Chapter 3 Group Theory p. 1 - Remark: This is only a brief summary of most important results of groups theory with respect

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 3 Group Theory p. 1 - Remark: This is only a brief summary of most important results of groups theory with respect"

Transcription

1 Chapter 3 Group Theory p Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For a more detaled descrpton see references. 3. Defnton of a group We assume a set of elements G {A, B, C,...}. Furthermore, we assume that there s a defnton of a combnaton of two elements AB, whch we denote as the product of two elements. G s a group f the followng condtons are satsfed:. Closure: AB G The product of two elements of the group s also an element of the group.. Identty element: The s an element E G, such that AE EA A for all A G. A neutral element exsts, whch has no effect on the other elements f the group. 3. Assocate law: A ( BC ) ( AB ) C for all A, B, C G. 4. Inerse element: The s an element A G, such that A A E for all A G. In nerse element exsts for all elements of the group, whch nerts the acton of a gen element. For some, but not for all groups, the commutate law holds (commutate law: AB BA for all A G ). These groups are called Abelan groups. Wthout proof (see e.g. F. A. Cotton): C ( ABC ) A B. (3.: Example for a group) 3. rder of a group The number of elements of a group s called the order of the group.

2 Chapter 3 Group Theory p ultplcaton table For any group, we can set up a multplcaton table, whch tabulates the results of the products of two elements. Wthout proof: Eery lne and column contans eery group element exactly once. No lne or column s dentcal to another one. (3.: How many types of groups are there wth three elements? Dere the multplcaton tables.) 3.4 Cyclc groups In a cyclc group all n elements are generated by powers of the frst element n n G {E( A ), A, A,...A }. (3.3: Example of a cyclc group) An mportant property of cyclc groups s that they are Abelan (as n m n+ m A A A A m A n ). 3.5 Subgroups A subset of the elements of the group G can tself form a group U. We call U a subgroup of G. (3.4: Example of a subgroup) 3.6 Symmetry groups The complete set of symmetry elements of a molecule, surface or crystal has the mathematc structure of a group. The set s called the symmetry group. (3.5: Example of a symmetry group. H molecule: show that the symmetry elements behae lke a group). 3.7 Classes We defne a smlarty transformaton B X AX

3 Chapter 3 Group Theory p. 3 - whch transform some element A by means of another element X nto some other element B. If A and X are elements of the group G, the elements are called conugated elements. A complete set of elements, whch s conugated to one another s called a class of elements of the group. The classes hae a fgurate meanng: Those symmetry operatons belong to the same class, whch can be reached by a transformaton of the coordnate system, whch s part of the symmetry group. (3.6: Example: Dde the elements of the symmetry group C 4 nto classes). (The defnton of classes wll greatly smplfy the work wth symmetry groups). 3.8 epresentaton of symmetry operatons by matrces We can represent all symmetry operatons dscussed so far n the form of a matrx. In the smplest case, these matrces act on ponts X r n three-dmensonal space and assgn a new poston bass coordnated r r X X (Note: f nstead we consder a bass transformaton defnng the new B r n terms of the old one as X r are r r - X A X ): (3.7: Examples for matrx representatons of symmetry operatons). r r B AB, the coordnates of the pont n the new 3.9 epresentatons of a group A set of matrces whch upon multplcaton behaes analogous to the elements of a group s called a representaton of the group. Example: We consder the transformaton of a pont X r n three dmensonal space accordng to the symmetry operatons of group C V.

4 Chapter 3 Group Theory p. 4 - E ; C ; ; Wth respect to matrx multplcaton, these matrces follow the multplcaton table of group C V. 3. educble and rreducble representatons As specfc case of matrces are so called block-dagonal matrces. Block-dagonal matrces are multpled accordng to the scheme,.e. the multplcaton can be reduced to the multplcatons of the sub-matrces of lower dmenson: y b x a y x b a For the specfc example consdered, the matrces are completely dagonal,.e. all blocks are of dmenson. Accordngly we can reduce the three dmensonal representaton gen aboe nto three one-dmensonal representatons, whch agan are representatons of the symmetry group C V : E E E ; C C C ; ; We consder a representaton of a group by a set of matrces of dmenson n. Addtonally, we consder a bass transformaton to a new coordnate system B B r r A, wth the coordnates of a ector n the new bass n terms of the old coordnates X X r - A. The representaton of the group n the new bass s A A A -. For any representaton, we can search for the bass transformaton, whch yelds a set of representatons wth lowest possble dmenson. We

5 Chapter 3 Group Theory p. 5 - denote a representaton wth the lowest possble dmenson as an rreducble representaton and a representaton wth hgher than mnmum dmenson as a reducble representaton. The example shows that there are rreducble representatons (bref: rreps) of dfferent type,.e. behang dfferently wth respect to the symmetry operatons contaned n the group. 3. Character of a matrx We defne the character χ of a matrx Γ as the sum oer the dagonal elements: Γ χ Γ. Γ (3.8: Character of matrces). The character of a matrx has an mportant property: It s narant upon a transformaton of the bass. (3.9: Character of matrces). Ths s qute handy, as n the followng t allows us to work wth characters nstead of the full representaton matrces, rrespecte of a specfc choce of the bass. 3. Propertes of rreducble representatons: GT great orthogonalty theorem (for proof see textbooks) Γ ( ) mnγ ( ) m n * δ δ δ mm nn h l l wth h : order of the group : symmetry operaton of the group Γ ( ) : matrx representaton for operaton of the rreducble representaton of type l : dmenson of the -th type of rreducble representaton

6 Chapter 3 Group Theory p. 6 - The ectors consstng of correspondng elements of the representaton matrces are orthogonal and normalzed. There are a number of smpler conclusons followng from the GT, whch can be easly proen (see e.g. A. F. Cotton), e.g: l χ ( E) h ; sum oer the dmenson squares of the rreps (sum oer the character squares of the dentty element) s equal to the order of the group. χ ( ) h ; sum oer character sqares oer all symmetry operaton for a gen type of representaton s equal to the order of the group. ( ) ( ) χ χ hδ : Character ectors of dfferent rreps are orthogonal. The characters of representaton matrces for a gen type of rrep for operatons belongng to a common class are dentcal. The number of classes s equal to the number of rreps. (3.: Deelop the characters and representaton matrces for the symmetry group C V from the aboe statements). 3.3 Analyss of reducble representatons The followng dea s a key pont for a large number of applcatons n the next chapters of ths course. We assume that Γ ( ) s a reducble representaton of the symmetry group G wth the correspondng characters χ ( ). We would lke to know, how many rreducble representatons of symmetry type are contaned n Γ ( ). For ths reason we assume that we hae transformed Γ ( ) to ts blockdagonal form ( ) As the characters are narant wth respect to ths transformaton, we obtan: χ ( ) χ ( ) a χ ( ) wth ( ) χ : character of -th rrep of group Γ. a : number of tmes that -th rrep s contaned n Γ ( ) By multplyng wth χ ( ) and summng oer all operatons of the group:

7 χ ( ) χ ( ) a χ ( ) χ ( ) a a χ ha h χ ( ) χ ( ) ( ) χ ( ) hδ a χ( ) χ ( ) Chapter 3 Group Theory p. 7 - Here, all we need as an nput s the characters of the rreps of the group. These are lsted n the so called character tables. 3.4 Character tables ost mportant nformaton whch s requred to work wth a gen symmetry group s summarzed n the co called character table. Example: C 4V group name (Schoenfless) symmetry operatons ordered by classes symmetry propertes of some functons and ther classfcaton by rreps C 4V E C 4 C d A z x +y, z A - - z B - - x -y B - - (x, y) xy E - ( x, y ) (xz, yz)

8 Chapter 3 Group Theory p. 8 - st of rreducble representatons: ullken notaton: () dm. rreps: A, B dm. rreps: E 3 dm. rreps: T 4 dm. rreps: G 5 dm. rreps: H () A/B: symmetrc / antsymmetrc wth respect to rotaton by π/n around prncple axs C n. (3) Index /: symmetrc / antsymmetrc wth respect to rotaton by π around C axs (perpendcular to C n ). (4) or : symmetrc / antsymmetrc wth respect to h. (5) g/u: symmetrc / antsymmetrc wth respect to.

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

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

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

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

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

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

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

+ + + - - 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

More information

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

21 Vectors: The Cross Product & Torque

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

More information

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

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

The complex inverse trigonometric and hyperbolic functions

The complex inverse trigonometric and hyperbolic functions Physcs 116A Wnter 010 The complex nerse trgonometrc and hyperbolc functons In these notes, we examne the nerse trgonometrc and hyperbolc functons, where the arguments of these functons can be complex numbers

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

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

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

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

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

IT09 - Identity Management Policy

IT09 - Identity Management Policy IT09 - Identty Management Polcy Introducton 1 The Unersty needs to manage dentty accounts for all users of the Unersty s electronc systems and ensure that users hae an approprate leel of access to these

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

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

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

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

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

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

V-1 V. Electronic Spectroscopy. Eigen value: lowest U el is the ground state by electronic dipole selection rule;

V-1 V. Electronic Spectroscopy. Eigen value: lowest U el is the ground state by electronic dipole selection rule; V-1 V. Electronc Spectroscopy What we have done so far s use B O approxmaton to elmnate φ el (q,q) For example dd datomc, put asde φ el and focused on vbratonal & rotatonal soluton Now must consder electronc

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More 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

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

Traffic State Estimation in the Traffic Management Center of Berlin

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

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More 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

The Full-Wave Rectifier

The Full-Wave Rectifier 9/3/2005 The Full Wae ectfer.doc /0 The Full-Wae ectfer Consder the followng juncton dode crcut: s (t) Power Lne s (t) 2 Note that we are usng a transformer n ths crcut. The job of ths transformer s to

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

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

A Performance Analysis of View Maintenance Techniques for Data Warehouses

A Performance Analysis of View Maintenance Techniques for Data Warehouses A Performance Analyss of Vew Mantenance Technques for Data Warehouses Xng Wang Dell Computer Corporaton Round Roc, Texas Le Gruenwald The nversty of Olahoma School of Computer Scence orman, OK 739 Guangtao

More information

A Simple Approach to Clustering in Excel

A Simple Approach to Clustering in Excel A Smple Approach to Clusterng n Excel Aravnd H Center for Computatonal Engneerng and Networng Amrta Vshwa Vdyapeetham, Combatore, Inda C Rajgopal Center for Computatonal Engneerng and Networng Amrta Vshwa

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation of Deutsch's Algorithm Using Mathcad Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

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

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

Politecnico di Torino. Porto Institutional Repository

Politecnico di Torino. Porto Institutional Repository Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC

More 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

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

APPLICATIONS OF VARIATIONAL PRINCIPLES TO DYNAMICS AND CONSERVATION LAWS IN PHYSICS

APPLICATIONS OF VARIATIONAL PRINCIPLES TO DYNAMICS AND CONSERVATION LAWS IN PHYSICS APPLICATIONS OF VAIATIONAL PINCIPLES TO DYNAMICS AND CONSEVATION LAWS IN PHYSICS DANIEL J OLDE Abstract. Much of physcs can be condensed and smplfed usng the prncple of least acton from the calculus of

More information

A Bonus-Malus System as a Markov Set-Chain

A Bonus-Malus System as a Markov Set-Chain A Bonus-alus System as a arov Set-Chan Famly and frst name: emec algorzata Organsaton: Warsaw School of Economcs Insttute of Econometrcs Al. epodleglosc 64 2-554 Warszawa Poland Telephone number: +48 6

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

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

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

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Study on Model of Risks Assessment of Standard Operation in Rural Power Network

Study on Model of Risks Assessment of Standard Operation in Rural Power Network Study on Model of Rsks Assessment of Standard Operaton n Rural Power Network Qngj L 1, Tao Yang 2 1 Qngj L, College of Informaton and Electrcal Engneerng, Shenyang Agrculture Unversty, Shenyang 110866,

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

QUESTIONS, How can quantum computers do the amazing things that they are able to do, such. cryptography quantum computers

QUESTIONS, How can quantum computers do the amazing things that they are able to do, such. cryptography quantum computers 2O cryptography quantum computers cryptography quantum computers QUESTIONS, Quantum Computers, and Cryptography A mathematcal metaphor for the power of quantum algorthms Mark Ettnger How can quantum computers

More information

Study on CET4 Marks in China s Graded English Teaching

Study on CET4 Marks in China s Graded English Teaching Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes

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

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng

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

Forecasting the Direction and Strength of Stock Market Movement

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

More information

Small pots lump sum payment instruction

Small pots lump sum payment instruction For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested

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

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

MONITORING METHODOLOGY TO ASSESS THE PERFORMANCE OF GSM NETWORKS

MONITORING METHODOLOGY TO ASSESS THE PERFORMANCE OF GSM NETWORKS Electronc Communcatons Commttee (ECC) wthn the European Conference of Postal and Telecommuncatons Admnstratons (CEPT) MONITORING METHODOLOGY TO ASSESS THE PERFORMANCE OF GSM NETWORKS Athens, February 2008

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 Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

Performance Analysis and Coding Strategy of ECOC SVMs

Performance Analysis and Coding Strategy of ECOC SVMs Internatonal Journal of Grd and Dstrbuted Computng Vol.7, No. (04), pp.67-76 http://dx.do.org/0.457/jgdc.04.7..07 Performance Analyss and Codng Strategy of ECOC SVMs Zhgang Yan, and Yuanxuan Yang, School

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

LECTURE 2: CRYSTAL BASES

LECTURE 2: CRYSTAL BASES LECTURE 2: CRYSTAL BASES STEVEN SAM AND PETER TINGLEY Today I ll defne crystal bases, and dscuss ther basc propertes. Ths wll nclude the tensor product rule and the relatonshp between the crystals B(λ)

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

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

More information

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

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

More information

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer

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

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

Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers

Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers Multplcaton Algorthms for Radx- RN-Codngs and Two s Complement Numbers Jean-Luc Beuchat Projet Arénare, LIP, ENS Lyon 46, Allée d Itale F 69364 Lyon Cedex 07 jean-luc.beuchat@ens-lyon.fr Jean-Mchel Muller

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

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

I. INTRODUCTION. 1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes

I. INTRODUCTION. 1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes he Knematc Analyss of a Symmetrcal hree-degree-of-freedom lanar arallel Manpulator Damen Chablat and hlppe Wenger Insttut de Recherche en Communcatons et Cybernétque de Nantes, rue de la Noë, 442 Nantes,

More information

copyright 1997 Bruce A. McCarl and Thomas H. Spreen.

copyright 1997 Bruce A. McCarl and Thomas H. Spreen. Appendx I: Usng Summaton Notaton Wth GAMS... AI-1 AI.1 Summaton Mechancs... AI-1 AI.1.1 Sum of an Item.... AI-1 AI.1.2 Multple Sums... AI-2 AI.1.3 Sum of Two Items... AI-2 AI.2 Summaton Notaton Rules...

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

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and POLYSA: A Polynomal Algorthm for Non-bnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n

More information

LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit

LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE

More information

An Integrated Semantically Correct 2.5D Object Oriented TIN. Andreas Koch

An Integrated Semantically Correct 2.5D Object Oriented TIN. Andreas Koch An Integrated Semantcally Correct 2.5D Object Orented TIN Andreas Koch Unverstät Hannover Insttut für Photogrammetre und GeoInformaton Contents Introducton Integraton of a DTM and 2D GIS data Semantcs

More information

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

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

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

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

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

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Rotation Kinematics, Moment of Inertia, and Torque

Rotation Kinematics, Moment of Inertia, and Torque Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute

More information

TENSOR GAUGE FIELDS OF DEGREE THREE

TENSOR GAUGE FIELDS OF DEGREE THREE TENSOR GAUGE FIELDS OF DEGREE THREE E.M. CIOROIANU Department of Physcs, Unversty of Craova, A. I. Cuza 13, 2585, Craova, Romana, EU E-mal: manache@central.ucv.ro Receved February 2, 213 Startng from a

More information