On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions

In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for ${\mathbb{E}^{3}_{1}}$ by using eigenvectors of a rotation matrix.     

A numerical approach to solve the model for HIV infection of CD4+T cells

In this study, we will obtain the approximate solutions of the HIV infection model of CD4⁺T by developing the Bessel collocation method. This model corresponds to a class of nonlinear ordinary differential equation systems. Proposed scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are computed using the matrix operations of derivatives together with the collocation method. The reliability and efficiency of the proposed approach are demonstrated in the different time intervals by a numerical example. All computations have been made with the aid of a computer code written in Maple 9.     

A note on “A LP-based heuristic for a time-constrained routing problem”

In their paper, Avella et al. (2006) investigate a time-constrained routing problem. The core of the proposed solution approach is a large-scale linear program that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row generation algorithm is proposed to solve this linear program optimally, and an optimality condition is presented to terminate the column-and-row generation algorithm. We demonstrate by using Lagrangian duality that this optimality condition is incorrect and may lead to a suboptimal solution at termination.     

Bessel collocation approach for solving continuous population models for single and interacting species

In this paper, we present a collocation method to obtain the approximate solutions of continuous population models for single and interacting species. By using the Bessel polynomials and collocation points, this method transforms population model into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Bessel coefficients. The reliability and efficiency of the proposed scheme are demonstrated by two numerical examples and performed on the computer algebraic system Maple.     

Locally Optimal Risk-Sensitive Controllers for Strict-Feedback Nonlinear Systems1,2

For a class of risk-sensitive nonlinear stochastic control problems with dynamics in strict-feedback form, we obtain through a constructive derivation state-feedback controllers which (i) are locally optimal, (ii) are globally inverse optimal, and (iii) lead to closed-loop system trajectories that are bounded in probability. The first feature implies that a linearized version of these controllers solve a linear exponential-quadratic Gaussian (LEQG) problem, and the second feature says that there exists an appropriate cost function according to which these controllers are optimal.     

On the problem of starting points for iterative methods

Numerical Algorithms - We propose an algorithm to find a starting point for iterative methods. Numerical experiments show empirically that the algorithm provides starting points for different...     

Pulling rate, thermal and capillary conditions setting for rod growth

The purpose of this paper is to elaborate a procedure, based on a mathematical model, for the setting of the pulling rate, capillary and thermal conditions, in order to grow a cylindrical rod with prescribed radius and length, by edge-defined film-fed growth (EFG) method. First, in the case of an axisymmetric meniscus, we use simultaneously the catching and the angle fixation conditions in order to find a formula, which describes the fluctuation of the angle between the horizontal axis Or and the tangent line to the free surface of the meniscus at the three phase point. This angle appears in the system of differential equations which describes the evolution of the radius of the rod. During the growth this angle can fluctuate due to the fluctuations of the crystal radius, or crystallization front level, or pressure, respectively. In the second part of the paper it is shown in which kind this formula together with the energy balance equation at the crystallization front level can be used for setting the pulling rate, the thermal and capillary conditions to grow a cylindrical rod with prescribed radius and length. Numerical illustration and simulation are presented for rods having thermo-physical properties similar to NdYAG and InSb. This type of results can be useful for the experiment planning, since personal computer simulation is less expensive than experiment. With this aim the present study was undertaken.     

Mathematical models and solution algorithms for computational design of RC piles under structural effects

The paper presents mathematical models and solution algorithms for RC pile design, through scanning soil stratums from top to downwards with an interactive scanner band. The equilibrium of transferred loads from the superstructure, friction forces and tip bearing forces are considered for the design, which leads to optimum pile length. The most important contribution of this research for designers is supplying an efficient tool to obtain optimum pile length and reinforced concrete design of pile foundation systems. A program package has been developed in MATLAB depending on the proposed algorithm. Soil behaviors depending on external effects, active and passive zone distributions are considered. All possible effects in all freedom degrees are taken into account in design process. Stress and strain distributions due to axial loads, bending moments, shear forces and torsional moments may be monitored. The optimum pile length, cross section dimension and reinforcement details may be found by using developed algorithm.     

Multiscale expansions in discrete world

In this paper, we show the attainability of KdV equation from some types of nonlinear Schrödinger equation by using multiscale expansions discretely. The power of this manageable method is confirmed by applying it to two selected nonlinear Schrödinger evolution equations. This approach can also be applied to other nonlinear discrete evolution equations. All the computations have been made with Maple computer packet program.