Mathematical, Computational, Physical, Electrical and Computer Engineering

Commenced in January 1999 | Frequency: Monthly | Edition: International | Paper Count: 9 |

9

10003438

A Generalization of Planar Pascal’s Triangle to Polynomial Expansion and Connection with Sierpinski Patterns

The very well-known stacked sets of numbers referred
to as Pascal’s triangle present the coefficients of the binomial
expansion of the form (x+y)n. This paper presents an approach (the
Staircase Horizontal Vertical, SHV-method) to the generalization of
planar Pascal’s triangle for polynomial expansion of the form
(x+y+z+w+r+⋯)n. The presented generalization of Pascal’s triangle
is different from other generalizations of Pascal’s triangles given in
the literature. The coefficients of the generalized Pascal’s triangles,
presented in this work, are generated by inspection, using embedded
Pascal’s triangles. The coefficients of I-variables expansion are
generated by horizontally laying out the Pascal’s elements of (I-1)
variables expansion, in a staircase manner, and multiplying them with
the relevant columns of vertically laid out classical Pascal’s elements,
hence avoiding factorial calculations for generating the coefficients
of the polynomial expansion. Furthermore, the classical Pascal’s
triangle has some pattern built into it regarding its odd and even
numbers. Such pattern is known as the Sierpinski’s triangle. In this
study, a presentation of Sierpinski-like patterns of the generalized
Pascal’s triangles is given. Applications related to those coefficients
of the binomial expansion (Pascal’s triangle), or polynomial
expansion (generalized Pascal’s triangles) can be in areas of
combinatorics, and probabilities.

8

10003500

Computing Transition Intensity Using Time-Homogeneous Markov Jump Process: Case of South African HIV/AIDS Disposition

This research provides a technical account of estimating Transition Probability using Time-homogeneous Markov Jump Process applying by South African HIV/AIDS data from the Statistics South Africa. It employs Maximum Likelihood Estimator (MLE) model to explore the possible influence of Transition Probability of mortality cases in which case the data was based on actual Statistics South Africa. This was conducted via an integrated demographic and epidemiological model of South African HIV/AIDS epidemic. The model was fitted to age-specific HIV prevalence data and recorded death data using MLE model. Though the previous model results suggest HIV in South Africa has declined and AIDS mortality rates have declined since 2002 – 2013, in contrast, our results differ evidently with the generally accepted HIV models (Spectrum/EPP and ASSA2008) in South Africa. However, there is the need for supplementary research to be conducted to enhance the demographic parameters in the model and as well apply it to each of the nine (9) provinces of South Africa.

7

10003593

Combined Effect of Moving and Open Boundary Conditions in the Simulation of Inland Inundation Due to Far Field Tsunami

Tsunami and inundation modelling due to far field tsunami propagation in a limited area is a very challenging numerical task because it involves many aspects such as the formation of various types of waves and the irregularities of coastal boundaries. To compute the effect of far field tsunami and extent of inland inundation due to far field tsunami along the coastal belts of west coast of Malaysia and Southern Thailand, a formulated boundary condition and a moving boundary condition are simultaneously used. In this study, a boundary fitted curvilinear grid system is used in order to incorporate the coastal and island boundaries accurately as the boundaries of the model domain are curvilinear in nature and the bending is high. The tsunami response of the event 26 December 2004 along the west open boundary of the model domain is computed to simulate the effect of far field tsunami. Based on the data of the tsunami source at the west open boundary of the model domain, a boundary condition is formulated and applied to simulate the tsunami response along the coastal and island boundaries. During the simulation process, a moving boundary condition is initiated instead of fixed vertical seaside wall. The extent of inland inundation and tsunami propagation pattern are computed. Some comparisons are carried out to test the validation of the simultaneous use of the two boundary conditions. All simulations show excellent agreement with the data of observation.

6

10003600

Modeling Bessel Beams and Their Discrete Superpositions from the Generalized Lorenz-Mie Theory to Calculate Optical Forces over Spherical Dielectric Particles

In this work, we propose an algorithm developed under Python language for the modeling of ordinary scalar Bessel beams and their discrete superpositions and subsequent calculation of optical forces exerted over dielectric spherical particles. The mathematical formalism, based on the generalized Lorenz-Mie theory, is implemented in Python for its large number of free mathematical (as SciPy and NumPy), data visualization (Matplotlib and PyJamas) and multiprocessing libraries. We also propose an approach, provided by a synchronized Software as Service (SaaS) in cloud computing, to develop a user interface embedded on a mobile application, thus providing users with the necessary means to easily introduce desired unknowns and parameters and see the graphical outcomes of the simulations right at their mobile devices. Initially proposed as a free Android-based application, such an App enables data post-processing in cloud-based architectures and visualization of results, figures and numerical tables.

5

10003656

A Survey on Positive Real and Strictly Positive Real Scalar Transfer Functions

Positive real and strictly positive real transfer functions are important concepts in the control theory. In this paper, the results of researches in these areas are summarized. Definitions together with their graphical interpretations are mentioned. The equivalent conditions in the frequency domain and state space representations are reviewed. Their equivalent electrical networks are explained. Also, a comprehensive discussion about a difference between behavior of real part of positive real and strictly positive real transfer functions in high frequencies is presented. Furthermore, several illustrative examples are given.

4

10003676

Exploring Counting Methods for the Vertices of Certain Polyhedra with Uncertainties

Vertex Enumeration Algorithms explore the methods and procedures of generating the vertices of general polyhedra formed by system of equations or inequalities. These problems of enumerating the extreme points (vertices) of general polyhedra are shown to be NP-Hard. This lead to exploring how to count the vertices of general polyhedra without listing them. This is also shown to be #P-Complete. Some fully polynomial randomized approximation schemes (fpras) of counting the vertices of some special classes of polyhedra associated with Down-Sets, Independent Sets, 2-Knapsack problems and 2 x n transportation problems are presented together with some discovered open problems.

3

10003807

Highly Conductive Polycrystalline Metallic Ring in a Magnetic Field

Electrical conduction in a quasi-one-dimensional
polycrystalline metallic ring with a long electron phase coherence
length realized at low temperature is investigated. In this situation, the
wave nature of electrons is important in the ring, where the electrical
current I can be induced by a vector potential that arises from a static
magnetic field applied perpendicularly to the ring’s area. It is shown
that if the average grain size of the polycrystalline ring becomes
large (or comparable to the Fermi wavelength), the electrical current
I increases to ~I0, where I0 is a current in a disorder-free ring. The
cause of this increasing effect is examined, and this takes place if the
electron localization length in the polycrystalline potential increases
with increasing grain size, which gives rise to coherent connection
of tails of a localized electron wave function in the ring and thus
provides highly coherent electrical conduction.

2

10004024

Robust Variogram Fitting Using Non-Linear Rank-Based Estimators

In this paper numerous robust fitting procedures are considered in estimating spatial variograms. In spatial statistics, the conventional variogram fitting procedure (non-linear weighted least squares) suffers from the same outlier problem that has plagued this method from its inception. Even a 3-parameter model, like the variogram, can be adversely affected by a single outlier. This paper uses the Hogg-Type adaptive procedures to select an optimal score function for a rank-based estimator for these non-linear models. Numeric examples and simulation studies will demonstrate the robustness, utility, efficiency, and validity of these estimates.

1

10004793

Existence and Uniqueness of Positive Solution for Nonlinear Fractional Differential Equation with Integral Boundary Conditions

By using fixed point theorems for a class of
generalized concave and convex operators, the positive solution of
nonlinear fractional differential equation with integral boundary
conditions is studied, where n ≥ 3 is an integer, μ is a parameter
and 0 ≤ μ < α. Its existence and uniqueness is proved, and an
iterative scheme is constructed to approximate it. Finally, two
examples are given to illustrate our results.