On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

On π-Solvable and Locally π-Solvable Groups with Factorization

We prove that, in a locally -solvable group G = AB with locally normal subgroups A and B, there exist pairwise-permutable Sylow - and p-subgroups A , A _p and B , B _p , p , of the subgroups A and B, respectively, such that A B is a Sylow -subgroup of the group G and, for an arbitrary nonempty set ,

A Remark on Subgroup Separability in the Class of Finite π-Groups

We prove that if a group G is residually , then for every -subgroup of the group G, the set of -roots from this subgroup is a -separable -subgroup.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Das System Totalpolarer und eine Kennzeichnung des Winkels von Hyperebenen in Cayley/Klein-Räumen

In this paper at first we introduce thesystem of total polar subspaces of an arbitrary k-diniensional plane with respect to the absolute (configuration) of an arbitrary n- dimensional CAYLEY/KLEIN space as a generalization of the total polar set of a regular k-plane of . Using the system of total polar sets of the intersection (n–2)-plane of two hyperplanes and we give the followingnew characterization of the angle of . and : for any straight line g with and g=Ø the angle of and is equal to the distance of the two intersection points of g with and .

---

---

Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

Lower bounds for heights on finitely generated groups

LetG be a finitely generated subgroup of rankr of the multiplicative groupK ^* of a number field. We construct a basis ₁,..., _r ofG such that for G, writing where is a root of unity and where thex _i are integers, we get a lower bound for the absolute heighth() in terms ofr and max .     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Some Ramsey-Turán type results for hypergraphs

To everyk-graphG let(G) be the minimal real number such that for every>0 andn>n ₀(,G) everyk-graphH withn vertices and more than (+) ( ) edges contains a copy ofG. The real number (G) is defined in the same way adding the constraint that all independent sets of vertices inH have sizeo(n). Answering a problem of Erds and Sós it is shown that there exist infinitely manyk-graphs with 0<(G)<(G) for everyk3. It is worth noting that we were unable to find a singleG with the above property.This paper was written while the authors were visiting AT&T Bell Laboratories, Murray Hill, NJ 07974.     

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.