Continued Fraction Research Articles

An Algorithm for Computing the n^th Root of Real Numbers using a General Method

Abstract: Based on an old article about extracting the ࢔ ࢎ࢚ of real numbers using a general method, we introduce an algorithm for computing such roots, using a different approach, through introducing an operator which we define as the inverse binomial operator .We illustrate the procedure by computing ࢔ ࢎ࢚ roots of some ,randomly, selected real numbers. To show the beauty of using such a new procedure, we compare our calculations with the calculations obtained, for the a certain example, via well-known iteration methods such as the bisection and Newton-Raphson methods; moreover a comparison is made for same purpose using continued fractions method. Finally, a conclusion is given to clarify the advantages of the inverse binomial operator.

Mitigation of numerical issues appearing in transient analyses when applying fractional derivative approximations

Due to the potential decrease of the computation time for problems with fractional order derivatives, and due to the extension of the range of applicable solvers for a given problem, the approximation of fractional derivatives (e.g., using Oustaloup’s method) is an important and frequently discussed topic. A significant problem that can occur (although one that is not discussed very often) when approximations are applied concerns the potential numerical issues that can arise when applying these approximations in a transient analysis, e.g., as shown in the paper, in a time-stepping solver. This paper examines five different methods for approximating the fractional derivative (Oustaloup’s method, a refined Oustaloup method, a modified continued fraction expansion method, Matsuda’s method and Charef’s method). An important preliminary step in the analysis was the establishment of the general form of FDAE (fractional differential-algebraic equations), the general form of the applied approximations and the general form of the DAE (differential-algebraic equations) resulting from the approximation. This allowed for a precise description of the equations according to which the transformation from the FDAE to the approximating DAE takes place. Three problems derived from studies in electrical engineering have been introduced into the analysis (a linear circuit problem, a circuit problem with nonlinear fractional elements, and a problem featuring nonlinearities and pregenerated noise). For these problems, the conditions in which numerical issues appear have been studied in detail. They are the main motivation of this study as these issues are so significant that they often result in zero-solutions or solutions tending to infinity (giving an impression as if the system is unstable). Modifications and alternatives of the transformation into the DAE, that aim at the mitigation of these numerical errors, are mentioned later. The final result is very satisfactory, where the algorithm for the transformation of the FDAE to the approximating DAE practically eliminates the most important barriers (which was the impossibility of using approximations above a certain order due to the mentioned numerical issues). The study presented in this paper is motivated by problems in electrical engineering but due to its generality it is also applicable in other fields where fractional calculus is used.

New dynamical system for circular satellites relative motion

In this work, a new perturbed dynamical system to describe the satellites relative motion is derived considering that the gravitational attractive force is generated by the continued fraction potential. In this context, the parameter of continuation fraction creates an additional force which is the source of perturbation for motion. Aiming to find a simple and precise mathematical model to characterize this motion while maintaining high accuracy results, a new circular linear perturbed system is also derived under the effect of this parameter. Furthermore, the perturbed linear dynamical system of satellites relative motion is studied, along with finding the conditions of stable and periodic solutions. Finally, these solutions are calculated and their graphical simulation is introduced. We emphasize that the new proposed system could be applied in space rendezvous missions aiming at getting precise descriptions of satellites relative motion.

General Integral Transform Performance for Space-Time Fractional Telegraph Equations

The development of technology has supported effective tools in industrial machines and set up the remarkable phase that serves well-being such as kinetic energy, kinetic movement, and nuclear energy. Applied mathematics has also contributed valuable procedures in various fields of these sciences, especially the creation of transformation. With practical relevance, a new general integral (NGI) transform has also shown a crucial role in the same pragmatic methods. In this paper, the NGI transform using the combination of Padé approximation including continued fraction expansions (CFE) has been used to attain approximate solutions of space-time fractional telegraph equations by directly getting the inverse transform.

Multiple orthogonal polynomials, 𝑑-orthogonal polynomials, production matrices, and branched continued fractions

I analyze an unexpected connection between multiple orthogonal polynomials, d d -orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot’s combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.

Fractionation of edible fats—Separation of fat‐covered particles by decanter centrifuge

AbstractThis work investigates the feasibility of using a decanter centrifuge to separate solid particles coated with a high melting fat from a liquid oil. The process involves feeding cold silica particles into a melt, upon which the hardstock fat crystallizes, and the particles are subsequently separated using a decanter centrifuge. Being the first attempt for such a design, only the separation step is studied, and a model system is used. It comprised canola oil (CO), fully hydrogenated rapeseed oil a priori crystallized on silica gel particles. Different particle sizes and accelerations of gravity were studied. To measure the success of this separation process, the data from light microscopy, and differential scanning calorimetry of the feed, oleins, and stearins were used. The findings show that the separation of the fatty particles used in this work, and liquid oil in decanter centrifuges is feasible while future research should focus on the crystallization part of the process. The model developed to describe the oil‐holding capacity of the cake focuses on the capillary liquid entrapped in the spaces between the particles and sufficiently represents the experimental findings which show that separation efficiencies increase with increasing centrifugal acceleration.Practical Applications: This work introduces an alternative fat fractionation process by separating a model system of CO, FHRO, and solid entrainers through a decanter centrifuge. The results can be potentially used for the development of a fully continuous dry fractionation process, which can achieve in one step higher SE than the conventional ones and can be tripalmitate‐selective.

Quantum statistical mechanics and the boundary of modular curves

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

Successful cryptanalysis on RSA type modulus [formula omitted]

As internet technology advances and our interactions increasingly take place online, cryptography emerges as a valuable tool to address security concerns. Cryptography serves as a means to guarantee the protection of privacy and confidential information, thereby instilling confidence when sharing and exchanging such data with other parties. One of the benefits of cryptography is providing confidentiality, which protects our information; either data in transit or data at ease. Three new attacks have been proposed on RSA type modulus N=p2q. The equation eX−NY=Z−(p2k+q2m)Y is the basis for the first attack involves random values of k and m such that k being a multiple of 2 and m being a multiple of 3, both being integers with |p2k+q2m|<N1/2 and gcd(X,Y)=1. If Z<|p2k−q2m|3(p2k+q2m|)N1/3Y and XY<N4|p2k+q2m|, then by using continued fractions, factoring N can be accomplished within polynomial time.This paper also suggested the vulnerabilities t RSA cryptosystem moduli Ns=ps2qs for t≥2 and s=1,...,t for the second and third attack. The attacks are effective if there exists a relationship between t RSA public keys (Nt,et) expressed as esx−Nsys=zs−(ps2k+qs2m)ys or esxs−Nsy=zs−(ps2k+qs2m)y, where the variables x, xs, y, ys, and zs are sufficiently small. The importance lies in the presence of an algorithm that operates within probabilistic polynomial time, which is capable of accepting public parameters as input and yield the factors p and q as output. By employing this algorithm, one can verify whether a key falls within the vulnerable class or not. This characteristic can be valuable when designing a cryptosystem during the key generation process, as it helps prevent the inadvertent creation of weak keys.

Stability Analysis of Linear Control Systems by Wall’s Continued Fraction Expansion

Stability Analysis of Linear Control Systems by Wall’s Continued Fraction Expansion

Nature inspired algorithm based design of near ideal fractional order low pass Chebyshev filters and their realization using OTAs and CCII

Fractional order filters offer greater freedom of design and a precise control over stopband attenuation in electronic circuits and systems. This paper presents the design of a fractional order low pass Chebyshev filter (FOLCF) that achieves near-ideal response characteristics. The methodology introduced utilizes metaheuristic optimization methods, including particle swarm optimization, firefly algorithm, and grey wolf optimization. These techniques are employed to precisely adjust the filter coefficients for the orders (1+α), (2+α), and (3+α). The adjustment is carried out by comparing the desired behaviour of the FOLCF with generalized fractional order low pass transfer functions. Throughout these instances, the parameter α is varied within the range of (0, 1). The designed filters are then tested and compared on the basis of various factors. Simulation results demonstrate that the designed filters closely follow the behaviour of an ideal Chebyshev filter with maximum passband and stopband magnitude errors being −31.93 dB and −52.74 dB respectively for (1+α) order filters. These values for (2+α) and (3+α) order FOLCF have been observed to be −30.04 dB and −55.91 dB; −17.72 dB and −49.52 dB respectively. Furthermore, it has been observed that the proposed work outperforms existing state-of-the-art approaches in various aspects, including magnitude error, stopband attenuation, and cut-off frequency. The stability of the designed filters has been verified through stability analysis. Additionally, practical feasibility of the proposed FOLCF is demonstrated through SPICE simulations for α = [0.2,0.5,0.8] using second generation current conveyor (CCII) and operational transconductance amplifier (OTA) based topologies while approximating the constant phase element using fifth order continued fraction expansion. The SPICE implementations closely follow the behaviour of ideal filter with −48.67 dB and −62.8 dB as mean square errors for CCII and OTA circuits respectively, showcasing the proposed filters' superiority and practical applicability in advanced electronic design.

THE SUMMED PAPERFOLDING SEQUENCE

Abstract The sequence $a( 1) ,a( 2) ,a( 3) ,\ldots, $ labelled A088431 in the Online Encyclopedia of Integer Sequences, is defined by: $a( n) $ is half of the $( n+1) $ th component, that is, the $( n+2) $ th term, of the continued fraction expansion of $$ \begin{align*} \sum_{k=0}^{\infty }\frac{1}{2^{2^{k}}}. \end{align*} $$ Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding sequence, A014577. This paper proves several results for this summed paperfolding sequence and confirms Hendriks’s conjecture.

On the analytic extension of the Horn's hypergeometric function $H_4$

The paper establishes new convergence domains of branched continued fraction expansions of Horn's hypergeometric function $H_4$ with real and complex parameters. These domains enabled the PC method to establish the analytical extension of analytical functions to their expansions in the studied domains of convergence. A few examples are provided at the end to illustrate this.

Optimal and typical $$L^2$$ discrepancy of 2-dimensional lattices

AbstractWe undertake a detailed study of the $$L^2$$ L 2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal $$L^2$$ L 2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler’s number e. In the metric theory, we find the asymptotics of the $$L^2$$ L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.

Multifractal analysis of convergence exponents for products of consecutive partial quotients in continued fractions

Multifractal analysis of convergence exponents for products of consecutive partial quotients in continued fractions

Zaremba’s Conjecture for Geometric Sequences: An Algorithm

Even though Zaremba’s conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba’s conjecture.

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

In the paper, the problems of convergence and relative stability to perturbations of a branched continued fraction with positive elements and a fixed number of branching branches are investigated. The conditions under which the sets of elements \[\Omega_0 = ( {0,\mu _0^{(2)}} ] \times [ {\nu _0^{(1)}, + \infty } ),\quad \Omega _{i(k)}=[ {\mu _k^{(1)},\mu _k^{(2)}} ] \times [ {\nu _k^{(1)},\nu _k^{(2)}} ],\]\[i(k) \in {I_k}, \quad k = 1,2,\ldots,\] where $\nu _0^{(1)}&gt;0,$ $0 &lt; \mu _k^{(1)} &lt; \mu _k^{(2)},$ $0 &lt; \nu _k^{(1)} &lt; \nu _k^{(2)},$ $k = 1,2,\ldots,$ are a sequence of sets of convergence and relative stability to perturbations of the branched continued fraction \[\frac{a_0}{b_0}{\atop+}\sum_{i_1=1}^N\frac{a_{i(1)}}{b_{i(1)}}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{b_{i(2)}}{\atop+}\ldots{\atop+} \sum_{i_k=1}^N\frac{a_{i(k)}}{b_{i(k)}}{\atop+}\ldots\] have been established. The obtained conditions require the boundedness or convergence of the sequences whose members depend on the values $\mu _k^{(j)},$ $\nu _k^{(j)},$ $j=1,2.$ If the sets of elements of the branched continued fraction are sets ${\Omega _{i(k)}} = ( {0,{\mu _k}} ] \times [ {{\nu _k}, + \infty } )$, $i(k) \in {I_k}$, $k = 0,1,\ldots,$ where ${\mu _k} &gt; 0$, ${\nu _k} &gt; 0$, $k = 0,1,\ldots,$ then the conditions of convergence and stability to perturbations are formulated through the convergence of series whose terms depend on the values $\mu _k,$ $\nu _k.$ The conditions of relative resistance to perturbations of the branched continued fraction are also established if the partial numerators on the even floors of the fraction are perturbed by a shortage and on the odd ones by an excess, i.e. under the condition that the relative errors of the partial numerators alternate in sign. In all cases, we obtained estimates of the relative errors of the approximants that arise as a result of perturbation of the elements of the branched continued fraction.

Metrical properties of finite product of partial quotients in arithmetic progressions

Metrical properties of finite product of partial quotients in arithmetic progressions

A characterization of quasipositive two-bridge knots

We prove a simple necessary and sufficient condition for a two-bridge knot [Formula: see text] to be quasipositive, based on the continued fraction expansion of [Formula: see text]. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.

Assouad type dimensions of infinitely generated self-conformal sets

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.