Use Of Transcendental Functions Research Articles

Higher order investigation on modulated waves in the Peyrard–Bishop–Dauxois DNA model

Despite the widespread use of transcendental functions in the modeling of the dynamics of DNA, most research efforts are limited in their analytical studies of this enthralling system to cubic order polynomial approximations of the corresponding equations of motion. In this paper, we present an investigation of waves in the Peyrard–Bishop–Dauxois model of DNA; while extending the polynomial approximation of the Morse potential up to the sixth order. We show that, within a generalized version of the reductive perturbation method that we have adopted, the equations governing the envelop consist of the standard cubic nonlinear Schrödinger equation and its non homogeneous linearizations. Exact and explicit analytical solutions that correspond to bright solitary waves are obtained for these coupled amplitude equations. A notable qualitative feature of these solutions is the dependence of their propagation speeds and frequencies on their amplitudes. Our approach additionally unveils that these solutions contain some harmonic terms; which are missed in existing works. A very good agreement is found between our analytical analysis and the numerical simulations of the full discrete nonlinear equation of the lattice which use these solutions as initial conditions.

Orientation Modeling Using Quaternions and Rational Trigonometry

In recent years, the recreational and commercial use of flight and driving simulators has become more popular. All these applications require the calculation of orientation in either two or three dimensions. Besides the Euler angles notation, other alternatives to represent rigid body rotations include axis-angle notation, homogeneous transformation matrices, and quaternions. All these methods involve transcendental functions in their calculations, which represents a disadvantage when these algorithms are implemented in hardware. The use of transcendental functions in software-based algorithms may not represent a significant disadvantage, but in hardware-based algorithms, the potential of rational models stands out. Generally, to calculate transcendental functions in hardware, it is necessary to utilize algorithms based on the CORDIC algorithm, which requires a significant amount of hardware resources (parallel) or the design of a more complex control unit (pipelined). This research presents a new procedure for model orientation using rational trigonometry and quaternion notation, avoiding trigonometric functions for calculations. We describe the orientation of a gimbal mechanism presented in many applications, from autonomous vehicles such as cars or drones to industrial manipulators. This research aims to compare the efficiency of a rational implementation to classical modeling using the techniques mentioned above. Furthermore, we simulate the models with software tools and propose a hardware architecture to implement our algorithms.

Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Padé approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.

Efficient FPGA Hardware Implementation for Robot Manipulator Kinematic Modeling Using Rational Trigonometry

A common approach to model kinematics for robot manipulators uses functions such as sin(), cos() and atan(),however, the rational trigonometry allows to exclude the use of these transcendental functions. Most of the computing processing to control a manipulator is dedicated to solving its kinematics, therefore, reduce the complexity of kinematic equations leads to a more compact and efficient model. In this paper, we propose a geometric analysis to solve the kinematics of robot manipulator of up to three degrees of freedom, avoiding the use of transcendental functions in the kinematic equations. We present three kinematic models for the mechanism positions and theirimplementations in an FPGA. The experimental results show that our proposed models: 1) use less logic elements compared with an implementation based on parallel CORDIC and 2) has lower latency respect with iterative CORDIC implementation.

Optimization method of a six-bar steering mechanism formulated with natural coordinates

In this work, an optimization method of a six - bars mechanism is proposed, which is formulated with natural coordinates easing the kinematic modeling, since it avoids use of transcendental functions. After the kinematic model is established, the optimization problem is formulated to be solved by using a genetic algorithm with real or continuous coding; thus, avoiding computation of highly complex derivatives. The relevance of this work lies on fact that an alternative method is offered to those dependent on angular parameters, where equations that model the mechanism, are complicated, even in the simplest mechanism.

APPROXIMATE SOLUTION OF ONE PROBLEM ON ELECTRICAL OSCILLATIONS IN WIRES WITH THE USE OF POLYLOGARITHMS

The article considers a mixed problem with homogeneous boundary conditions for onedimensional homogeneous wave equation. Such a problem can arise, for example, when studying oscillations of current and voltage in the conductor through which electric current flows, while the line is free from distortion. The solution can be found with the use of the Fourier method in the form of trigonometric series. This representation is of purely theoretical interest, because the real calculation should be, first, to find a large number of coefficients of the integrals, which in itself is not a trivial task and, second, it is almost impossible to assess the error of the calculations. An alternative way of solving this problem based on the use of transcendental functions i. e. polylogarithms that represent complex power series of a special kind. The exact solution of the problem is expressed through the imaginary part of a polylogarithm of the first order on the single circle and the approximate one – via the real part of the dilogarithm. In addition, if the initial conditions in the problem are elementary functions, then the solution is also computed using elementary functions. A simple and effective error estimate of the approximate solution has been found. It does not depend on time and it has the first-order of accuracy regarding the step of a partitioning segment of the numerical axis on which the problem is considered. This valuation is uniform with respect to the variables of the problem – both spatial and temporal.

A submicron DC MOSFET model for simulation of analog circuits

This paper presents an efficient dc MOSFET model for accurate simulation of analog circuits. A new approach to model channel length modulation is presented. An empirical expression for channel length modulation is derived from measurements. This is used to model the observed behavior of g/sub D/ with gate, drain, and substrate bias. Some of the models commonly used for circuit simulation do not predict the effects of gate and substrate bias adequately. A new smoothing function is used to unify the linear and saturation regions in a single expression. Continuity of transconductance is maintained between the weak and strong inversion regions. Model efficiency is maintained by avoiding the use of transcendental functions in the smoothing techniques. We demonstrate >

GENERAL COMPUTER METHOD OF ANALYSIS OF CONDUCTION AND DIFFUSION IN BIOLOGICAL SYSTEMS WITH DISTRIBUTIVE SOURCES.

A method of analysis by use of the computer has been developed which permits the physical laws governing conduction, or diffusion, to be expressed in first-order finite form. Solutions are obtained in the one-dimensional case for rectangular, cylindrical, and spherical geometries without the use of transcendental functions or other conventional mathematical iterative methods. The case of the cylindrical calorimeter with concentric insulating and conducting layers of varying chemical properties and with distributive heat sources arising from biochemical reactions at the center is treated in the transient and steady-state case.