The two-loop (order $\alpha'$) $\beta$-function equations, which are equivalent to the equations of motion of $\alpha'$-corrected string effective action, are considered for anisotropic homogeneous space-times. These equations are solved for all Bianchi-type models in two schemes of effective action, namely $R^2$ and Gauss-Bonnet schemes with zero cosmological constant and then the metric, dilaton and $B$-field are found at $\alpha'$ perturbative corrections.Comment: 33 page

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Abstract. In this paper we extend Takahashi's fixed point theorem on discrete semigroups to general semi-topological semigroups. Next we define the semi-asymptotic non-expansive action of semi-topological semigroups to give a partial affirmative answer to an open problem raised by A.T-M. Lau.

It has been asked by Lau several times whether a Banach space with weak fixed point property has weak fixed point property for left reversible semigroups. This problem is known as Bruck generalized conjecture (BGC). The aim of this note is to propose a new approach to tackle the BGC. Our approach uses the order structure of the semigroup for the first time in literature to construct an ultra-product structure. Then, we use this ultra-product structure to give an affirmative answer to BGC for the case of nearly uniformly convex (NUC) Banach spaces. One should note that alternatives proofs are available in the case of NUC Banach spaces, but what we hope for is that the originality of our method could pave the way for studying the BGC in its utmost generality.

We learn mathematics subjectively and must apply it objectively. But sometimes, we apply it subjectively by using wrong intuitions which may be elusive to our eyes. The aim of this note is to disclose the secretes of two kinds of these false intuitions and the opportunities they may provide. We first discuss the wrong assumption which says that each topology is uniquely determined by studying a very bad phenomenon happening in dealing with weak* topology. Then, we consider the problem of completing a normed space under two comparable norms, one being smaller than the other. Here, we show that the common belief contending that smaller norms give rise to larger completions is wrong. We then pose some serious questions arising from these wrong intuitions. As we will see finding fallacies are as important as major mathematical activities like proving and disproving.2010 Mathematics Subject Classification. 22A25; 46L05 and 46L10.