On the continued fraction and Berlekamp's algorithm (Corresp.)

Let q, fj, t2, be a sequence of elements of a field F. We give a 2 3 continued fraction algorithm for t^x + txx + t2x + .If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence.More generally the algorithm produces a nontrivial solution of the system s ',+/*/.0</<*-l, /=0 for every positive integer s. 1.Let t0, tx, t2, be a sequence of elements of a field F. Set T-\ttft./=0 Let d be a nonnegative integer.We say that T* is an approximation of T of degree d if there exist polynomials V and W such that T* = V/W, deg V < d, deg W < d, x\w, and x2d\WT -V.We shall give a continued fraction expansion for xT.This yields polynomials V-, W, and integers d-, 0 = dx < d2 < d3 < , such that (V, W) = 1 and V,/W-is an approximation of T of degree d.Suppose T* is any approximation of T of some degree d.Then T* = V/W for that value of i such that d < d<di+x.If the sequence of the t satisfies a linear recurrence of degree d, but not one of smaller degree, then there is an m such that dm = d and the linear recurrence is given by the polynomial Wm.In this case, WmT = Vm, the continued fraction expansion, terminates at i = m, and we can determine Wm from the first 2d of the t:, i.e., from those t such that 0 </ < 2d.Our algorithm is closely related to Zierler's version of Berlekamp's algorithm [1].2. We consider continued fraction expansions of the form a = A, +-l A, + 2 A3+-where A,, A2, A3, are elements from some field E. We can write

Motivated by problems in random continued fraction expansions, this research describes the design and implementation on θ-expansions of a real number, x in (0, θ) where 0 < θ < 1. Initially, for such a number θ, the convergent of θ-expansions is computed with comparison to the regular continued fraction (RCF)-convergent based on the samples of the approximated value, x. θ-expansions algorithm and continued fraction algorithm are applied in a Maple software to compute the θ-convergent and RCF-convergent. In this respect, the purpose of this research is to examine the growth rate of convergent for both expansions and provide the theoretical behaviours of θ-convergent in θ-expansions. Numerical results based on these samples show 3 conditions of convergence growth rate, which are C2 and C4 give the best decimal approximations in θ-expansions, whereas, Cn be the best approximations in RCF expansions. Therefore, this research reveals that various value of θ affect the performance on their growth rate of convergent for these two expansions. As θ approaching to 1, it reduces their convergence errors. The analysis of the experimental results reveals the RCF-convergent give a better performance compared to θ-convergent as its yield to less convergence errors. In addition, the value of convergent do converge to the approximated value, x and that is how the name of convergent comes.

Optimal continued fractions

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity. It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization. Nevertheless, recently Berthe, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982). Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincare section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well. It is the aim of my series of lectures to review the above results.

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

The expected cell count for a 2 x 2 contingency table, governed by the noncentral (extended) hypergeometric distribution, is expressed as a terminating continued fraction. The coefficients in the continued fraction are better behaved than the multinomial coefficients required for the usual moment calculation. The expected cell count must be calculated repeatedly in a conditional maximum likelihood analysis of K2 x 2 contingency tables. Since the continued fraction can be easily evaluated, a rapid and numerically stable computational algorithm results. Once this first moment is known, higher moments can be obtained as shown by Harkness (1965, Annals of Mathematical Statistics 36, 938-945). A BASIC program to implement the continued fraction algorithm is given in an appendix.

RSA is a well-known and widely used cryptosystem. One major limitation of RSA is its maximum input length depends on the chosen key size and the selected padding. This paper proposed a hybrid data compression algorithm to increase the input data length to be encrypted by RSA. Firstly, a plain text is compressed by the LZW algorithm. The compressed data is padded. After that, the size of the compressed data is further reduced by using continued fraction. Double-compressed data is, then, used as an input of RSA. When using a combination of continued fraction and LZW algorithm, the results showed that, on average, the input data length is increased by 18.91 per cent. In most cases, the processing time is much better than not doing any compression. In the case of decryption, a cipher text is decrypted by RSA first. The padding is removed. Then, the Euclidean algorithm and LZW are used respectively to restore the original data. The hybrid data compression is not only relieved the limitation of RSA but also enhances the security level.

In order to implement the continued fraction algorithm on a highly parallel computer, like the Massively Parallel Processor, it is necessary to be able to compute certain numbers which occur at widely-spaced intervals within the continued fraction expansion of f~, where N is the number to be factored.In this paper several properties of the continued fraction expansion of a quadratic irrational are developed.These results are then applied to the development of a very simple algorithm for finding the widely-spaced numbers referred to above.

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ ≥1 → : ℝ ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q | q ) has infinitely many solutions in rational numbers p/q ? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions. Dirichlet and Liouville Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b , there exists an integer a with |ξ – a/b | ≤ 1/(2 b ), and one may hope that there are infinitely many integers b for which |ξ – a/b | is in fact much smaller than 1/(2 b ). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.

This thesis uses dynamical systems to study two types of objects: number expansions (Chapters 1 and 2) and translation surfaces (Chapter 3). Our first chapter builds a broad, unifying theory for a large class of continued fraction algorithms producing what we call contracted Farey expansions. These algorithms are based on three ideas: (i) contraction of generalised continued fractions, (ii) induced transformations, and (iii) the natural extension of the Farey tent map. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties; and a unifying framework to prove several old and new results in Diophantine approximation. Chapter 2 introduces a new, one-parameter family of functions called skewed symmetric golden maps. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in number expansions produced by these maps. The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions. Our final chapter deals with translation surfaces, i.e., surfaces obtained by gluing pairs of parallel, equal-length, and oppositely oriented edges of planar polygons. The Veech group of a translation surface is the group of Jacobians of its orientation-preserving affine automorphisms. We develop a novel algorithm to construct translation surfaces with prescribed lattice Veech groups in given strata. Our ideas are also used to obtain obstructions for the realisability of certain Veech groups in certain strata. In particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is the modular group.

In order to implement the continued fraction algorithm on a highly parallel computer, like the Massively Parallel Processor, it is necessary to be able to compute certain numbers which occur at widely-spaced intervals within the continued fraction expansion of N \sqrt N . where N is the number to be factored. In this paper several properties of the continued fraction expansion of a quadratic irrational are developed. These results are then applied to the development of a very simple algorithm for finding the widely-spaced numbers referred to above.

In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T^slow and T^fast: these maps are defined for (x,y)∈[0,1], where x is the number for which a semi- regular continued fraction representation is developed by T^slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map T^fast is a “sped up" version of the map T^slow, and we show that T^fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, T^slow preserves no such measure, but does preserve an infinite, σ-finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y under rotation by x modulo one. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada’s α-continued fractions.

Part I: Synthesis of Immittances with Two Poles and Two Zeros. Cauer's continued-fraction technique is generalized for use in realizing RLC network immittances. Using this method, element values are determined by simple processes of forward and reverse division. Immittances F(p) with two poles and zeros may be grouped in three classes according to the frequency j?m at which Re F(j?) is a minimum. For Class I, wm= 0; for Class II, ?m = ?; and for Class III, ?m is finite. Class I and II F(p) can be realized immediately by making a continued-fraction expansion. If both F(p) and 1/F(p) are Class III, to obtain a realization without unity-coupled coils the function must be split into two terms which may then be expanded in continued fractions. Simple formulas are presented which enable one to easily classify F(p) and determine an appropriate realization method. Part II: A Constant-Resistance Ladder for Transfer Function Synthesis. The physical factors that determine the poles and zeros of a transfer function are examined. By use of physical insight, a design procedure for a constant-resistance ladder network is arrived at. This ladder network is found to have the same realm of application as the conventional, RLC, constant-resistance bridged-T. However, the ladder has the advantages of: fewer elements, more flexibility, and requiring less flat loss if the bridged-T requires flat loss. Complicated transfer functions may be realized in a chain of ladder sections having arm immittances with only two poles and two zeros.

In this research work, aHOS(high order system)remains decreased into LOS (lower order system) by usage of MCF (modified cauer form) method. The ROM is founded upon CF (Continued fraction) technique also find comes out to be LOM (low-order model). For the reduction of HOS indirect method is used. Presents article, show the graphical comparison of simulation results between MCF ROM and another diminution technique presented in literature. In the simulation, the response comparison displays improved performance of proposed ROM than further approaches obtainable in literature and discuss the result technics of peak time, settling time, rise time also peak amplitude information for higher order and ROM.

Schweiger introduced the notion of a subtractive algorithm, to classify certain types of multidimensional continued fractions. We study the limit behaviour of one particular subtractive algorithm, which generalises a continued fraction algorithm that was originally proposed by Selmer. The algorithm that we study depends on two parameters a and b. We first find a Markov partition if a ≥ b. Using inducing techniques, we then prove the existence of an ergodic absolutely continuous invariant probability measure a ≥ b for the quality of the rational approximations for Lebesgue-typical multidimensional vectors.