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Relative geometries

Thomas Blossier, Amador Martin-Pizarro, Frank Olaf Wagner (2015)

Journal of the European Mathematical Society

In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators.

Relatively recursive expansions

C. Ash, J. Knight (1992)

Fundamenta Mathematicae

In this paper, we consider the following basic question. Let A be an L-structure and let ψ be an infinitary sentence in the language L∪R, where R is a new relation symbol. When is it the case that for every B ≅ A, there is a relation R such that (B,R) ⊨ ψ and R ≤ T D ( B ) ? We succeed in giving necessary and sufficient conditions in the case where ψ is a “recursive” infinitary Π 2 sentence. (A recursive infinitary formula is an infinitary formula with recursive disjunctions and conjunctions.) We consider also...

Relatively recursive expansions II

C. Ash, J. Knight, Theodore Slaman (1993)

Fundamenta Mathematicae

In [AK], we asked when a recursive structure A and a sentence φ, with a new relation symbol, have the following property: for each ℬ≅ A there is a relation S such that S is recursive relative to ℬ and ℬ,S)⊨ φ. Here we consider several related properties, in which there is a uniform procedure for determining S from ℬ ≅A, or from ℬ,¯b)≅(A,ā), for some fixed sequence of parameters ā from A; or in which ℬ and S are required to be recursive. We investigate relationships between these properties, showing...

Representation-directed algebras form an open scheme

Stanislaw Kasjan (2002)

Colloquium Mathematicae

We apply van den Dries's test to the class of algebras (over algebraically closed fields) which are not representation-directed and prove that this class is axiomatizable by a positive quantifier-free formula. It follows that the representation-directed algebras form an open ℤ-scheme.

Representation-finite triangular algebras form an open scheme

Stanisław Kasjan (2003)

Open Mathematics

Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.

Rich families and elementary submodels

Marek Cúth, Ondřej Kalenda (2014)

Open Mathematics

We compare two methods of proving separable reduction theorems in functional analysis - the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system and in spaces of density ℵ1. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality...

Rigid ℵ ε -saturated models of superstable theories

Ziv Shami, Saharon Shelah (1999)

Fundamenta Mathematicae

In a countable superstable NDOP theory, the existence of a rigid ﬡ ε -saturated model implies the existence of 2 λ rigid ﬡ ε -saturated models of power λ for every λ > 2 ﬡ 0 .

Robust neural network control of robotic manipulators via switching strategy

Lei Yu, Shumin Fei, Jun Huang, Yongmin Li, Gang Yang, Lining Sun (2015)

Kybernetika

In this paper, a robust neural network control scheme for the switching dynamical model of the robotic manipulators has been addressed. Radial basis function (RBF) neural networks are employed to approximate unknown functions of robotic manipulators and a compensation controller is designed to enhance system robustness. The weight update law of the robotic manipulator is based on switched multiple Lyapunov function method and the periodically switching law which is suitable for practical implementation...

Rosenthal compacta and NIP formulas

Pierre Simon (2015)

Fundamenta Mathematicae

We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about ϕ-types for ϕ NIP. In particular, we show that if M is a countable model, then an M-invariant ϕ-type is Borel-definable. Also, the space of M-invariant ϕ-types is a Rosenthal compactum, which implies a number of topological tameness properties.

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