Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Universality of the automorphism group of the real line

We discuss the question, whether each automorphism group (of cardinality at most ) of a linear order is embeddable into the automorphism group of the real line. We show that the answer to this question is independent of the axioms of ZFC: the answer is positive, if we assume < and Souslin's hypothesis; the answer is negative, if we assume orV=L.     

K-theory of twisted differential operators on flag varieties

Let be a semisimple Lie algebra overk, an algebraically closed field of characteristic zero, and let be a Cartan subalgebra inside a Borel subalgebra of . LetU be the enveloping algebra of . For letM() denote the corresponding Verma modúle and letU _u=U/AnnM(). LetW be the Weyl group and letW ⁰ be the stabiliser of inW. We prove the following theorem, which affirms a conjecture of T.J. Hodges.Oblatum 16-XII-1994     

Sylow subgroups of finitary linear groups

In this paper we describe the structure and the conjugacy classes of Sylow p-subgroups of FGL(V, ), the group of finitary -automorphisms of the -vector space V.The Author is member of the GNSAGA.     

On causality in commutant lifting theorem. I

Two functionals (A) and for an operatorA were introduced in [11] for the study of causality in commutant lifting theory. In this paper we give sufficient and necessary conditions for in a special case. We prove that in this case , and we show by some examples related to nonlinear system control that is the best constant in our inequality.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Operators of fibered groups

Let (G, ) denote a finite groupG with fibration . The group Aut (G, ) of operators of (G, ) is closely related to the group of collineations of (G, ). In this paper we investigate the geometric properties imposed on (G, ) by requiring that Aut (G, )=AutG. We find that in many instances this algebraic property restricts the geometry to a very special form.Dedicated to Professor Helmut Karzel on the occasion of his 60th birthday.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

On some properties of the metric subalgebras ofℓ ∞

The objects studied are the subalgebras of which contain c_o. These are isometrically isomorphic to the algebras C( ) where is a compactification of a discrete denumerable set N . It is shown: 1) If is metric then there is a projection of norm 1, P: C( ) C( ) with kernel c_o defined by PF = f o where is a retraction of onto = – N . 2) If is metric, then the group of homeomorphisms of is isomorphic to a complete group of permutations of the natural numbers . 3) The group of homeomorphisms of a compact metric space is the homomorphic image of a complete group of permutations of ("complete" means "no outer automorphisms, trivial center").     

Sechzehndimensionale lokalkompakte Translationsebenen,deren Kollineationsgruppe G2 enthält

New results of Salzmann and Hubig say that a 16-dimensional (locally) compact topological projective plane in which the group of continuous collineations has dimension 40 is a translation plane. It is therefore important to determine all 16-dimensional locally compact translation planes with dim 40. From previous work of the author ([10]), it is known that such a plane is either the classical octonion plane, or dim =40 and contains a subgroup isomorphic to the compact exceptional group G₂, but no larger compact simple subgroup. In the present paper, all planes satisfying the latter property more generally with dim 38 are explicitly determined. Together with the classification of all 16-dimensional locally compact translation planes in which contains Spin(7) given by the author in [8], one thus knows all 16-dimensional locally compact translation planes with containing G₂ and dim 38. Via suitable Baer subplanes, the classification makes use of analogous results for 8-dimensional planes ([7]).

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Meinem verehrten Lehrer Helmut Salzmann zum 60. Geburtstag     

Spreads and group divisible designs

By removing the components of at-spread of a finite projective spacePG(d, q) from each hyperplane ofPG(d, q), the blocks of a regular group divisible design are obtained We characterize geometrict-spreads as thoset-spreads which are such that the dual of is also a group divisible design.     

An infinitesimal characterization of the complexity of homogeneous spaces

A characterization of the complexity of a homogeneous space of a reductive groupG is given in terms of the mutual position of the tangent Lie algebra of the stabilizer of a generic point of and the (–1)-eigenspace of a Weyl involution of .Supported by Russian Foundation for Basic Research, Grant No. 98-01-00598.Supported by the U. S. Civilian Research and Development Foundation, Award No. 206, and Russian Foundation for Basic Research, Grant No. 98-01-00598.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

Polynomial invariants of graphs II

Negami and Kawagoe has already defined a polynomial associated with each graphG as what discriminates graphs more finely than the polynomialf(G) defined by Negami and the Tutte polynomial. In this paper, we shall show that the polynomial includes potentially the generating function counting the independent sets and the degree sequence of a graphG, which cannot be recognized fromf(G) in general, and discuss on of treesT with observations by computer experiments.     

Chu-Dualität und zwei Klassen maximal fastperiodischer Gruppen

This is a continuation of the paper Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen, [6]. In [6], the classes and were introduced. We give sufficient conditions to conclude thatG is in if one knows thatG/G ₀ is in . If a groupG is in and ifG satisfies the Chu-duality then all closed subgroups ofG satisfy the Chu-duality. The Chu-quasi-dual of the Heisenberg groupH with integral coefficients is computed. It is shown thatH does not satisfy the Chu-duality, thatH is in , and thatH is not in .