Continued Fraction Research Articles

Public key exponent attacks on multi-prime power modulus using continued fraction expansion method

This paper proposes three public key exponent attacks of breaking the security of the prime power modulus 𝑁=𝑝2𝑞2 where 𝑝 and 𝑞 are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus 𝑁=𝑝2𝑞2 for q<𝑝<2q using key equation 𝑒𝑑−𝑘𝜙(𝑁)=1 where 𝜙(𝑁)= 𝑝2𝑞2(𝑝−1)(𝑞−1) can be broken by recovering the secret keys 𝑘 /𝑑 from the convergents of the continued fraction expansion of e/𝑁−2𝑁3/4 +𝑁1/2 . The paper also reports the second and third approaches of factoring 𝑛 multi-prime power moduli 𝑁𝑖=𝑝𝑖2 𝑞𝑖2 simultaneously through exploiting generalized system of equations 𝑒𝑖𝑑−𝑘𝑖𝜑(𝑁𝑖)=1 and 𝑒𝑖𝑑𝑖−𝑘𝜑(𝑁𝑖)=1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for 𝑖=1,2,…,𝑛.

On the growth behavior of partial quotients in continued fractions

On the growth behavior of partial quotients in continued fractions

Lochs-type theorems beyond positive entropy

Lochs’ theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. In its original version, Lochs’ theorem related decimal expansions with continued fraction expansions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. This is the viewpoint we develop here. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature.

Slowly rotating solution of quadratic gravity: An analytical approximation method

As we know, it is a nontrivial task to obtain analytical solutions for various gravitational theories. In this work, we obtain an analytical approximate rotating black hole solution for quadratic gravity (QG). To get a complete solution, we construct near-horizon and asymptotic solutions and then use these limited cases to obtain an approximate analytic solution through a continued fraction method. By checking the first law of thermodynamics and the Smarr formula, we studied the thermodynamics of the black hole solution. Finally, we investigate various physical properties of the solution including its photon sphere, and shadow.

High order coupled microring resonator filters for on-chip optical interconnects with steep edge response

ABSTRACT In this paper, high-order microring resonator filters coupled serially up to the 7th order are proposed for on-chip optical interconnects. This work mainly focuses on achieving a flat-top, higher out-of-band rejection ratio, and enhanced group delay value than previously demonstrated microring filters. Further, the effect of varying inter-resonator coupling coefficients on the filtering operation of wavelength 1552 nm is investigated, using the continued fraction method, and optimum conditions for coupling coefficients between ring-ring waveguides are realized to obtain steep edge response. The proposed design exhibits a minimum group delay of 7.457 ps at a coupling value of 0.04.

Applications of Continued Fractions

Many continuing fraction applications from different mathematical contexts and levels are the topic of this investigation. Many continuing fraction qualities are examined as the first step in the investigation. When looking for the best rational approximations of a real number x, the continuous fraction expansion is a particularly effective method. In addition, continuing fractions are a remarkably flexible tool for handling issues involving motions spanning more than one time period. In mathematics, continued fractions play a crucial role. The fact that they may be used in so many different areas of pure and practical research is a major reason for their significance. Although though many people have heard of them, Continuous Fractions (CF) are an old topic. Algebra and other areas of study including arithmetic, physics, and chemistry all make use of repeated fractions.

Natural extensions and entropy of $ \alpha $-continued fraction expansion maps with odd partial quotients

In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $ \alpha\in [g, G] $, where $ g = \tfrac{1}{2}(\sqrt{5}-1) $ and $ G = g+1 = 1/g $ are the two golden mean numbers. Using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [1] are obtained, and it is shown that for each $ \alpha, \alpha^*\in [g, G] $ the natural extensions from [1] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $ \alpha\in [g, G] $, a fact already observed in [1]. Furthermore, it is shown that this approach can be extended to values of $ \alpha $ smaller than $ g $, and that for values of $ \alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g] $ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $ \alpha\in [0, G] $. It is shown that if there exists an ergodic, absolutely continuous $ T_{\alpha} $-invariant measure, in any neighborhood of $ 0 $ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. Moreover, we identify the largest interval on which the entropy is constant. In order to prove this we use a phenomenon called matching.

Transcendence and continued fraction expansion of values of Hecke–Mahler series

Let $\theta $ and $\rho $ be real numbers with $0 \le \theta , \rho \lt 1$ and $\theta $ irrational. We show that the Hecke–Mahler series $$ F_{\theta , \rho } (z_1, z_2) = \sum _{k_1 \ge 1} \, \sum _{k_2 = 1}^{\lfloor k_1 \theta + \rho \rfloor } z_1^{k_1

Design and implementation of fractional-order controller in delta domain

In this work, a fractional-order controller (FOC) is designed in a discrete domain using delta operator parameterization. FOC gets rationally approximated using continued fraction expansion (CFE) in the delta domain. Whenever discretization of any continuous-time system takes place, the choice of sampling time becomes the most critical parameter to get most accurate results. Obtaining a higher sampling rate using conventional shift operator parameterization is not possible and delta operator parameterized discretize time system takes the advantages to circumvent the problem associated with the shift operator parameterization at a high sampling limit. In this work, a first-order plant with delay is considered to be controlled with FOC, and is implemented in discrete delta domain. The plant model is designed using MATLas well as in hardware. The fractional-order controller is tuned in the continuous domain and discretized in delta domain to make the discrete delta FOC. Continuous time fractional order operator (s??) is directly discretized in delta domain to get the overall FOC in discrete domain. The designed controller in implemented using MATLABSimulink and dSPACE board such that dSPACEboard acts as the hardware implemented FOC. The step response characteristics of the closed-loop system using delta domain FOC resembles to that of the results obtained by continuous time controller. It proves that at a high sampling rate, the continuous-time result and discrete-time result are obtained hand to hand rather than the two individual cases. Therefore, the analysis and design of FOC parameterized with delta operator opens up a new area in the design and implementation of discrete FOC, which unifies both continuous and discrete-time results. The discrete model performance characteristics are evaluated in software simulation using MATLAB, and results are validated through the hardware implementation using dSPACE.

Diophantine exponents of lattices and the growth of higher-dimensional analogues of partial quotients

A three-dimensional analogue of the connection between the exponent of the irrationality of a real number and the growth of the partial quotients of its expansion in a simple continued fraction is investigated. As a multidimensional generalization of continued fractions, Klein polyhedra are considered. Bibliography: 12 titles.

Model of money income diffusion in the European integration context

A model of money income diffusion is constructed taking into account the processes of social comparison and the development of the income formation spatial structure. To implement such model, the analytical theory of continued fractions is used and the corresponding differential equation solution is found in the form of a formal functional continued fraction. The values of the approach fractions of a continuous fraction describing the dynamics of changes in the level of money income give an approximation of the real values with almost predetermined, arbitrarily high accuracy. This allowed us to qualitatively describe the general dynamics of the money income diffusion process.

Continued Fractions of Higher Order Polygonal Numbers with Respect to Order and Rank

Developing number sequence based on polygonal numbers is an enthusiastic field in number theory. As tetrahedral numbers are similar to pyramids, one of the Seven wonders of the World, yields a unique copiousness in its suitability. In number theory study of pyramidal numbers vary in richness and variety. Also the study of continued fractions is a fastly developing field.

Forbidden conductors of $L$-functions and continued fractions of particular form

In this paper we study the forbidden values of the conductor $q$ of the $L$-functions of degree 2 in the extended Selberg class by a novel technique, linking the problem to certain continued fractions and to their weight $w_q$. Our basic result states tha

Diophantine exponents of lattices and the growth of higher-dimensional analogues of partial quotients

Изучается трехмерный аналог связи экспоненты иррациональности вещественного числа с ростом его неполных частных при разложении в обыкновенную цепную дробь. В качестве многомерного обобщения цепных дробей рассматриваются полиэдры Клейна. Библиография: 12 названий.

New proofs of some Dedekind η-function identities of level 6

Recently, Shaun Cooper proved several interesting ?-function identities of level 6 while finding series and iterations for 1/?. In this sequel, we present some new proofs of the ?-function identities of level 6 discovered by Cooper. Here, in this article, we make use of the modular equation of degree 3 in two methods. We further give some interesting combinatorial interpretations of colored partitions. We also briefly describe a potential direction for further researches based upon some related recent developments involving the Jacobi?s triple-product identity and the theta-function identities as well as on several other q-functions which emerged from the Rogers-Ramanujan continued fraction R(q) and its such associates as G(q) and H(q). We point out the importance of the usage of the classical q-analysis and we also expose the current trend of falsely-claimed ?generalization? by means of its trivial and inconsequential (p, q)-variation by inserting a forced-in redundant (or superfluous) parameter p.

Continued fractions for bi-periodic Fibonacci sequence

In this paper, we study generalized continued fractions for the expression of bi-periodic Fibonacci ratios.

Collision detection for N-body Kepler systems

Context. In a Keplerian system, a large number of bodies orbit a central mass. Accretion disks, protoplanetary disks, asteroid belts, and planetary rings are examples. Simulations of these systems require algorithms that are computationally efficient. The inclusion of collisions in the simulations is challenging but important. Aims. We intend to calculate the time of collision of two astronomical bodies in intersecting Kepler orbits as a function of the orbital elements. The aim is to use the solution in an analytic propagator (N-body simulation) that jumps from one collision event to the next. Methods. We outline an algorithm that maintains a list of possible collision pairs ordered chronologically. At each step (the soonest event on the list), only the particles created in the collision can cause new collision possibilities. We estimate the collision rate, the length of the list, and the average change in this length at an event, and study the efficiency of the method used. Results. We find that the collision-time problem is equivalent to finding the grid point between two parallel lines that is closest to the origin. The solution is based on the continued fraction of the ratio of orbital periods. Conclusions. Due to the large jumps in time, the algorithm can beat tree codes (octree and k-d tree codes can efficiently detect collisions) for specific systems such as the Solar System with N < 108. However, the gravitational interactions between particles can only be treated as gravitational scattering or as a secular perturbation, at the cost of reducing the time-step or at the cost of accuracy. While simulations of this size with high-fidelity propagators can already span vast timescales, the high efficiency of the collision detection allows many runs from one initial state or a large sample set, so that one can study statistics.

On symmetries of 3-dimensional algebraic continued fractions

On symmetries of 3-dimensional algebraic continued fractions

Application of Continued Fraction in Pell's Equation

This paper uses a continued fraction to explain various approaches to solving Diophantine equations. It first examines the fundamental characteristics of continued fractions, such as convergent and approximations to real numbers. Using continued fractions, we can solve the Pell's equation. Certain theorems have also been discussed for how to expand quadratic irrational integers into periodic continued fractions. Finally, the relationship between convergent and best approximations and use of continuous fraction in calendar construction has-been investigated. The analytical theory of continued fractions is a significant generalization of continued fractions and represents a large field for current and future research.

Discretization of Fractional Order Operator in Delta Domain

The fractional order operator is the backbone of the fractional order system (FOS). The fractional order operator (FOO) is generally represented as s^(±μ) (0<μ<1). Discrete time FOS can be obtained through the discretization of the fractional order operator. The FOO is the general form of either fractional order differentiator (FOD) or integrator (FOI) depending upon the values of μ. Out of the two discretization methods, direct discretization outperforms the method of indirect discretization. The mapping between the continuous time and discrete time domain is done with the development of generating function. Continuous fraction expansion (CFE) is used expand the generating function for the rational approximation of the FOO. There is an inherent problem associated with the discretization of FOO in discrete z-domain particularly at very fast sampling rate. In the other hand, discretization using delta operator parameterization provides the continuous time and discrete time results in hand to hand, when the continuous time systems are sampled at very fast sampling rate and circumventing the problem with shift operator parameterization at fast sampling rate. In this work, a new generating function is proposed to discretize the FOO using the Gauss-Legendre 3-point quadrature rule and generating function is expanded using the CFE to form rational approximation of the FOO in delta domain. The benchmark fractional order systems are considered in this work for the simulation purpose and comparison of results are made to prove the efficacy of the proposed method using MATLAB.