of rank 2; and are designated as SO(2) atrices. Operators with det Φ 1 are called iproper rotations. They can always be expressed as a proper rotation followed by a reflection. Theore 6.1: There is a one-to-one correspondence between rotations in a plane and SO(2) atrices. The proof of this theore is left as an exercise. [Proble 6.1] This correspondence is a general one, applicable to SO n atrices and rotations in the Euclidean space of diension n for any n. O(2) Group The O 2 atrices with Φ 0, for a group called O 2. The law of coposition is (6.1-7) R Φ 2 R Φ 1 R Φ 2 Φ 1 with the understanding that any Φ outside 0, is brought back to Φ 0, using (6.1-8) R Φ R Φ n n integer Obviously R Φ 0 E and R Φ 1 R Φ R Φ Note that O 2 is not a Lie group since the iproper rotations are not continuously related to E. Theore 6.2 (2-D Rotation Group): The 2-D proper rotations R Φ for a Lie group called the rotation group R 2 or SO 2 group.

4 4 6._1DContinuousGroups.nb The set Φ 0, corresponds to all points on the unit circle which defines the "topology" of the group paraeter space ( anifold ) [ cf Fig. 6.2 ]. Obviously, this paraeterization is not unique. Any onotonic function Ξ Φ of Φ over the above doain can serve as an alternative label for the group eleent. The structure of the group and its reps should not be affected by the labelling schee. We shall coe back to the "naturalness" of the variable Φ and the question of topology in a later section. Both issues are iportant for the analysis of general continuous groups. Note that Eq. (6.1-7) iplies R Φ 1 R Φ 2 R Φ 2 R Φ 1 Φ 1, Φ 2 Thus, the group SO(2) is abelian. Fig. 6.2 SO(2) group anifold. 6.2 Generator of SO(2) Consider an infinitesial SO(2) rotation by the angle dφ. Differentiability of R Φ in Φ requires that R dφ differs fro R 0 E by only a quantity of order dφ which we define by the relation (6.2-1) R E i dφ J where the factor i is included by convention and for later convenience. The quantity J is independent of the rotation angle dφ. Next, consider the rotation R Φ dφ, which can be evaluated in two ways: (6.2-2) R Φ dφ R Φ R dφ R Φ i dφ R Φ J R Φ dφ R Φ dφ d R Φ Coparing the two equations, we obtain the differential equation, d R Φ (6.2-3) i R Φ J with the boundary condition R 0 E. The solution to Eq. (6.2-3) is therefore unique, we present it in the for of a theore.

5 6._1DContinuousGroups.nb Theore 6.3 (Generator of SO(2)): All 2-D proper rotations can be expressed in ters of the operator J as: (6.2-4) R Φ e i Φ J J is called the generator of the group. Thus the general group structure and its reps are deterined ostly by J which, in turn, is specified by local behavior of the group near the identity. This is typical of Lie groups. Note however that certain global properties of the group, such as Eq. (6.1-8), cannot be deduced fro (6.2-3). These global properties, ostly topological in nature, also play a role in deterining the IRs of the group, as we shall see in the next section. Let us turn fro this abstract discussion to the explicit representation of R Φ given by (6.1-3). We have, to first order in dφ, dφ 1 dφ dφ 1 Coparing with Eq. (6.2-1), we deduce (6.2-5) 0 i i 0 Thus is a traceless heritian atrix. It is easy to show that 2, 3, etc. Therefore, e i Φ i Φ 1 2 Φ2 i Φ3 cosφ i sinφ which reproduces Eq. (6.1-3). cosφ sinφ sinφ cosφ IRs of SO(2) Consider any rep of SO(2) defined on a finite diensional vector space V. Let U Φ be the operator on V which corresponds to R Φ. Then, according to Eq. (6.1-7), we ust have U Φ 2 U Φ 1 U Φ 2 Φ 1 U Φ 1 U Φ 2 with the sae understanding that U Φ U Φ n. For an infinitesial transforation, we can again define an operator corresponding to the generator J in Eq. (6.2-1). In order to avoid a proliferation of sybols, we use the sae letter J to denote this operator, (6.3-1) U dφ E i dφ J Repeating the arguents of the last section, we obtain (6.3-2) U Φ e i Φ J which is now an operator equation on V. If U Φ is to be unitary for all Φ, J ust be heritian with real eigenvalues.

6 6 6._1DContinuousGroups.nb Since SO(2) is an abelian group, all its IRs are 1-D. This eans that given any Α in a inial invariant subspace under SO(2) we ust have: (6.3-3) J Α Α Α (6.3-4) U Φ Α Α e i Φ Α where Α is an eigenvalue of J. It is easy to show that the for given by Eq. (6.3-4) autoatically satisfies the group ultiplication rule Eq. (6.1-7) for an arbitrary Α. However, in order to satisfy the global constraint Eq. (6.1-8), a restriction ust be placed on the eigenvalue Α. Indeed, we ust have e i n Α 1 n integer which iplies that Α is an integer. We denote this integer by, and the corresponding representation by U : (6.3-5) J (6.3-6) U Φ e i Φ Let us take a closer look at these reps: (i) When 0, This is the identity rep; (ii) When 1, R Φ U 0 Φ 1. R Φ U 1 Φ e i Φ This is an isoorphis between SO(2) group eleents and ordinary nubers on the unit circle in the coplex nuber plane. As R Φ ranges over the group space, U 1 Φ covers the unit circle once, in the clockwise sense; (iii) When 1, R Φ U 1 Φ e i Φ The situation is the sae as above, except that the unit circle is covered once in the counter- clockwise direction; (iv) When ±2, R Φ U ±2 Φ e 2 i Φ These are appings of the group paraeter space to the unit circle on the coplex nuber plane covering the latter twice in opposite directions. The general case follows in an obvious anner fro these exaples. We suarize these results in the for of a theore. Theore 6.4 ( IRs of SO(2) ): The single valued IRs of SO(2) are given by J where is any integer, and (6.3-7) U Φ e i Φ Of these, only the ±1 ones are faithful reps. The defining equation for R Φ, Eq. (6.1-3), is a 2-D rep of the group. It has to be reducible. Indeed, it is equivalent to a direct su of the ±1 representations. To see this. we note that due to Eq. (6.3-2), it suffices to diagonalize the atrix corresponding to the generator. J 0 i i 0

7 6._1DContinuousGroups.nb It is obvious that J has two eigenvalues, ±1; and the corresponding eigenvectors are: e ± 1 2 e 1 i e 2 [ Proble 6.2 ] Thus, with respect to the new basis. (6.3-8) J e ± ± e ± R Φ e ± e ± e i Φ 6.4 Invariant Integration Measure, Orthonorality & Copleteness Relations We now derive the orthonorality and copleteness relations for the representation functions U Φ e i Φ. Because Φ is a continuous variable, the suation over group eleents ust be replaced by an integration, and the integration easure ust be well defined. Now, any function Ξ Φ, onotonic in 0 Φ, can also serve as label. However, for an arbitrary function f of the group eleents, d Ξ f R Ξ Ξ ' Φ f R Φ f R Φ [ Ξ ' d Ξ ] Hence, "integration" of f over the group anifold is not well defined a priori. Now, the Rearrangeent Lea lies at the heart of the proof of ost iportant results of the representation theory. Therefore we seek to define an integration easure such that, (6.4-1) dτ R f R dτ R f S 1 R S G dτ SR f R If the group eleents are labelled by the paraeter Ξ, then dτ R Ρ R Ξ dξ where Ρ R Ξ is soe appropriately defined "weight function". Definition 6.1 (Invariant Integration Measure): A paraeterization R Ξ in group space with an associated weight function Ρ R Ξ is said to provide an invariant integration easure if Eq. (6.4-1) holds. The validity of Eq. (6.4-1) requires dτ R dτ SR which iposes the condition on the weight function, Ρ R Ξ (6.4-2) Ρ SR Ξ d Ξ SR d Ξ R This condition is autoatically satisfied if we define (6.4-3) Ρ R Ξ d Ξ E d Ξ R where Ξ E is the group paraeter around the identity eleent E and Ξ R Ξ ER is the corresponding paraeter around R. 2 In evaluating the right- hand side of the above equation, R is to be regarded as fixed; the dependence of Ξ ER on Ξ E is deterined by the group ultiplication rule.

8 8 6._1DContinuousGroups.nb The situation is siplest when Ξ SR linear in Ξ R,. This is the case when Ξ Φ is the rotation angle. The group ultiplication rule, Eq. (6.1-7), requires Φ ER Φ E Φ R Ρ R E ER R 1 Theore 6.5 (Invariant Integration Measure of SO(2) ): The rotation angle Φ, Fig. 6.1, and the volue easure dτ R dφ, provide the proper invariant integration easure over the SO(2) group space. If Ξ is a general paraeterization of the group eleent, then dτ R Ρ R Ξ dξ Ρ R Φ dφ dφ We ust have, therefore, Ρ R Ξ d Ξ The above discussion ay appear to be rather long-winded just to arrive at a relatively obvious conclusion. The otivation for including so uch detail is to set up a line of reasoning which can be applied to general continuous groups in later Chapters. Once an invariant easure is found, it is straightforward to write down the expected orthonorality and copleteness relations. Theore 6.6: The SO(2) representation functions U n Φ of Theore 6.4 satisfy the following orthonorality and copleteness relations: (6.4-4) 1 0 dφ U n Φ U Φ n (orthogonality) U n Φ U n Φ' Φ Φ' n (copleteness) Three siple rearks of general iportance are in order here: (i) These relations are natural generalizations of Theore 3.5 and 3.6 (for finite groups) to a continuous group; the only change is the replaceent of the finite su over group eleents by the invariant integration over the continuous group paraeter; (ii) Theore 6.6. with U n Φ given by Eq. (6.3-7), is equivalent to the classical Fourier Theore for periodic functions, the continuous group paraeter Φ and the discrete representation label n are the "conjugate variables" ; (iii) These relations are also identical to the results encountered in Chap.1, Eqs. (1.4-1) (1.4-2), in connection with the discrete translation group T d. Note, however, the roles of the group eleent label (discrete) and the representation label (continuous) are exactly reversed in coparison to the present case. 6.5 Multi-Valued Reps For later reference, we ention here a new feature of continuous groups the possibility of having ulti- valued representations. To introduce the idea, consider the apping (6.5-1) R Φ U 1 2 Φ e i 2 Φ

9 6._1DContinuousGroups.nb This is not a unique representation of the group, as (6.5-2) U 1 2 Φ e i Π i 2 Φ U 1 2 Φ whereas, we expect, on physical grounds, R 2 n Φ R Φ. However, since U Π Φ U 1 2 Φ Eq.(6.5-1) does define a one-to-two apping where each R Φ is assigned to two coplex nubers e i 2 Φ differing by a factor of 1. This is a two- valued representation in the sense that the group ultiplication law for SO(2) is preserved if either one of the two nubers corresponding to R Φ can be accepted. Clearly, we can also consider general appings, (6.6-3) R Φ U n Φ e i n Φ where n and are integers with no coon factors. For any pair n, this apping defines a "-valued representation" of SO(2) in the sae sense as described above. The following questions naturally arise: (i) Do continuous groups always have ulti-valued irreducible representation; (ii) How do we know whether (and for what values of do) ulti-valued iepresentations exist; (iii) When ulti-valued representations exist, are they realized in physical systes? In other words, does it ake sense to restrict our attention to solutions of classical and/or quantu- echanical systes only to those corresponding to singlevalued representations of the appropnate syetry groups? It turns out that the existence of ulti- valued representations is intiately tied to connectedness" a global topological property of the group paraeter space. In the case of SO(2), the group paraeter space (the unit circle) is "ultiplyconnected" 3, which iplies the existence of ulti- valued representations. Thus, it is possible to deterine the existence and the nature of ulti- valued representations fro an intrinsic property of the group. In regard to the last question posed above, so far as we know, both single- and double- valued representations, but no others, are realized in quantu echanical systes, and only single- valued representations appear in classical solutions to physical probles. The occurrence of double- valued representations can be traced to the connectedness of the group anifolds of syetries associated with the physical 3- and 4- diensional spaces. This observation will becoe clearer after we discuss the full rotation group and the Lorentz group in the next few chapters. 6.6 T 1 Rotations in the 2-diensional plane (by the angle Φ) can be interpreted as translations on the unit circle (by the arc length Φ). This fact accounts for the siilarity in the for of the irreducible representation function, U n Φ e i n Φ, in coparison to the case of discrete translation, t k n e i n k b, discussed in Chap. 1. The "copleentary" nature of these results has been noted in Sec We now extend the investigation to the equally iportant and basic continuous translation group in one diension, which we shall refer to as T 1. Let the coordinate axis of the one- diensional space be labelled x. An arbitrary eleent of the group T 1 corresponding to translation by the distance x will be denoted by T x. Consider "states" x 0 4 of a "particle" localized at the position x 0. 5 The action of T x on x is: (6.6-1) T x x 0 x x 0 It is easy to see that T x ust have the following properties: (6.6-2a) T x 1 T x 2 T x 1 x 2 (6.6-2b) T 0 E (6.6-2c) T x 1 T x

10 10 6._1DContinuousGroups.nb These are just the properties that are required for T x, x to for a group [ cf. Eqs. (1.2-1abc) ]. For an infinitesial displaceent denoted by dx, we have (6.6-3) T dx E i dx P which defines the (displaceent-independent) generator of translation P. Next, we write T x dx in two different ways: d T x (6.6-4a) T x dx T x dx d x and (6.6-4b) T x dx T dx T x Substituting (6.6-3) in (6.6-4b), and coparing with (6.6-4a), we obtain d T x (6.6-5) i P T x d x Considering the boundary condition (6.6-2b), this differential equation yields the unique solution, (6.6-6) T x e i P x It is straightforward to see that with T x written in this for, all the required group properties, (6.6-2a,b,c), are satisfied. This derivation is identical to that given for the rotation group SO(2). [ cf. Theore 6.3 ] The only difference is that the paraeter x in T x is no longer restricted to a finite range as for Φ in R Φ. As before, all irreducible representations of the translation group are onediensional. For unitary representations, the generator P corresponds to a heritian operator with real eigenvalues, which we shall denote by p. For the representation T x U p x, We obtain: (6.6-7) P p p p (6.6-8) U p x p p e i p x It is easy to see that all the group properties, Eqs. (6.6-2a,b,c), are satisfied by this representation function for any given real nuber p. Therefore the value of p is totally unrestricted. Coparing these results with those obtained in Chap. 1 for the discrete translation group T d and in Sections ( ) for SO(2), we reark that: (i) The representation functions in all these cases take the exponential for [ cf. Eqs. (1.3-3), (6.3-6), (6.6-8) ], reflecting the coon group ultiplication rule [ cf Eqs. (1.2-la),(6.1-7),(6.6-2a) ]; (ii) For T d, the group paraeter (n in Eq. (1.3-3)) is discrete and infinite, the representation label (k) is continuous and bounded. For SO(2), the forer (Φ in Eq. (6.3-6)) is continuous and bounded, the latter () is discrete and infinite. Finally, for T 1 the forer (x in Eq. (6.6-8)) is continuous and unbounded, so is the latter (p). The conjugate role of the group paraeter and the representation label in the sense of Fourier analysis was discussed in Sec The case is strengthened ore by applying the orthonorality and copleteness relations of representation functions to the present case of full one- diensional translation. For this purpose we ust again define an appropriate invariant easure for integration over the group eleents. Just as in the case of SO(2), one needs only to pick the natural Cartesian displaceent x as the integration variable. Because the range of integration is now infinite, not all integrals are strictly convergent in the classical sense. But for our purposes, it suffices to say that all previous results hold in the sense of generalized functions. We obtain: (6.6-9) (6.6-10) dx U p x U p ' x N p p' dp U p x U p x ' N x x '

11 6._1DContinuousGroups.nb where N is a yet unspecified noralization constant. Since U p x e i p x, these equations represent a stateent of the Fourier theore for arbitrary (generalized) functions. This correspondence also gives the correct value of N, i.e. N. 6.7 Conjugate Basis Vectors In Chap. 1 we described two types of basis vectors: x, defined by Eq. (1.1-6), and u E, k, defined by Eq. (1.3-3). The first represents "localized states" at soe position x ; the second corresponds to noral odes which fill the entire lattice and have siple translational properties. Each one has its unique features and practical uses. State functions expressed in ters of these two bases are related by a Fourier expansion. Analogous procedures can be applied in the representation space of the rotation group SO(2) and the continuous translation group. We describe the in turn. Consider a particle state localized at a position represented by polar coordinates r, Φ On the 2-diensional plane. The value of r will not be changed by any rotation; therefore we shall not be concerned about it in subsequent discussions. Intuitively, (6.7-1) U Φ Φ 0 Φ Φ 0 so that, (6.7-2) Φ U Φ 0 0 Φ where 0 represents a "standard state" aligned with a pre-chosen x-axis. How are these states related to the eigenstates of J defined by Eqs. (6.3-5) and (6.3-6)? If we expan in ters of the vectors ; 0, ±1,, Φ Φ then Φ U Φ 0 U Φ 0 0 e i Φ States with different values of are unrelated by rotation, and we can choose their phases (i.e. a ultiplicative factor e i Α for each ) such that 0 1 for all thus obtaining (6.7-3) Φ e i Φ To invert this equation, ultiply by e i Φ and integrate over Φ. We obtain (6.7-4) Φ Φ ei 0 We see that by this convention, the "transfer atrix eleents" functions. Φ between the two are just the group representation An arbitrary state Ψ in the vector space can be expressed in either of the two bases: (6.7-5) Ψ Ψ Φ Ψ Φ 0 The "wave functions" Ψ and Ψ Φ are related by (6.7-6) Ψ Φ Φ Ψ Φ Φ e i Φ Ψ and (6.7-7) Ψ 0 Ψ Φ e i Φ

12 12 6._1DContinuousGroups.nb Let us exaine the action of the operator J on the states (6.7-8) J Φ J e i Φ e i Φ d i Φ Φ. Fro Eq. (6.7-3), we obtain For an arbitrary state, we have: (6.7-9) Φ J Ψ J Φ Ψ 1 i 1 i d Φ Ψ d Ψ Φ Readers who have had soe quantu echanics [ Messiah, Schiff ] will recognize that J is the angular oentu operator (easured in units of ħ ). The above purely group- theoretical derivation underlines the general, geoetrical origin of these results. The above discussion can be repeated for the continuous translation group. The "localized states" "translationally covariant" states p, Eq. (6.6-7) are related by (6.7-10) x and d p p x p e i (6.7-11) p d x x e i p x x, Es. (6.6-1), and the where the noralization of the states is chosen, by convention, as x ' x x x ' p ' p p p ' The transfer atrix eleents are, again, the group representation functions [ Eq. (.6-8) ], (6.7-12) p x e i p x As before, if (6.7-13) Ψ x Ψ x d x p Ψ p d p then (6.7-14) Ψ x Ψ p e i p x d p (6.7-15) Ψ p Ψ x e i p x d x and (6.7-16) x P Ψ P x Ψ 1 i d d x Ψ x Thus, the generator P can be identified with the linear oentu operator in quantu echanical systes. [ Messiah, Schiff ]

13 6._1DContinuousGroups.nb Probles 6.1 Show that the rotation atrix R Φ, Eq. (6.1-3), is an orthogonal atrix and prove that every SO(2) atrix represents a rotation in the plane. 6.2 Show that e ± 1 2 e 1 i e 2 are eigenvectors of J with eigenvalues ±1 respectively [ cf. Eq. (6.3-8) ]. FootNotes Notes 1: Matrices satisfying Eq. (h.1-5) but with deterinant equal to -1 correspond physically to rotations cobined with spatial inversion or irror reflection. This set of atrices is not connected to the identity transforation by a continuous change of paraeters. We shall include spatial inversion in our group theoretical analysis in Chap. 11. Notes 2: We note that Ρ R d Ξ E d Ξ ER d Ξ E d Ξ SR d Ξ SR d Ξ R Ρ SR d Ξ SR d Ξ R Notes 3: This eans that there exist closed "paths" on the unit circle which wind around it ties (for all Integers ) and which cannot be continuously defored into each other. Notes 4: The "state" can be interpreted in the sense of either classical echanics or quantu echanics. We use the state- vector convention of quantu echanics only for the sake of clarity in notation. By "particle" we siply ean an entity with no spatial extension, which can be represented by a atheati- Notes 5: cal point.

i ij j ( ) sin cos x y z x x x interchangeably.)

i ij j ( ) sin cos x y z x x x interchangeably.) Tensor Operators Michael Fowler,2/3/12 Introduction: Cartesian Vectors and Tensors Physics is full of vectors: x, L, S and so on Classically, a (three-diensional) vector is defined by its properties under

More information

Physics 215 Winter The Density Matrix

Physics 215 Winter The Density Matrix Physics 215 Winter 2018 The Density Matrix The quantu space of states is a Hilbert space H. Any state vector ψ H is a pure state. Since any linear cobination of eleents of H are also an eleent of H, it

More information

Four-vector, Dirac spinor representation and Lorentz Transformations

Four-vector, Dirac spinor representation and Lorentz Transformations Available online at www.pelagiaresearchlibrary.co Advances in Applied Science Research, 2012, 3 (2):749-756 Four-vector, Dirac spinor representation and Lorentz Transforations S. B. Khasare 1, J. N. Rateke

More information

G G G G G. Spec k G. G Spec k G G. G G m G. G Spec k. Spec k

G G G G G. Spec k G. G Spec k G G. G G m G. G Spec k. Spec k 12 VICTORIA HOSKINS 3. Algebraic group actions and quotients In this section we consider group actions on algebraic varieties and also describe what type of quotients we would like to have for such group

More information

3.8 Three Types of Convergence

3.8 Three Types of Convergence 3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to

More information

Physics 139B Solutions to Homework Set 3 Fall 2009

Physics 139B Solutions to Homework Set 3 Fall 2009 Physics 139B Solutions to Hoework Set 3 Fall 009 1. Consider a particle of ass attached to a rigid assless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about

More information

Block designs and statistics

Block designs and statistics Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent

More information

Quantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search

Quantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths

More information

Feature Extraction Techniques

Feature Extraction Techniques Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that

More information

. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respe

. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respe PROPERTIES OF MULTIVARIATE HOMOGENEOUS ORTHOGONAL POLYNOMIALS Brahi Benouahane y Annie Cuyt? Keywords Abstract It is well-known that the denoinators of Pade approxiants can be considered as orthogonal

More information

First of all, because the base kets evolve according to the "wrong sign" Schrödinger equation (see pp ),

First of all, because the base kets evolve according to the wrong sign Schrödinger equation (see pp ), HW7.nb HW #7. Free particle path integral a) Propagator To siplify the notation, we write t t t, x x x and work in D. Since x i, p j i i j, we can just construct the 3D solution. First of all, because

More information

The Weierstrass Approximation Theorem

The Weierstrass Approximation Theorem 36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fitting of Data David Eberly, Geoetric Tools, Redond WA 98052 https://www.geoetrictools.co/ This work is licensed under the Creative Coons Attribution 4.0 International License. To view a

More information

The Fundamental Basis Theorem of Geometry from an algebraic point of view

The Fundamental Basis Theorem of Geometry from an algebraic point of view Journal of Physics: Conference Series PAPER OPEN ACCESS The Fundaental Basis Theore of Geoetry fro an algebraic point of view To cite this article: U Bekbaev 2017 J Phys: Conf Ser 819 012013 View the article

More information

The Transactional Nature of Quantum Information

The Transactional Nature of Quantum Information The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property.

More information

Lecture 13 Eigenvalue Problems

Lecture 13 Eigenvalue Problems Lecture 13 Eigenvalue Probles MIT 18.335J / 6.337J Introduction to Nuerical Methods Per-Olof Persson October 24, 2006 1 The Eigenvalue Decoposition Eigenvalue proble for atrix A: Ax = λx with eigenvalues

More information

Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom

Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Ato Thoas S. Kuntzlean Mark Ellison John Tippin Departent of Cheistry Departent of Cheistry Departent

More information

Solutions of some selected problems of Homework 4

Solutions of some selected problems of Homework 4 Solutions of soe selected probles of Hoework 4 Sangchul Lee May 7, 2018 Proble 1 Let there be light A professor has two light bulbs in his garage. When both are burned out, they are replaced, and the next

More information

Lecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum

Lecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum Lecture 8 Syetries, conserved quantities, and the labeling of states Angular Moentu Today s Progra: 1. Syetries and conserved quantities labeling of states. hrenfest Theore the greatest theore of all ties

More information

Chaotic Coupled Map Lattices

Chaotic Coupled Map Lattices Chaotic Coupled Map Lattices Author: Dustin Keys Advisors: Dr. Robert Indik, Dr. Kevin Lin 1 Introduction When a syste of chaotic aps is coupled in a way that allows the to share inforation about each

More information

THE POLYNOMIAL REPRESENTATION OF THE TYPE A n 1 RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n

THE POLYNOMIAL REPRESENTATION OF THE TYPE A n 1 RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n THE POLYNOMIAL REPRESENTATION OF THE TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n SHEELA DEVADAS AND YI SUN Abstract. We study the polynoial representation of the rational Cherednik algebra

More information

12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015

12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015 18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.

More information

arxiv: v2 [hep-th] 16 Mar 2017

arxiv: v2 [hep-th] 16 Mar 2017 SLAC-PUB-6904 Angular Moentu Conservation Law in Light-Front Quantu Field Theory arxiv:70.07v [hep-th] 6 Mar 07 Kelly Yu-Ju Chiu and Stanley J. Brodsky SLAC National Accelerator Laboratory, Stanford University,

More information

Ph 20.3 Numerical Solution of Ordinary Differential Equations

Ph 20.3 Numerical Solution of Ordinary Differential Equations Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v20170314- This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing

More information

Partial traces and entropy inequalities

Partial traces and entropy inequalities Linear Algebra and its Applications 370 (2003) 125 132 www.elsevier.co/locate/laa Partial traces and entropy inequalities Rajendra Bhatia Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi

More information

Force and dynamics with a spring, analytic approach

Force and dynamics with a spring, analytic approach Force and dynaics with a spring, analytic approach It ay strie you as strange that the first force we will discuss will be that of a spring. It is not one of the four Universal forces and we don t use

More information

The Hilbert Schmidt version of the commutator theorem for zero trace matrices

The Hilbert Schmidt version of the commutator theorem for zero trace matrices The Hilbert Schidt version of the coutator theore for zero trace atrices Oer Angel Gideon Schechtan March 205 Abstract Let A be a coplex atrix with zero trace. Then there are atrices B and C such that

More information

Polygonal Designs: Existence and Construction

Polygonal Designs: Existence and Construction Polygonal Designs: Existence and Construction John Hegean Departent of Matheatics, Stanford University, Stanford, CA 9405 Jeff Langford Departent of Matheatics, Drake University, Des Moines, IA 5011 G

More information

Partial Differential Equations of Physics

Partial Differential Equations of Physics Partial Differential Equations of Physics arxiv:gr-qc/9602055v1 27 Feb 1996 Robert Geroch Enrico Feri Institute, 5640 Ellis Ave, Chicago, Il - 60637 1 Introduction February 3, 2008 The physical world is

More information

About the definition of parameters and regimes of active two-port networks with variable loads on the basis of projective geometry

About the definition of parameters and regimes of active two-port networks with variable loads on the basis of projective geometry About the definition of paraeters and regies of active two-port networks with variable loads on the basis of projective geoetry PENN ALEXANDR nstitute of Electronic Engineering and Nanotechnologies "D

More information

Singularities of divisors on abelian varieties

Singularities of divisors on abelian varieties Singularities of divisors on abelian varieties Olivier Debarre March 20, 2006 This is joint work with Christopher Hacon. We work over the coplex nubers. Let D be an effective divisor on an abelian variety

More information

Support Vector Machine Classification of Uncertain and Imbalanced data using Robust Optimization

Support Vector Machine Classification of Uncertain and Imbalanced data using Robust Optimization Recent Researches in Coputer Science Support Vector Machine Classification of Uncertain and Ibalanced data using Robust Optiization RAGHAV PAT, THEODORE B. TRAFALIS, KASH BARKER School of Industrial Engineering

More information

Supporting Information for Supression of Auger Processes in Confined Structures

Supporting Information for Supression of Auger Processes in Confined Structures Supporting Inforation for Supression of Auger Processes in Confined Structures George E. Cragg and Alexander. Efros Naval Research aboratory, Washington, DC 20375, USA 1 Solution of the Coupled, Two-band

More information

An Exactly Soluble Multiatom-Multiphoton Coupling Model

An Exactly Soluble Multiatom-Multiphoton Coupling Model Brazilian Journal of Physics vol no 4 Deceber 87 An Exactly Soluble Multiato-Multiphoton Coupling Model A N F Aleixo Instituto de Física Universidade Federal do Rio de Janeiro Rio de Janeiro RJ Brazil

More information

arxiv: v1 [math.nt] 14 Sep 2014

arxiv: v1 [math.nt] 14 Sep 2014 ROTATION REMAINDERS P. JAMESON GRABER, WASHINGTON AND LEE UNIVERSITY 08 arxiv:1409.411v1 [ath.nt] 14 Sep 014 Abstract. We study properties of an array of nubers, called the triangle, in which each row

More information

Generalized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.

Generalized eigenfunctions and a Borel Theorem on the Sierpinski Gasket. Generalized eigenfunctions and a Borel Theore on the Sierpinski Gasket. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz May 26, 2006 1 Introduction There is a well developed theory (see [5,

More information

Lecture 21 Principle of Inclusion and Exclusion

Lecture 21 Principle of Inclusion and Exclusion Lecture 21 Principle of Inclusion and Exclusion Holden Lee and Yoni Miller 5/6/11 1 Introduction and first exaples We start off with an exaple Exaple 11: At Sunnydale High School there are 28 students

More information

RECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS MEMBRANE

RECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS MEMBRANE Proceedings of ICIPE rd International Conference on Inverse Probles in Engineering: Theory and Practice June -8, 999, Port Ludlow, Washington, USA : RECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS

More information

Fast Montgomery-like Square Root Computation over GF(2 m ) for All Trinomials

Fast Montgomery-like Square Root Computation over GF(2 m ) for All Trinomials Fast Montgoery-like Square Root Coputation over GF( ) for All Trinoials Yin Li a, Yu Zhang a, a Departent of Coputer Science and Technology, Xinyang Noral University, Henan, P.R.China Abstract This letter

More information

OBJECTIVES INTRODUCTION

OBJECTIVES INTRODUCTION M7 Chapter 3 Section 1 OBJECTIVES Suarize data using easures of central tendency, such as the ean, edian, ode, and idrange. Describe data using the easures of variation, such as the range, variance, and

More information

Fourier Series Summary (From Salivahanan et al, 2002)

Fourier Series Summary (From Salivahanan et al, 2002) Fourier Series Suary (Fro Salivahanan et al, ) A periodic continuous signal f(t), - < t

More information

Kernel Methods and Support Vector Machines

Kernel Methods and Support Vector Machines Intelligent Systes: Reasoning and Recognition Jaes L. Crowley ENSIAG 2 / osig 1 Second Seester 2012/2013 Lesson 20 2 ay 2013 Kernel ethods and Support Vector achines Contents Kernel Functions...2 Quadratic

More information

ADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE

ADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE ADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE CHRISTOPHER J. HILLAR Abstract. A long-standing conjecture asserts that the polynoial p(t = Tr(A + tb ] has nonnegative coefficients whenever is

More information

Data-Driven Imaging in Anisotropic Media

Data-Driven Imaging in Anisotropic Media 18 th World Conference on Non destructive Testing, 16- April 1, Durban, South Africa Data-Driven Iaging in Anisotropic Media Arno VOLKER 1 and Alan HUNTER 1 TNO Stieltjesweg 1, 6 AD, Delft, The Netherlands

More information

RESTARTED FULL ORTHOGONALIZATION METHOD FOR SHIFTED LINEAR SYSTEMS

RESTARTED FULL ORTHOGONALIZATION METHOD FOR SHIFTED LINEAR SYSTEMS BIT Nuerical Matheatics 43: 459 466, 2003. 2003 Kluwer Acadeic Publishers. Printed in The Netherlands 459 RESTARTED FULL ORTHOGONALIZATION METHOD FOR SHIFTED LINEAR SYSTEMS V. SIMONCINI Dipartiento di

More information

(a) As a reminder, the classical definition of angular momentum is: l = r p

(a) As a reminder, the classical definition of angular momentum is: l = r p PHYSICS T8: Standard Model Midter Exa Solution Key (216) 1. [2 points] Short Answer ( points each) (a) As a reinder, the classical definition of angular oentu is: l r p Based on this, what are the units

More information

Modern Physics Letters A Vol. 24, Nos (2009) c World Scientific Publishing Company

Modern Physics Letters A Vol. 24, Nos (2009) c World Scientific Publishing Company Modern Physics Letters A Vol. 24, Nos. 11 13 (2009) 816 822 c World cientific Publishing Copany TOWARD A THREE-DIMENIONAL OLUTION FOR 3N BOUND TATE WITH 3NFs M. R. HADIZADEH and. BAYEGAN Departent of Physics,

More information

Reed-Muller Codes. m r inductive definition. Later, we shall explain how to construct Reed-Muller codes using the Kronecker product.

Reed-Muller Codes. m r inductive definition. Later, we shall explain how to construct Reed-Muller codes using the Kronecker product. Coding Theory Massoud Malek Reed-Muller Codes An iportant class of linear block codes rich in algebraic and geoetric structure is the class of Reed-Muller codes, which includes the Extended Haing code.

More information

26 Impulse and Momentum

26 Impulse and Momentum 6 Ipulse and Moentu First, a Few More Words on Work and Energy, for Coparison Purposes Iagine a gigantic air hockey table with a whole bunch of pucks of various asses, none of which experiences any friction

More information

On the approximation of Feynman-Kac path integrals

On the approximation of Feynman-Kac path integrals On the approxiation of Feynan-Kac path integrals Stephen D. Bond, Brian B. Laird, and Benedict J. Leikuhler University of California, San Diego, Departents of Matheatics and Cheistry, La Jolla, CA 993,

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 5 Lecture Oscillations (Chapter 6) What We Did Last Tie Analyzed the otion of a heavy top Reduced into -diensional proble of θ Qualitative behavior Precession + nutation Initial condition

More information

Proc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES

Proc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co

More information

Curious Bounds for Floor Function Sums

Curious Bounds for Floor Function Sums 1 47 6 11 Journal of Integer Sequences, Vol. 1 (018), Article 18.1.8 Curious Bounds for Floor Function Sus Thotsaporn Thanatipanonda and Elaine Wong 1 Science Division Mahidol University International

More information

Physics 221A: HW3 solutions

Physics 221A: HW3 solutions Physics 22A: HW3 solutions October 22, 202. a) It will help to start things off by doing soe gaussian integrals. Let x be a real vector of length, and let s copute dxe 2 xt Ax, where A is soe real atrix.

More information

Hee = ~ dxdy\jj+ (x) 'IJ+ (y) u (x- y) \jj (y) \jj (x), V, = ~ dx 'IJ+ (x) \jj (x) V (x), Hii = Z 2 ~ dx dy cp+ (x) cp+ (y) u (x- y) cp (y) cp (x),

Hee = ~ dxdy\jj+ (x) 'IJ+ (y) u (x- y) \jj (y) \jj (x), V, = ~ dx 'IJ+ (x) \jj (x) V (x), Hii = Z 2 ~ dx dy cp+ (x) cp+ (y) u (x- y) cp (y) cp (x), SOVIET PHYSICS JETP VOLUME 14, NUMBER 4 APRIL, 1962 SHIFT OF ATOMIC ENERGY LEVELS IN A PLASMA L. E. PARGAMANIK Khar'kov State University Subitted to JETP editor February 16, 1961; resubitted June 19, 1961

More information

On Nonlinear Controllability of Homogeneous Systems Linear in Control

On Nonlinear Controllability of Homogeneous Systems Linear in Control IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 1, JANUARY 2003 139 On Nonlinear Controllability of Hoogeneous Systes Linear in Control Jaes Melody, Taer Başar, and Francesco Bullo Abstract This work

More information

The Distribution of the Covariance Matrix for a Subset of Elliptical Distributions with Extension to Two Kurtosis Parameters

The Distribution of the Covariance Matrix for a Subset of Elliptical Distributions with Extension to Two Kurtosis Parameters journal of ultivariate analysis 58, 96106 (1996) article no. 0041 The Distribution of the Covariance Matrix for a Subset of Elliptical Distributions with Extension to Two Kurtosis Paraeters H. S. Steyn

More information

Implementing Non-Projective Measurements via Linear Optics: an Approach Based on Optimal Quantum State Discrimination

Implementing Non-Projective Measurements via Linear Optics: an Approach Based on Optimal Quantum State Discrimination Ipleenting Non-Projective Measureents via Linear Optics: an Approach Based on Optial Quantu State Discriination Peter van Loock 1, Kae Neoto 1, Willia J. Munro, Philippe Raynal 2, Norbert Lütkenhaus 2

More information

Finite fields. and we ve used it in various examples and homework problems. In these notes I will introduce more finite fields

Finite fields. and we ve used it in various examples and homework problems. In these notes I will introduce more finite fields Finite fields I talked in class about the field with two eleents F 2 = {, } and we ve used it in various eaples and hoework probles. In these notes I will introduce ore finite fields F p = {,,...,p } for

More information

A remark on a success rate model for DPA and CPA

A remark on a success rate model for DPA and CPA A reark on a success rate odel for DPA and CPA A. Wieers, BSI Version 0.5 andreas.wieers@bsi.bund.de Septeber 5, 2018 Abstract The success rate is the ost coon evaluation etric for easuring the perforance

More information

All you need to know about QM for this course

All you need to know about QM for this course Introduction to Eleentary Particle Physics. Note 04 Page 1 of 9 All you need to know about QM for this course Ψ(q) State of particles is described by a coplex contiguous wave function Ψ(q) of soe coordinates

More information

Entangling characterization of (SWAP) 1/m and Controlled unitary gates

Entangling characterization of (SWAP) 1/m and Controlled unitary gates Entangling characterization of (SWAP) / and Controlled unitary gates S.Balakrishnan and R.Sankaranarayanan Departent of Physics, National Institute of Technology, Tiruchirappalli 65, India. We study the

More information

Kinetic Theory of Gases: Elementary Ideas

Kinetic Theory of Gases: Elementary Ideas Kinetic Theory of Gases: Eleentary Ideas 17th February 2010 1 Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion

More information

A NOTE ON HILBERT SCHEMES OF NODAL CURVES. Ziv Ran

A NOTE ON HILBERT SCHEMES OF NODAL CURVES. Ziv Ran A NOTE ON HILBERT SCHEMES OF NODAL CURVES Ziv Ran Abstract. We study the Hilbert schee and punctual Hilbert schee of a nodal curve, and the relative Hilbert schee of a faily of curves acquiring a node.

More information

Geometrical approach in atomic physics: Atoms of hydrogen and helium

Geometrical approach in atomic physics: Atoms of hydrogen and helium Aerican Journal of Physics and Applications 014; (5): 108-11 Published online October 0, 014 (http://www.sciencepublishinggroup.co/j/ajpa) doi: 10.11648/j.ajpa.014005.1 ISSN: 0-486 (Print); ISSN: 0-408

More information

DERIVING PROPER UNIFORM PRIORS FOR REGRESSION COEFFICIENTS

DERIVING PROPER UNIFORM PRIORS FOR REGRESSION COEFFICIENTS DERIVING PROPER UNIFORM PRIORS FOR REGRESSION COEFFICIENTS N. van Erp and P. van Gelder Structural Hydraulic and Probabilistic Design, TU Delft Delft, The Netherlands Abstract. In probles of odel coparison

More information

Lecture #8-3 Oscillations, Simple Harmonic Motion

Lecture #8-3 Oscillations, Simple Harmonic Motion Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.

More information

NUMERICAL MODELLING OF THE TYRE/ROAD CONTACT

NUMERICAL MODELLING OF THE TYRE/ROAD CONTACT NUMERICAL MODELLING OF THE TYRE/ROAD CONTACT PACS REFERENCE: 43.5.LJ Krister Larsson Departent of Applied Acoustics Chalers University of Technology SE-412 96 Sweden Tel: +46 ()31 772 22 Fax: +46 ()31

More information

Scattering and bound states

Scattering and bound states Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states

More information

}, (n 0) be a finite irreducible, discrete time MC. Let S = {1, 2,, m} be its state space. Let P = [p ij. ] be the transition matrix of the MC.

}, (n 0) be a finite irreducible, discrete time MC. Let S = {1, 2,, m} be its state space. Let P = [p ij. ] be the transition matrix of the MC. Abstract Questions are posed regarding the influence that the colun sus of the transition probabilities of a stochastic atrix (with row sus all one) have on the stationary distribution, the ean first passage

More information

Direct characterization of quantum dynamics: General theory

Direct characterization of quantum dynamics: General theory PHYSICAL REVIEW A 75, 062331 2007 Direct characterization of quantu dynaics: General theory M. Mohseni 1,2 and D. A. Lidar 2,3 1 Departent of Cheistry and Cheical Biology, Harvard University, 12 Oxford

More information

A := A i : {A i } S. is an algebra. The same object is obtained when the union in required to be disjoint.

A := A i : {A i } S. is an algebra. The same object is obtained when the union in required to be disjoint. 59 6. ABSTRACT MEASURE THEORY Having developed the Lebesgue integral with respect to the general easures, we now have a general concept with few specific exaples to actually test it on. Indeed, so far

More information

Extension of CSRSM for the Parametric Study of the Face Stability of Pressurized Tunnels

Extension of CSRSM for the Parametric Study of the Face Stability of Pressurized Tunnels Extension of CSRSM for the Paraetric Study of the Face Stability of Pressurized Tunnels Guilhe Mollon 1, Daniel Dias 2, and Abdul-Haid Soubra 3, M.ASCE 1 LGCIE, INSA Lyon, Université de Lyon, Doaine scientifique

More information

List Scheduling and LPT Oliver Braun (09/05/2017)

List Scheduling and LPT Oliver Braun (09/05/2017) List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)

More information

Kinetic Theory of Gases: Elementary Ideas

Kinetic Theory of Gases: Elementary Ideas Kinetic Theory of Gases: Eleentary Ideas 9th February 011 1 Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion of

More information

USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta

USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta 1 USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS By: Ian Bloland, Augustana Capus, University of Alberta For: Physics Olypiad Weeend, April 6, 008, UofA Introduction: Physicists often attept to solve

More information

Probability Distributions

Probability Distributions Probability Distributions In Chapter, we ephasized the central role played by probability theory in the solution of pattern recognition probles. We turn now to an exploration of soe particular exaples

More information

13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization

13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization 3 Haronic oscillator revisited: Dirac s approach and introduction to Second Quantization. Dirac cae up with a ore elegant way to solve the haronic oscillator proble. We will now study this approach. The

More information

Comparison of Stability of Selected Numerical Methods for Solving Stiff Semi- Linear Differential Equations

Comparison of Stability of Selected Numerical Methods for Solving Stiff Semi- Linear Differential Equations International Journal of Applied Science and Technology Vol. 7, No. 3, Septeber 217 Coparison of Stability of Selected Nuerical Methods for Solving Stiff Sei- Linear Differential Equations Kwaku Darkwah

More information

13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices

13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay

More information

1 Bounding the Margin

1 Bounding the Margin COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost

More information

Computational and Statistical Learning Theory

Computational and Statistical Learning Theory Coputational and Statistical Learning Theory Proble sets 5 and 6 Due: Noveber th Please send your solutions to learning-subissions@ttic.edu Notations/Definitions Recall the definition of saple based Radeacher

More information

The Lagrangian Method vs. other methods (COMPARATIVE EXAMPLE)

The Lagrangian Method vs. other methods (COMPARATIVE EXAMPLE) The Lagrangian ethod vs. other ethods () This aterial written by Jozef HANC, jozef.hanc@tuke.sk Technical University, Kosice, Slovakia For Edwin Taylor s website http://www.eftaylor.co/ 6 January 003 The

More information

A1. Find all ordered pairs (a, b) of positive integers for which 1 a + 1 b = 3

A1. Find all ordered pairs (a, b) of positive integers for which 1 a + 1 b = 3 A. Find all ordered pairs a, b) of positive integers for which a + b = 3 08. Answer. The six ordered pairs are 009, 08), 08, 009), 009 337, 674) = 35043, 674), 009 346, 673) = 3584, 673), 674, 009 337)

More information

Non-Parametric Non-Line-of-Sight Identification 1

Non-Parametric Non-Line-of-Sight Identification 1 Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,

More information

Intelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines

Intelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines Intelligent Systes: Reasoning and Recognition Jaes L. Crowley osig 1 Winter Seester 2018 Lesson 6 27 February 2018 Outline Perceptrons and Support Vector achines Notation...2 Linear odels...3 Lines, Planes

More information

3.3 Variational Characterization of Singular Values

3.3 Variational Characterization of Singular Values 3.3. Variational Characterization of Singular Values 61 3.3 Variational Characterization of Singular Values Since the singular values are square roots of the eigenvalues of the Heritian atrices A A and

More information

Optical Properties of Plasmas of High-Z Elements

Optical Properties of Plasmas of High-Z Elements Forschungszentru Karlsruhe Techni und Uwelt Wissenschaftlishe Berichte FZK Optical Properties of Plasas of High-Z Eleents V.Tolach 1, G.Miloshevsy 1, H.Würz Project Kernfusion 1 Heat and Mass Transfer

More information

On the summations involving Wigner rotation matrix elements

On the summations involving Wigner rotation matrix elements Journal of Matheatical Cheistry 24 (1998 123 132 123 On the suations involving Wigner rotation atrix eleents Shan-Tao Lai a, Pancracio Palting b, Ying-Nan Chiu b and Harris J. Silverstone c a Vitreous

More information

Lectures 8 & 9: The Z-transform.

Lectures 8 & 9: The Z-transform. Lectures 8 & 9: The Z-transfor. 1. Definitions. The Z-transfor is defined as a function series (a series in which each ter is a function of one or ore variables: Z[] where is a C valued function f : N

More information

2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all

2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either

More information

Complex Quadratic Optimization and Semidefinite Programming

Complex Quadratic Optimization and Semidefinite Programming Coplex Quadratic Optiization and Seidefinite Prograing Shuzhong Zhang Yongwei Huang August 4 Abstract In this paper we study the approxiation algoriths for a class of discrete quadratic optiization probles

More information

Research Article Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials

Research Article Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials Discrete Dynaics in Nature and Society Volue 202, Article ID 927953, pages doi:055/202/927953 Research Article Soe Forulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynoials Yuan He and

More information

Principal Components Analysis

Principal Components Analysis Principal Coponents Analysis Cheng Li, Bingyu Wang Noveber 3, 204 What s PCA Principal coponent analysis (PCA) is a statistical procedure that uses an orthogonal transforation to convert a set of observations

More information

Optimal quantum detectors for unambiguous detection of mixed states

Optimal quantum detectors for unambiguous detection of mixed states PHYSICAL REVIEW A 69, 06318 (004) Optial quantu detectors for unabiguous detection of ixed states Yonina C. Eldar* Departent of Electrical Engineering, Technion Israel Institute of Technology, Haifa 3000,

More information

FAST DYNAMO ON THE REAL LINE

FAST DYNAMO ON THE REAL LINE FAST DYAMO O THE REAL LIE O. KOZLOVSKI & P. VYTOVA Abstract. In this paper we show that a piecewise expanding ap on the interval, extended to the real line by a non-expanding ap satisfying soe ild hypthesis

More information

Linear Transformations

Linear Transformations Linear Transforations Hopfield Network Questions Initial Condition Recurrent Layer p S x W S x S b n(t + ) a(t + ) S x S x D a(t) S x S S x S a(0) p a(t + ) satlins (Wa(t) + b) The network output is repeatedly

More information

E0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis

E0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds

More information

Soft Computing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis

Soft Computing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis Soft Coputing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis Beverly Rivera 1,2, Irbis Gallegos 1, and Vladik Kreinovich 2 1 Regional Cyber and Energy Security Center RCES

More information

The proofs of Theorem 1-3 are along the lines of Wied and Galeano (2013).

The proofs of Theorem 1-3 are along the lines of Wied and Galeano (2013). A Appendix: Proofs The proofs of Theore 1-3 are along the lines of Wied and Galeano (2013) Proof of Theore 1 Let D[d 1, d 2 ] be the space of càdlàg functions on the interval [d 1, d 2 ] equipped with

More information