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One of the uses of the Chinese Remainder Theorem (CRT) is to shorten the time in the RSA Cryptosystem decryption process. However, in applying a combination of the CRT method and the RSA Algorithm with Python programming modules, recursion errors often occurred when determining the inverse modulus to establish the private key. The error was caused by limitation iterations of the related algorithm. Some Python library needs more iteration processes to calculate the inverse modulus for larger key size. This is a pure computational problem. The standard RSA algorithm uses two prime numbers to generate their key pairs. The multi-prime RSA modification based on more than two prime numbers can be applied to solve the problem. A modified RSA was applied in Python programming that generates keys based on 2, 4, 8, 16, and 32 prime numbers. Regarding on our experiments, the combination of the CRT method and RSA algorithm based on four prime numbers can be applied without recursion error up to a key length of 6280 bits, with eight primes up to 12,560 bits, with sixteen primes up to 25,120 bits, and with thirty-two primes up to 50,240 bits. Implementation of more multi-prime numbers can expand to combine the CRT method and RSA Algorithm for longer key sizes by approximation relation y = 1567,8x + 63,333, which y is the key size and x is the number of primes numbers. The method also can increase key generation and decryption rate significantly. [ABSTRACT FROM AUTHOR]

In computer science, divide-and-conquer is an important algorithm. The literal interpretation is "divide-and-conquer," in which a complex problem is divided into two or more identical or similar sub-problems, and then sub-problems are divided into smaller sub-problems, until the final sub-problem can be solved simply and directly. The combination of solutions of the solution of the original problem. This technique is the foundation of many efficient algorithms, such as sort algorithm (fast sort, merge sort), Fourier transform (fast Fourier transform) and so on. When it comes to divide-and-conquer algorithms, it's necessary to say recursion. They're like twin brothers, often used in algorithm design at the same time, and As a result, many efficient algorithms are generated. The direct or indirect call of its own algorithm is called a recursive algorithm. The function defined by the function itself is called recursive function. The sub-problem generated by divide-and- conquer method is often the smaller pattern of the original problem, which makes it convenient to use recursive technique. In this case, the repeated application of divide-and-conquer means can make the sub-problem consistent with the original problem type but its scale is constantly reduced, and finally the sub-problem can be reduced to a very easy solution. This naturally leads to recursion [ABSTRACT FROM AUTHOR]

In this paper the Noncentral Pólya-Aeppli distribution is introduced. It is a sum of two independent random variables, one that is a Poisson and another one, a Pólya-Aeppli distributed. The probability mass function, moments, recursion formulas and some properties of the defined distribution are derived. [ABSTRACT FROM AUTHOR]

Separate Multinode Ascending Derivatives Expansion (SMADE) is a recently developed function representation method based on "Separate Node Ascending Derivatives Expansion (SNADE)" which was proposed by Prof. Demiralp. For this reason we call this method in this work "Demiralp's SMADE". The basic difference between two methods is that SNADE uses one separate node for each derivative to construct the expansion while SMADE uses multinodes for the same entities even though the separate nature of the nodes is not mandatory. This study focuses on SMADE both to present all important details of the method including its formulation and its basis polynomials. [ABSTRACT FROM AUTHOR]

The determination of the energy states is highly studied issue in the quantum mechanics. Based on expectation values dynamics, energy states can be observed. But conditions and calculations vary depending on the created system. In this work, a symmetric exponential anharmonic oscillator is considered and development of a recursive approximation method is studied to find its ground energy state. The use of majorant values facilitates the approximate calculation of expectation values. [ABSTRACT FROM AUTHOR]

The fast progress in recent years in symbolic computation and variable-precision arithmetic provide a possibility for generating the recursion coefficients in the three-term recurrence relation for orthogonal polynomials with respect to several nonclassical weight functions, as well as the construction of the corresponding quadrature rules of Gaussian type. Such quadratures are very important in many applications in engineering (fracture mechanics, damage mechanics, etc.), as well as in other computational and applied sciences. The boundary element method (BEM), finite element method (FEM), methods for solving integral equations, etc. very often require the numerical evaluation of one dimensional or multiple integrals with singular or near singular integrands with a high precision. In this paper we give some improvements of quadrature rules of Gaussian type with logarithmic and/or algebraic singularities. A numerical examples is included. [ABSTRACT FROM AUTHOR]

We present a flexible and robust system for implementing one-dimensional coined quantum walks. A recursion loop in the optical network together with a translation of the spatial into the time domain ensures the possible increment of the step number without need of additional optical elements. An intrinsic phase stability assures a high degree of coherence and hence guarantees a good scalability of the system. We performed a quantum walk over 27 steps and analyzed the 54 output modes. Furthermore, we estimated that up to 100 steps can be realized with only minor changes in the used components. [ABSTRACT FROM AUTHOR]

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket SGb(n) at stage n with b equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants. [ABSTRACT FROM AUTHOR]

In this paper, we consider the manifold of covariance matrices of order n parametrized by reflection coefficients which are derived from Levinson's recursion of autoregressive model. The explicit expression of the reparametrization and its inverse are obtained. With the Riemannian metric given by the Hessian of a Kähler potential, we show that the manifold is in fact a Cartan-Hadamard manifold with lower sectional curvature bound -4. The explicit expressions of geodesics are also obtained. After that we introduce the notion of Riemannian median of points lying on a Riemannian manifold and give a simple algorithm to compute it. Finally, some simulation examples are given to illustrate the applications of the median method to radar signal processing. [ABSTRACT FROM AUTHOR]

Dubois coined the term incursion, for an inclusive or implicit recursion, to describe a discrete-time anticipatory system which computes its future states by reference to its future states as well as its current and past states. In this paper, we look at a model which has been proposed in the context of a social system which has functionally differentiated subsystems. The model is derived from a discrete-time compartmental SIS epidemic model. We analyse a low order instance of the model both in its form as a recursion with no anticipatory capacity, and also as an incursion with associated anticipatory capacity. The properties of the incursion are compared and contrasted with those of the underlying recursion. [ABSTRACT FROM AUTHOR]

This work focuses on the construction of a new method to solve the ordinary differential equations. The method presented here uses space extension concept. Main purpose of this work is to construct an approximation sheme to the solutions of the ordinary differential equations via two-term recursion formulae. In order to do this, the equation is converted to a new form which is called “Okubo Form” by introducing new unknowns into equation. Additionally, this work concerns with the series solution of the resulting Okubo Form and attempts to construct approximate solutions by truncating this series solution. [ABSTRACT FROM AUTHOR]

We review the recently developed bootstrap method for the compulation of high-multiplicity QCD amplitudes at one loop. The method combines (generalized) unitarity with on-shell recursion relations to determine the not cut-constructible, rational terms of these amplitudes. The bootstrap approach works for arbitrary configurations of gluon helicities and arbitrary numbers of external legs. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

The study of anticipatory systems derived from causal systems or recursions has been well established. This paper investigates what happens to the fixed points of a system modelled by a first order recursion when the recursion is replaced by an associated incursion. Results obtained by Dubois with this technique applied to the logistic map are extended to other unimodal maps, typical of a class of population models. Other incursions associated with these maps are also studied, and a paradigm is given for generating a strong anticipatory system from a recursion whose map is of bounded variation. This is done by replacing the recursion with an incursion defined in terms of the variation of the map. The effect of this replacement on the stability properties of the fixed points of the map is examined. [ABSTRACT FROM AUTHOR]

The coupled KdV-mKdV system arises as the classical part of one of superextensions of the KdV equation. For this system, we prove its complete integrability, i.e., existence of a recursion operator and of infinite series of symmetries. After giving a short introduction into the theory of symmetries, coverings, and the notion of Cartan-covering, the recursion operator will be constructed as a symmetry in the Cartan covering of the KdV-mKdV system. [ABSTRACT FROM AUTHOR]

We develop a first-principles calculation method for the quantum transport through single molecules between semiconductor electrodes by using the recursion-transfer-matrix (RTM) method combined with non-equilibrium Green’s function (NEGF) method. We describe the procedure of the present RTM/NEGF method with the boundary conditions to connect to semiconductor electrodes and show electronic states of a single molecule bridged between Si electrodes. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]