Algebraic Structures of B-series

B-series are a fundamental tool in practical and theoretical aspects of numerical integrators for ordinary differential equations. A composition law for B-series permits an elegant derivation of order conditions, and a substitution law gives much insight into modified differential equations of backward error analysis. These two laws give rise to algebraic structures (groups and Hopf algebras of trees) that have recently received much attention also in the non-numerical literature. This article emphasizes these algebraic structures and presents interesting relationships among them.     

代数结构上的模糊拟伪$b$-度量

研究代数结构上的模糊(拟)伪$b$-度量. 主要结论有: (1)设$G$是一个抽象群, $\tau$是$G$上一个左不变模糊拟伪$b$-度量(伪$b$-度量)诱导的拓扑,如果$(G,\tau)$是右拓扑群,那么$(G,\tau)$是一个仿拓扑群(拓扑群); (2)设$S$是一个半群,如果$\tau$是$S$上一个不变模糊拟伪$b$-度量诱导的拓扑,那么$(S,\tau)$是一个拓扑半群.     

On Priestley Spaces of Lattice-Ordered Algebraic Structures

The laws defining many important varieties of lattice-ordered algebras, such as linear Heyting algebras, MV-algebras and l-groups, can be cast in a form which allows dual representations to be derived in a very direct, and semi-automatic, way. This is achieved by developing a new duality theory for implicative lattices, which encompass all the varieries above. The approach focuses on distinguished subsets of the prime lattice filters of an implicative lattice, ordered as usual by inclusion. A decomposition theorem is proved, and the extent to which the order on the prime lattice filters determines the implicative structure is thereby revealed.     

Countably Categorical Structures with n-Degenerate Algebraic Closure

We study the class of ω-categorical structures with n-degenerate algebraic closure for some n ε ω, which includes ω-categorical structures with distributive lattice of algebraically closed subsets (see [4]), and in particular those with degenerate (trivial) algebraic closure. We focus on the models of ω-categorical universal theories, absolutely ubiquitous structures, and ω-categorical structures generated by an indiscernible set. The assumption of n-degeneracy implies total categoricity for the first class, stability for the second, and ω-stability for the third.     

Operator Algebraic Quotient Structures Induced by Directed Graphs

Our result is about inclusions for (finite or infinite) countable directed graphs. In the proof, we use Free Probability Theory, groupoids, and algebras of operators (von Neumann algebras). We show that inclusions of directed graphs induce quotients for associated groupoid actions. With the use of operator thechniques, we then establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is that each connected graph induces a quotient generated by a free group. This is a generalization of the notion of induced representations in the context of unitary representations of groups, i.e., the induction from the representations of a subgroup of an ambient group. The analogue is to systems of imprimitivity based on the homogeneous space. The parallel of this is the more general context of graphs (extending from groups to groupoids): We first prove that inclusions for connected graphs correspond to free group quotients, and we then build up the general case via connected components of given graphs.     

Computability of String Functions Over Algebraic Structures Armin Hemmerling

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a BSS-like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper.     

Punctual definability on structures

We study punctual categoricity on a cone and intrinsically punctual functions and obtain complete structural characterizations in terms of model-theoretic notions. As a corollary, we answer a question of Bazhenov, Downey, Kalimullin, and Melnikov by showing that relational structures are not punctually universal. We will also apply this characterisation to derive an algebraic characterisation of relatively punctually categorical mono-unary structures.     

Elementary Equivalence of Categories of Modules and other Algebraic Structures

This paper is a review of some recent results of elementary equivalence of linear and algebraic groups and our recent new results of elementary equivalence of categories of modules, endomorphism rings of modules, and automorphism groups of modules. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 13, Algebra, 2004.     

Operator Algebraic Quotient Structures Induced by Direct Graphs II

The main purpose of this paper is to investigate the operator algebraic quotient structures induced by directed graphs. We enlarge our study of Cho (Compl Anal Oper Theory, 2008) to the general case. This can be done by constructing new graphs from given graphs called the pull-back graphs. We consider the corresponding groupoids, and von Neumann algebras of pull-back graphs.     

A New Approach to the Algebraic Structures for Integration Methods

The analysis of compositions of Runge-Kutta methods involves manipulations of functions defined on rooted trees. Existing formulations due to Butcher [1972], Hairer and Wanner [1974], and Murua and Sanz-Serna [1999], while equivalent, differ in details. The subject of the present paper is a new recursive formulation of the composition rules. This both simplifies and extends the existing approaches. Instead of using the order conditions based on trees, we propose the construction of the order conditions using a suitably chosen basis on the tree space. In particular, the linear structure of the tree space gives a representation of the C and D simplifying assumptions on trees which is not restricted to Runge-Kutta methods. A proof of the group structure of the set of elementary weight functions satisfying the D simplifying assumptions is also given is this paper.     

完备格上模糊粗近似算子的代数结构

模糊粗集是粗集的模糊化。现有模糊粗集大多建立在t-范数、模糊（t-）相似关系以及对偶原理基础之上。本文将模糊知识（模糊数据）的属性值集[0,1]拓广到一般完备格,基于一般二元模糊关系、模糊合取算子以及模糊合取和模糊蕴涵间的“伴随”关系研究一类模糊粗近似算子。本文对一般模糊粗集的代数结构做了详尽的探讨,并研究了新的模糊粗集与经典粗集和模糊粗集之间的联系。结论表明：粗集的这种模糊化方法保持了Pawlak粗集的代数性质;所提出的模糊粗集是现有典型模糊粗集的一般化;而且,模糊粗近似的贴近度得以提高。     

转动系统相对论性动力学方程的代数结构与Poisson积分

研究转动相对论系统动力学方程的代数结构,得到了完整保守转动相对论系统与特殊非完整转动相对论系统动力学方程具有Lie代数结构;一般完整转动相对论系统、一般非完整转动相对论系统动力学方程具有Lie容许代数结构。并给出转动相对论系统动力学方程的Poisson积分。     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

On P Versus NP for Parameter‐Free Programs Over Algebraic Structures

Based on the computation mode introduced in [13], we deal with the time complexity of computations over arbitrary first‐order structures.The main emphasis is on parameter‐free computations. Some transfer results for solutions of P versus NP problems as well as relationships to quantifier elimination are discussed. By computation tree analysis using first‐order formulas, it follows that P versus NP solutions and other results of structural complexity theory are invariant under elementary equivalence of structures.