In this paper, we consider the generalized Kropina conformal change of m-th root metric and for this, prove a necessary and su, cient condition of Locally projectively atness. Also we proved a necessary and su, cient condition for the generalized Kropina conformal change of m-th root metric is Locally dually at.

In this paper, we consider the square metric of the form $F=(\alpha+\beta)^2/\alpha$ and study the change of square metrics of the $m$-th root Finsler metrics. We also investigate and study the necessary and sufficient conditions for this type of Finsler metrics to be Locally projectively flat and Locally dually flat.

In this paper‎, ‎we study the Kropina transformation of exponential (α,β)-metric F=α\exp(s),  s:=β/α‎. ‎We characterize the conditions under which this class of (α,β)-metric is Locally projectively flat‎, ‎Locally dually flat‎, ‎and Douglas metric‎. ‎Based on‎, ‎we show that the Kropina transformation of an exponential (α,β)-metric is Locally projectively flat‎, ‎Locally dually flat and Douglas metric if and only if the exponential (α,β)-metric is Locally projectively flat‎, ‎Locally dually flat and Douglas metric‎, ‎respectively.

In this paper, we first study a special class of (α,β)-metrics in the form F = α + εβ + k β2/α , where α is Riemannian metric, β is a 1-form, and ε,k(≠ 0) are constant. We give a complete classification for such metrics to be Locally dually flat. By assumption β is a conformal 1-form, we show that the metric is Locally dually flat if and only if α is a Euclidean metric and β is a constant 1-form. Further, we classify Locally dually flat of a class of Finsler metric in the form F = α exp( α/β ) + εβ, where ε is constant.

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended V-De Morgan law introduced in [20]. Then, using this result and the results of [20], we prove our main result which gives an explicit description of simple algebras (=subdirectly irreducibles) in the variety of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety.In Part II, we prove, using the description of simples obtained in this Part, that the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element RDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras {the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of RDQDStSH1. The Part II concludes with some open problems for further investigation.

This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element RDQDStSH1-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras -the latter is known to be generated by the expansions of the three 4-element Boolean semi-Heyting algebras. As consequences of our main theorem, we present (equational) axiomatizations for several subvarieties of RDQDStSH1. The paper concludes with some open problems for further investigation.

The competition among firms is a basic argument in microeconomics. Firms in economy, behave as profit or non-profit ones. The active firms in service industries such as education, health care, drama products, consorts, sports, and etc, don’t pursue the profit such as profit firms. The competition model among non profit firms is similar to competition model among profit ones, but some differences are observed. This paper seeks to design a dually competition model for Iranian universities (public and private universities) by using of optimization approach in mathematical economics and microeconomics principles. The model, is discussed includes different assumptions about teaching and research excellence in absolute and comparative frame. The results show that the support costs for universities in both dually competition model Follower- Follower and Leader- Follower are different. The costs are determined in relation to conditions for universities in both models. Due to the finding of this research in the Leader- Follower model, Leader University doesn’t decline less costs than Follower University. The support costs of Leader University would be less than Follower University, when the relationship mentioned in paper, has been appointed.