For a non-zero normed linear space A, we consider Cb (K), the complex-valued, bounded and continuous functions space on K with ∥, ·, ∥, ∞, , where K = B(0) 1 (the closed unit ball of A). Also for a non-zero element φ,∈,A ∗,with ∥, φ, ∥,≤,1, we consider the space Cbφ,(K) as the linear space Cb (K) with the new norm ∥, f∥,φ,= ∥, fφ, ∥, ∞,for all f ∈,Cb (K). Some basic properties such as, PROXIMINALITY, E-PROXIMINALITY, strongly PROXIMINALITY and quasi Chebyshev for certain subsets of Cb (K) are characterized with the norms ∥, ·, ∥,φ,and ∥, ·, ∥, ∞, . Some examples for illustration and for comparison between the norms ∥, ·, ∥,φ,and ∥, ·, ∥, ∞,on Cb (K) are presented.

In this paper, we introduce the notions of (Î-, Î- Vq-) -fuzzy bi-ideals, (Î, Î Vq) -antifuzzy bi-ideals and (Î-, Î- Vq-) -antifuzzy bi-ideals of an ordered semigroup. We characterize (Î, Î Vq) -antifuzzy bi-ideals by (Î-, Î- Vq-) -fuzzy bi-ideals, lower level sets and upper level sets. We characterize (Î-, Î- Vq-) -antifuzzy bi-ideals by (Î, Î Vq) -fuzzy bi-ideals, lower level sets and upper level sets. We characterize regular ordered semigroups through (Î-, Î- Vq-) -antifuzzy bi-ideals and (Î, Î Vq) -fuzzy bi-ideals.

We prove some results on characterization of e-simultaneous approximations of downward sets in vector lattice Banach spaces. Also, we give some results about simultaneous approximations of normal sets.

This paper studies some existence results for generalized e-vector equilibrium problems and generalized e-vector variational inequalities. The existence results for solutions are derived by using the celebrated KKM theorem. The results achieved in this paper generalize and improve the works of many authors in references.

The commonly used linear k-e turbulence model is shown to be incapable of accurate prediction of turbulent flows, where non-isotropy is dominant. Two examples of non-isotropic flows, which have a wide range of applications in marine waters, are saline water flow and the stratified flows due to temperature gradients. These relate to stratification and consequently, variation of density through vertical layers. In this paper, a nonlinear k-e turbulence model, firstly presented by Speziale (1987) and was implemented in the existing hydrodynamic model. The energy equation has been also added and solved in the hydrodynamic model. The hydrodynamic model solves the fully nonlinear Navier-Stokes equations based on an ALE (Arbitrary Lagrangian Eulerian) description. The model is an extension to WISE (Width Integrated Stratified Environments) 2DV numerical model, originally developed by Hejazi (2002). The simulated values have been compared with the experimental data and have shown acceptable agreements. The predictions are also compared with the results of the original model employing a standard buoyant k-e turbulence model, which showed the advantage of the new turbulence model in prediction of non-isotropic flows.