The global phase portrait describes the qualitative behaviour of the solution
set of a nonlinear ordinary differential equation, for all time. In general, this
is as close as we can come to solving nonlinear systems. In this research work
we study the dynamics of a bead sliding on a wire with a given specified
shape. A long wire is bent into the shape of a curve with equation z = f (x)
in a fixed vertical plane. We consider two cases, namely without friction and
with friction, specifically for the cubic shape f (x) = x^{3}−x . We derive the
corresponding differential equation of motion representing the dynamics of
the bead. We then study the resulting second order nonlinear ordinary differential
equations, by performing simulations using MathCAD 14. Our main
interest is to investigate the existence of periodic solutions for this dynamics
in the neighbourhood of the critical points. Our results show clearly that periodic
solutions do indeed exist for the frictionless case, as the phase portraits
exhibit isolated limit cycles in the phase plane. For the case with friction, the
phase portrait depicts a spiral sink, spiraling into the critical point.

The
relation between noncommutative (or quantum) geometry and themathematics of spacesis in many ways
similar to the relation between quantum physicsand classical physics. One moves from the commutative algebra of
functions on a space (or a commutative algebra of classical observable in
classical physics) to a noncommutative algebra representing a noncommutative
space (or a noncommutative algebra of
quantum observables in quantum physics). The object of this paper is to study
the basic rules governing q-calculus
as compared with the classical Newton-Leibnitz calculus.

In this research article, we investigate the stability of a complex dynamical
system involving coupled rigid bodies consisting of three equal masses joined
by three rigid rods of equal lengths, hinged at each of their bases. The system
is free to oscillate in the vertical plane. We obtained the equation of motion
using the generalized coordinates and the Euler-Lagrange equations. We then
proceeded to study the stability of the dynamical systems using the Jacobian
linearization method and subsequently confirmed our result by phase portrait
analysis. Finally, we performed MathCAD simulation of the resulting ordinary
differential equations, describing the dynamics of the system and obtained
the graphical profiles for each generalized coordinates representing the
angles measured with respect to the vertical axis. It is discovered that the
coupled rigid pendulum gives rise to irregular oscillations with ever increasing
amplitude. Furthermore, the resulting phase portrait analysis depicted spiral
sources for each of the oscillating masses showing that the system under investigation
is unstable.

In this work, we developed a compartmental bio-mathematical model to
study the effect of treatment in the control of malaria in a population with
infected immigrants. In particular, the vector-host population model consists
of eleven variables, for which graphical profiles were provided to depict their
individual variations with time. This was possible with the help of MathCAD
software which implements the Runge-Kutta numerical algorithm to solve
numerically the eleven differential equations representing the vector-host
malaria population model. We computed the basic reproduction ratio R_{0} following the next generation matrix. This procedure converts a system of
ordinary differential equations of a model of infectious disease dynamics to
an operator that translates from one generation of infectious individuals to
the next. We obtained R_{0} = , i.e., the square root of the product of
the basic reproduction ratios for the mosquito and human populations respectively. R_{0m} explains the number of humans that one mosquito can infect
through contact during the life time it survives as infectious. R_{0h} on the
other hand describes the number of mosquitoes that are infected through
contacts with the infectious human during infectious period. Sensitivity
analysis was performed for the parameters of the model to help us know
which parameters in particular have high impact on the disease transmission,
in other words on the basic reproduction ratio R_{0}.

Abstract:
The global phase portrait describes the
qualitative behaviour of the solution set for all time. In general, this is as
close as we can get to solving nonlinear systems. The question of particular
interest is: For what parameter values does the global phase portrait of a
dynamical system change its qualitative structure? In this paper, we attempt to
answer the above question specifically for the case of certain third order
nonlinear differential equations of the form . The linear case where ？is also
considered. Our phase portrait analysis shows that under certain conditions on
the coefficients as well as the function , we have asymptotic stability of
solutions.

Abstract:
We present here asymptotic solutions of equations of the type , where is a large parameter. The Bessel differential equation is considered as a typical example of the above and the solutions are provided as . Furthermore, the behaviour of the solutions as well as the stability of the Bessel ode is investigated numerically as the parameter grows indefinitely.

Abstract:
In recent times, mathematical models have been developed to describe various scenarios obtainable in the management of inventories. These models usually have as objective the minimizing of inventory costs. In this research work we propose a mathematical model of an inventory system with time-dependent three-parameter Weibull deterioration and a stochastic type demand in the form of a negative exponential distribution. Explicit expressions for the optimal values of the decision variables are obtained. Numerical examples are provided to illustrate the theoretical development.

Abstract:
Concrete research is gradually shifting from the conventional strength-based approach to durability-centred in the past decade. Durability is the measure of the robustness of constructed facilities against deterioration tendencies. The rate of deterioration is affected by the loading condition, and more importantly the physical and chemical nature of the host environments. This paper reports the experimental investigation of unstressed concrete substructure in the natural (uncontaminated) and cassava’s hydrocyanide effluent-polluted soils on the compressive and flexural strengths of buried concrete specimens for a maximum of 84 days. The compressive strengths of the cubes were tested every 7 days until the 84th day, while the beams were only subjected to third-point loading flexural tests at age 84 days. The compressive strength of concrete specimens in the two soil environments increased, though the trend was lower in the polluted soil. The strength reduced by 2.50% to 9.47% between the 7th and 28th days, but steadily between the 28th and 84th days with strength loss of 9.95% (COV = 2.64%). The load-deflection curves were quadratic for the beams in the two geo-environments. The beams in cyanide-polluted soil lost 34.5% of its flexural stiffness, while its loss of load-carrying capacities at the first crack and ultimate failure was 15.8% and 20% respectively. Higher degree of deterioration is certain for loaded concrete substructures in similar conditions. Hence, prior knowledge of soil chemistry is crucial to determining suitable concrete grade and nominal cover for durable substructural elements.

Abstract:
This study comparatively evaluated the flexural performance and deformation characteristics of concrete elements reinforced with bamboo (Bambusa vulgaris), rattan (Calamuc deerratus) and the twisted steel rebars. The yield strength (YS), ultimate tensile strength (UTS) and the elongation of 50 specimens of the three materials were determined using a universal testing machine. Three beams of concrete strength 20 N/mm^{2} at age 28 days were separately reinforced with bamboo, rattan and steel bars of same percentage, while the stirrups were essentially mild steel bars. The beams were subjected to centre-point flexural loading according to BS 1881 to evaluate the flexural behaviour. The YS of bamboo and rattan bars were 13% and 45% of that of steel respectively, while their UTS were 16% and 62% of that of steel in the same order. The elongation of bamboo, rattan and steel were 7.42%, 10% and 14.7% respectively. The natural rebars were less than the 12% minimum requirement of BS 4449. The load-deflection plots of bamboo and steel RC beams were quadratic, while rattan RC beams had curvilinear trend. The stiffness of bamboo RC beams (BB) and rattan RC beams (RB) were 32% and 13.5% of the stiffness of steel RC beams (SB). The post-first crack residual flexural strength was 41% for BB and SB, while RB was 25%. Moreover, the moment capacities of BB and RB corresponded to 51% and 21% respectively of the capacity of steel RC beams. The remarkable gap between the flexural capacities of the natural rebars and that of steel can be traced not only to the tensile strength but also the weak bonding at the bar-concrete interface. It can be concluded that the bamboo bars are suitable rebars for non-load bearing and lightweight RC flexural structures, while more pre-strengthening treatment is required more importantly for rattan for improved interfacial bonding and load-carrying capacity.

Abstract:
The synthesis, structural characterization, and amplified spontaneous emission spectroscopy of dye-scattering particles in inorganic medium based on Rhodamine 610-TiO2 nanoparticles confined in silica xerogel matrix have been reported. Optimum concentrations have been determined depending on the normal fluorescence spectra for laser dye, in order to provide amplification, and TiO2 nanoparticals as scatter center. Random Laser has been studied under second harmonic Nd: YAG laser excitation. At the optimum concentrations, the results show that the values of bandwidth at full width half-maximum (FWHM) and the threshold energy are about 11 nm and 3 mJ respectively. The scattered and amplified probe light has been collected on a PC-interfaced CCD camera system.