##### Research on the Dual Problem of Trust Region Bundle Method

Jie Shen, Ya-Li Gao

Journal of Advances in Mathematics and Computer Science, Page 1-6
DOI: 10.9734/BJMCS/2017/33880

With the rapid development of science and technology as well as the cross-integration between the various disciplines, the nonsmooth optimization problem plays an increasingly important role in operational research. In this paper, we use the trust region method to study nonsmooth unconstrained optimization problems. Trust region subproblem is constructed to produce the next iteration point by using feasible set as constraint condition. As the number of iterations increases, the compression principle is used to control the elements in a bundle of information. And then the subproblem is studied by Lagrangian function and penalized bundle method [1]. The optimal solution and the relevant derivative conclusion are obtained by transforming the primal problem and dual problem into each other.

##### Variable Viscosity and Thermal Conductivity Effect of Soret and Dufour on Inclined Magnetic Field in Non-Darcy Permeable Medium with Dissipation

R. A. Kareem, S. O. Salawu

Journal of Advances in Mathematics and Computer Science, Page 1-12
DOI: 10.9734/BJMCS/2017/33669

The analysis of thermal-diffusion (Soret) and diffusion-thermo (Dufour) effects on variable thermal conductivity and viscosity in a dissipative heat and mass transfer of an inclined magnetic field in a permeable medium past a continuously stretching surface for power-law difference in the concentration and temperature are examined. The flow is incompressible with the thermal conductivity and fluid viscosity are assumed to be temperature dependent. The local similarity variables for various values of the parameters are considered for the momentum, heat and mass equations. The dimensionless equations are solved numerically using fourth order Runge-Kutta scheme coupled with shooting method. It was noticed that an increase in the values of  enhances the temperature profiles as heat moves from the plate surface to the ambient medium when , otherwise it flows away from the medium to the stretching sheet. Finally, the influences of Skin friction, Nusselt and Sherwood numbers are also presented and discussed.

##### Viscosity Approximation Methods in Reflexive Banach Spaces with a Sequence of Contractions

K. Piesie Frimpong, E. Prempeh

Journal of Advances in Mathematics and Computer Science, Page 1-10
DOI: 10.9734/BJMCS/2017/33414

The aim of this paper is to study viscosity approximation methods in re exive Banach spaces. Let
E be a re exive Banach space which admits a weakly sequentially continuous duality mapping  a nonempty closed convex subset of   a sequence of contractions on C and Tn, n = 1, 2, 3, · · ·N a nite family of nonexpansive mappings on C.  We show that under
appropriate conditions on κ the implicit iterative sequence τ dened by

where κ ∈ (0, 1)  converges strongly to a common xed point  . We further show that
the results hold for an innite family   of nonexpansive mappings.

##### Computation of k-out-of-n System Reliability via Reduced Ordered Binary Decision Diagrams

Journal of Advances in Mathematics and Computer Science, Page 1-9
DOI: 10.9734/BJMCS/2017/33642

A prominent reliability model is that of the partially-redundant (k-out-of-n) system. We use algebraic as well as signal-flow-graph methods to explore and expose the AR algorithm for computing k-out-of-n reliability. We demonstrate that the AR algorithm is, in fact, both a recursive and an iterative implementation of the strategy of Reduced Ordered Binary Decision Diagrams (ROBDDs). The underlying ROBDD for the AR recursive algorithm is represented by a compact Signal Flow Graph (SFG) that is used to deduce AR iterative algorithms of quadratic temporal complexity and linear spatial complexity. Extensions of the AR algorithm for (single or scalar) threshold, double-threshold, vector-threshold, and k-to-l-out-of-n systems have similar ROBDD interpretations.