Generalized Principal Extensions in Topological Algebras

The ??generalized principal extension?? of a subset of a topological algebra is, by definition, a subset of another topological algebra. Hence, the generalized spectrum of an element, ??local spectrum??, of a topological algebra is thus a subset of a topological algebra, not necessarily of ${{\mathbb C}}$ . We also give a criterion for the continuity of the (generalized) Newburg map (Theorem 3.1).     

Zero-Dimensional Imbeddability

In this article, we try to understand which generically complete intersection monomial ideals with fixed radical are Cohen–Macaulay. We are able to give a complete characterization for a special class of simplicial complexes, namely the Cohen–Macaulay complexes without cycles in codimension 1. Moreover, we give sufficient conditions when the square-free monomial ideal has minimal multiplicity.     

Strongly Zero-Dimensional Locales

New kinds of strongly zero-dimensional locales are introduced and characterized, which are different from Johnstone's, and almost all the topological properties for strongly zero-dimensional spaces have the pointless localic forms. Particularly, the Stone-Čech compactification of a strongly zero-dimensional locale is strongly zero-dimensional. Received January 21, 1999, Accepted February 1, 2000     

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

For an ideal I??[x] given by a set of generators, a new semidefinite characterization of its real radical I(V _?(I)) is presented, provided it is zero-dimensional (even if I is not). Moreover, we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V _?(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.     

一种几何定理机器证明的零维化方法

本文对一类初等几何定理的证明给出了一种机械化方法，利用这种方法，可计算出一个由有限个素理想组成的集合，所有属于假设部分对应的某一扩域上的理想的素理想都在这个集合中出现并且可以挑选出来．因而一个几何定理一般真确，当且仅当终结多项式属于全部的这种素理想，即对其不可约特征列的余式为零．     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

Functional Analysis and Its Applications - In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things,...     

Clifford Extensions

Formulating the construction of Clifford algebras, we introduce the notion of Clifford extensions and show that Clifford extensions are Frobenius extensions. Consequently, Clifford extensions of Auslander–Gorenstein rings are Auslander–Gorenstein rings.     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.     

Convex-Concave Extensions

In this paper we introduce a new notion which we call convex-concave extensions. Convex-concave extensions provide for given nonlinear functions convex lower bound functions and concave upper bound functions, and can be viewed as a generalization of interval extensions. Convex-concave extensions can approximate the shape of a given function in a better way than interval extensions which deliver only constant lower and upper bounds for the range. Therefore, convex-concave extensions can be applied in a more flexible manner. For example, they can be used to construct convex relaxations. Moreover, it is demonstrated that in many cases the overestimation which is due to interval extensions can be drastically reduced. Applications and some numerical examples, including constrained global optimization problems of large scale, are presented.     

Ulyanov-Type Embedding Theorems for Functions on Zero-Dimensional Locally Compact Groups

Siberian Mathematical Journal - We establish some analogs of Ulyanov’s and Andrienko’s Theorems on the embedding of the Hölder spaces of integrable functions on zero-dimensional...     

Isomorphism and Measure Rigidity for Algebraic Actions on Zero-Dimensional Groups

We consider mixing ^d-actions on compact zero-dimensional abelian groups by automorphisms. Rigidity of invariant measures does not hold for such actions in general; we present conditions which force an invariant measure to be Haar measure on an affine subset. This is applied to isomorphism rigidity for such actions. We develop a theory of halfspace entropies which plays a similar role in the proof to that played by invariant foliations in the proof of rigidity for smooth actions.     

Simultaneous Extensions Revisited

After an introduction presenting the problem of simultaneous extensions, some concrete examples show the difficulties in examining this problem.     

Covariant Projective Extensions

The theory of crossed products of C~*-algebras by groups of automorphisms is a well-developed area of the theory of operator algebras. Given the importance and the success ofthat theory, it is natural to attempt to extend it to a more general situation by, for example,developing a theory of crossed products of C~*-algebras by semigroups of automorphisms, or evenof endomorphisms. Indeed, in recent years a number of papers have appeared that are concernedwith such non-classical theories of covariance algebras, see, for instance [1-3].