scholarly journals Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras

1996 ◽  
Vol 37 (4) ◽  
pp. 613-624 ◽  
Author(s):  
Oleg V. Belegradek
Keyword(s):  
General Setting ◽  
Embedding Theorem ◽  
Existentially Closed

scholarly journals HNN-extensions of algebras and applications

1984 ◽  
Vol 29 (2) ◽  
pp. 215-229
Author(s):  
Hans-Christian Mez
Keyword(s):  
Conjugacy Class ◽  
Additional Condition ◽  
Main Part ◽  
Regular Ring ◽  
Embedding Theorem ◽  
Closed Group ◽  
Existentially Closed ◽  
Hnn Extensions ◽  
Associative Rings ◽  
Order Sequence

The classic HNN-embedding theorem for groups does not transfer to associative rings or algebras. In its first part this paper presents constructions which provide such a theorem if an additional condition is put on the isomorphic subalgebras or if one restricts to algebras over fields and drops the associativity. The main part of the paper deals with applications of these results. For example, it is known that every existentially closed group is ω-homogeneous. It is shown that the corresponding is false for existentially closed associative Δ-algebras but true for existentially universal nonassociative K-algebras. Further-more, orthogonal sequences of idempotents in existentially closed associative Δ-algebras over a regular ring Δ are investigated. It is shown that the conjugacy class of such a sequence depends only on a corresponding order sequence. In particular, in every existentially closed K-algebra all idempotents different from 0 and 1 are conjugated.

Valued Abelian Groups

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Abelian Groups ◽  
Logarithmic Derivative ◽  
General Setting ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Ordered Sets ◽  
Valued Fields ◽  
Differential Field ◽  
Basic Facts

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more detail. Ordered abelian groups occur as value groups of valued fields, whereas valued abelian groups arise because the logarithmic derivative map on a valued differential field like induces a valuation on the value group that turns out to be very useful. Furthermore, the notion of a pseudocauchy sequence makes perfect sense in the general setting of valued abelian groups, and the basic facts about these sequences yield a natural proof of a generalized Hahn Embedding Theorem. The chapter also considers valued vector spaces, including spherically complete valued vector spaces, and proves a version of the Hahn Embedding Theorem for valued vector spaces. Special attention is given to particularly well-behaved valued vector spaces known as Hahn spaces.

scholarly journals Existentially closed Leibniz algebras and an embedding theorem

2021 ◽  
Vol 29 (2) ◽  
pp. 163-170
Author(s):  
Chia Zargeh
Keyword(s):  
Embedding Theorem ◽  
Leibniz Algebras ◽  
Existentially Closed ◽  
Hnn Extensions

Abstract In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

scholarly journals High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

2021 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Multistep Methods ◽  
General Setting ◽  
Reaction Diffusion ◽  
Strong Stability ◽  
High Order ◽  
Iterative Solvers ◽  
Time Dependent ◽  
Linear Multistep Methods ◽  
Advection Diffusion Equation ◽  
Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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