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The purpose of this paper is to illustrate the many aspects of Poincare recurrence time theorem for an archetype of a complex system, the logistic map.

At the beginning of the twentieth century, Poincare's recurrence theorem had revolutionized modern mechanics and statistical physics. However, this theorem did not attract considerable attention, at least from a numerical and computational point of view. In a series of relatively recent papers, Balakrishnan, Nicolis and Nicolis have addressed the recurrence time problem in a firm basis, introducing notation, theory, and numerical studies. Motivated by this call, the paper proposes to illustrate the many aspects of Poincare recurrence time theorem for an archetype of a complex system, the logistic map. The authors propose here in different tests and computations, each one illuminating the many aspects of the problem of recurrence. The paper ends up with a short discussion and conclusions.

In this paper, the authors obtain new results on computations, each one illuminating the many aspects of the problem of recurrence. One striking aspect of this detailed work, is that when the sizes of the cells in the phase space became considerable, then the recurrence times assume ordinary values.

The paper extends previous results on chaotic maps to the logistic map, enhancing comprehension, making possible connections with number theory, combinatorics and cryptography.

This paper aims to analyze transient and steady state processes in series resistive capacitive-circuits when the supplied voltage is a sequence of periodic rectangular pulses. The purpose is to obtain an analytical formula for the capacitor voltage at any instant of time, both for the transient and the steady-state processes.

The main approach is to use a combination of differential and recurrence equations.

An analytical expression (formula) for obtaining the capacitor voltage at any instant of time has been found.

The results obtained are new. They are much more convenient to use than the results obtained by the Fourier series. The exposed approach can also be used in other circuits and other forms of the supplied voltage.

In most commercial processors, enhancing the speed of multiplication using radix-8 booth encoding is the preferred option. In radix-8 architecture, the 3X(= 2X + X) multiple generation is a major bottleneck. This paper aims to propose a parallel implementation scheme recognizing the symmetry in the carry recurrence equations of 3X multiples. The proposed architecture evaluates the odd (H) and even (K) carry signals separately. As prefix tree structure offers fast carry propagation, the parallel implementation is based on a hybrid style of two popular prefix architectures.

The performance of the proposed architecture is evaluated using Cadence TSMC 180 nm library. A comparison of performance parameters with other architectures has been carried out to highlight the architectural advantages of the proposed architecture.

A comparison of performance parameters with others shows that the proposed architecture has a reduced critical path and a commensurate improvement in delay for a bit width of 64. It is shown that up to 32 bits, this parallel architecture has a superior performance and would be the appropriate choice for Application Specific Integrated Circuit (ASIC) implementation. It has also been suggested that higher-order bit widths could be implemented using a modular arrangement.

This paper proposes a new parallel architecture for hard multiple (3X) generation in Radix-8 Booth encoding. As the multiplication is the key operation in digital signal processors, this type of high-speed architectures gains importance in the future processor design. Defence applications such as target finding and multiple target recognitions and image processing applications necessitate this type of high-speed multipliers. Also, it is appropriate for the ASIC implementation. The authors would like to mention that this paper is not yet published anywhere, and it is the research paper of Dr R. Nirmaladevi.

The paper examines the convergence characteristics of the “minimized residual method based on the MRTR (three‐term recurrence formula of CG‐type) method” for solving large linear simultaneous equations.

The paper uses an example of magnetic field analysis of permanent magnet type of MRI taking account of the minor loop and eddy current.

It is shown that the preconditioned MRTR method can get a stable and quick convergence for such a relatively ill‐conditioned problem.

Illustrates that the convergence of the Incomplete Cholesky Conjugate Gradient method is one of the important issues in the practical 3D magnetic field analysis.

The purpose of this paper is to introduce an innovative strategy for the approximate solution of the heat flow problems in two- and three-dimensional spaces. This new strategy is very easy to implement and handles the restrictive variable that may ruin the physical nature of the problem.

This study combines Sawi transform (ST) and the homotopy perturbation method (HPM) to formulate the idea of Sawi homotopy perturbation transform method (SHPTM). First, this study implements ST to handle the recurrence relation and then incorporates HPM to derive the series solutions of this recurrence relation. ST has the advantage in that it does not require any assumptions or hypothesis for the evaluation of series solutions.

This strategy finds the results very accurate and close to the precise solution. The graphical observations and the surface solution demonstrate that SHPTM is a reliable and powerful scheme for finding the approximate solution of heat flow problems.

The study presents an original work. This study develops SHPTM for the approximate solution of two- and three-dimensional heat flow problems. The obtained results and graphical representation demonstrate that SHPTM is a very authentic and reliable approach.

This paper sets out to study a system of fourth‐order obstacle boundary value problems associated with obstacle, unilateral and contact problems.

Differential transform method was used to solve the system.

It is demonstrated that the proposed scheme validates for this type of problems.

It is the first time, to the best of one's knowledge, that the method is applied to obstacle boundary value problems. Also, the technique implemented in this study can be used for this type of physical length sensitive problems.

Optimizing an electromechanical device often requires a significant number of evaluations of the winding inductance. In order to reduce drastically the computing costs associated with the calculation of inductances, the purpose of this paper is to propose a semi-analytical toolbox to calculate inductances in any winding made of axial and azimuthal wires and lying in the air.

First, this paper presents a typical rectangular, spiral winding and the way its geometry is approximated for inductance calculations. Second, the basic formulas to calculate inductances of various windings arrangements are provided. The analytical model of the inductances is exposed, and the formulas for the inductances are derived. Finally, a validation is proposed by comparing analytical predictions to 3D FE simulations results and experimental measurements.

The semi-analytical predictions agree with the finite element methods (FEM) and experimental data. Furthermore, the calculation of the inductances was done using much fewer resources with the semi-analytical model than with FEM.

The analytical formula for the mutual inductance between coaxial circular arcs is a series with an infinite number of terms which should be truncated appropriately. This is necessary because the term are found using a recurrence formula which may be unstable for a high number of terms.

The paper includes implications for the optimization of electromechanical devices comprising windings made of axial and azimuthal pieces of wires.

The main original result resides in the analytical expression of Neumann’s integral for the inductance between two coaxial circular arcs.

The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method to find approximate analytical solutions to fractional oscillators.

The modified differential transform method is used to obtain the solutions of the systems. This approach rests on the recently developed modification of the differential transform method. Some examples are given to illustrate the ability and reliability of the modified differential transform method for solving fractional oscillators.

The main conclusion is that the proposed method is a good way for solving such problems. The results are compared with those obtained by the fourth‐order Runge‐Kutta method. It is shown that the results reveal that the modified differential transform method in many instances gives better results.

The paper demostrates that a hybrid method of differential transform method, Laplace transform and Padé approximations provides approximate solutions of the oscillatory systems.

In this contribution, the solution of Mass-Spring-Damper Systems in the fractional context by using Caputo derivative and Orthogonal Collocation Method is investigated. For this purpose, different case studies considering constant and periodic sources are evaluated. The dimensional consistency of the model is guaranteed by introducing an auxiliary parameter. The obtained results are compared with those found by using both the analytical solution and the predictor-corrector method of Adams–Bashforth–Moulton type. The influence of the fractional order on the mechanical system is evaluated.

In the present contribution, an extension of the Orthogonal Collocation Method to solve fractional differential equations is proposed.

In general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics.

The development of a new numerical method to solve fractional differential equations is the major contribution.

Mathematical models of biofeedback learning provide empirical laws of these processes and could be used in order to improve experiments. The systemic approach provides a well‐adapted framework, in which learning is studied as a control problem. The model proposed here is a system of recurrence equations involving the successive measurements of the subject's physiological activity collected by the biofeedback device, and also the performance index presented to the subject. A parameter identification method is described and tested on a computer simulated process.