Theta Subgroups and the Solvability of FiniteGroups

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＤｅｆｉｎｉｔｉｏｎＡｌｌｇｒｏｕｐｓｔｒｅａｔｅｄａｒｅｆｉｎｉｔｅ．Ｉｎ［１］，Ｎ．Ｐ．ＭｕｋｈｅｒｊｅｅａｎｄＰ．Ｂｈａｔｔａｃｈａｒｙａｄｅｆｉｎｅｄｔｈｅｃｏｎｃｅｐｔｏｆθ－ｐａｉｒｓａｓｓｏｃｉａｔｅｄｔｏａｍａｘｉｍａｌｓｕｂｇｒｏｕｐＭｏｆａｇｒｏｕｐＧａｓｆｏｌｌｏｗｓ：ａθ－ｐａｉｒｆｏｒＭｉｓａｎｙｐａｉｒｏｆｓｕｂｇｒｏｕｐｓ（Ｃ，Ｄ）ｏｆＧｓｕｃｈｔｈａｔ（１）ＤＧ，Ｄ＜Ｃ，（２）〈Ｍ，Ｃ〉＝Ｇ，〈Ｍ，Ｄ〉＝Ｍａｎｄ（３）Ｃ／Ｄ…     

Saks空间中（A）型相容映射的叠合点及公共不动点（英）

ＣｏｉｎｃｉｄｅｎｃｅＰｏｉｎｔｓａｎｄＣｏｍｍｏｎＦｉｘｅｄＰｏｉｎｔｓｆｏｒＣｏｍｐａｔｉｂｌｅＭａｐｓｏｆＴｙｐｅ（Ａ）ｏｎＳａｋｓＳｐａｃｅｓＰ．Ｐ．Ｍｕｒｔｈｙ，Ｂ．Ｋ．Ｓｈａｒｔｒｉａ，Ｙ．Ｊ．ＣｈｏＣｏｉｎｃｉｄｅｎｃｅＰｏｉｎｔｓａｎ...     

关于布尔关系友阵（英）

ＯｎＣｏｍｐａｎｉｏｎＢｏｏｌｅａｎＲｅｌａｔｉｏｎＭａｔｒｉｃｅｓＣｈａｏＣｈｏｎｇｙｕｎ（Ｄｅｐｔ．ｏｆＭａｔｈＵｎｉｖｏｆＰｉｔｔｓｂｕｒｇｈＰｉｔｔｓｂｕｒｇｈ，ＰＡ１５２６０）ＷａｎｇＴｉａｎｍｉｎｇ（Ｉｎｓｔ．ｏｆＭａｔｈ．Ｓｃｉｅｎｃｅ...     

Separateness in Topological Molecular Lattices

ＳｅｐａｒａｔｅｎｅｓｓｉｎＴｏｐｏｌｏｇｉｃａｌＭｏｌｅｃｕｌａｒＬａｔｔｉｃｅｓＬｉＬｅｉ（李雷）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＨｕａｉｂｅｉＣｏａｌＴｅａｃｈｅｒｓＣｏｌｌｅｇｅ，２３５０００）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｄｉｓ...     

On Bourque-Ligh Conjecture of LCM Matrices

ＯｎＢｏｕｒｑｕｅ－ＬｉｇｈＣｏｎｊｅｃｔｕｒｅｏｆＬＣＭＭａｔｒｉｃｅｓ￥ＨｏｎｇＳｈａｏｆａｎｇ（洪绍方）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，Ｓｉｃｈｕａｎ，６１００６４）（Ｃｏｍｍｕｎｉｃａｔｅｄｂｙ...     

N-Centralizing Mappings in Semiprime Rings

Ｎ┐ＣｅｎｔｒａｌｉｚｉｎｇＭａｐｐｉｎｇｓｉｎＳｅｍｉｐｒｉｍｅＲｉｎｇｓＭａｊｅｅｄ，Ａ．Ｈ．ａｎｄＮｉｕＦｅｎｇｗｅｎ（牛凤文）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＪｉｌｉｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈａｎｇｃｈｕｎ，１３００２３）...     

Invariant Kahler Metrics and Curvatures on a Class of Pseudoconvex Domains

（童武）（殷慰萍）ＩｎｖａｒｉａｎｔＫａｈｌｅｒＭｅｔｒｉｃｓａｎｄＣｕｒｖａｔｕｒｅｓｏｎａＣｌａｓｓｏｆＰｓｅｕｄｏｃｏｎｖｅｘＤｏｍａｉｎｓ￥ＴｏｎｇＷｕＹｉｎＷｅｉｐｉｎｇ（Ｄａｐｔ．ｏｆＭａｔｈ．，ＣａｐｉｔａｌＮｏｒｍａｌＵｎｉｖｅｒｓｉ...     

Some New Results in Waring's Problem

ＳｏｍｅＮｅｗＲｅｓｕｌｔｓｉｎＷａｒｉｎｇ’ｓＰｒｏｂｌｅｍＭｅｎｇＺａｉｚｈａｏ（蒙在照）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙｏｆＣｈｉｎａ）Ｈｅｆｅｉ，ａｎｈｕｉ，２３００２６Ｃｏｍｍｕｎ...     

A Comtrol Model of Linear Fuzzy Silf-Regression

ＡＣｏｎｔｒｏｌＭｏｄｅｌｏｆＬｉｎｅａｒＦｕｚｚｙＳｅｌｆ－ＲｅｇｒｅｓｓｉｏｎＬｕＦｅｎｇ，ＸｕＸｉａｏｇｕａｎｇ（Ｄｅｐ．ｏｆＭａｔｈ．Ｎｏｒｔｈｅａｓｔ（ＢａｓｉｃＢｒａｎｃｈＣｈａｎｇｃｈｕｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ）ＴａｘＣｏｌｌ...     

MAXIMUM TREES OF SUBSETS

ＭＡＸＩＭＵＭＴＲＥＥＳＯＦＳＵＢＳＥＴＳＷｕＳｈｉｑｕａｎ（巫世权）（Ｉｎｓｔ．ｏｆＡｐｐｌ．Ｍａｔｈ．，ＣｈｉｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，ＰＯＢｏｘ２７３４，Ｂｅｉｊｉｎｇ１０００８０，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：ＬｅｔＸｂｅａｆ...     

Abelianization and orthogonal complements of additive categories

An abelian completion of an additive regular category is constructed and in­vestigated. It is applied to give an axiomatic characterization of the categories which appear as categories of torsion free objects in abelian categories with pre-radicals, radicals, hereditary radicals.     

On Matrix Rings and Subhereditary Radicals

Abstract

We study some one-sided ideals of row bounded and of column bounded matrix rings over free algebras. Obtained results are applied to answer several open problems on subhereditary radicals of associative rings. In particular we show that the right strongly prime radical is not left subhereditary and that the lattice of left (right) subhereditary radicals is not complete.     

Lattices of Radicals of Involution Rings

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We show that certain sublattices are isomorphic to sublattices of the lattice of radicals of rings. We characterize the atoms of certain lattices of radicals of involution rings.1991 Mathematics Subject Classification: 16W10, 16N99, 06B99     

Using restrictions to accept or reject solutions of radical equations

The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original equation. We propose alternative solution procedures that are rigorous and simple to execute where the extraneous solutions can be identified without verification against the original equation. In this article, we review previous literature, establish and illustrate rigorous solution procedures for radical equations of depth 1 (i.e. equations where all radicals can be eliminated in one step), and deal with an ambiguity concerning the definition of real-valued solutions to radical equations. An application to defining the inverse function, resulting in a parametric radical equation, is also explained.     

On Extensions of Chain Algebras and Atoms of the Lattice of Radicals

In this article we study the problem when a left distributive algebra over a field determines an atom of the lattice of all radicals of rings. We show that this problem is strictly connected with the problem of describing extensions of left chain algebras and give a characterization of left chain algebras determining atoms.     

On Principally Left Strong Radicals

Principally left strong radicals were introduced in [4]. Here, we continue to study such radicals. Principally left strong radicals and their semisimple classes are characterized in Section 2. Sections 3 and 4 are devoted to the lower principally left strong radical construction and to the upper principally left strong radical construction, respectively. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the lattice of matric-extensible radicals. II

Summary We continue our study of the lattice of matric-extensible radicals of associative rings. We find some atoms generated by simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate atoms of these lattices.     

On some infinite sums of iterated radicals

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals.     

A New Class of Infinite Products Generalizing Viète's Product Formula for π

We show how functions F(z) which satisfy an identity of the form F(α z) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π. Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained. 2000 Mathematics Subject Classification: Primary—40A20; Secondary—33E05