A b s t r a c t. In this paper we study some model theory for Gaif-man probability structures. A classical result of Horn-Tarski concerning the extension of probabilities on Boolean algebras will allow us to prove some preservation theorems for probability structures, the model-companion of logical probability, etc. extending some classical results in eastern model theory.

We introduce a new class of functions called almost gs-closed and use the functions to improve several preservation theorems of normality and regularity and also their generalizations. The main result of the paper is that normality and weak normality are preserved under almost gs-closed continuous surjections.

In this paper, we present results concerning orthogonality in Hilbert C∗-modules. Moreover, for a C∗-algebra A, prove theorems the multi-A-linearity and its preservation byA-linear operators. New version of solution equation on C∗-modules mappings preserving are also investigated.

. . . . . . . . . . . . . . . . . . . . . . ix Possible lecture schedules . . . . . . . . . . . . . . x Acknowledgments . . . . . . . . . . . . . . . . . . xi Chapter

We give a relatively short and self contained proof of Ratner’s theorem in the special case of SL(2, R)-invariant measures.

A class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (γ, β and α-acyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theore...

We show that the P−Glivenko property of classes of functions F1, . . . ,Fk is preserved by a continuous function φ from R to R in the sense that the new class of functions x → φ(f1(x), . . . , fk(x)), fi ∈ Fi, i = 1, . . . , k is again a Glivenko-Cantelli class of functions if it has an integrable envelope. We also prove an analogous result for preservation of the uniform Glivenko-Cantelli prop...

Formulated Atomization Theorems extend the theory of Atomic AString Functions evolving since 1970s allowing representation polynomials, complex analytic functions, and solutions linear nonlinear differential equations via Series over smooth finite Splines. Noting preservation analyticity for Ricci Einstein tensors, special new theorems are formulated General Relativity representing spacetime fi...

In this paper we continue work from a previous paper on the fragility and indestructibility of the tree property. We present the following: (1) A preservation lemma implicit in Mitchell’s PhD thesis, which generalizes all previous versions of Hamkins’ Key lemma. (2) A new proof of theorems the ‘superdestructibility’ theorems of Hamkins and Shelah. (3) An answer to a question from our previous p...