T h e preface paradox strikes us as puzzling because we feel that if a person holds a set of inconsistent beliefs, i.e. beliefs such that at least one of them must be incorrect, then he should give at least one of them up. Equally, if a person’s belief is rational, then he has a right to hold it. Yet the preface example is prima facie a case in which a person holds an inconsistent set of belief...

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear spaces. This resolves famous Tingley's problem in class

This paper is going to discuss the following three questions from the view of Whitehead philosophy of education: (1) what is the idea of integrated education, (2) why should we have the integrated education, and (3) how can we achieve it? The concept of integrated education emphasizes methods which concentrate on viewing the student as a whole person. The goal is not about how to find a good jo...

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm for factoring tropical polynomials of a single variable. A central idea is the concept of least-coefficient polynomials as representatives for classes of funct...

Abstract It is well known that Nash equilibria may not be Pareto-optimal; worse, a unique equilibrium Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove previously conjectured result: every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic (NME) pure strategies, which define illustrate. The outcome it gives, depends on ...