1. ### The Bifurcation and Stability of the Solutions of the Boussinesq Equations

Mohammad Yahia Ibrahim Awajan, Ruwaidiah Idris

School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu (UMT), 21030 Kuala Nerus, Malaysia Terengganu.

Abstract: The study determined the bifurcation and stability of the solutions of the Boussinesq equations as well as the onset of the Rayleigh-Benard convection. The article established the nonlinear theory for this problem using a new notion of bifurcation known as attractor bifurcation. This article considered the theory that comprises the following three perspectives. We initially deal with the problem that bifurcates from the trivial solution an attractor  while the Rayleigh number  intersects the ﬁrst critical Rayleigh number  for all physically boundary conditions, despite the multiplicity of the eigenvalue  for the linear problem. Hence, secondly, the study considered the bifurcated attractor  as asymptotically stable. Lastly, the bifurcated solutions are also structurally stable when the spatial dimension is two, and are classiﬁed as a bifurcated solution as well. Furthermore, the technical method explained here provides a means, which can be adopted for many different problems in bifurcation and other pattern formation that are related.

Keywords: Rayleigh-Benard Convection, Dynamic bifurcation, Boussinesq equation, Rayleigh number, Attractor bifurcation.

2. ### A Random Walk From the Landau to the Riemann Hypothesis

Thomas J. Osler, Marcus Wright

Mathematics Department, Rowan University, Glassboro, NJ 08028.

Abstract: The Riemann hypothesis is arguably the most important unsolved problem in mathematics. It is even difficult to state to beginning students since it requires a knowledge of the zeta function for complex values of the argument. However, there are at least 23 equivalent statements of this hypothesis that are much easier to state. In this paper we examine one such idea called the Landau hypothesis. This Landau hypothesis has a very simple interpretation in terms of  the prime factorization of integers. Surprisingly this hypothesis also has a useful description in terms of a one dimensional random walk. We show by means of a known, but not often seen, integral representation for any Dirichlet series that the Landau hypothesis implies the Riemann  hypothesis.

Keywords: Riemann hypothesis, Landau hypothesis, random walk, number theory.

3. ### Second Order Optimality Conditions for a Semilinear Elliptic Control Problem of Infinite Order with Pointwise Control Constraints

S. A. El-Zahaby and Samira El-Tamimy

Department of Mathematics, Faculty of Science, Al-Azhar University [For Girls],Nasr City, Cairo, Egypt.

Abstract: An optimal control problem for a semilinear elliptic equation with infinite order is investigated, where pointwise constraints are given on the control. First order necessary optimality conditions are derived, second-order sufficient optimality condition is established that consider strongly active constraints.

Keywords: Distributed control, semilinear elliptic equation, infinite order operator, pointwise control constraint, necessary optimality conditions, second order sufficient optimality conditions.