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Thinking in Problems: How Mathematicians Find Creative - download pdf or read online

Introduces key problem-solving concepts in depth
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This concise, self-contained textbook supplies an in-depth examine problem-solving from a mathematician’s point-of-view. every one bankruptcy builds off the former one, whereas introducing a number of equipment that may be used whilst imminent any given challenge. inventive pondering is the most important to fixing mathematical difficulties, and this publication outlines the instruments essential to enhance the reader’s technique.

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The matter of enumerating maps (a map is a collection of polygonal "countries" on a global of a definite topology, no longer inevitably the airplane or the sector) is a vital challenge in arithmetic and physics, and it has many functions starting from statistical physics, geometry, particle physics, telecommunications, biology, .

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4 All Genus It is convenient to define the generating function of maps regardless of their genus. Thanks to Eq. e. an equality between the coefficients of terms with equal powers of t. To any order in t, the sum over g is finite. Therefore the sum over g is NOT a large N expansion, it is a small t expansion. 6 We will see later in this book, that each series Wk has a finite radius of convergency, and is in fact an algebraic function of t. But the all genus generating function Wk is not algebraic, and may have a vanishing radius of convergency.

A formal integral is a formal series (an asymptotic series) whose coefficients are Gaussian integrals, it is not necessarily a convergent series (in fact in our case it is always not convergent, it has a vanishing radius of convergency). Wick’s theorem [86] gives a method to compute Gaussian matrix integrals in a combinatorial way, it relates formal matrix integrals to generating functions for maps. The relationship between formal matrix integrals and maps, was first noticed in 1974 by ’t Hooft (1999 physics Nobel prize) in the context of the study of strong nuclear interactions [48], and then really introduced as a tool for studying maps by physicists Brezin-Itzykson-Parisi-Zuber in 1978 [74].

G/ tj Tl1 Cj 1;K jD3 which we call “Loop equation” or “higher genus Tutte’s equation”. 2) 24 1 Maps and Discrete Surfaces Here, we have presented only an intuitive derivation, and we present a more rigorous derivation in Chap. 2, with a very different technique, called loop equations for formal matrix integrals. 4 Exercises Exercise 1 Count all connected quadrangulations with n4 D 1 and n4 D 2 quadrangles, count them with their symmetry factors and according to their topology. Answer: There is one planar quadrangulation with n4 D 1 quadrangle, and it has symmetry factor 2, and one quadrangulation of genus g D 1 with n4 D 1 quadrangle, and it has symmetry factor 4.

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Analyse Combinatoire by Louis Comtet


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