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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 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 real numbers. A complex vector space s one n whch the scalars are complex numbers. Thus, f v 1, v 2,..., v m are vectors n a complex vector space, then a lnear combnaton s of the form where the scalars c are complex numbers. The complex verson of R n 1, c 2,..., c s the complex vector space m consstng of ordered n-tuples of complex numbers. Thus, a vector n has the form c 1 v 1 c 2 v 2 It s also convenent to represent vectors n v a 1 b 1 a 2 b 2. a n b n. c m v m v a 1 b 1, a 2 b 2,..., a n b n. by column matrces of the form As wth R n, the operatons of addton and scalar multplcaton n are performed component by component. EXAMPLE 1 Vector Operatons n Let v 1 2, 3 and u 2, 4 be vectors n the complex vector space C 2, and determne the followng vectors. (a) v u (b) 2 v (c) 3v 5 u Soluton (a) In column matrx form, the sum v u s 1 2 v u (b) Snce and 2 3 7, we have 2 v 2 1 2, 3 5, 7. (c) 3v 5 u 3 1 2, 3 5 2, 4 (c) 3 6, , 2 4 (c) 12, 11

2 456 CHAPTER 8 COMPLEX VECTOR SPACES Many of the propertes of R n are shared by. For nstance, the scalar multplcatve dentty s the scalar 1 and the addtve dentty n s,,,...,. The standard bass for s smply e 1 1,,,..., e 2, 1,,...,.. e n,,,..., 1 R n whch s the standard bass for. Snce ths bass contans n vectors, t follows that the dmenson of s n. Other bases exst; n fact, any lnearly ndependent set of n vectors n can be used, as we demonstrate n Example 2. EXAMPLE 2 Verfyng a Bass Show that v 1 v 2 v 3 S,,,,,,,, s a bass for C 3. Soluton Snce C 3 has a dmenson of 3, the set v 1, v 2, v 3 wll be a bass f t s lnearly ndependent. To check for lnear ndependence, we set a lnear combnaton of the vectors n S equal to as follows. c 1,, c 2, c 2,,, c 3,, Ths mples that { { { c 1 c 2 c 2 c 3. c 1 v 1 c 2 v 2 c 3 v 3,, c 1 c 2, c 2, c 3,, Therefore, c 1 c 2 c 3, and we conclude that v 1, v 2, v 3 s lnearly ndependent. EXAMPLE 3 Representng a Vector n by a Bass Use the bass S n Example 2 to represent the vector v 2,, 2.

3 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 457 Soluton By wrtng v c 1 v 1 c 2 v 2 c 3 v 3 c 1 c 2, c 2, c 3 2,, 2 we obtan c 1 c 2 2 c 2 c 3 2 whch mples that c 2 1 and c and Therefore, c 3 2 v 1 2 v 1 v v Try verfyng that ths lnear combnaton yelds 2,, 2. Other than, there are several addtonal examples of complex vector spaces. For nstance, the set of m n complex matrces wth matrx addton and scalar multplcaton forms a complex vector space. Example 4 descrbes a complex vector space n whch the vectors are functons. EXAMPLE 4 The Space of Complex-Valued Functons Consder the set S of complex-valued functons of the form where and are real-valued functons of a real varable. The set of complex numbers form the scalars for S and vector addton s defned by It can be shown that S, scalar multplcaton, and vector addton form a complex vector space. For nstance, to show that S s closed under scalar multplcaton, we let c a b be a complex number. Then s n S. f x) f 1 x f 2 x f 1 f 2 f x g x f 1 x f 2 x g 1 (x g 2 x f 1 x g 1 x f 2 x g 2 x. cf x a b f 1 x f 2 x af 1 x bf 2 x bf 1 x af 2 x

4 458 CHAPTER 8 COMPLEX VECTOR SPACES The defnton of the Eucldean nner product n s smlar to that of the standard dot product n R n, except that here the second factor n each term s a complex conjugate. Defnton of Eucldean Inner Product n Let u and v be vectors n. The Eucldean nner product of u and v s gven by u v u 1 v 1 u 2 v 2 u n v n. REMARK: Note that f u and v happen to be real, then ths defnton agrees wth the standard nner (or dot) product n R n. EXAMPLE 5 Fndng the Eucldean Inner Product n C 3 Soluton Determne the Eucldean nner product of the vectors u 2,, 4 5 and v 1, 2,. u v u 1 v 1 u 2 v 2 u 3 v Several propertes of the Eucldean nner product are stated n the followng theorem. Theorem 8.7 Propertes of the Eucldean Inner Product Let u, v, and w be vectors n and let k be a complex number. Then the followng propertes are true. 1. u v v u 2. u v w u w v w 3. ku v k u v 4. u kv k u v 5. u u 6. u u f and only f u. Proof We prove the frst property and leave the proofs of the remanng propertes to you. Let u u 1, u 2,..., u n and v v 1, v 2,..., v n. Then v u v 1 u 1 v 2 u 2... v n u n v 1 u 1 v 2 u 2... v n u n v 1 u 1 v 2 u 2... v n u n

5 SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 459 u 1 v 1 u 2 v 2 u v. u n v n We now use the Eucldean nner product n to defne the Eucldean norm (or length) of a vector n and the Eucldean dstance between two vectors n. Defnton of Norm The Eucldean norm (or length) of u n s denoted by u and s and Dstance n u u u 1 2. The Eucldean dstance between u and v s d u, v u v. The Eucldean norm and dstance may be expressed n terms of components as u u 1 2 u 2 2 u n d u, v u 1 v 1 2 u 2 v 2 2 u n v n EXAMPLE 6 Fndng the Eucldean Norm and Dstance n Determne the norms of the vectors u 2,, 4 5 and and fnd the dstance between u and v. v 1, 2, Soluton The norms of u and v are gven as follows. u u 1 2 u 2 2 u v v 1 2 v 2 2 v The dstance between u and v s gven by d u, v u v 1, 2,

6 46 CHAPTER 8 COMPLEX VECTOR SPACES Complex Inner Product Spaces The Eucldean nner product s the most commonly used nner product n. However, on occason t s useful to consder other nner products. To generalze the noton of an nner product, we use the propertes lsted n Theorem 8.7. Defnton of a Complex Inner Product Let u and v be vectors n a complex vector space. A functon that assocates wth u and v the complex number u, v s called a complex nner product f t satsfes the followng propertes. 1. u, v v, u 2. u v, w u, w v, w 3. ku, v k u, v 4. u, u and u, u f and only f u. A complex vector space wth a complex nner product s called a complex nner product space or untary space. EXAMPLE 7 A Complex Inner Product Space Let u u and be vectors n the complex space C 2 1, u 2 v v 1, v 2. Show that the functon defned by u, v u 1 v 1 2u 2 v 2 s a complex nner product. Soluton We verfy the four propertes of a complex nner product as follows. 1. v, u v 1 u 1 2v 2 u 2 u 1 v 1 2u 2 v 2 u, v 2. u v, w u 1 v 1 w 1 2 u 2 v 2 w 2 u 1 w 1 2u 2 w 2 v 1 w 1 2v 2 w 2 u, w v, w 3. ku, v ku 1 v 1 2 ku 2 v 2 k u 1 v 1 2u 2 v 2 k u, v 4. u, u u 1 u 1 2u 2 u 2 u u 2 2 Moreover, u, u f and only f u 1 u 2. Snce all propertes hold, u, v s a complex nner product.

7 SECTION 8.4 EXERCISES 461 SECTION 8.4 EXERCISES In Exercses 1 8, perform the ndcated operaton usng u, 3, v 2, 3, and w 4, u 2. 4w w 4. v 3w 5. u 2 v v 2 2 w 7. u v 2w 8. 2v 3 w u In Exercses 9 12, determne whether S s a bass for. 9. S 1,,, 1 1. S 1,,, S,,,,,,,, S 1,, 1, 2,, 1, 1, 1, 1 In Exercses 13 16, express v as a lnear combnaton of the followng bass vectors. (a),,,,,,,, (b) 1,,, 1, 1,,,, v 1, 2, 14. v 1, 1, v, 2, v,, In Exercses 17 24, determne the Eucldean norm of v. 17. v, 18. v 1, 19. v 3 6, 2 2. v 2 3, v 1, 2, 22. v,, In Exercses 25 3, determne the Eucldean dstance between u and v v 1 2,, 3, 1 v 2, 1, 2, 4 u 1,, v, u 2, 4,, v 2, 4, u, 2, 3, v, 1, u 2, 2,, v,, u 1,, v, 1 u 1, 2, 1, 2, v, 2,, 2 In Exercses 31 34, determne whether the set of vectors s lnearly ndependent or lnearly dependent ,,, , 1, 1,,, 1, 2, 1, 33. 1,, 1,,,,,, , 1,, 1,,,, 1, 1 In Exercses 35 38, determne whether the gven functon s a complex nner product, where u u 1, u 2 and v v 1, v u, v 4u 1 v 1 6u 2 v u, v u 1 v 1 u 2 v Let v 1,, and v 2,,. If v 3 z 1, z 2, z 3 and the set {v 1, v 2, v 3 } s not a bass for C 3, what does ths mply about z 1, z 2, and z 3? 4. Let v 1,, and v 2 1,, 1. Determne a vector v 3 such that v s a bass for C 3 1, v 2, v 3. In Exercses 41 45, prove the gven property where u, v, and w are vectors n and k s a complex number. 41. u v w u w v w u kv k u v u u f and only f u. 46. Let u, v be a complex nner product and k a complex number. How are u, v and u, kv related? In Exercses 47 and 48, determne the lnear transformaton T : C m that has the gven characterstcs In Exercses 49 52, the lnear transformaton T : C m s gven by T v Av. Fnd the mage of v and the premage of w u, v u 1 u 2 v 2 u, v u 1 v 1 2 u 2 v 2 T 1, 2, 1, T, 1, T, ) 2, 1, T,, A 1 A A 1 A, v 1 1, v , v 1, w 1 1 2, v 5 1, w 1 1 1, w 2 2, w 2 3 ku) u v k u v u 53. Fnd the kernel of the lnear transformaton gven n Exercse 49.

8 462 CHAPTER 8 COMPLEX VECTOR SPACES 54. 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 matrces. A 1 T 1 and T 2 A T 2 T T 1 T Determne whch of the followng sets are subspaces of the vector space of 2 2 complex matrces. (a) The set of 2 2 symmetrc matrces. (b) The set of 2 2 matrces A satsfyng A T A. (c) The set of 2 2 matrces n whch all entres are real. (d) The set of 2 2 dagonal matrces. 58. Determne whch of the followng sets are subspaces of the vector space of complex-valued functons (see Example 4). (a) The set of all functons f satsfyng f. (b) The set of all functons f satsfyng f 1. (c) The set of all functons f satsfyng f f. 8.5 UNITARY AND HERMITIAN MATRICES Problems nvolvng dagonalzaton of complex matrces, and the assocated egenvalue problems, requre the concept of untary and Hermtan matrces. These matrces roughly correspond to orthogonal and symmetrc real matrces. In order to defne untary and Hermtan matrces, we frst ntroduce the concept of the conjugate transpose of a complex matrx. Defnton of the Conjugate Transpose of a Complex Matrx The conjugate transpose of a complex matrx A, denoted by A*, s gven by A* A T where the entres of A are the complex conjugates of the correspondng entres of A. Note that f A s a matrx wth real entres, then A* A T. To fnd the conjugate transpose of a matrx, we frst calculate the complex conjugate of each entry and then take the transpose of the matrx, as shown n the followng example. EXAMPLE 1 Fndng the Conjugate Transpose of a Complex Matrx Determne A* for the matrx A

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

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

QUANTUM MECHANICS, BRAS AND KETS

QUANTUM MECHANICS, BRAS AND KETS PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

More information

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

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

More information

Linear Algebra for Quantum Mechanics

Linear Algebra for Quantum Mechanics prevous ndex next Lnear Algebra for Quantum Mechancs Mchael Fowler 0/4/08 Introducton We ve seen that n quantum mechancs, the state of an electron n some potental s gven by a ψ x t, and physcal varables

More information

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

where the coordinates are related to those in the old frame as follows. Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

The eigenvalue derivatives of linear damped systems

The eigenvalue derivatives of linear damped systems Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

More information

PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

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

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)

5. Simultaneous eigenstates: Consider two operators that commute:  η = a η (13.29) 5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let  satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB  η = a η (13.29) ˆB [  η ] = ˆB [a η ] = a

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

A Note on the Decomposition of a Random Sample Size

A Note on the Decomposition of a Random Sample Size A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton

More information

1 Example 1: Axis-aligned rectangles

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

More information

Formula of Total Probability, Bayes Rule, and Applications

Formula of Total Probability, Bayes Rule, and Applications 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

L10: Linear discriminants analysis

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

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem. Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set

More information

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

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Ring structure of splines on triangulations

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

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 Contents lsts avalable at ScVerse ScenceDrect Journal of Computatonal and Appled Mathematcs journal homepage: wwwelsevercom/locate/cam The nverse

More information

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

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

More information

CHAPTER 7 VECTOR BUNDLES

CHAPTER 7 VECTOR BUNDLES CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching) Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

More information

Generalizing the degree sequence problem

Generalizing the degree sequence problem Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

More information

Matrix Multiplication I

Matrix Multiplication I Matrx Multplcaton I Yuval Flmus February 2, 2012 These notes are based on a lecture gven at the Toronto Student Semnar on February 2, 2012. The materal s taen mostly from the boo Algebrac Complexty Theory

More information

The Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance

The Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance JOURNAL OF RESEARCH of the Natonal Bureau of Standards - B. Mathem atca l Scence s Vol. 74B, No.3, July-September 1970 The Dstrbuton of Egenvalues of Covarance Matrces of Resduals n Analyss of Varance

More information

AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES

AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES MATHEMATICS OF COMPUTATION Volume, Number, Pages S 5-578(XX)- AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI Abstract. We

More information

Non-degenerate Hilbert Cubes in Random Sets

Non-degenerate Hilbert Cubes in Random Sets Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Non-degenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne

More information

2. Linear Algebraic Equations

2. Linear Algebraic Equations 2. Lnear Algebrac Equatons Many physcal systems yeld smultaneous algebrac equatons when mathematcal functons are requred to satsfy several condtons smultaneously. Each condton results n an equaton that

More information

Nonbinary Quantum Error-Correcting Codes from Algebraic Curves

Nonbinary Quantum Error-Correcting Codes from Algebraic Curves Nonbnary Quantum Error-Correctng Codes from Algebrac Curves Jon-Lark Km and Judy Walker Department of Mathematcs Unversty of Nebraska-Lncoln, Lncoln, NE 68588-0130 USA e-mal: {jlkm, jwalker}@math.unl.edu

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

Math 131: Homework 4 Solutions

Math 131: Homework 4 Solutions Math 3: Homework 4 Solutons Greg Parker, Wyatt Mackey, Chrstan Carrck October 6, 05 Problem (Munkres 3.) Let {A n } be a sequence of connected subspaces of X such that A n \ A n+ 6= ; for all n. Then S

More information

Loop Parallelization

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

More information

IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1

IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1 Nov Sad J. Math. Vol. 36, No. 2, 2006, 0-09 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned

More information

The descriptive complexity of the family of Banach spaces with the π-property

The descriptive complexity of the family of Banach spaces with the π-property Arab. J. Math. (2015) 4:35 39 DOI 10.1007/s40065-014-0116-3 Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the π-property Receved: 25 March 2014

More information

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!

More information

Yves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 4-6

Yves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 4-6 Submtted to ECC 99 as a regular paper n Lnear Systems Postve transfer functons and convex optmzaton 1 Yves Genn, Yur Nesterov, Paul Van Dooren CESAME, Unverste Catholque de Louvan B^atment Euler, Avenue

More information

DiVA Digitala Vetenskapliga Arkivet

DiVA Digitala Vetenskapliga Arkivet DVA Dgtala Vetenskaplga Arkvet http://umudva-portalorg Ths s a book chapter publshed n Hgh-performance scentfc computng: algorthms and applcatons (ed Berry, MW; Gallvan, KA; Gallopoulos, E; Grama, A; Phlppe,

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

More information

Systematic Circuit Analysis (T&R Chap 3)

Systematic Circuit Analysis (T&R Chap 3) Systematc Crcut Analyss TR Chap ) Nodeoltage analyss Usng the oltages of the each node relate to a ground node, wrte down a set of consstent lnear equatons for these oltages Sole ths set of equatons usng,

More information

Lecture 18: Clustering & classification

Lecture 18: Clustering & classification O CPS260/BGT204. Algorthms n Computatonal Bology October 30, 2003 Lecturer: Pana K. Agarwal Lecture 8: Clusterng & classfcaton Scrbe: Daun Hou Open Problem In HomeWor 2, problem 5 has an open problem whch

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Questions that we may have about the variables

Questions that we may have about the variables Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato

More information

Chapter 7 Symmetry and Spectroscopy Molecular Vibrations p. 1 -

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

More information

Supplementary material: Assessing the relevance of node features for network structure

Supplementary material: Assessing the relevance of node features for network structure 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 information

Compiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing

Compiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing Complng for Parallelsm & Localty Dependence Testng n General Assgnments Deadlne for proect 4 extended to Dec 1 Last tme Data dependences and loops Today Fnsh data dependence analyss for loops General code

More information

The k-binomial Transforms and the Hankel Transform

The k-binomial Transforms and the Hankel Transform 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 9 (2006, Artcle 06.1.1 The k-bnomal Transforms and the Hankel Transform Mchael Z. Spvey Department of Mathematcs and Computer Scence Unversty of Puget

More information

Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering

Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering Out-of-Sample Extensons for LLE, Isomap, MDS, Egenmaps, and Spectral Clusterng Yoshua Bengo, Jean-Franços Paement, Pascal Vncent Olver Delalleau, Ncolas Le Roux and Mare Oumet Département d Informatque

More information

Point cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors

Point cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000-014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and

More information

Solutions to the exam in SF2862, June 2009

Solutions to the exam in SF2862, June 2009 Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

More information

Properties of American Derivative Securities

Properties of American Derivative Securities Capter 6 Propertes of Amercan Dervatve Securtes 6.1 Te propertes Defnton 6.1 An Amercan dervatve securty s a sequence of non-negatve random varables fg k g n k= suc tat eac G k s F k -measurable. Te owner

More information

Fisher Markets and Convex Programs

Fisher Markets and Convex Programs Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and

More information

Extending Probabilistic Dynamic Epistemic Logic

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

More information

REGULAR MULTILINEAR OPERATORS ON C(K) SPACES

REGULAR MULTILINEAR OPERATORS ON C(K) SPACES REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of

More information

Level Annuities with Payments Less Frequent than Each Interest Period

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

More information

On Lockett pairs and Lockett conjecture for π-soluble Fitting classes

On Lockett pairs and Lockett conjecture for π-soluble Fitting classes On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

More information

SOME MATHEMATICAL FOUNDATIONS OF CRYPTOGRAPHY

SOME MATHEMATICAL FOUNDATIONS OF CRYPTOGRAPHY SOME MATHEMATICAL FOUNDATIONS OF CRYPTOGRAPHY PARKER HAVIZA Abstract. Ths paper ntroduces three mathematcal methods of sharng encrypted nformaton: the Pohlg-Hellman Key Exchange and the ElGamal and RSA

More information

HÜCKEL MOLECULAR ORBITAL THEORY

HÜCKEL MOLECULAR ORBITAL THEORY 1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ

More information

FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER

FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 27695-7909 Introducton he convectve heat

More information

Fuzzy Regression and the Term Structure of Interest Rates Revisited

Fuzzy Regression and the Term Structure of Interest Rates Revisited Fuzzy Regresson and the Term Structure of Interest Rates Revsted Arnold F. Shapro Penn State Unversty Smeal College of Busness, Unversty Park, PA 68, USA Phone: -84-865-396, Fax: -84-865-684, E-mal: afs@psu.edu

More information

Basic Image Compression Algorithm and Introduction to JPEG Standard

Basic Image Compression Algorithm and Introduction to JPEG Standard Basc Image Compresson Algorthm and Introducton to JPEG Standard Pao-Yen Ln E-mal: r97942117@ntu.edu.tw Graduate Insttute of Communcaton Engneerng atonal Tawan Unversty, Tape, Tawan, ROC Abstract Because

More information

High Order Reverse Mode of AD Theory and Implementation

High Order Reverse Mode of AD Theory and Implementation Hgh Order Reverse Mode of AD Theory and Implementaton Mu Wang and Alex Pothen Department of Computer Scence Purdue Unversty September 30, 2016 Mu Wang and Alex Pothen Hgh Order Reverse AD September 30,

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

A Probabilistic Theory of Coherence

A 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 information

Software Alignment for Tracking Detectors

Software Alignment for Tracking Detectors Software Algnment for Trackng Detectors V. Blobel Insttut für Expermentalphysk, Unverstät Hamburg, Germany Abstract Trackng detectors n hgh energy physcs experments requre an accurate determnaton of a

More information

Nonlinear data mapping by neural networks

Nonlinear data mapping by neural networks Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

Logistic Regression. Steve Kroon

Logistic Regression. Steve Kroon Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro

More information

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

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

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz

More information

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

More information

LECTURE 1: MOTIVATION

LECTURE 1: MOTIVATION LECTURE 1: MOTIVATION STEVEN SAM AND PETER TINGLEY 1. Towards quantum groups Let us begn by dscussng what quantum groups are, and why we mght want to study them. We wll start wth the related classcal objects.

More information

SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar

SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

The Noether Theorems: from Noether to Ševera

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

More information

Some geometric probability problems involving the Eulerian numbers

Some geometric probability problems involving the Eulerian numbers Some geometrc probablty problems nvolvng the Euleran numbers Frank Schmdt Rodca Smon Department of Mathematcs The George Washngton Unversty Washngton, DC 20052 smon@math.gwu.edu Dedcated to Herb Wlf on

More information

Regression Models for a Binary Response Using EXCEL and JMP

Regression Models for a Binary Response Using EXCEL and JMP SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STAT-TECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal

More information

A CHARACTERIZATION FOR THE LENGTH OF CYCLES OF THE N - NUMBER DUCCI GAME

A CHARACTERIZATION FOR THE LENGTH OF CYCLES OF THE N - NUMBER DUCCI GAME A CHARACTERIZATION FOR THE LENGTH OF CYCLES OF THE N - NUMBER DUCCI GAME NEIL J. CALKIN DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-1907 AND JOHN G. STEVENS AND DIANA M. THOMAS

More information

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

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

More information

A Computer Technique for Solving LP Problems with Bounded Variables

A Computer Technique for Solving LP Problems with Bounded Variables Dhaka Unv. J. Sc. 60(2): 163-168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1 (4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

More information

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space. Algebrac Quantum Mechancs, Algebrac Spnors and Hlbert Space. B. J. Hley. Theoretcal Physcs Research Unt, Brkbeck, Malet Street, London WCE 7HX. b.hley@bbk.ac.uk Abstract. The orthogonal Clfford algebra

More information