Ergodicity for [formula omitted]-adic continued fraction algorithms

The problem of simultaneously approximating a vector of irrational numbers with rationals is analyzed in a geometrical setting using notions of dynamical systems theory. We discuss here a (vectorial) multidimensional continued-fraction algorithm (MCFA) of additive type, the generalized mediant algorithm (GMA), and give a geometrical interpretation to it. We calculate the invariant measure of the GMA shift as well as its Kolmogorov-Sinai (KS) entropy for arbitrary number of irrationals. The KS entropy is related to the growth rate of denominators of the Euclidean algorithm. This is the first analytical calculation of the growth rate of denominators for any MCFA.

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.

Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.

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.

The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. In this paper, we focus on the normal stress difference responses to large-amplitude oscillatory shear (LAOS) flow. The nonlinearity of the polymeric liquids, triggered by LAOS, causes these responses at even multiples of the test frequency. We call responses at a frequency higher than twice the test frequency higher harmonics. The normal stress difference responses for the Oldroyd 8-constant framework has recently yielded to the exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within summations to infinity. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 17th power of the shear rate amplitude. Our main result reduces to the well-known expression for the special cases of the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8-constant framework does not. We use Ewoldt grids to show our main result to be highly accurate for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions as long as Wi/De&amp;lt;3310. We compare our main result, for the special case of the Johnson–Segalman fluid, with measurements on dissolved polyisobutylene in the isobutylene oligomer. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measurements.

We consider the multi-point Schwarz-Pick Lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point Schwarz-Pick Lemma give the Schur’s continued fraction algorithm and several inequalities for bounded analytic functions on the unit disk.

We study the effect of coupling between the superconducting current and\nmagnetization in the superconductor/ferromagnet/superconductor Josephson\njunction under an applied circularly polarized magnetic field. Manifestation of\nferromagnetic resonance in the frequency dependence of the amplitude of the\nmagnetization and the average critical current density is demonstrated\nnumerically. The IV-characteristics show subharmonic steps that form devil's\nstaircases, following a continued fraction algorithm. The origin of the found\nsteps is related to the effect of the magnetization dynamics on the phase\ndifference in the Josephson junction. The dynamics of our system is described\nby a generalized RCSJ model coupled to the Landau-Lifshitz-Gilbert equation. In\nthe suplement we justify analytically the appearance of the fractional steps in\nIV-characteristics of the superconductor/ferromagnet/superconductor Josephson\njunction.\n

While quantum computers might speed up in principle certain computations dramatically, in practice, though quantum computing technology is still in its infancy. Even we cannot clearly envision at present what the hardware of that machine will be like. Nevertheless, we can be quite confident that it will be much easier to build any practical quantum computer operating on a few number of quantum bits rather than one operating on a huge number of quantum bits. It is therefore of big practical impact to use the resource of quantum bits very spare, i.e., to find quantum algorithms which use as few as possible quantum bits. Here, we present a method to reduce the number of actually needed qubits in Shor's algorithm to factor a composite number N. Exploiting the inherent probabilism of quantum computation we are able to substitute the continued fraction algorithm to find a certain unknown fraction by a simultaneous Diophantine approximation. While the continued fraction algorithm is able to find a Diophantine approximation to a single known fraction with a denominator greater than N2, our simultaneous Diophantine approximation method computes in polynomial time unusually good approximations to known fractions with a denominator of size N1+?, where ? is allowed to be an arbitrarily small positive constant. As these unusually good approximations are almost unique we are able to recover an unknown denominator using fewer qubits in the quantum part of our algorithm.

In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems.KeywordsProduct FamilyIterate Function SystemMarkov PartitionSturmian SequenceDiscrete PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Continued fraction techniques are equivalent to Berlekamp's algorithm. The sequence D(k), k \geq 0 , in Berlekamp's algorithm provides the information about when Berlekamp's algorithm completes one iterative step of the continued fraction. In fact, this happens when D(K) ; and when D(k) \neq D(k + 1) , it implies that Berlekamp's algorithm begins the next iterative step of the continued fraction.

This thesis falls naturally into two distinct parts. Both come under the general heading of the theory of conformal mapping though the later part incorporates work in potential theory. We first study the growth of means of the logarithmic derivative of a univalent function in the disc. Here results have been obtained by Hayman and by Baernstein and Brown. Hayman has shown that an elementary upper bound for the growth of these means is best possible in general. Later on, Baernstein and Brown showed that the means of certain classes of monotone slit mappings, including support points of the class S of normalised univalent functions, grow no faster than those of the Koebe function up to a multiplicative constant. The question remained open for unrestricted monotone slit mappings. We settle this question by constructing a monotone slit mapping the mean of whose logarithmic derivative grows faster than that of the Koebe function. Following this, we discuss some recent work by Burdzy on the boundary behaviour of positive harmonic functions in Lipschitz domains and applications of this work to the angular derivative problem. Burdzy obtains his results on the angular derivative by probabilistic methods. Rodin and Warschawski later gave a classical proof of part of Burdzy’s main result and related his criteria for the existence of an angular derivative to criteria which they had used previously. They were, however, unable to obtain a non-probabilistic proof of the full theorem. Using a new non-probabilistic method, we prove a theorem on the growth of positive harmonic functions vanishing near a boundary point of a Lipschitz domain. The plane case of this result and some special cases in space were proved by Burdzy in a series of articles. He went on to prove the full result in space in a later paper with R. J. Williams. Our result enables us to give an elementary proof of the remainder of Burdzy’s theorem on the angular derivative and so complements Rodin and Warschawski’s work. We complete our study of this problem by proving two further related results on the boundary behaviour of positive harmonic functions in Lipschitz domains. It is likely that the methods used will be helpful in problems of a similar nature.

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.