Continued Fractions Research Papers

Möbius transformations mapping the unit disk into itself

2007, The Ramanujan Journal

We consider linear fractional transformations T n which map the unit disk U into itself with the property that T n (U) ⊆ T n−1 (U) ⊆ U for all n. Clearly, the closed sets T n (U) form a nested sequence of circular disks, and thus has a... more

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Two theorems concerning rational approximations

by Norman Carey and

2012, Journal of Mathematics and Music

The two theorems of the title constitute the mathematical results underlying well-formed scale theory. This paper includes the purely mathematical portion of a manuscript from 1988, which the authors cited the following year in N. Carey... more

Since there must be two intervals having the same span and different sizes, we must be able to choose an interval r such that r = (sj, 5(¢4a) has size ag and r’ = mod n) (Si+1 mod N> $(i+d+1) moa w) BAS Size Ba. See Figure 1. Interval r’ is simply r plus one step, and minus one step. By construction, r’ contains more 3; steps than r, so the only possibility is that r’ contains one more 3 step, i.e., 8g = @gt+1—a1, or ag—Bg = 1 —f 1.

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Aspects of well-formed scales

by Norman Carey and

1989, Music Theory Spectrum

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and... more

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A new Fortran 90 program to compute regular and irregular associated Legendre functions

by Klaus Bartschat

2010

We present a modern Fortran 90 code to compute the regular P m l (x) and irregular Q m l (x) associated Legendre functions for all x ∈ (−1, +1) (on the cut) and |x| > 1 and integer degree (l) and order (m).

Regular associated Legendre functions for selected (x,1,m). Table 1

Irregular associated Legendre functions for selected (x,1,m). The results using the Miller algorithm are displayed with the differences using the Wronskian in parenthes This provides a check on the accuracy of the calculation for a particularly difficult case.

Convergence of two-electron Coulomb potential 1/|r; — 1r2| with respect to €max in the Neumann expansion in prolate spheroidal coordinates. Internuclear separation R is 1.4. All the quantities are given in atomic units.

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A theorem of Evariste Galois on continued fractions

by Marco Debe

This thesis is aimed to study a significant part of the history of continued fractions: the Lagrange's theorem, on the development of the roots of quadratic equations with coefficients integers, and the theorem of Galois on irrational... more

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Matematica e Musicologia: la costruzione dei temperamenti.

by Silvio Gregorini

Capitolo 1: Introduzione all'argomento 1.1 Breve introduzione storica 1 1.2 Il suono 5 Capitolo 2: Gli strumenti matematici 2.1 Morfismi e Teoria Gruppale 11 2.2 Frazioni continue 15 2.3 Convessità e compattezza 18 Capitolo 3:... more

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The bernoulli map

by Linas Vepstas

2004

This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another definition, it is the transfer operator (the... more

Then, equating coefficients of powers of t of the generating function, one obtains the de- sired result.

notation, one may write the Taylor’s series as This notation allows the matrix elements Uj, = (m|Z|n) of the transfer operator to be articulated as well. Starting with the matrix notation of equation 4.1, one has

The eigenvalues may be promptly read off the diagonal; these eigenvalues are 1, = 2~". Because the matrix is upper-triangular, it is easily solvable for both the left and right eigenvectors, which agree perfectly with those given by Driebe[6]. Visually, the upper-left of this matrix looks like The right and left eigenvectors are developed in the following sections. Note that Uj», is very well-behaved: all entries remain small. For large k, each column resembles a Gaussian (this follows from Stirling’s approximation for the factorial), and specifically, this Gaussian has a peak at k = 2m. That is, one has Ux < Un2m © 1/2./mm for large m.

where the substitution j = m—k was made to obtain the last sum. The vanishing of the last sum is a well-known identity on the Bernoulli numbers. The n 4 0 case requires more Proof. Consider first the n = 0 case. One has and furthermore, the duality is algebraic, in that no infinite sums need be performed tc demonstrate duality.

The other left eigenvectors can also be made more concrete by looking at how they act on some function f(x); this may be written as

The last case can be shown to vanish, by again making the substitution 7 = m—k, to get

where the second step makes use of equation 5.5. This is remarkable, as any periodic wave constructed from sines and cosines seems to be in the kernel. By contrast, I does not behave this way: which is just a re-derivation of equation 4.5. Thus, Ig and Iy are inequivalent when acting Monomial Bernoulli on sine functions; so and are inequivalent as well. Since the sine function is the limit of a polynomial sequence, it seems strange or some- how contradictory that Bernoulli has only a trivial kernel on the space of polynomials, while killing odd-period sine and cosine functions. In order to define limits, or more pre- cisely, in order to define sin 27x as the limit of a sequence of polynomials, one must define the manner in which a polynomial sequence can converge to a function, and, for that, one must have a topology. This conundrum is revisited, and solved, in section 7.1. Since the sine function is the limit of a polynomial sequence, it seems strange or some-

so that u(m) = [Zp] by one, so that im 1S the traditional Moebius function, while further rows are offse It’s perhaps not obvious from casual examination of the above, but each row is just an elongated copy of the moebius function itself. That is,

where one has, as before, the sum-of-divisors function: By analogy, the generalized divisor operator corresponding to the sigma function is then 7.7. More Number Theory. Further well-known identities from Dirichlet convolution follow in a similar fashion. Let 1 = (1,1,1,---) be the row vector of all-ones; let ¢ = (1,0,0,---) be the row vector with one in the first position, and then all zeros. Then clearly, 1.4p = € is the operator equation analog of the Dirichlet convolution | * u = €. The Moebius inversion formula, with states that g = f «1 if and only if f = g*p has a clear operator analog: Y = ¥ Zp if and only if ¥ = YMp which follows trivially from the identity %p.4p = 1 which was obtained above. As before, this Dirichlet convolu- tion identity holds only on the space of /;-summable series, and it holds on the space of l5-summable series, if RH holds. Clearly, the identities cannot hold when the space of functions is expanded by analytic continuation, as the kernels of and Y would be, by construction, become tainted by the zeros of Zp and Lp.

It is straight-forward to verify that the column sums give the Euler totient function: viz:

the zeta eigenfunctions, which are defined for the entire complex A -plane (although these are not integrable when |A| > 1). This implies that the construction of eigenfunctions by means of lattice shifts is somehow innately limited: it allows some, but not all possible eigenfunctions to be found. The true nature of this obstruction is unclear. The proximal cause of the obstruction is that Y,, A” = 1/(1—A) has a pole at A = 1, thus preventing the convergence of the analytic series beyond that point. However, in the more global view, one has an operator acting on an infinite-dimensional topological vector space; whether the obstruction is due to the operator or the structure of the space itself is unclear.

This figure shows a graph of the eigenfunction cy (x) for A = 0.4. As can be clearly seen, it takes on values in the Cantor set.

For concreteness, consider first the action of the right-sift operators Sp and S,, defined by equation 8.4, acting on the k = 0 dyadic eigenfunction 8.9 built from the Haar wavelet. One has (dropping the subscript k fromy): with the last step following because the Haar wavelet is identically -1 on the interval 0 < x < 1/2. To emphasize this, write it more simply as

As the set B is taken to be smaller and smaller, the sets A; will become smaller as well. Denote by x; the corresponding limit point of each Aj, so that g(x;) = y and the pre-image of y consists of these points: g~'(y) = {x1,x2,-+- | (xj) =y}. One now combines these provisions to write The limit in the last line of this sequence of manipulations may be interpreted in two ways, depending on whether one wants to define the measure v on Y to be the push-forward of UL, or not. If one does take it to be the push-forward, so that v = g,U, then one has

The above made use of the usual expansion of the Dirac delta as a Fourier series: These are also complete over coordinate space, in that, for any arbitrary square-integrable coordinate-space function f(x) = (x|f) , one has

while (n|e0) = 6,0. Essentially, this is nothing more than a plain-old Taylor’s series expan sion of the exponential function; reasonable, insofar as the exponential function is an entir function on the complex plane. The first step in the above was to insert the coordinate-spac identity operator: Begin by writing the components of the vector |em) in the polynomial-space represen- tation. For m 4 0, one has

FIGURE F.3. The Non-Singular Part of the Divergent Eigenfunctions

This figure shows ringing/Gibbs phenomenon as s approaches zero. In the limit of s = 0, we expect the real part of B to approach the trivial eigenfunction lim,_,9+ KB (x;s) = —Bo(x) = —1. As this graph shows, the function is indeed trying very desperately to get flat, with not much success. The ringing occurs only at s = 0; there is no problem with convergence near larger integers, where lims_,, R(—i)*B (x;s) = —B,(x) converges very smoothly and cleanly.

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STRUCTURAL PROPERTIES OF MUSICAL SCALES

by Norman Carey and

Here the elements of P are arranged linearly, each element together with its corresponding note and pair of exponents: Similarly we can consider P in this two-dimensional setting, represented by the positive quadrant of the lattice Z x Z. This two-dimensional version of P was known in antiquity and is referred to as the Nicomachus Triangle. Of course, P can also be ordered in a linear sequence. The interplay between the linear (one-dimensional) sequence of P and the two-dimensional display of P provides the basis for much of what follows. The accompanying diagrams show P in a two-dimensional array and in a linear sequence. The reader will note that every pure power of 3 presents a new tone, while every pure power of 2 re-presents the origin Fp in different registers.

Gis,3) consists of the 15 factors of 144 = 23° and contains the 11 elements of S(5,3) plus the four elements 144 (243*), 72 (2337), 48 (2431), and 36 (273°). The complements of these four are the first four elements of (5,3), 1 (= 2°3°), 2 (= 213°), 3 (= 2°3'), and 4 (= 273°). The remaining 7 elements form Ri5,3), the largest connected subset of complements in G(s 3). In order to show the musical significance of regions we develop the next two examples, the Structural Region and the Pentatonic Region.

This region embraces the key proportion of Greek musical theory 6: 8 :: 9: 12 taken over two octaves. This proportion has the property that 8 and 9 are the har- monic and arithmetic means, respectively, between 6 and 12. Musically, of course, this proportion corresponds to the structure, Tonic, Sub-Dominant, Dominant, and Tonic, which forms the framework of fixed tones for many scales throughout the world. All of this leads us to call Ris5.3) the Structural Region and the elements of Ris5,3) the structurals.

A finite continued fraction is a number of the form 10G.D. Olds Continued Fractions Random House 1963; Hardy and Wright An Introduction to the Theory of Numbers Oxford University Press 1979. 3.2. Continued Fractions. The region R,,4,) is defined in terms of the connected pair of elements 24 and 3%. So far the only information we have about these numbers is that A and B must be relatively prime. We do not know anything about the ordered pairs as such, how to compute them, even whether or not the number of such pairs is infinite. The branch of mathematics which provides the tools for answering these questions is the branch of number theory referred to as the theory of continued fractions. Therefore, we will give a brief treatment of continued fractions, referring the reader unfamiliar with the topic to introductory texts in number theory for proofs and more detailed discussion.1° Y an, an oF 7PfP Y a 7 re 7

The continued fraction algorithm illustrated above may be applied to an irra- tional number, but in this case the process continues indefinitely, generating an infinite sequence of integers. In fact, though again we do not prove it, the con- tinued fraction algorithm sets up a one-to-one correspondence between the set of all irrationals and the set of all infinite sequences of integers, to, t1,...t,... where t1,te,... are positive. We do show that any infinite sequence of integers to,t1,...t,... with t1,to,... positive defines an infinite continued fraction [to;t1,...]: If cx is defined as before. the question is whether cy, approaches a limit as k increases without bound, and we appeal to the fundamental theorem of analysis that a bounded, monotonically increasing or decreasing sequences converges to a limit. Theorems 1 through 5 may be applied to the cz, so we immediately have the result that the even convergents are strictly increasing and the odd convergents are strictly decreasing. Moreover,

Continuing in this way it is possible to compute the continued fraction expansion of logs 3 = [1;1,1, 2, 2,3,1,5, 2, 23, 2,...].

From a musical point of view the most interesting family is the family whose final region is the Diatonic Region R(11,7). The interval which relates these regions is the whole step, 37/2°. The Structural, Pentatonic, and Diatonic Regions all contain whole steps, which fill in the initial octave of the Octave Region. This family is depicted below: That the diatonic region R(11,7), is the final member of this family is related to the coincidence up unti the convergents to the are [1] = +, [1; 1] = 2 and the associated wel to the golden number. 11/7 of the convergents and semi-convergents of log, 3 with golden number, ¢. Since the continued fraction for log, 3 begins [1;1,1,2,...], the first five elements in the sequence of (semi-)convergents 1; 1,1) = 3,151, 1,1] = §, and [1;1,1,2] = [1;1,1,1,1) = 8, coinciding with the first five convergents to 6 = [1;1,1,1,1,...].. The Diatonic Region is then the last region where both of the intervals which make up the region -formed Pythagorean scale are determined by convergents

2.4. Definition of the Region R(4 3). At this point the framework for the theory of well-formed Pythagorean scales is in lace. We proceed to offer a definition of the region Ria4,p), an operational definition in the sense that it gives a procedure for constructing all regions, making use of the set P and two properties, connectedness and complementarity.

Assuming 2“ and 3” are connected in P, the region R:,,p) is defined to be the largest non-empty connected set of complements in G,4,g). Table 2.4 gives the situation for the first four prime ratios: In the case (A, B) = (1,0), 2! = 2 and 3° = 1 are connected in P, but Ria,p) = S(4,B) = G(a,B) = 0. To avoid trivial complications later we will add the further restriction that R(4,g) be non-empty. As long as B > 1, Rya.p) # 0: Certainly s = 2938-1 and s = 24~13° are in S(4,,) as noted above. Then at least these elements (or this element, if A = B = 1) belong to R:,,g). If B > 1, then there are elements in G(4,g) that are not in S(4,g), hence elements in $4.) not in Ri 4,2). Rawpntns, OA and SP! ava anancsatad fh DP. tha san Bex xe a dabwad ta ha tha

Table 3.2.4 points to the importance of the semi-convergents to logs 3. We will show that the correspondence conjectured above holds true, but to do this we will need one more result to characterize semi-convergents in general. This motivates a closer study of Continued Fraction Theorem 8, to reinforce this result and to attempt to prove a converse theorem. onvergents “” for even. After each full convergent the value of t, is written.’

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The Harmonic Math, Chromatic numbers and sounds

by Paolo Di Pasquale

https://pdpseven.wixsite.com/sound-color The Harmonic Math, Chromatic numbers and sounds" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic... more

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On differences of zeta values

by Linas Vepstas

2008, Journal of Computational and Applied Mathematics

Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri-Lagarias, Maślanka, Coffey,... more

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by Norman Carey and

1996, Perspectives of new music

SELF -SIMILAR STRUCTURE is one that exhibits parallel construction at different levels of scale. Notions of self-similarity have often been invoked in organicist explanations of the evolution and unity of musical compositions. At around... more

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On a certain invariant for real quadratic fields

by Seokmin Lee

2003, Proceedings of the Japan Academy, Series A, Mathematical Sciences

Let K = Q(√ m) be a real quadratic field, O K its ring of integers and G = Gal(K/Q). For γ ∈ H 1 (G, O × K), we associate a module M c /P c for γ = [c]. It is known that M c /P c ≈ Z/∆ m Z where ∆ m = 1 or 2 and we will determine ∆ m .

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Sur l'accélération de la convergence de la série de Madhava-Leibniz (On the acceleration of the convergence of the Madhava-Leibniz series)

by David POUVREAU

2015, Quadrature

This paper expounds very innovative results achieved between the mid-14th century and the beginning of the 16th century by Indian astronomers belonging to the so-called "Mādhava school". These results were in keeping with researches in... more

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THE MINKOWSKI QUESTION MARK, GL(2,Z) AND THE MODULAR GROUP (EXPOSITORY)

by Linas Vepstas

2008

Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both are described by certain subsets of the modular group GL... more

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Continued fractions and the Riemann zeta: connections with string theory

by Michele Nardelli

In this research thesis, the authors in this article show how, starting from the simple continued fractions, one can reach the most advanced theories of physics, as the connections between the prime numbers and the strings adic, adelic... more

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The rogers-ramanujan continued fraction and a quintic iteration for 1/π

by Shaun Cooper

2007, Proceedings of the American Mathematical Society

Properties of the Rogers-Ramanujan continued fraction are used to obtain a formula for calculating 1/π with quintic convergence.

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Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials

by Qing-hu Hou

2006, Discrete Applied Mathematics

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and... more

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On the Minkowski Measure

by Linas Vepstas

2008

Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to... more

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VERIŽNI RAZLOMCI I PROBLEM KALENDARA

by Andrej Dujella

1999, Matematika i škola

U ovomčlanku razmotrit´cemo matematičku pozadinu problema kalendara. Postoje dvije očite prirodne jedinice vremena: dan i (Sunčeva) godina. Nažalost ove dvije jedinice nije jednostavno medusobno uskladiti. Problem je u tomě sto Sunčeva... more

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A generated approximation of the gamma function related to Windschitl's formula

by Lixin Song

2014, Journal of Number Theory

In this paper, based on Windschitl's formula, a generated approximation of the factorial function and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over Windschitl's... more

School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, Chink Dawei Lu*, Lixin Song, Congxu Ma

Simulations for an, Bn, Un; 9n, Yn and Vy.

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A series representation for the riemann zeta derived from the gauss-kuzmin-wirsing operator

by Linas Vepstas

2005, Preprint

A series representation for the Riemann zeta function in terms of the falling Pochhammer symbol is derived from the polynomial representation of the Gauss-Kuzmin-Wirsing (GKW) operator.

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Optimal approximation by continued fractions

by W. Bosma

1991, Journal of the Australian Mathematical Society

Among all possible semiregular continued fraction expansions of an irrational number the one with the best approximation properties, in a well-defined and natural sense, is determined. Some properties of this so called optimal continued... more

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Frações Contínuas no Ensino Médio

by wmpommer W A G N E R M A R C E L O Pommer

2009, Wagner Marcelo Pommer

Apesar dos avanços das recentes propostas curriculares, alguns temas matemáticos, geralmente abordados no Ensino Superior, ainda se encontram distantes do cotidiano da sala de aula do ciclo básico. Assim, este texto propõe uma breve... more

Apesar dos avanços das recentes propostas curriculares, alguns temas matemáticos, geralmente abordados no Ensino Superior, ainda se encontram distantes do cotidiano da sala de aula do ciclo básico. Assim, este texto propõe uma breve reflexão envolvendo aspectos curriculares epistemológicos e didáticos para a discussão da temática das Frações Contínuas do ciclo básico. O tratamento deste tema, não como componente curricular, mas através de situações de ensino permite a re-valorização dos atuais temas do currículo, ligados a Teoria dos Números e realça as conexões internas e externas aos conhecimentos matemáticos. Introdução: Inicialmente, observando a própria denominação, as Frações Contínuas apresentam, por um lado, um aspecto associado ao discreto, pelo fato de representar uma fração (relação de números inteiros) e, ainda, o segundo termo remonta as grandezas de natureza contínua. Assim, as: (...) frações contínuas constituem um exemplo interessante de procedimento que é finito, quando operado sobre números racionais, e infinito, quando o número dado é irracional. A origem das frações contínuas está na Grécia, onde as frações, para efeito de comparações, eram todas escritas com numerador '1'(CUNHA, 2007, p. 3). Com relação a introdução de temas mais atuais no ensino da Matemática, faço referência a atual Proposta Curricular do Estado de São Paulo (2008). Este documento apresenta uma lista de conteúdos para a Matemática do ciclo básico sem alterações apreciáveis em relação ao que se veicula nos diversos sistemas de ensino existentes atualmente, levando-se em consideração "o princípio de que os conteúdos são meios para o desenvolvimento das competências pessoais" (SÃO PAULO, 2008, p. 48). A questão da atualização dos temas matemáticos do currículo é valorizada por muitos autores e é inclusive discutida aqui no SEMA/FEUSP. De modo geral, são apontados a Programação Linear, o Cálculo Diferencial e Integral, os Fractais e as Frações Contínuas, dentre outros, como temas propícios para serem abordados no ciclo básico, numa abordagem adequada a este nível de ensino. Assim, há a necessidade da escola estar sempre alerta para mudanças de conteúdos e metodologias em face de novas realidades do mundo atual, propiciando condições para que os indivíduos se adaptem a estas mudanças e sejam cidadãos conscientes de seu papel na sociedade. Esta proposta de atualização de temas do currículo de matemática e a nova Proposta Curricular podem ser conciliadas, como, por exemplo, acontece com inserção no ensino da Biologia e a nova revolução da genética, assunto complexo, mas que se mostrou passível de ser abordado e trazido para o cotidiano da sala de aula numa abordagem acessível a alunos do ensino básico. Também, o movimento para a renovação do Ensino de Física está discutindo a inserção de tópicos usualmente abordados somente no Ensino Superior, como alguns resultados da Física Moderna. Dentro dessa visão: (...) o conteúdo matemático a ser tratado na escola básica é apenas um veículo para o desenvolvimento das idéias fundamentais de cada disciplina, que devem ser convenientemente articuladas tendo em vista as funções a serem desempenhadas no currículo. É a forma de abordagem dos diferentes assuntos que distingue diferentes propostas, dando-lhes cor e substância (MACHADO, 1990, p. 22). Assim, os pressupostos delineados me alertaram a possibilidade do repensar a nível curricular para a existência de outros temas do ensino superior que poderiam ser abordados neste nível de ensino, com uma transposição didática adequada, como geradores de situações de ensino propícias. Atentamos para a possibilidade de um novo olhar para o currículo, concebendo a metáfora do conhecimento como rede, conforme Machado (1995) e a busca de temas associados à própria disciplina de Matemática, que permite uma articulação e integração entre os próprios conceitos matemáticos vigentes no atual currículo. A idéia de rede constitui uma imagem, onde conhecer é como:

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On continued fraction expansion of real roots of polynomial systems, complexity and condition numbers

by Angelos Mantzaflaris

2011, Theoretical Computer Science

We elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only... more

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Dynamics of continued fractions and kneading sequences of unimodal maps

by Carlo Carminati

2010

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies... more

(b) The quadratic minor lamination QML is the col- lection of all minor leaves. Real leaves are drawn in red, and the set FR is the set of all endpoints of red leaves. A thicker red line represents the minor leaf corresponding to the lamination to the left. We will consider the set RQML C QML of real leaves inside the quadratic minor lamination: under the previous correspondence, they correspond to real points on the boundary of the Man- delbrot set. Moreover, let R = RQML1S' be the set of endpoints of all leaves in RQML. Since RMQL is symmetric, it is enough to consider the “upper half” RM [0, 4).

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The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions

by Fabrice Guillemin

2000, Advances in Applied Probability

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Thesis defense

by David Arroyo

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Distribution modulo 1 and musical scales

by Norman Carey

1998

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The minkowski question mark, gl (2, z) and the modular group

by Linas Vepstas

2004

Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in... more

importantly, its self-similarity properties. These will be seen to be the self-similarities of the infinite binary tree.

has a unit determinant, and so A € SL(2,Z), where SL(2,Z) is the group of 2x2 matrices with unit determinant. Although its appearance here seems arbitrary, this group of matrices will recur more deeply throughout the theory of binary trees. The unimodular property then allows one to prove that

whereas the right move is defined similarly, with the role of even and odd reversed. That is, a left move either increases the trailing entry in the continued fraction by one, or it inserts a comma, to start a new entry for the continued fraction. The seemingly strange appearance of the “-1,2” in the last is simply a byproduct of the identity [a1,a2,--- ,an] = [a1,42,---,ay — 1,1]. That is, the left move adds a one to the last entry, but only after normalizing the continued fraction so that it has an odd length. The right move does the same, after first normalizing to an even length. Thus, to understand the general case, its sufficient to contemplate the value of [a1,a2,--- ,ay + 1]. Tr

2.3. Blow-ups. The self-similarity maps above were expressed in the form of mapping the whole interval to a sub-interval. Of course, the direction of this map can be turned around, so that sub-invervals are mapped to the whole interval. Things become interesting when one concatenates shrinks with blow-ups that are valid only on a sub-interval. This results in identities that are valid only on a sub-interval. However, the range of possibilities is much greater, and some of these identities take on new, unusual forms. Examples include XXX finish this section.

FIGURE 3.2. A three-point turn forward L, turning, then back up R™!, turning, and again pull forward L, straightening. The net result is that the car has been turned by 180 degrees!

Figure above shows a tree with p branches, made from a binary tree by equating nodes. position. Clearly, the third strand can wind around in every which way, thus forming the free group. In terms of the group generators, these moves correspond to X = ba = i and Y = bt = R~*. Because only the squares appear, the previously established identity LR~'L = R7'!LR™! cannot be applied to limit which combinations of X and Y are unique.

It is straightforward to verify that this gives the expression 2.9 for the action of a general monoid element on x. Although Yp is thus representable as a two-by-two matrix, it is utterly different from yc. In particular, the determinant of Ypis not 1. The Yp are lower-triangular, and thus generate a “Borel monoid” , in analogy to the name “Borel group” given to the eroup of upper-triangular matrices.

of all possible de Rham curves generated by two-dimensional linear affine transformations is five-dimensional (excluding degeneracies due to rotations, translations and rescaling), and is explored as a picture gallery in [20]. Several famous fractal curves fall in this class, including the Koch snowflake curve, the Peano curve and the Lévy C curve.

Proof. To show this, one need only find a counter-example. This may be found by exam- ining the various cases. There are nine distinct arrangements of the posets a,b,c, ignoring permutations of b and c. These are shown in the figure below. The identity aA (bV c) = (aAb) V (aA c) holds for all of the arrangements except for that in arrangement 6. To illustrate that example, consider first arrangement |. There, one has bV c =a since a is the smallest common tree that contains both tree b and tree c. Thus, clearly a (bV c) =a. On the other side, one has a b = b, since b is the largest tree that is contained inside of both a and b. Similarly, aAc =c and so (aA b)V (aAc) =bVc=a and so the distributive law holds for arrangement 1.

The rooted binary tree may be converted into a more symmetrical graph by removing the root. The process is shown here, graphically. In the upper image, the root at 1/2 is removed and replaced with a single arc joining the left and right subtrees. This may be mapped to the Poincaré disk D, as shown in the three middle images. The location of the original root is marked by a small black square. The Poincaré disk may be mapped to the upper half-plane H; one such mapping is shown in the bottom image. The small black square is located at z = i; the leaves of the tree run to the real axis $z = 0. The arced segments are circle arcs, so that, for instance, the arcs along the top are parts of circles of an: fA

This unrooted dyadic tree may be embedded into the Poincaré disk using the mapping Consider next the embedding of the unrooted dyadic tree into to upper half-plane, as depicted in figure 8.1. This mapping takes the left half of the binary tree, and maps it into the left quadrant of the complex plane, and the right half of the tree to the right quadrant. If the dyadic tree is labelled as in figure 8.1, then one has a map 6 : [0,1] + Rgiven by This result is essentially a variant of the mapping between the Farey tree and the Stern- Brocot tree, given in eqn. 1.1. It is noteworthy, in that it provides a rapid algorithm to compute the inverse of the question mark function.

This figure shows a graph of the quesiton mark function ?(x), and the inverses of the functions 6(x)/2z and $(x)/2z discussed in the text. Note the similarity of the overall features, together with localized differences in slope.

8.2. Hyperbolic Rotations. The geometric embedding of the (unlabelled) binary tree into the hyperbolic plane is invariant the action of PSL(2,Z). However, if the nodes are labelled, then clearly, the action of PSL(2,Z) permutes these labels. Insofar as the tips of the tree can be identified with the irrational numbers on the unit interval, so one has an action of PSL(2,Z) on the unit interval. This action is breifly explored in this section.

A homotpically equivalent representation of fundamental domains, mapped to a binary tree. The arithmetic significance of the cusps becomes clearer when the geometry is deformed.

This figure illustrates the traditional fundamental domains as used in number theory. Each is in the shape of a triangle, with two points in the upper-half plane, connected by a solid blue line, and a third point, the cusp, located at infinity, or, equivalently, on the real axis. The dashed yellow lines run from the interior points to the cusp. Note that the solid blue lines form the un-rooted binary tree, as demonstrated above.

Consider a finite subtree of the infinite tree, shown in bold in the above figures. To the left and right of each segment lies a domain: these can be paired together, and, in a sense, “represent” that segment. These can be used to tile the hyperbolic plane. For example, the rightmost figure, consisting of four cusps, can be used to tile the plane. Note also that such finite tree can be used to represent a finite state machine. For example, in the rightmost figure, the point “a” represents the initial state of the machine. The machine accepts strings in the two letters, “L” and “R”’. The input letter “R” takes the machine to state “e’, which, if the tiling is to succeed, must be the same state as state “a” The input letter “L” rakes one to state “b”, while the input strings “LL” and “LR” take the machine to states “c” and “d’”, respectively, which, again, are exactly the same state as “‘a’”’.

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A multidimensional continued fraction based on a high-order recurrence relation

by David Ricardo Quimbaya Trujillo

2007

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to... more

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Measures on the unit circle and unitary truncations of unitary operators

by Luis Velázquez

2006, Journal of Approximation Theory

In this paper we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the... more

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Addition theorems via continued fractions

by Mourad Ismail

2009, Transactions of the American Mathematical Society

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for... more

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Continued Fractions and Dynamics

by Stefano Isola

2014, Applied Mathematics

Several links between continued fractions and classical and less classical constructions in dynamical systems theory are presented and discussed.

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A CONVERGENCE THEOREM FOR CONTINUED FRACTIONS OF THE FORM K ∞ n=1 a n /1

by James Mc Laughlin

In this paper we present a convergence theorem for continued fractions of the form K ∞ n=1 an/1. By deriving conditions on the an which ensure that the odd and even parts of K ∞ n=1 an/1 converge, these same conditions also ensure that... more

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The Early History of the Moment Problem

by Tinne Hoff Kjeldsen

1993, Historia Mathematica

In the present study it is discussed how the moment problem naturally arose within Stieltjes' creation of the analytical theory of continued fractions. Further it is shown how the moment problem in the work of Hamburger came to be... more

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Aspects of Well-Formed Scales

by David Clampitt

1989, Music Theory Spectrum

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and... more

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by David Clampitt

1996, Perspectives of New Music

SELF -SIMILAR STRUCTURE is one that exhibits parallel construction at different levels of scale. Notions of self-similarity have often been invoked in organicist explanations of the evolution and unity of musical compositions. At around... more

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Efficient and Reliable Multiprecision Implementation of Elementary and Special Functions

by Cuyt Annie

2006, SIAM Journal on Scientific Computing

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The Minkowski Question Mark, GL(2,Z) and the Modular Group (Expository)

by Linas Vepstas

2008

Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in... more

Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained by the fact that both can be represented with the infinite binary tree, which in turn describes the structure of the Cantor set. The infinite binary tree can be viewed as a certain subset of the modular group PSL (2, Z). The subset is essentially the dyadic groupoid or dyadic monoid. It provides the natural setting for the symmetry and self-similarity of many fractals, including those associated with period-doubling maps, with phase-locking maps, and with various dynamical systems in general. The aim of this text is to provide a simple exposition of the symmetry and its articulation. In the process, this paper attempts to clarify the relationships between a cluster of interrelated ideas from number theory: those surrounding the modular group, elliptic curves and the Cantor Set. It has long been widely known that the modular group PSL (2, Z) (the general linear group of 2 by 2 matrices over the integers) is the symmetric group of elliptic curves. Connections between this group, and the rational numbers are commonly presented in books and coursework; the connection to the dyadic subsets is rarely, if ever, mentioned -this paper attempts to correct this shortcoming. Likewise, the Cantor set plays an important role in many areas of mathematics; among others, it is a superset of the real numbers, and more general a universal cover for compact metric spaces. Its symmetry properties are rarely discussed, but are in fact described by the very same dyadic subsets of PSL (2, Z). This paper shows how all of these are related -the infinite binary tree, the Cantor set, the set of binary numbers (the set of infinitely long strings of 1's and 0's), the rational numbers, the Farey and Stern-Brocot trees, continued fractions, the set of quadratic irrationals and the Minkowski Question Mark function: these are all shown to be inter-related aspects of the same underlying structure, a structure having dyadic fractal self-symmetry. This paper is written at an expository level, and should be readily accessible to advanced undergraduates and all graduate students. XXX This paper is in a perpetual state of being unfinished. Although this version corrects a number of serious errors in the previous drafts, it is surely still misleading and confusing in many ways. The second half, in particular must surely contain errors and mis-statements! Caveat emptor! XXX

Note that c =?~! 0 bo? although the tools to prove this won’t be available until the next section, when self-similarity is discussed.

In the right-hand tree, the central edge is marked with an arrowhead. This singles it out as a special edge: the arrow is a device to keep track of one’s position in the tree, as well as one’s orientation, as one moves about. Without it, things can get confusing. The left hand tree shows the effect of the moves e, L, R, L~! and R~!. The identity e is, of course, no move at all. The four neighboring edges are marked with L, R, L~! and R7!: the arrowheads show how the central arrowhead travels, as the moves are applied.

This figure provides a visualization of SL(2,Z) as Mobius transforms acting on the unit square. Specifically, every possible matrix A given by eqn 3.3 (with no initial P and with a trailing V) is generated, corresponding to all dyadics with denominator less than or equal to 2!*. Eac density scatterplot is then taken of the points (x, | y|) with uniform distribution for x combined with a unform distribution for y. The brighter/redder the color, the greater the density. Note h such A is then treated as a Mobius transform y = (ax+b)/(cx+d). A how the diagonals resemble those of figure 8.4. The same scatterplot, using elements starting with P looks the same, but with the diagonals missing. It should be clear that the intersection of a horizonatal or a vertical line with this plot will have the d istribution shown in figure XXX (wtf, where is the measure?)

It is straightforward to verify that this gives the expression 2.10 for the action of a general monoid element on x. Although Yp is thus representable as a two-by-two matrix, it is utterly different from Yc. In particular, the determinant of Ypis not 1. The Yp are lower-triangular, and thus generate a “Borel monoid” , in analogy to the name “Borel group” given to the group of upper-triangular matrices. Affine transformations can always be written as a matrix equation by taking x to be a vector, and bumping up the dimension of the vector by one. Thus, for example, one writes

By contrast, its left-right symmetry just means that rt, = ty. It is not hard to deduce that the action of a general element y= g“'rg”r...rg@" on ty will be of the form a+ bx+ ct, for some constants a,b,c. This may be recognized immediately as a three-dimensional vector, and so the action of g and r can be given as operators acting on a three-dimensional space. Its also not hard to see that the action is linear, and so g and r are given by 3x3 matrices. Their explicit form may be given by making the identification of 1, x and f, as the basis vectors of the three dimensional space:

of the other). Thus, the collection of interval maps can be understood to be a groupoid; we call this the dyadic groupoid. It contains both the shrinking and the expanding maps.

The rooted binary tree may be converted into the unrooted tree of figure 3.1 by unfolding it, and removing the root, as shown above. In the upper image, the root at 1/2 is removed and replaced with a single arc joining the left and right subtrees. This may be mapped to the Poincaré disk D, as shown in the three middle images. The location of the original root is marked by a small black square. The third disk also shows the fundamental domains (see ahead to figure 8.6). The Poincaré disk may be mapped to the upper half-plane H; one such mapping is shown in the bottom image. The small black square is located at z = i; the leaves of the tree run to the real axis $z = 0. The arced segments are circle arcs, so that, for instance, the arcs along the top are parts of circles of radius 1, centered at z =n. They meet at the intersections z =n+ 5 +iV3 /2. These figures make clear that the rooted binary tree, after removal of the root at at 1/2, is a Cayley tree (also called a Bethe lattice) with coordination number 3.

the left quadrant of the complex plane, and the right half of the tree to the right quadran If the dyadic tree is labeled as in figure 8.1, then one has a map 6 : [0,1] + Rgiven by This unrooted dyadic tree may be embedded into the Poincaré disk using the mapping which places z = i at the center of the disk. Here, z € H is a point in the upper-half-plane, and w € D is the corresponding point in the Poincaré disk. This mapping wraps the unit interval around the perimeter of the disk in a counter-clockwise fashion, starting with x = 0 at the right-most edge of the circle. Again, this is illustrated in figure 8.1. The mapping is not linear. A little bit of work shows that it is given by

This figure shows a graph of the question mark function ?(x), and the inverses of the functions 0(x)/2z and $(x)/2z discussed in the text. Note the similarity of the overall features, together with localized variations in slope.

To obtain this last form, one makes use of the identities given in section 2.3. It is perhaps curious that these rotations are linear and fractional-linear, respectively, and do not have a more complex form. Since both are monotonically increasing, both are invertible. A few of the compositions of these maps are shown in figure 8.4

A homotopically equivalent representation of fundamental domains, mapped to a binary tree. The arithmetic significance of the cusps becomes clearer when the geometry is deformed.

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Beauty in Motion through the Lens of Doppler's Formula

by Ramzi Suleiman

2020, Beauty in Motion through the Lens of Doppler’s Formula

Many founding fathers of science have underscored the importance of beauty in mathematical representations of natural phenomena and their connection with the beauty of the objects they represent. Paul Dirac, for example, believed that the... more

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Solving a family of quartic Thue inequalities using continued fractions

by Andrej Dujella

Rocky Mountain J. Math., to appear

In this paper we find all primitive solutions of Thue inequality

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On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

by James Mc Laughlin

2004, … -AMERICAN MATHEMATICAL SOCIETY.

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Continued Fractions \\ A Brief Note

by Dr. J. M. Ashfaque (MInstP)

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Continued Fractions and Gaps

by Linas Vepstas

2005

Given a continued fraction, we construct a certain function hat is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is... more

For reference, overdrawn over the image is the parabola. The vertical black stripes occur at the Farey numbers, and appear to be ordered in size according to the Farey tree (i.e. the largest stripe is at x = 1/2, the next largest at x = 1/3, x = 2/3 and the next smaller ones at x = 1/4 and x = 2/5 etc.). The progression of the horizontal bands are Clearly related to the Farey tree through the parabola. Since the parabola gives a rational out when fed one in, we see that we’ve discovered a different tree of rationals that’s related to the Farey tree. One can see that any ratio of polynomials will this give a tree of rationals. We discuss the Farey Tree (or Stern-Brocot Tree) in a different chapter.

However, closer examination reveals some surprising details. Here’s the same pic- ture, but this time showing only the rationals p/q = [1..719]/720 (all rationals with numerator between | and 719 and denominator of 720:

Now that we know what sort of pattern to look for, we can find the cross-bars in this picture, for a denominator of 1024: Notice that a Moire pattern of fringes seems to be forming at x = 1/2. The ap- pearance of various parabolas at various locations seems to indicate that there are some sort of curious algebraic relationships between the rationals that are worth exploring. The factorial in the denominator seems important: the picture for the denominator 2-3-5-7-11= 2310 blurs out the features, and mostly looks much more random, as shown below:

The analogous picture for a denominator consisting of powers of 3 is less dis- tinct. Pictures for prime denominators appear considerably more random, although containing hints of a different kind of structure. For example, here is a picture with denominator 1023:

The next picture below is a scatter-plot showing all rationals with denominators less than 200. This now clearly exhibits a fantastic self-similar fractal latticework:

We will recognize in later chapters that these curves correspond to hyperbolic maps of the unit interval back onto itself, generated by elements of the modular group SL(2,Z) acting on binary trees. It would be interesting to specify the shapes of all the various bifurcating curves that seem to appear in this latticework, and to give the algebraic equations whose solutions are plotted above. Note that because a curve can seemingly bifurcate into another curve in many places, enumerating all of them would be a trick.

Note that 720 = 6! (six factorial) and that similar but denser images appear for higher factorials: Here’s one for 7! that is, p/g = [1..5039]/5040:

This picture tells us something that is surprising: this function is space-fil dense; it has no ho es or islands; it has a fractal measure of 2.0. Even in the ing and land of fractals, this is fairly unusual, as this function is, after all, not some self-similar curve, but rather a ’plain-old’ map from the unit interval to the unit interval. By ’dense’, I Cauchy-sequence notion of density: for any real values x, y € [0, 1] and positive € > 0, we can find a rational p/g such that |x — p/q| < € and [y — Ty/q| <e€ mean the traditiona . Of course, a computer generated picture is not a proof, but a proof is straight- and is discussed be ow. I am not aware of any other map that has this property forward,

directly compare different values of x. So instead, we introduce a ’normalized’ variant: Vv.(x = p/q,w) =x+4q?[xe(w) —x] and similarly v,(x,w). This is done below, for 200 evenly spaced values of x. On the horizontal axis is x running from 0 to 1, and along the vertical axis, w, from 0 at bottom to | at top: Clearly, it can be seen that v.(x,w) is a monotonically increasing function of w which follows easily by examining the partial sums: if w < z then

picture that the leading w and w” terms dominate. We can guess at the form of these terms: we define ue(x,w) = V.(x,w) — w+w? — w?/2 and graph it below, exposing the randomness of the cubic and higher terms directly. The seeming parallel-ness of some trajectories is an optical illusion, due to the pix- elization of the picture. There are more pictures, with a greater number of strands, at the Gap Room http://www. linas.org/art-gallery/farey/gap- room/gap- room.html.

The figure above shows the distribution of the ratio of the final two convergent de- nominators gv—1/gw for all rationals with denominators less than 200. Note that the scale along the x-axis runs from 0 to 1 but along the y axis, from 0 to 1/2. To create this distribution, we create an empty grid that is MxM=600x600 pixels in size. We then consider a fraction x = p/g and its denominator ratio y = gy_1/qn. If we have that i << Mx <i+1 andj < My < j+1 for some integers i, 7, then we increment the value at pixel (i, 7) by one. When we are done, we visualize the grid by assigning black to empty pixels, blue to pixels with a very small count, green to pixels with a medium-small count, yellow to pixels with a medium count, and red to pixels with a large count.

The figure above shows the distribution of the ratio of the final two convergent denom- inators gv—1/gqw for all rationals with denominators less than 1800. Note that the scale along the x-axis runs from 0 to | but along the y axis, from 0 to 1/2. Thus, the largest horizontal line is at gv_1/qn = 1/3, with the remaining lines distributed at the Farey Fractions. The uniform blue color indicates a fairly even distribution, with the coloring as before: red indicating an excess, and black a deficit. As one goes to larger denomi- nators, the uniformity domintes, with the distribution tending towards even-ness.

The figure above shows the distribution of the ratio of the pre-final two convergent de- nominators qy—2/qn—1 for all rationals with denominators less than 1800. Note that the scale along the x and y-axis runs from 0 to 1. Note that this figure is fundamentally different, in a certain sense, than the previous figures. Here, as one goes to the higher denominators, one seems to develop both a uniform background, and a filigree super- imposed on top. This behaviour is qualitatively different than what one sees when one just considers the final ratios.

The figure above shows the distribution of the ratio of the two convergent denominators qi/qz for all rationals with denominators less than 1800, and which have at least two terms (so that gz 4 q1).

The figure above shows the distribution of the ratio of the two convergent denominators q2/q3 for all rationals with denominators less than 1800, and which have at least three terms (so that g3 4 q2).

The figure above shows the distribution of the ratio of the two convergent denominators q3/qa for all rationals with denominators less than 1800, and which have at least four terms (so that ga 4 q3).

The figure above shows the distribution of the ratio of the two convergent denominators qa/qs for all rationals with denominators less than 1800, and which have at least five terms (so that gs # qa).

In this expression, the value of gy—3/gn—2 = r was fixed by our initial choice of p'/q'. However, we are free to pick any ay—1 > k and any ay. Note that by picking ay = 2 and ay- very large, we can approach the upper limit of the interval, and by picking ay very large, we can approach the lower limit of the interval. We see that for values of y close to 1 /m for some positive integer m, we can also approach arbitrarily close, thus completing the proof for these values. However, there are still *holes’ that cannot be approached without jiggering the value of r. For that, we need to adjust our initial pick of p’/q’ to get the r that we want. We then find that we have to apply induction, in reverse, to get to there. I believe this completes the proof. qn—1/qn to satisfy the desired result for many values of y. That is, we use the recurrence relation gy = anqn—1 + Gn-—2 to deduce that

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Aggregation matrix for the reduced-order continued fraction expansion model of Chen and Shieh

by Vittal Rao

1978, IEEE Transactions on Automatic Control

Let us consider two cases separately.

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Generalized Brjuno functions associated to -continued fractions

by Stefano Marmi

2010, Journal of Approximation Theory

For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), the nearest integer (α = 1/2) and by-excess (α = 0) continued fraction maps. To each of these expansions and to... more

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The Size of the Fundamental Solutions of Consecutive Pell Equations

by Michael John Jacobson, Jr.

2000, Experimental Mathematics

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Newton's formula and continued fraction expansion of sqrt d

by Andrej Dujella

2001, Experiment. Math. 10 (2001), 125-131.

It is known that if the period s(d) of the continued fraction expansion of √ d satisfies s(d) ≤ 2, then all Newton's approximants R n = 1 2 ( pn qn + dqn pn ) are convergents of √ d, and moreover we have R n = p2n+1 q2n+1 for all n ≥ 0.... more

Table 2: upper bounds for sz, We know that s; = 1, s2 = 2, s3 = 5, s4 = 6 and sg = 8. In Table 2 we lis upper bounds for sy obtained by experiments. Example 2 Let d= 25[(102 + 1)? +4]. Then

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Continued Fractions

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