In this paper, we choose an approach to generalize the idea  of usual norm keeping up  analogous structure and introduce a  generalized norm, called  $ \Phi $-norm using a  function $ \Phi $ which satisfies some special condition. We show that the induced metric of it is $ \phi $-metric. We also study some basic properties and obtain some fundamental results in such  SPACEs.

In this article, the ant colony method is utilized for topology optimization of SPACE STRUCTURES. Strain energy of the structure is minimized while the material volume is limited to a certain amount. In other words, the stiffest possible structure is sought when certain given materials are used. In addition, a noise cleaning technique is addressed to prevent undesirable members in optimum topology. The performance of the method for topology optimization of SPACE STRUCTURES are demonstrated by three numerical examples.

A set of locally product STRUCTURES on tangent manifold TM of a SPACE form M is pointed out. This is found in a study of a type of Sasaki metric whose second term is a special deformation of the first one. Introducing an adequate almost product structure we find at first a large class of locally almost product STRUCTURES on TM for a (pseudo)-Riemannian manifold M. When M is a SPACE form, a subset of it is made of locally product STRUCTURES.

1. Introduction: Optimization by explicit and deterministic variables is called deterministic optimization. It is known that there are wide varieties of uncertainties which are originated from the errors in modeling and simulation and condition of services such as changes in loading. Moreover, we must consider these uncertainties in different stages of design process of STRUCTURES. Some of the most important factors which should be considered in structural designing are variation of material properties, geometrical dimensions of the STRUCTURES, and the loading patterns. On the other hand, deterministic optimization on STRUCTURES uses the strong safety factors instead of uncertainties. Thus, the resultant designed-STRUCTURES are leading to the excessive conservative STRUCTURES that would not be economical and optimal. According to, existing some uncertainties in each level of engineering design process; it is required to consider safety factors in computation of the deterministic optimization. In order to resolve this problem, an efficient approach is recently developed which is called reliability based optimal design. In this method the safety of STRUCTURES are evaluated based on the failure probability and the aforementioned uncertainties are modeled by the probability distribution of random variables. Finally, the investigated components embedded in the assessed acceptable ranges of safety which is introduces with the failure definition. This practical method can be applied to different STRUCTURES...