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This chapter mathematically analyzes the behavior of a card trick described by an undergraduate math major at the University of California, Los Angeles, named Norman Gilbreath. Stated succinctly, this trick can be performed by handing an audience member a deck of cards, letting them cut the deck several times, and then having them deal a number of cards from the top into a pile. The audience member takes the two piles (the cards in hand and the set now piled on the table) and riffle-shuffles them together. The magician now hides the deck (under a cloth, perhaps, or behind the back) and proceeds to produce one pair of cards after another in which one card is black and the other is red.

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by Arabic and Greek, mathematicians. The chapter then addresses two questions about continued fractions of algebraic functions. The first concerns the behavior of the solvability of the polynomial Pell equation for families of polynomials. It must be noted that these questions are related to problems of unlikely intersections in families of Jacobians of hyperelliptic curves (or generalized Jacobians). The chapter also reviews several classical definitions and results related to the continued fraction expansion of real numbers and illustrates them by examples.

The semantic theory developed for a formal language in the previous chapters is now formulated in the language itself. Applying the semantic theory to this formulation, one finds that the theory of truth itself turns out to be neither true nor false. An analogy to the theory of continued fractions is proposed.

The projected system matrix in Krylov subspace methods consists of moments of the original system matrix with respect to the initial residual(s). This hints that Krylov subspace methods can be viewed as matching moments model reduction. Through the simplified Stieltjes moment problem, orthogonal polynomials, continued fractions, and Jacobi matrices, we thus obtain the Gauss–Christoffel quadrature representation of the conjugate gradient method (CG). It is described how generalisations to the non-Hermitian case can easily be achieved using the Vorobyev method of moments. Finally, the described results and their historical roots are linked with the model reduction of large-scale dynamical systems. The chapter demonstrates the strong connection between Krylov subspace methods used in state-of-the-art numerical calculations and classical topics of analysis and approximation theory. Since moments represent very general objects, this suggests that Krylov subspace methods might have much wider applications beyond their immediate context of solving algebraic problems.

This introductory chapter provides an overview of the three topics discussed in this book: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. These topics were covered during the Alpbach Summerschool 2016, the celebration of the tenth session with outstanding speakers covering very different research areas in arithmetic and Diophantine geometry. The first course was given by Peter Scholze on local Shimura varieties and features recent results concerning the local Langlands conjecture. It considers the unpublished theorem which states that for each local Shimura datum, there exists a so-called local Shimura variety, which is a (pro-)rigid analytic space. The second course was given by Umberto Zannier and deals with a rather classical theme but from a modern point of view. His course is on hyperelliptic continued fractions and generalized Jacobians, using the classical Pell equation as the starting point. The third course was given by Shou-Wu Zhang and originates in the famous Chowla–Selberg formula, which was taken up by Gross and Zagier in 1984 to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Building on this work, X. Yuan, Shou-Wu Zhang, and Wei Zhang succeeded in proving the Gross–Zagier formula on Shimura curves and shortly later they verified the Colmez conjecture on average. In the course, Zhang presents new interesting aspects of the formula.

Basic theory of continued fractions: finite continued fractions (for rational numbers) and infinite continued fractions (for irrational numbers). This also includes computation of the quadratic number with a given periodic continued fraction, conjugate quadratic numbers, and approximation of reals and convergents of continued fractions. The chapter then takes on quadratic bounds for the error term and Legendre’s theorem, and reals having the same expansion up to rank n. Next, it discusses Lagrange number and its characterization as an upper limit, and equivalence of real numbers (equivalent numbers have the same Lagrange number). Finally, it covers ordering real numbers by alternating lexicographical order on continued fractions.

Throughout his life, the great mathematician Leonhard Euler spent most of his free time on music, to which he devoted his first book. This chapter discusses how he reformulated the ordering of musical intervals on a new mathematical basis. For this purpose, Euler devised a “degree of agreeableness” that numerically indexed musical intervals and chords, replacing ancient canons of numerical simplicity with a new criterion based on pleasure. Euler applied this criterion (and Aristotle’s teachings about the pleasure of tragedy) to argue that minor intervals and chords evoke sadness through their greater numerical complexity, hence lower degree of agreeableness than the major. This work involved extensive attention to ratios and numerical factorization immediately preceding his subsequent interest in continued fractions and number theory. Having devised a new kind of index, Euler was prepared to put forward indices that would address novel problems like the Königsberg bridge problem and the construction of polyhedra, basic concepts of what we now call topology. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).