On the computability-theoretic complexity of trivial, strongly minimal models

We show the existence of a trivial, strongly minimal (and thus uncountably categorical) theory for which the prime model is computable and each of the other countable models computes . This result shows that the result of Goncharov/Harizanov/Laskowski/Lempp/McCoy (2003) is best possible for trivial strongly minimal theories in terms of computable model theory. We conclude with some remarks about axiomatizability.

     

强可约极大三角代数的检测问题

对强可约极大三角代数的酉等价和代数同构回答了检测问题     

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behavior with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.     

On the number of noncritical vertices in strongly connected digraphs

By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.     

Operators with connected spectrum + compact operators = strongly irreducible operators

Herrero’s conjecture that each operator with connected spectrum acting on complex, separable Hilbert space can be written as the sum of a strongly irreducible operator and a compact operator is proved. Jiang, C. L., Power, S., Wang, Z. Y., Biquasitriangular + small compact = strongly irreducible,J. London Math., to be published.     

On minimal strongly prime ideals

In this paper we give some characterizations of a ring Rwhose unique maximal nil ideal N _r (R) coincides with the set of all its nilpotent elements N(R) by using its minimal strongly prime ideals.     

A normal countably compact not absolutely countably compact space

We construct an example of a normal countably compact not absolutely countably compact space. We also prove that every hereditarily normal countably compact space is absolutely countably compact and suggest a method for construction of hereditarily normal spaces without property .

     

Hodge decomposition theorem on strongly K(a)hler Finsler manifolds

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the (θ)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M, F) is a compact strongly K(a)hler Finsler manifold, we define a (θ)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in K(a)hler manifolds is still true in the more general compact strongly K(a)hler Finsler manifolds.     

On minimal non-MSN-groups

A finite group G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G are subnormal in G. In this paper, we determinate the structure of non-MSN-groups in which all of whose proper subgroups are MSN-groups.     

On minimal triangle-free 6-chromatic graphs

A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 24 vertices. This implies that Reed's conjecture holds for triangle-free graphs up to at least this order. We also establish that a smallest regular triangle-free 5-chromatic graph has 24 vertices. Finally, we show that the smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and that the smallest 4-chromatic graphs of girth at least 6 have at least 25 vertices.     

On quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

In this paper, by developing the techniques of F. Dillen and L. Vrancken in [6], we study quasi-umbilical locally strongly convex homogeneous unimodular-affine hypersurfaces. We will present a characterization of a certain subclass in all dimensions; finally, in dimension five, we will give a complete classification of all quasi-umbilical homogeneous unimodular-affine hypersurfaces.     

The translational hull of a strongly right type-A semigroup

It is proved that the translational hull of a strongly right type-A (C-rpp) semigroup is itself a strongly right type-A(C-rpp) semigroup.     

Spaces embeddable as regular closed subsets into acc spaces and -spaces

It is well known that some classes of spaces such as absolutely countably compact (abbreviated acc) spaces and (a)-spaces are not hereditary with respect to closed, and even regular closed subspaces. In this paper, we investigate conditions for spaces being regular closed embeddable into spaces with certain covering properties, in particular we characterize spaces which can be embedded as regular closed subsets into acc spaces and (a)-spaces. Some examples are presented as applications of the criteria. Two problems raised by Matveev [Topology Appl. 80 (1997) 169–175] are answered.     

On base-cover metacompact products

Generalizing results of Yang Gao, Lei Mou and Shangzhi Wang, as well as a result of the author, we prove that a topological space is locally compact and metacompact if and only if its product with every compact space is base-cover metacompact.     

强1-星紧空间的性质

探讨强1-星紧与可数紧、feebly紧、伪紧和DFCC之间的关系，给出它在具有分离性（Q）的T1空间类中的特征．     

SOME PROPERTIES OF STRONGLY MONOTONICALLY T2 SPACES

This paper investigates Buck＇s question about which class of spaces is strongly monotonically T2,and if other properties are combined with strongly monotonically T2,which class of spaces could be got. Based on having a cushioned pair-base space and compact strongly monotonically T2 space,some results （Theorems 1--3） are obtained.