内射余分解类与投射分解类

令是直和与直和项封闭的右R-模类.本文讨论了关于内射余分解类与投射分解类的左(右)-维数和左正合函子之间的关系,并由此得到一些应用.     

Exact Inclusions of Gehring Classes in Muckenhoupt Classes

An exact bound for exponents of Muckenhoupt classes of functions containing a given Gehring class is obtained in the one-dimensional case.     

格序群的C-根类和K-根类

本文研究了Ｃ-根类（即凸ｌ－子群格可分辨根类）和Ｋ－根类．证明了：投射ｌ－群、强投射ｌ－群、两两分裂ｌ－群、Ｆ－群等许多ｌ－群类都是ｌ－子群格可分辨的,但Ｈａｍｉｌｔｏｎｌ－群不是;Ｃ－根类格是根类格的一个有伪补的闭子格,而且是根类半群的一个子半群．每个凸Ｏ一子群都是闭的,而且给出了由一簇Ｋ一根类及一个根类所生成的Ｋ－根类的形式．     

Adic Foxby Classes

Algebras and Representation Theory - We continue our work on adic semidualizing complexes over a commutative noetherian ring R by investigating the associated Auslander and Bass classes...     

Generalized Green Classes

Generalized Green classes are introduced; some basic properties of members in a generalized Green class are studied. Finally, we apply the results to (Λ), the Ringel–Hall algebra of a finite-dimensional hereditary algebra Λ over a finite field. In particular, it is proved that (Λ) belongs to a suitable generalized Green class, and that there is direct decomposition of spaces (Λ) =  (Λ)  J, where (Λ) is the composition algebra of Λ and J is a twisted Hopf ideal of (Λ), which is exactly the orthogonal complement of (Λ).     

Orthogonality in Classes

 Let denote the von Neumann–Schatten class, its norm and let be an elementary operator defined by . We shall characterize those operators which are orthogonal to the range of in the sense that for all . The main results of this paper are: If and (i) if A, C, respectively B, D are commuting normal operators with , or (ii) if A, B are contractions and , then is orthogonal to the range of if and only of S is in the kernel of . Furthermore, in both cases, the algebraic direct sum satisfies . (Received 9 February 2000; in revised form 21 February, 2001)     

Even Linkage Classes

In this paper we generalize the and -type resolutions used by Martin-Deschamps and Perrin for curves in to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao's correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves satisfying and . Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in to subschemes of pure codimension two in . In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.

     

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

Special Classes of Snarks

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.     

Elementary Classes of Groups

Let be a class of groups. The elementary class with base is defined as the minimal class of groups containing and closed with respect to taking subgroups, quotient groups, group extensions, and direct limits. Properties of such classes are studied. Some applications to the theory of elementary amenable groups and a relation to the Kurosh--Chernikov classes of generalized solvable groups are considered.     

On Similar Special Classes

Two special classes are said to be similar if they determine the same upper radical. A necessary and sufficient condition for a special radical to be generated by distinct similar special classes is given and all similar special classes generating a given special radical are described.     

Classes of cycle bases

In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiries. At present, the complexity status of the MCB problem has been settled only for undirected, directed, and strictly fundamental cycle bases.In this paper, we offer an unitary classification accommodating these three classes and further including the following four relevant classes: 2-bases (or planar bases), weakly fundamental cycle bases, totally unimodular cycle bases, and integral cycle bases. The classification is complete in that, for each ordered pair (A,B) of classes considered, we either prove that A⊆B holds for every graph or provide a counterexample graph for which A?B. The seven notions of cycle bases are distinct (either A?B or B?A is exhibited for each pair (A,B)).All counterexamples proposed have been designed to be ultimately effective in separating the various algorithmic variants of the MCB problem naturally associated to each one of these seven classes. Furthermore, we provide a linear time algorithm for computing a minimum 2-basis of a graph. Finally, notice that the resolution of the complexity status of some of the remaining three classes would have an immediate impact on practical applications, as for instance in periodic railway timetabling, only integral cycle bases are of direct use.     

Classes of formal grammars

Generating grammars, recognition automata, and other facilities for the specification of languages are reviewed.Translated from Itogi Nauki i Tekhniki (Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika), Vol. 12, pp. 155–187, 1975.     

Torsion and Ortho-Slender Classes

This paper generalizes a number of results obtained by Dimitrić in (Glas. Mat. 21(41):327–329, 1986; Proceedings of Hobart Conference on Rings, Modules and Radicals 1987, 204:41–50, Gordon and Breach, 1989) and Dimitrić and Goldsmith in (Glas. Mat. 23(43):241–246, 1988). The original papers were restricted to the category of Abelian groups and orthogonality was to the group of integers ℤ. Here, we are in a general Abelian category with products and coproducts, with applications to module categories and further to modules over PID’s. Another generalization is in replacing ℤ by an entire class of subobjects of the underlying category. We examine properties of the torsion class , Hom(T,C)=0} in relation to purity, direct summands and indecomposability as well as commutation with direct products, for example. Of special interest are members of this class when is a class of slender objects in the ground category; in this case, members of are called ortho-slender objects. In a sense, ortho-slenderness represents complementary, if not dual, notion to slenderness.      

Classes of biorthonormal systems

Summary The biorthogonal systems which are studied in this paper are composed of an Appel set of polynomials and of a sequence of derivatives. Conditions on one sequence are given which ensure the existence of the other. Expansion theorems in terms of derivatives and by means of summability methods are then proved. This work has been sponsored by the European Office, Air Research and Development Command, U. S. Air Force, under Contract AF 61(052)-324.     

Classes of biorthonormal systems

Summary The (A)-summability and mean-convergence of Fourier series in terms of certain biorthonormal systems have been investigated. The problem of completing a set of polynomials to a biothonormal system has been solved in the case of certain sets of type zero (Sheffer’s classification). The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States air Force, through its European Office, under Contract N₀ AF 61 (052)-324.