Aims/ Objectives: To identify some new classes of graceful diameter six trees using component moving transformation techniques. Study Design: Literature Survey to our findings. Place and Duration of Study: Department of Mathematics, C.V. Raman College of Engineering, Bhubaneswar, India, between June 2014 and September 2016. Methodology: Component Moving Transformation. Results: Here a diameter six tree is denoted by (a0; a1; a2; ....; am; b1; b2; ....; bn; c1; c2; ....; cr) with a0 as the center of the tree, ai; i = 1; 2; ....;m, bj ; j = 1; 2; ....; n, and ck; k = 1; 2; ....; r are the vertices of the tree adjacent to a0; each ai is the center of some diameter four tree, each bj is the center of some star, and each ck is some pendant vertex. This article gives graceful labelings to a family of diameter six trees (a0; a1; a2; ....; am; b1; b2; ....; bn; c1; c2; ....; cr) with diameter four trees incident on ais possess an odd number of branches comprising of six different combinations of odd, even, and pendant branches. Here a star is called an odd branch if its center has an even degree, an even branch if its center has an odd degree, and a pendant branch if its center has degree one.
Conclusions: Our article finds many new graceful diameter six trees by component moving techniques. However, the problem that all diameter six trees are graceful is still open and we conclude that one can not give graceful labelings to all diameter six trees by component moving techniques.
This is about understanding in a mathematical perspective, how the introduction of antiretroviral therapy (ART) is shaping the spread of HIV. A non-linear mathematical model for HIV transmission in a variable size population is formulated in this paper. The model is about analysis and simulation of HIV spread along with treating infected individuals with ARV therapy. Thus, the model is constructed by including individuals who are under ARV therapy as transmitters of HIV. This paper includes studying the speed of spread and how best could be controlled by including the concept of doubling time. The model’s point of equilibria have been found and their stability have been investigated. The model has two points of equilibria, the disease free and the endemic equilibrium. It has been found that if basic reproduction number, R0< 1 the disease free equilibrium is asymptotically stable under some conditions. On the other hand if R0> 1 the disease free equilibrium is not stable. In addition, when R0 > 1 there exist a unique endemic equilibrium, which is found to be both locally and globally stable under some conditions.Simulations of the model have been conducted, taking Tanzania as a case study for the year 2015 onwards. Initial values for the population size start in 2015, and the endemic equilibrium has been estimated. The measures to control the spread of HIV have been suggested to ensure that R0 < 1. One of the case simulated is that R0 = 0.6875 < 1, in which the epidemic diminishes. When R0 > 1 the disease grows, and this has been simulated for R0 = 1.3 > 1.
Data exploration tasks often require inversion of large matrices. The paper presents a new method of matrices inversion, which uses the basis exchange algorithm controlled by the convex and piecewise linear (CPL) inversion criterion function. Using basis exchange algorithms might increase the dimension of the inverted matrices and computational efficiency of the inversion tasks. Basis exchange algorithms are based on the Gauss-Jordan transformation which is used e.g. in the famous Simplex algorithm applied in linear programming.
The forced convective boundary layer flow of electrically conducting micropolar fluids has been investigated in the presence of magnetic field applied in the normal direction of a sheet that shrinks or stretches horizontally and thus causes the fluid motion. Self-similar transforms have been employed to convert the governing partial differential equations into ordinary differential form. The resulting highly non-linear model has been solved numerically with coding in Mathematica. Rigorous computational work has been carried out for sufficient ranges of the parameters of the study namely suction parameter S, stretching/shrinking parameter ∈ , magnetic parameter M, material constants d1, d2, d3 involved in micromotion, heat source parameter B, radiation parameter Rn, heat flux parameter n and Prandtl number Pr. The effects of these parameters on the physical quantities like skin friction coefficient, velocity, temperature and micromotion are presented graphically.
We propose the Z distribution to tackle the Behrens-Fisher problem. First, we define the Z distribution which is a generalization of the t distribution, and then nd the pdf and cdf of the Z distribution. After that, we apply the Z distribution in the hypothesis testing of two normal means, where three different assumptions of the variances are considered. The Z distribution is very flexible in the applications in which one statistics that obeys the Z distribution is applicable to all the three assumptions of the variances. Finally, we provide two groups of simulation studies for the hypothesis testing problems of two normal means.