An essential base of the central types of the convex theory

In this paper, we consider the model-theoretical properties of the essential base of the central types of convex theory. Also shows the connection between the center and Jonsson theory in permissible enrichment signatures. Moreover, the theories under consideration are hereditary. This article is divided into 2 sections: 1) an essential types and an essential base of central types (in this case, the concepts of an essential type and an essential base are defined using the Rudin-Keisler order on the set of central types of some hereditary Jonsson theory in the permissible enrichment); 2) the atomicity and the primeness of ϕ(x)-sets. In this paper, new concepts are introduced: the ϕ(x)-Jonsson set, the AP A-set, the AP A-existentially closed model, the ϕ(x)-convex theory, the ϕ(x)-transcendental theory, the AP A-transcendental theory. One of the ideas of this article refers to the fact that in the work of Mustafin T.G. it was noticed that any universal model of a quasi-transcendental theory with a strong base is saturated, but we generalized this result taking into account that: the concept of quasi-transcendence will be replaced by the ϕ(x)-transcendence, where ϕ(x) defines some Jonsson set; and the notion of a strong base is replaced by the notion of an essential base, but in a permissible enrichment of the hereditary Jonsson theory. The main result of our work shows that the number of fragments obtained under a closure of an algebraic or definable type does not exceed the number of homogeneous models of a some Jonsson theory, which is obtained as a result of a permissible enrichment of the hereditary Jonsson theory.

STRONG DENSITY OF DEFINABLE TYPES AND CLOSED ORDERED DIFFERENTIAL FIELDS

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X ⊆ Mn, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

The canonical topology on dp-minimal fields

We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].

Preliminaries

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

Introduction

This book deals with non-archimedean tame topology and stably dominated types. It considers o-minimality as an analogy and reduces questions over valued fields to the o-minimal setting. A fundamental tool, imported from stability theory, is the notion of a definable type, which plays a number of roles, starting from the definition of a point of the fundamental spaces. One of the roles of definable types is to be a substitute for the classical notion of a sequence, especially in situations where one is willing to refine to a subsequence. To each algebraic variety V over a valued field K, the book associates in a canonical way a projective limit unit vector V of spaces, which is the stable completion of V. In case the value group is ℝ, the results presented in this book relate to similar tameness theorems for Berkovich spaces.

Valuation theoretic content of the Marker-Steinhorn theorem

The Marker-Steinhorn Theorem (cf. [2] and [3]), says the following. If T is an o-minimal theory and M ≺ N is an elementary extension of models of T such that M is Dedekind complete in N, then for every N-definable subset X of Nk, the trace X ∩ Mk is M-definable. The original proof in [2] gives an explicit method how to construct a defining formula of X ∩ Mk out of a defining formula of X. A geometric reformulation of the Marker-Steinhorn Theorem is the definability of Hausdorff limits of families of definable sets. An explicit construction of these Hausdorff limits for expansions of the real field has recently been achieved in [1]. Both proofs and also the treatment [3] are technically involved.Here we give a short algebraic, but not constructive proof, if T is an expansion of real closed fields. In fact we'll identify the statement of the Theorem with a valuation theoretic property of models of T (namely condition (†) below). Therefore our proof might be applicable to other elementary classes which expand fields, if a notion of dimension and a reasonable valuation theory are available.From now on, let T be an o-minimal expansion of real closed fields. We have to show the following (cf. [2], Th. 2.1. for this formulation). If M is a model of T and p is a tame n-type over M (i.e., M is Dedekind complete in M ⟨ᾱ⟩ := dcl(Mᾱ) for some realization ᾱ of p), then p is a definable type (cf. [4], 11 .b).

A Learning Shell for Iterative Design (L’SID): Concepts and Applications

A software shell called “Learning Shell for Iterative Design,” L’SID, has been developed in conjunction with a simple data matrix, the “learn table.” Histories of design are utilized in aiding the acceleration of routine design problems. The class of problems addressed are non-convex, noninvertible and with multiple performance criteria. The design parameters can be of any definable type; continuous, integer or nonordered feature based. L’SID is domain independent and highly modular. The ability of L’SID to aid deterministic methods is shown statistically with two example problems (extrusion die and airfoil). Results also show the ability of the technique to surmount nonconvexity in design space and computational noise related to roundoff.