Use Of Continued Fractions Research Articles

A novel improvement in RSA algorithm

In this study, we propose a new cryptographic scheme based on the textbook RSA algorithm associating the concept of continued fractions. The algorithm is designed under the three primary steps of key generation, encryption, and decryption. The key generation of the system is improved to generate a large encryption key without affecting the decryption key. The use of continued fractions gives an additional advantage as it encrypts a string of plaintexts by encrypting only two integers which optimizes the time and the memory consumption. The encryption process and the decryption process are similar to that of the standard RSA algorithm, insomuch the implemented algorithm does not affect the security of the original RSA algorithm which depends on the factoring problem. In addition, as the algorithm uses a large encryption key, it has been proven that the system is secure against the Wiener’s attack.

Some Topological Indices of Double Square Snake Graphs

In many areas of science, lattice structures are very useful phenomenons. In network sciences, in chemistry and in social sciences, we face them in the solution of many daily life problems. Several large lattice structures can also be thought as graphs and in that way, are useful in the study of large networks. A very recently defined and studied class of such networks are snake graphs which has close relations with number theory due to the use of continued fractions. Following the works on square snake graphs, in this work, we study some non-Zagreb type topological graph indices of some interesting lattice structures called as double square snake graphs. We first obtain the vertex and edge partitions of these graphs and calculate their indices by means of these partitions.

On a Multi-Channel Retrial Queueing System

The paper considers Markov multi-channel queueing systems with a single retrial attempt to begin service process. The authors establish the conditions of the existence of stationary mode and propose efficient algorithms for calculation of the stationary probabilities with the use of continued fractions and explicit vector-matrix formulas.

Pattern equivariant cohomology and theorems of Kesten and Oren

In 1966, Harry Kesten settled the Erdős–Szüsz conjecture on the local discrepancy of irrational rotations. His proof made heavy use of continued fractions and Diophantine analysis. In this paper, we give a purely topological proof Kesten's theorem (and Oren's generalization of it) using the pattern equivariant cohomology of aperiodic tiling spaces.

The use of continued fractions to describe the highly excited rotational states of a H2O molecule

Continued fractions D(a, x) were used in the analysis of the highly excited (J ≤ 42, Ka ≤ 32) rotational levels of the first eight vibrational states of a H216O molecule. The fractions depend on the rotational operator Jz, defined by the variable x, and the shape parameter a, which determines the asymptotic behavior of the calculated energy levels. Continued fractions D(a, x) are new forms of the generating functions for the effective rotational Hamiltonian of some nonrigid molecule of the H2X type.

High-order harmonic generation and dynamic localization in a driven two-level system, a non-perturbative solution using the Floquet–Green formalism

We apply the Floquet-Green operator formalism to the case of a harmonically-driven two-level system. We derive exact expressions for the quasi-energies and the components of the Floquet eigenstates with the use of continued fractions. We study the avoided crossings structure of the quasi-energies as a function of the strength of the driving field and give an interpretation in terms of resonant multi-photon processes. From the Floquet eigenstates we obtain the time-evolution operator. Using this operator we study Dynamic Localization and High-order Harmonic Generation in the non-perturbative regime.

Transient solutions of Markov processes and generalized continued fractions

In 1974 J. A. Murphy and M. R. O'Donohoe numerically approximated the minimal solution of the Kolmogorov forward equation for the generalized birth and death process by use of continued fractions. This paper generalizes this approach by suggesting an algorithm for q-matrices of lower band structure (n, 1). This is achieved by analogy with generalized continued fractions. Applications involving q-matrices of this type include, for example, many types of queueing systems with batch processing or birth-death-catastrophe population processes in biology.

Numerical analysis of single-variable problems with use of continued fractions

The results of this paper suggest that a continued fraction representation of single-variable problems provides certain advantages over more traditional methods of numerical analysis. The key advantage of continued fraction methods lies in the reduced number of mathematical operations, when compared to traditional methods, needed for data interpolation, root finding and extremum estimation. A systematic representation of continued fractions is presented which allows easy computer adaptation for any size problem.

Use of continued fractions for the calculation of higher-order terms in a perturbation system

Higher-order terms of eigenvalues in a perturbation system with discrete energy levels described by the Schrödinger equation are calculated using continued fractions. This method is used for calculating energy eigenvalues for a rigid rotator which has a dipole and interacts with an applied electric field. Also, a rigid plane rotator with a potential of the cosine type is considered.

On the Use of Continued Fractions for Digital Computer Arithmetic

Recently, there has been some interest in the use of continued fractions for digital hardware calculations. We require that the coefficients of the continued fractions be integral powers of 2 and, therefore, well-known continued fraction expansions of functions cannot be used. Methods of expansion of a large number of functions are presented. We show that the problem of selection of coefficients of the continued fractions does not have practical solution in most of the cases we have considered.

On a class of arithmetic codes and a decoding algorithm (Corresp.)

Multiple-error-correcting arithmetic codes which are nonlinear are constructed by residue encoding. A simple algorithm is given for correcting multiple errors which makes use of continued fractions.

Application of Continued Fractions for Fast Evaluation of Certain Functions on a Digital Computer

The purpose of this paper is to develop a method for evaluation of certain elementary functions on a digital computer by the use of continued fractions. The time required for this evaluation is drastically reduced by using short operations like shift and add, instead of multiplications. Functional consistency is the most important factor that aliows the expansion of a function into a continued fraction. Several cases are discussed; in particular the solution of the quadratic equation is discussed in more detail to demonstrate the convergence of the method.

The Status of Investigations into Computer Hardware Design Based on the Use of Continued Fractions

The purpose of this paper is to demonstrate that representations of numbers other than positional notation may lead to practical hardware realizations for digital calculation of classes of algorithms. This paper describes current research in the use of continued fractions. Although practicality has not been demonstrated, theoretical results are promising.

Continued Fractions

CONTINUED fractions have been known for centuries and, during the 1800's, were considered to be a standard part of higher algebra taught at the college level. For the first half of this century these fractions have received little attention. During the last ten or fifteen years, however, there has been a growing interest in the use of continued fractions.

Use of Mass as a Perturbation Parameter in Vibrations

Linear vibration problems involving harmonic excitation of discrete and continuous systems are solved by using the classical perturbation technique. The perturbation parameter is proportional to a mass and the square of the excitation frequency. The power series solution for the displacement of some point in the system is converted to the quotient of two polynomials by the use of continued fractions. The eigenvalues (natural frequencies) of the problem are calculated by finding the roots of the denominator polynomial. The situation wherein a quantity which cannot vanish at any frequency can be found is treated as a special case.

Computational Aspects of Three-Term Recurrence Relations

Computational Aspects of Three-Term Recurrence Relations

Numerical Evaluation of Continued Fractions

Numerical Evaluation of Continued Fractions

Bounds for Mills' Ratio for the Type III Population

Cohen [1] and Des Raj [2] have shown that in estimating the parameters of truncated type III populations, it is necessary to calculate for several values of $x$ the Mills ratio of the ordinate of the standardized type III curve at $x$ to the area under the curve from $x$ to $\infty$. Des Raj [3] has also noted that for large values of $x$ the existing tables of Salvosa [4] are inadequate for this purpose and he has found lower and upper bounds for the ratio. The object of this note is to improve these bounds, by obtaining monotonic sequences of lower and upper bounds through the use of continued fractions.

Use of continued fractions in high speed computing

Three numerical methods for evaluating continued fractions are discussed. An Arithmetic example is given for each method. A comparison is made of the speed of convergence of power series expansions and continued fraction expansions for several functions. Recommendations are made for the most efficient method of evaluating continued fractions for different types of computing machines.

2030. A practical use of continued fractions

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.