Robert W. Vallin
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0012
- Subject:
- Mathematics, History of Mathematics
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, ...
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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.Less
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.
Laura Capuano, Peter Jossen, Christina Karolus, and Francesco Veneziano
- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0003
- Subject:
- Mathematics, Geometry / Topology
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 ...
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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.Less
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.
Tim Maudlin
- Published in print:
- 2004
- Published Online:
- January 2005
- ISBN:
- 9780199247295
- eISBN:
- 9780191601781
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199247293.003.0004
- Subject:
- Philosophy, Philosophy of Language
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 ...
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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.Less
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.
Jörg Liesen and Zdenek Strakos
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199655410
- eISBN:
- 9780191744174
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655410.003.0003
- Subject:
- Mathematics, Applied Mathematics, Algebra
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 ...
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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.Less
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.
Gisbert Wüstholz and Clemens Fuchs (eds)
- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the ...
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This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. The first course contains recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. The second course addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. The third course originates in the Chowla–Selberg formula and relates values of the L-function for elliptic curves with the height of Heegner points on the curves. It proves the Gross–Zagier formula on Shimura curves and verifies the Colmez conjecture on average.Less
This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. The first course contains recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. The second course addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. The third course originates in the Chowla–Selberg formula and relates values of the L-function for elliptic curves with the height of Heegner points on the curves. It proves the Gross–Zagier formula on Shimura curves and verifies the Colmez conjecture on average.
Gisbert Wüstholz and Clemens Fuchs (eds)
- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0001
- Subject:
- Mathematics, Geometry / Topology
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 ...
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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.Less
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.
Christophe Reutenauer
- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.003.0006
- Subject:
- Mathematics, Pure Mathematics
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 ...
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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.Less
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.
Richard Evan Schwartz
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.001.0001
- Subject:
- Mathematics, Educational Mathematics
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane ...
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Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.Less
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.
Christophe Reutenauer
- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.001.0001
- Subject:
- Mathematics, Pure Mathematics
Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It ...
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Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.Less
Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.
Peter Pesic
- Published in print:
- 2014
- Published Online:
- May 2017
- ISBN:
- 9780262027274
- eISBN:
- 9780262324380
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262027274.003.0010
- Subject:
- Music, History, Western
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 ...
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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).Less
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).
Jörg Liesen and Zdenek Strakos
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199655410
- eISBN:
- 9780191744174
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655410.001.0001
- Subject:
- Mathematics, Applied Mathematics, Algebra
This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov ...
More
This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov subspace methods are characterised by their orthogonality and minimisation properties. Projections onto highly nonlinear Krylov subspaces can be linked with the underlying problem of moments, and therefore Krylov subspace methods can be viewed as matching moments model reduction. This allows enlightening reformulations of questions from matrix computations into the language of orthogonal polynomials, Gauss–Christoffel quadrature, continued fractions, and, more generally, of Vorobyev method of moments. Using the concept of cyclic invariant subspaces conditions are studied that allow generation of orthogonal Krylov subspace bases via short recurrences. The results motivate the practically important distinction between Hermitian and non-Hermitian problems. Finally, the book thoroughly addresses the computational cost while using Krylov subspace methods. The investigation includes effects of finite precision arithmetic and focuses on the method of conjugate gradients (CG) and generalised minimal residuals (GMRES) as major examples. The book emphasises that algebraic computations must always be considered in the context of solving real-world problems, where the mathematical modelling, discretisation, and computation cannot be separated from each other. Moreover, the book underlines the importance of the historical context and it demonstrates that knowledge of early developments can play an important role in understanding and resolving very recent computational problems. Many extensive historical notes are therefore included as an inherent part of the text. The book ends with formulating some omitted issues and challenges which need to be addressed in future work. The book is intended as a research monograph which can be used in a wide scope of graduate courses on related subjects. It can be beneficial also for readers interested in the history of mathematics.Less
This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov subspace methods are characterised by their orthogonality and minimisation properties. Projections onto highly nonlinear Krylov subspaces can be linked with the underlying problem of moments, and therefore Krylov subspace methods can be viewed as matching moments model reduction. This allows enlightening reformulations of questions from matrix computations into the language of orthogonal polynomials, Gauss–Christoffel quadrature, continued fractions, and, more generally, of Vorobyev method of moments. Using the concept of cyclic invariant subspaces conditions are studied that allow generation of orthogonal Krylov subspace bases via short recurrences. The results motivate the practically important distinction between Hermitian and non-Hermitian problems. Finally, the book thoroughly addresses the computational cost while using Krylov subspace methods. The investigation includes effects of finite precision arithmetic and focuses on the method of conjugate gradients (CG) and generalised minimal residuals (GMRES) as major examples. The book emphasises that algebraic computations must always be considered in the context of solving real-world problems, where the mathematical modelling, discretisation, and computation cannot be separated from each other. Moreover, the book underlines the importance of the historical context and it demonstrates that knowledge of early developments can play an important role in understanding and resolving very recent computational problems. Many extensive historical notes are therefore included as an inherent part of the text. The book ends with formulating some omitted issues and challenges which need to be addressed in future work. The book is intended as a research monograph which can be used in a wide scope of graduate courses on related subjects. It can be beneficial also for readers interested in the history of mathematics.