CHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION

Size: px
Start display at page:

Download "CHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION"

Transcription

1 CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr theor uses algorthms and algebra to analse data It s used b socal scentsts to analse nteractons between actors and can be used to complement analses carred out usng game theor or other analtcal tools The concepts of fuzz matr have been defned n chapter A standard fuzz matr s the fuzz matr of the followng form where all the entres are less than or equal to : = () 3 Operatons on Two Fuzz Matrces Let us defne two fuzz matrces and Y of order as ()

2 9 3 6 Y (3) Addton of Two Fuzz Matrces Two fuzz matrces and Y are compatble under matr addton f the are of same order For nstance for the fuzz matrces and Y gven b () and (3) we have, Clearl Y () Y s a matr, but not a fuzz matr ence we can conclude that addton of two fuzz matrces compatble under addton need not be a fuzz matr owever, addton of two standard fuzz matrces s a fuzz matr Mamum Operaton of Two Fuzz Matrces Two fuzz matres are conformable for mamum operaton f the are of the same order ence for two matrces and Y of order m n, mama of these two matrces s a matr c ma(, ) Ma (, Y) c of order m n, where ence for the matrces and Y gven b () and (3) we have, Ma,Y ()

3 Mnmum Operaton of Two Fuzz Matrces Two fuzz matres are conformable for mnmum operaton f the are of same order ence for two matrces these two matrces s a matr where mn(, ) and Y of order m n, fuzz mnma of Mn (, Y) c of order m n c ence for the matrces and Y gven b () and (3) we have, Mn (, Y) (6) In case of fuzz matrces, we have seen that the addton s not defned, where as the mama and mnma operatons are defned Clearl under the mamum and mnmum operatons the resultant matr s agan a fuzz matr of the same order and thus s n somewa analogous to our usual addton Product of Two Fuzz Matrces To fnd the product of two fuzz matrces wth Y gven b () and (3), where and Y are compatble under multplcaton; e the number of column of equal to the number of row of Y ; stll we ma not have the product Y to be a fuzz matr 38 6 Y (7) Clearl Y s not a fuzz matr Thus we need to defne a compatble operaton analogous to multplcaton of two fuzz matrces so that the product agan happens to be a fuzz matr owever, even for ths new operaton f the product Y s to be defned we need the number of columns

4 of s equal to the number of rows of Y e the fuzz matr should be compatble for multplcaton The two tpes of operatons whch we can have are ma-mn operaton and mn-ma operatonthese operatons are defned below: To fnd Y usng ma-mn operatons, we have c c 3 c c 3 c c c c c c c3 c C (8) c c c c 3 where, c = ma {mn (3, ), mn (7, 3), mn (8, 8),mn (9, 3)} = ma {3, 3, 8, 3}= 8 c = ma {mn (3, ), mn (7, 6), mn (8, 9),mn (9, ) = ma {, 6, 8, }= 8 and so on Thus, we get C (9) Now suppose that the fuzz matrces and Y are gven b () and (3), on applng mn-ma operaton, we get where, Y = mn {ma (3, ), ma (7, 3), ma (8, 8), ma (9, 3)} = mn {, 7, 8, 9} = 7 = mn {ma (3, ), ma (7, 6), ma (8, 9), ma (9, )} = mn {3, 7, 9, 9} = 3 and so on Thus, we have 3 () 3

5 () From (9) and (), t s clear that C Some eperts ma lke to work wth mama-mnma value and some wth the mnma-mama value and accordngl the can adopt t ence we can have the product of two fuzz matrces Y = C or as per our requrement It s also observed that Y s defned but Y ma not be defned Conugate of Fuzz Matr Let = ( ) m n be a fuzz matr, the matr obtaned b replacng each element of b ts dual - ) ( s called conugate fuzz matr of and s denoted as Conugate of fuzz matrces and Y fuzz matr product of ma-mn and mn-ma operatons as gven below: () ma mn Y mn - ma Y () () mn ma Y ma - mn Y (3) () and (3) can be llustrated b the followng eample: Illustraton() Let and Y are two fuzz matrces gven b () and (3) respectvel Then, 3 ma mn Y () 6 3 mn - ma Y ()

6 From ()and () we conclude () holds Illustraton: () Let and Y are two fuzz matrces gven b () and (3) respectvel Then, mn ma Y ma - mn Y From (6) and (7) we conclude () holds Fuzz matr Theor (FST) has been appled to man felds such as control, sgnal and mage processng, medcne, the econom, etc The results show that FMT elds effcent solutons to varous problems In crsp set theor, a member of a set s represented b or So, n a crsp matr, a member ether belongs or doesn't belong to a class owever, n FMT, a member of a set s represented b a degree between and The degree s called membershp degree whch shows belongng degree of the member to the class The membershp degree s computed usng the membershp functon obtaned b the eperts on the subect or a pror knowledge In the present chapter we prove that the set of fuzz matrces forms a lattce under matr mnma and matr mama bnar fuzz operatons n secton In secton 3 we ntroduce and characterze a new fuzz nformaton measure on fuzz matr and ts propertes have been studed n secton In secton we defne a new measure of nformaton on fuzz bnar relaton A Lattce of Fuzz Matrces A partal ordered set n whch ever par of elements has both least upper bound and greatest lower bound s called a lattce refer to Trembla and Manohar (997) or n

7 other words Algebra L,, s called a lattce f L s a non empt set, and are bnar operatons on L, satsfng (A-) Idempotent ( A A= A or A A = A), (A-) Commutatve ( A B = B A or A B = B A) (A-3) Assocatve ( A ( B C ) = ( A B ) C or A ( B C )=( A B ) C ) (A-) Absorpton law ( A ( A B ) = A or A ( A B ) = A) Theorem Let A s a set of m n fuzz matrces where J,,, then A J under matr mnma and matr mama operatons A forms a lattce Proof To prove that the gven set A of fuzz matrces forms a lattce, we shall show that A satsfes the four propertes (A-) to (A-) (A-) Idempotent Law For an fuzz matr A, the followng holds: mn( A, A ) A and ma( A, A ) A ence Idempotent Law s satsfed (A-) Commutatve Law It can be easl verfed that for all the fuzz matrces A anda A, the followng holds: mn( A, A ) mn( A, A ) and ma( A, A ) ma( A, A ) Ths proves that Commutatve Law s satsfed (A-3) Assocatve Law For an fuzz matrces A, A A A, t can be proved that, k mn ( A, A ), A mn( A, A, A ) and ma ( A, A ), A ma( A, A, A ) k k ence we can conclude that A s assocatve under matr mama and matr mnma operatons k k 6

8 (A-) Absorpton Law For an fuzz matrces A, A, we can prove that ma A,mn A, A A mn A,ma A, A A and ence Absorpton Law holds Snce t satsfes all the four propertes (A-) to (A-) of the lattces, therefore the set of the matrces s a lattce under matr mama and matr mnma Ma(A,A,A k ) Ma(A, A ) Ma(A, A k ) Ma(A, A k ) A A A k Mn(A, A ) Mn(A, A k Mn(A, A k ) Mn(A,A, A k ) Fgure A lattce of fuzz matrces under matr mama and matr mnma Fgure s the pctoral representaton of the lattce wth the fuzz matrces A, A and A k wth the least element mnma ( A, A, A k ) and the greatest element mama ( A, A, A k ) 3 Informaton Measure on Fuzz Matr Fuzz nformaton measures the degree of fuzzness of a fuzz set It s pecular to mathematcs, nformaton theor and computer scence It s an mportant concept n fuzz set theor and has been successfull appled to pattern recognton, mage processng, classfer desgn and neural network structure etc 7

9 The concept of nformaton measures was developed b Shannon (98) to measure the uncertant of a probablt dstrbuton The concept of fuzz set was ntroduced b Zadeh (966) who also developed hs own theor to measure the ambgut of a fuzz set Let,,, n be the unverse set of dscourse and ( ) A be membershp functon defned on A Then A ( ), A ( ), A ( n ) le between (, ) and these are not probabltes because ther sum s not unt owever, A( ) A ( ) n,,,, n, (3) A( ) s a probablt dstrbuton Thus Kaufman (98) defned entrop of a fuzz set A havng n support ponts as n ( A) A( ) log A( ) (3) log n Correspondng to entrop due to Shannon (98), eluca and Termn (97) suggested the followng measure of fuzz entrop: n ( A) ( )log ( ) ( ) log ( ) (33) A A Correspondng to (33) we propose the followng nformaton measure defned on fuzz matrces: A A m n ( ) log log, (3) where s the th (, ) element of the standard fuzz matr Theorem The fuzz nformaton measure gven b (3) s a vald measure Proof To prove that the gven measure s a vald measure of fuzz nformaton, we shall show that (3) satsfes the followng four propertes (P-) to (P-): 8

10 (P-) ( ) f onl f s non fuzz matr or crsp matr We know that log and log f onl f ether or, =,, It mples ( ) f onl f s non fuzz or crsp matr (P-) ( ) s mamum f and onl f s the most fuzz matr, e for all =,, fferentatng ( ) wth respect to, we have d d m n whch vanshes at log log (3) = Agan dfferentatng ( ) wth respect to, we have d d m n (36) Puttng n (36), we have d ( ) d ence ( ) s mamum f and onl f s most fuzz matr e : =,, (P-3) Sharpenng reduces the value of nformaton measure Let us consder, then d d m n log log (37) It mples ( ) s an ncreasng functon of n the regon Smlarl, we can prove that ( ) s a decreasng functon of n the regon ence we can conclude that ( ) s a concave functon 9

11 Let = ( such that ) m n then, * = (* ) m n : [,] such that be * for all =,, * Then * s called the sharpened verson of fuzz matr and snce ( ) s ncreasng functon of n the regon, therefore () f * ( *) ( ) n [,) (38) Smlarl f = ( such that then, * = n ) m n ( * ) m : [,] such that be * for all =,, * Then * s called the sharpened verson of fuzz matr and snce ( ) s decreasng functon of n the regon, therefore () f * ( *) ( ) (39) (38) and (39) together gves ( *) ( ) Thus (P-3) s proved (P-) ual propert e ( ) ( ) It s evdent from the defnton that ( ) ( ) ence ( ) satsfes all the essental four propertes of fuzz nformaton measure Thus t s a vald measure of fuzz nformaton We can call ths measure as fuzz matr nformaton measure Propertes of Fuzz Matr Informaton Measure Propert For an standard fuzz matrces and Y compatble under mama and mnma operatons, the followng holds: mn {, Y} ma {, Y} and mn {, Y} ma {, Y} Proof Let and Y are the two standard fuzz matrces Snce fuzz matr nformaton measure ( ) s an ncreasng functon of n the regon Therefore

12 mn {, Y} ma {, Y} and mn {, Y} ma {, Y} or mn {, Y} Y ma {, Y} Propert For an two fuzz matrces and Y, such that Y and Y ests, we have Y Y and Y Y Y Proof of ths s evdent, whch can be llustrated n table, consderng dfferent pars of fuzz matrces and Y Table Y Y Y Y Y Y Propert 3 For an fuzz matr we have, T, where T s the transpose of The followng table llustrates the proof for dfferent fuzz matr : Table T T Propert For an fuzz matr of order n n, let n be the set of square sub matrces, such that, e the nested form of the sub 3

13 matrces, where A B means order of B s less than order of A We have element 3 n and the greatest element The followng table3 verfes the result emprcall:, whch forms a lne lattce wth the least Table

14 Fgure s the pctoral representaton of the lne lattce / column lattce of the fuzz nformaton measures on fuzz matr of the nested matrces n Fgure The pctoral representaton of the lne lattce Propert For an fuzz matr of order m n and an arbtrar element a les between and, we have a a Fuzz Bnar Relaton Informaton Measure The word relaton suggests some famlar eamples of relatons such as the relaton of father to son, mother to son, etc Famlar eamples n arthmetc are the relatons such as greater than or less than and so on These eamples suggest that relatonshps est among two obects Such tpe of relatons s known as bnar relatons, between a par of obects e an set of order pars defnes a bnar relatons The relatonshp between three concdent lnes or a pont between two gven ponts s eamples of relatons among three obects Smlarl relatons can est among four or more obects 3

15 Fuzz Bnar Relaton Let,, 3,, and Y,, 3,, be two fnte sets, fuzz bnar relaton R whch ma assgn two or more elements of Y correspondng to each element of Let R be the fuzz bnar membershp relaton such as, 9,,,,,, 3, 3, 3,, 3,,,,,, () Operatons on Fuzz Bnar Relatons Fuzz bnar relaton s ver mportant because the can descrbe nteractons between varables () Intersecton of fuzz bnar relatons The ntersecton of two fuzz bnar relatons R and S s defned b, mn R S for each and () R S, () Unon of fuzz bnar relatons The unon of two fuzz bnar relatons R and S s defned b, ma R S for each and (3) R S, Fuzz bnar relatons pla an mportant role n fuzz modellng, fuzz dagnoss, and fuzz control These also have wde applcatons n felds such as pscholog, medcne, economcs, and socolog Fuzz Bnar Relaton Matr A Fuzz bnar relaton R from a fnte set to a fnte set Y can also be represented b a fuzz matr called the fuzz bnar relaton matr of R Fuzz bnar relaton matr s obtaned usng the membershp functon and the membershp degree Let,, 3,, and Y,, 3,, be two sets and the fuzz bnar

16 relaton defned on (,Y ) s gven b (), then the fuzz bnar relaton matr s as follows: R, Y () The nverse fuzz bnar relaton of R s the relaton R from Y to, and the correspondng matr s equal to the transpose of the matr R, Y, e R Y, R Y, T () Analogous to (3) we can defne the followng fuzz nformaton measure on fuzz bnar relaton matr (): m n R,Y ( R )ln ( R ) ( R ) ln ( R ), (6) th where, (R) s the, element of the bnar fuzz relaton matr R, Y On the lnes of proof of theorem, t can easl be verfed that (6) s a vald measure of fuzz nformaton owever, ths measure wll be called as fuzz bnar relaton nformaton measure

17 6 Concluson In the present chapter we have defned two bnar operatons on fuzz matrces We have also proved that the set of fuzz matrces forms a lattce under these bnar fuzz operatons Further we have ntroduced and characterzed a new nformaton measure on fuzz matr Propertes of ths proposed measure have also been studed Fuzz relaton plas an mportant role n fuzz modellng, fuzz dagnoss and fuzz control The also have applcatons n felds such as pscholog, medcne, economcs and socolog A fuzz bnar relaton R from a fnte set to a fnte set Y can also be represented b a fuzz matr called the fuzz bnar relaton matr of R The fuzz nformaton measure thus defned on fuzz bnar relaton matr s a vald measure of fuzz nformaton and can further be generalzed We can also stud ts applcaton; however, t s an open problem 6

CHAPTER 4. Vector Spaces

CHAPTER 4. Vector Spaces man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear

More information

CHAPTER 4 MAX-MIN AVERAGE COMPOSITION METHOD FOR DECISION MAKING USING INTUITIONISTIC FUZZY SETS

CHAPTER 4 MAX-MIN AVERAGE COMPOSITION METHOD FOR DECISION MAKING USING INTUITIONISTIC FUZZY SETS 56 CHAPER 4 MAX-MIN AVERAGE COMPOSIION MEHOD FOR DECISION MAKING USING INUIIONISIC FUZZY SES 4.1 INRODUCION Intutonstc fuzz max-mn average composton method s proposed to construct the decson makng for

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Complement of Type-2 Fuzzy Shortest Path Using Possibility Measure

Complement of Type-2 Fuzzy Shortest Path Using Possibility Measure Intern. J. Fuzzy Mathematcal rchve Vol. 5, No., 04, 9-7 ISSN: 30 34 (P, 30 350 (onlne Publshed on 5 November 04 www.researchmathsc.org Internatonal Journal of Complement of Type- Fuzzy Shortest Path Usng

More information

A New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems

A New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems Appled Mathematcal Scences, Vol. 4, 200, no. 2, 79-90 A New Algorthm for Fndng a Fuzzy Optmal Soluton for Fuzzy Transportaton Problems P. Pandan and G. Nataraan Department of Mathematcs, School of Scence

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

THE SUMMATION NOTATION Ʃ

THE SUMMATION NOTATION Ʃ Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the

More information

Double Layered Fuzzy Planar Graph

Double Layered Fuzzy Planar Graph Global Journal of Pure and Appled Mathematcs. ISSN 0973-768 Volume 3, Number 0 07), pp. 7365-7376 Research Inda Publcatons http://www.rpublcaton.com Double Layered Fuzzy Planar Graph J. Jon Arockaraj Assstant

More information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification

2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

Solutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1

Solutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1 Solutons to Homework 7, Mathematcs 1 Problem 1: a Prove that arccos 1 1 for 1, 1. b* Startng from the defnton of the dervatve, prove that arccos + 1, arccos 1. Hnt: For arccos arccos π + 1, the defnton

More information

Matrix-Norm Aggregation Operators

Matrix-Norm Aggregation Operators IOSR Journal of Mathematcs (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. PP 8-34 www.osrournals.org Matrx-Norm Aggregaton Operators Shna Vad, Sunl Jacob John Department of Mathematcs, Natonal Insttute of

More information

Power law and dimension of the maximum value for belief distribution with the max Deng entropy

Power law and dimension of the maximum value for belief distribution with the max Deng entropy Power law and dmenson of the maxmum value for belef dstrbuton wth the max Deng entropy Bngy Kang a, a College of Informaton Engneerng, Northwest A&F Unversty, Yanglng, Shaanx, 712100, Chna. Abstract Deng

More information

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS

A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs

More information

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented

More information

Interactive Bi-Level Multi-Objective Integer. Non-linear Programming Problem

Interactive Bi-Level Multi-Objective Integer. Non-linear Programming Problem Appled Mathematcal Scences Vol 5 0 no 65 3 33 Interactve B-Level Mult-Objectve Integer Non-lnear Programmng Problem O E Emam Department of Informaton Systems aculty of Computer Scence and nformaton Helwan

More information

Perfect Competition and the Nash Bargaining Solution

Perfect Competition and the Nash Bargaining Solution Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange

More information

The Minimum Universal Cost Flow in an Infeasible Flow Network

The Minimum Universal Cost Flow in an Infeasible Flow Network Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

A Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach

A Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach A Bayes Algorthm for the Multtask Pattern Recognton Problem Drect Approach Edward Puchala Wroclaw Unversty of Technology, Char of Systems and Computer etworks, Wybrzeze Wyspanskego 7, 50-370 Wroclaw, Poland

More information

Module 9. Lecture 6. Duality in Assignment Problems

Module 9. Lecture 6. Duality in Assignment Problems Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept

More information

Xiangwen Li. March 8th and March 13th, 2001

Xiangwen Li. March 8th and March 13th, 2001 CS49I Approxaton Algorths The Vertex-Cover Proble Lecture Notes Xangwen L March 8th and March 3th, 00 Absolute Approxaton Gven an optzaton proble P, an algorth A s an approxaton algorth for P f, for an

More information

Fuzzy Boundaries of Sample Selection Model

Fuzzy Boundaries of Sample Selection Model Proceedngs of the 9th WSES Internatonal Conference on ppled Mathematcs, Istanbul, Turkey, May 7-9, 006 (pp309-34) Fuzzy Boundares of Sample Selecton Model L. MUHMD SFIIH, NTON BDULBSH KMIL, M. T. BU OSMN

More information

The Second Anti-Mathima on Game Theory

The Second Anti-Mathima on Game Theory The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,

More information

Dynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)

Dynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence) /24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes

More information

Mathematical Preparations

Mathematical Preparations 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More information

International Journal of Mathematical Archive-3(3), 2012, Page: Available online through ISSN

International Journal of Mathematical Archive-3(3), 2012, Page: Available online through   ISSN Internatonal Journal of Mathematcal Archve-3(3), 2012, Page: 1136-1140 Avalable onlne through www.ma.nfo ISSN 2229 5046 ARITHMETIC OPERATIONS OF FOCAL ELEMENTS AND THEIR CORRESPONDING BASIC PROBABILITY

More information

New Method for Solving Poisson Equation. on Irregular Domains

New Method for Solving Poisson Equation. on Irregular Domains Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad

More information

Solution Thermodynamics

Solution Thermodynamics Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs

More information

Research Article Relative Smooth Topological Spaces

Research Article Relative Smooth Topological Spaces Advances n Fuzzy Systems Volume 2009, Artcle ID 172917, 5 pages do:10.1155/2009/172917 Research Artcle Relatve Smooth Topologcal Spaces B. Ghazanfar Department of Mathematcs, Faculty of Scence, Lorestan

More information

Volume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].

Volume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2]. Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland

More information

On Similarity Measures of Fuzzy Soft Sets

On Similarity Measures of Fuzzy Soft Sets Int J Advance Soft Comput Appl, Vol 3, No, July ISSN 74-853; Copyrght ICSRS Publcaton, www-csrsorg On Smlarty Measures of uzzy Soft Sets PINAKI MAJUMDAR* and SKSAMANTA Department of Mathematcs MUC Women

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducible Nonnegative Interval Matrix Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

Assortment Optimization under MNL

Assortment Optimization under MNL Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.

More information

Lecture 3. Ax x i a i. i i

Lecture 3. Ax x i a i. i i 18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Neryškioji dichotominių testo klausimų ir socialinių rodiklių diferencijavimo savybių klasifikacija

Neryškioji dichotominių testo klausimų ir socialinių rodiklių diferencijavimo savybių klasifikacija Neryškoj dchotomnų testo klausmų r socalnų rodklų dferencjavmo savybų klasfkacja Aleksandras KRYLOVAS, Natalja KOSAREVA, Julja KARALIŪNAITĖ Technologcal and Economc Development of Economy Receved 9 May

More information

Unit 5: Quadratic Equations & Functions

Unit 5: Quadratic Equations & Functions Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and

More information

Lecture 5 Decoding Binary BCH Codes

Lecture 5 Decoding Binary BCH Codes Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

General theory of fuzzy connectedness segmentations: reconciliation of two tracks of FC theory

General theory of fuzzy connectedness segmentations: reconciliation of two tracks of FC theory General theory of fuzzy connectedness segmentatons: reconclaton of two tracks of FC theory Krzysztof Chrs Ceselsk Department of Mathematcs, West Vrgna Unversty and MIPG, Department of Radology, Unversty

More information

CHALMERS, GÖTEBORGS UNIVERSITET. SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS. COURSE CODES: FFR 135, FIM 720 GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET. SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS. COURSE CODES: FFR 135, FIM 720 GU, PhD CHALMERS, GÖTEBORGS UNIVERSITET SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS COURSE CODES: FFR 35, FIM 72 GU, PhD Tme: Place: Teachers: Allowed materal: Not allowed: January 2, 28, at 8 3 2 3 SB

More information

The Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices

The Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan

More information

find (x): given element x, return the canonical element of the set containing x;

find (x): given element x, return the canonical element of the set containing x; COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:

More information

Lecture 10: Dimensionality reduction

Lecture 10: Dimensionality reduction Lecture : Dmensonalt reducton g The curse of dmensonalt g Feature etracton s. feature selecton g Prncpal Components Analss g Lnear Dscrmnant Analss Intellgent Sensor Sstems Rcardo Guterrez-Osuna Wrght

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for

More information

An efficient algorithm for multivariate Maclaurin Newton transformation

An efficient algorithm for multivariate Maclaurin Newton transformation Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,

More information

NP-Completeness : Proofs

NP-Completeness : Proofs NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

For all questions, answer choice E) NOTA" means none of the above answers is correct.

For all questions, answer choice E) NOTA means none of the above answers is correct. 0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For

More information

A REVIEW OF ERROR ANALYSIS

A REVIEW OF ERROR ANALYSIS A REVIEW OF ERROR AALYI EEP Laborator EVE-4860 / MAE-4370 Updated 006 Error Analss In the laborator we measure phscal uanttes. All measurements are subject to some uncertantes. Error analss s the stud

More information

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product 12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton

More information

The lower and upper bounds on Perron root of nonnegative irreducible matrices

The lower and upper bounds on Perron root of nonnegative irreducible matrices Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College

More information

Linear Regression Analysis: Terminology and Notation

Linear Regression Analysis: Terminology and Notation ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented

More information

CS 468 Lecture 16: Isometry Invariance and Spectral Techniques

CS 468 Lecture 16: Isometry Invariance and Spectral Techniques CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that

More information

FREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,

FREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced, FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then

More information

Measurement Indices of Positional Uncertainty for Plane Line Segments Based on the ε

Measurement Indices of Positional Uncertainty for Plane Line Segments Based on the ε Proceedngs of the 8th Internatonal Smposum on Spatal ccurac ssessment n Natural Resources and Envronmental Scences Shangha, P R Chna, June 5-7, 008, pp 9-5 Measurement Indces of Postonal Uncertant for

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776

More information

SIMPLE LINEAR REGRESSION

SIMPLE LINEAR REGRESSION Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two

More information

A Novel Feistel Cipher Involving a Bunch of Keys supplemented with Modular Arithmetic Addition

A Novel Feistel Cipher Involving a Bunch of Keys supplemented with Modular Arithmetic Addition (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, A Novel Festel Cpher Involvng a Bunch of Keys supplemented wth Modular Arthmetc Addton Dr. V.U.K Sastry Dean R&D, Department of Computer

More information

e i is a random error

e i is a random error Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown

More information

9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers

9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers 9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw

More information

Feb 14: Spatial analysis of data fields

Feb 14: Spatial analysis of data fields Feb 4: Spatal analyss of data felds Mappng rregularly sampled data onto a regular grd Many analyss technques for geophyscal data requre the data be located at regular ntervals n space and/or tme. hs s

More information

On the Multicriteria Integer Network Flow Problem

On the Multicriteria Integer Network Flow Problem BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

arxiv:cs.cv/ Jun 2000

arxiv:cs.cv/ Jun 2000 Correlaton over Decomposed Sgnals: A Non-Lnear Approach to Fast and Effectve Sequences Comparson Lucano da Fontoura Costa arxv:cs.cv/0006040 28 Jun 2000 Cybernetc Vson Research Group IFSC Unversty of São

More information

Computing Correlated Equilibria in Multi-Player Games

Computing Correlated Equilibria in Multi-Player Games Computng Correlated Equlbra n Mult-Player Games Chrstos H. Papadmtrou Presented by Zhanxang Huang December 7th, 2005 1 The Author Dr. Chrstos H. Papadmtrou CS professor at UC Berkley (taught at Harvard,

More information

Exponential Type Product Estimator for Finite Population Mean with Information on Auxiliary Attribute

Exponential Type Product Estimator for Finite Population Mean with Information on Auxiliary Attribute Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10, Issue 1 (June 015), pp. 106-113 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Exponental Tpe Product Estmator

More information

Design and Optimization of Fuzzy Controller for Inverse Pendulum System Using Genetic Algorithm

Design and Optimization of Fuzzy Controller for Inverse Pendulum System Using Genetic Algorithm Desgn and Optmzaton of Fuzzy Controller for Inverse Pendulum System Usng Genetc Algorthm H. Mehraban A. Ashoor Unversty of Tehran Unversty of Tehran h.mehraban@ece.ut.ac.r a.ashoor@ece.ut.ac.r Abstract:

More information

Subset Topological Spaces and Kakutani s Theorem

Subset Topological Spaces and Kakutani s Theorem MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered

More information

Solving Fuzzy Linear Programming Problem With Fuzzy Relational Equation Constraint

Solving Fuzzy Linear Programming Problem With Fuzzy Relational Equation Constraint Intern. J. Fuzz Maeatcal Archve Vol., 0, -0 ISSN: 0 (P, 0 0 (onlne Publshed on 0 Septeber 0 www.researchasc.org Internatonal Journal of Solvng Fuzz Lnear Prograng Proble W Fuzz Relatonal Equaton Constrant

More information

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class

More information

Some modelling aspects for the Matlab implementation of MMA

Some modelling aspects for the Matlab implementation of MMA Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Pattern Classification

Pattern Classification Pattern Classfcaton All materals n these sldes ere taken from Pattern Classfcaton (nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wley & Sons, 000 th the permsson of the authors and the publsher

More information

A CLASS OF RECURSIVE SETS. Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA

A CLASS OF RECURSIVE SETS. Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA A CLASS OF RECURSIVE SETS Florentn Smarandache Unversty of New Mexco 200 College Road Gallup, NM 87301, USA E-mal: smarand@unmedu In ths artcle one bulds a class of recursve sets, one establshes propertes

More information

Group Theory Worksheet

Group Theory Worksheet Jonathan Loss Group Theory Worsheet Goals: To ntroduce the student to the bascs of group theory. To provde a hstorcal framewor n whch to learn. To understand the usefulness of Cayley tables. To specfcally

More information

The binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence

The binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the

More information

Edge Isoperimetric Inequalities

Edge Isoperimetric Inequalities November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary

More information

INTERVAL SEMIGROUPS. W. B. Vasantha Kandasamy Florentin Smarandache

INTERVAL SEMIGROUPS. W. B. Vasantha Kandasamy Florentin Smarandache Interval Semgroups - Cover.pdf:Layout 1 1/20/2011 10:04 AM Page 1 INTERVAL SEMIGROUPS W. B. Vasantha Kandasamy Florentn Smarandache KAPPA & OMEGA Glendale 2011 Ths book can be ordered n a paper bound reprnt

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

Neutrosophic Bi-LA-Semigroup and Neutrosophic N-LA- Semigroup

Neutrosophic Bi-LA-Semigroup and Neutrosophic N-LA- Semigroup Neutrosophc Sets Systems, Vol. 4, 04 9 Neutrosophc B-LA-Semgroup Neutrosophc N-LA- Semgroup Mumtaz Al *, Florentn Smarache, Muhammad Shabr 3 Munazza Naz 4,3 Department of Mathematcs, Quad--Azam Unversty,

More information

SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N)

SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N) SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) S.BOUCKSOM Abstract. The goal of ths note s to present a remarably smple proof, due to Hen, of a result prevously obtaned by Gllet-Soulé,

More information

System in Weibull Distribution

System in Weibull Distribution Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co

More information

Probability-Theoretic Junction Trees

Probability-Theoretic Junction Trees Probablty-Theoretc Juncton Trees Payam Pakzad, (wth Venkat Anantharam, EECS Dept, U.C. Berkeley EPFL, ALGO/LMA Semnar 2/2/2004 Margnalzaton Problem Gven an arbtrary functon of many varables, fnd (some

More information