LEADER 03158nam 2200493Ia 450 001 9910438146303321 005 20200520144314.0 010 $a3-319-00888-9 024 7 $a10.1007/978-3-319-00888-2 035 $a(OCoLC)853659945 035 $a(MiFhGG)GVRL6UQV 035 $a(CKB)2670000000533733 035 $a(MiAaPQ)EBC1398612 035 $a(EXLCZ)992670000000533733 100 $a20111102d2013 uy 0 101 0 $aeng 135 $aurun|---uuuua 181 $ctxt 182 $cc 183 $acr 200 10$aMarkov's theorem and 100 years of the uniqueness conjecture $ea mathematical journey from irrational numbers to perfect matchings /$fMartin Aigner 205 $a1st ed. 2013. 210 $aCham $cSpringer$d2013 215 $a1 online resource (x, 257 pages) $cillustrations, portraits 225 0 $aGale eBooks 300 $aDescription based upon print version of record. 311 $a3-319-00887-0 311 $a3-319-03309-3 320 $aIncludes bibliographical references and index. 327 $aApproximation of Irrational Numbers -- Markov's Theorem and the Uniqueness Conjecture -- The Markov Tree -- The Cohn Tree -- The Modular Group SL(2,Z) -- The Free Group F2 -- Christoffel Words -- Sturmian Words -- Proof of Markov's Theorem -- The Uniqueness Conjecture.  . 330 $aThis book takes the reader on a mathematical journey, from a number-theoretic point of view, to the realm of Markov?s theorem and the uniqueness conjecture, gradually unfolding many beautiful connections until everything falls into place in the proof of Markov?s theorem. What makes the Markov theme so attractive is that it appears in an astounding variety of different fields, from number theory to combinatorics, from classical groups and geometry to the world of graphs and words. On the way, there are also introductory forays into some fascinating topics that do not belong to the standard curriculum, such as Farey fractions, modular and free groups, hyperbolic planes, and algebraic words. The book closes with a discussion of the current state of knowledge about the uniqueness conjecture, which remains an open challenge to this day. All the material should be accessible to upper-level undergraduates with some background in number theory, and anything beyond this level is fully explained in the text. This is not a monograph in the usual sense concentrating on a specific topic. Instead, it narrates in five parts ? Numbers, Trees, Groups, Words, Finale ? the story of a discovery in one field and its many manifestations in others, as a tribute to a great mathematical achievement and as an intellectual pleasure, contemplating the marvellous unity of all mathematics. 606 $aMarkov processes 606 $aStochastic processes 615 0$aMarkov processes. 615 0$aStochastic processes. 676 $a519.233 700 $aAigner$b Martin$f1942-$0281930 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910438146303321 996 $aMarkov's theorem and 100 years of the uniqueness conjecture$91469554 997 $aUNINA