01235nam--2200397---450-99000099352020331620050927105433.00-582-74608-60099352USA010099352(ALEPH)000099352USA01009935220020226d1985----km-y0itay0103----baengGB||||||||001yyTeaching English through Englisha course in classroom language and techniquesJane WllisHarlowLongman1985XVI, 192 p25 cmLongman handbooks for language teachers2001Longman handbooks for language teachersLingua ingleseInsegnamento428WILLIS,Jane553047ITsalbcISBD990000993520203316VII.3.D. 454(IL i II 84)67045 ECIL i IIBKECOPATTY9020020226USA01110320020403USA011741PATRY9020040406USA011708COPAT59020050519USA010854COPAT59020050927USA011054Teaching English through English975086UNISA03304oam 2200565I 450 991078757770332120230803194840.00-429-07388-71-4665-9019-X10.1201/b15455 (CKB)2670000000395032(EBL)1375223(SSID)ssj0000876645(PQKBManifestationID)11495020(PQKBTitleCode)TC0000876645(PQKBWorkID)10905952(PQKB)11113029(MiAaPQ)EBC1375223(OCoLC)858282234(EXLCZ)99267000000039503220180331d2014 uy 0engur|n|---|||||txtccrAlgebras, graphs and their applications /Ilwoo Cho ; edited by Palle E.T. JorgensenBoca Raton :CRC Press,[2014]1 online resource (442 p.)Description based upon print version of record.1-4665-9020-3 Includes bibliographical references and index.Front Cover; Dedication; Preface; Contents; Chapter 1: Algebra on Graphs; Chapter 2: Representations and Operator Algebras of Graph Groupoids; Chapter 3: Operator Theory on Graphs; Chapter 4: Fractals on Graph Groupoids; Chapter 5: Entropy Theory on Graphs; Chapter 6: Jones Index Theory on Graph Groupoids; Chapter 7: Network Theory on Graphs; Chapter 8: K-Theory on GraphsPreface In this book, we consider algebra on directed graphs. From combinatorial objects, direct graphs, we establish corresponding algebraic objects which become groupoids. We call such groupoids graph groupoids. Connected with groupoid theory, we investigate the properties of graph groupoids. From this investigation, we can realize that graph groupoids act like the free groups in group theory. In other words, the study of graph groupoids is understood as groupoidal version of free-group theory. As application, we observe how graph groupoids are playing their role in different mathematical and scientific areas, including general groupoid theory, representation theory, automata theory, operator algebra (von Neumann algebra theory, C*-algebra theory, free probability, and index theory), noncommutative dynamical systems (groupoid dynamical systems), operator theory (spectral theory), fractal theory, information theory (entropy theory), and network theory, etc. We can check all operated groupoids (for instance, groupoid sums, product groupoids, quotient groupoids, etc) of graph groupoids are graph groupoids, too. This means that the study of operated groupoids of graph groupoids becomes nothing but studying other graph groupoids. It makes us easy to handle graph-groupoid related structures--Provided by publisher.GroupoidsOperator theoryGroupoids.Operator theory.511.54511/.54MAT002000MAT037000bisacshCho Ilwoo1548445Jorgensen Palle E. T.1947-MiAaPQMiAaPQMiAaPQBOOK9910787577703321Algebras, graphs and their applications3805477UNINA