Abstract. We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c 0 (Γ ) for any set Γ . The first one is [0, ω 1 ] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω 0 + 1.Foreword. We present two results of Jan Pelant. He presented them at seminars in the Mathematical Institute of Czech Academy of Sciences during the last two years. The example described in Theorem 4.1 below is quite recent. Unfortunately, Jan Pelant died before the results were prepared for publication. We decided to reconstruct them using his hand-written notes.1. Introduction. We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c 0 (Γ ) for any set Γ . This contributes to the study of nonlinear embeddings of (real) Banach spaces into other ones. This investigation is related to the study of the question which topological (or metric) properties enable one to reconstruct the linear structure of a Banach space.The well known Mazur-Ulam theorem says that the existence of an isometry of two Banach spaces implies their linear isometry. On the other hand, the result of H. Toruńczyk [17] (of M. I. Kadec [10] for separable spaces) shows that homeomorphism of two infinite-dimensional Banach spaces gives no information about their linear structure. This is the reason why uniform
We study interfaces between two coexisting stable phases for a general class of lattice models. In particular, we are dealing with the situation where several different interface configurations may enter the competition for the ideal interface between two fixed stable phases. A general method for constructing the phase diagram is presented. Namely, we give a prescription determining which of the phases and which of the interfaces are stable at a given temperature and for given values of parameters in the Hamiltonian. The stability here means that typical configurations of the limiting Gibbs state constructed with the corresponding interface boundary conditions differ only on a set consisting of finite components ("islands") from the corresponding ideal interface.
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