Inzidenzräume mit Süss'scher Kollineationsgruppe

The following theorem will be proved: THEOREM. Let (P,) be a linear space with dimension dim(P,) 2, such that each line contains at least three points. Let further Aut(P,) be a two — transitive group of collineations on the point set P, such that every automorphism fixing one flag (p,L), fixes all lines through p. Then (P,) is a desarguesian affine space and contains the whole group of dilatations of (P,). Moreover one can define an addition and a multiplication on the point set, such that (P,+,-) is a nearfield, the line joining the neutral elements is a normal subfield and consists of the affine transformations x a + mx, m 0 and= (K).

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Herrn Professor Dr. Helmut Karzel zum 60. Geburtstag     

Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

Über ebene affine Inzidenzgruppen

For a translation plane P with respect to f we consider the group of collineations generated by all elations fixing f and a point F of f. All subgroups or are determined which operate regularly on the points of the affine plane P. Group-theoretic and operating properties of the groups are stated especially for the finite and the Desarguesian cases. In the latter case the companion NL-near modules are constructed. Finally we characterize the groups within PGL(3, K) with commutative field K of finite characteristic.     

The thinnest holding-lattice of a set

In the present note we are going to study the following question proposed by L. Fejes Toth. In the Euclidean plane a lattice is called a holding-lattice of a planar setS if any set congruent toS contains at least one lattice point of . The density of equals (2)^–1, where denotes the area of a fundamental triangle of . A holding-lattice ofS of least possible density is said to be the thinnest holding-lattice ofS. The problem: Find the sets whose thinnest holding-lattice is a regular trianglelattice.To Professor L. Fejes Toth on his 70th Birthday     

On Distance-Regular Graphs with Height Two

Let be a distance-regular graph with diameter at least three and height h = 2, where . Suppose that for every in and in _d(), the induced subgraph on _d() ₂() is a clique. Then is isomorphic to the Johnson graph J(8, 3).     

Une généralisation d'un résultat de J. Aczél et M. Hosszú sur l'équation de translation

Résumé En généralisant un résultat de J. Aczél et M. Hosszú on donne des conditions nécessaires et suffisantes pour qu'une solution de l'équation de translationF(F(, x), y) = F(, xy), oùF: × G , est un ensemble arbitraire,G forme un groupe, soit de la formeF(, x) = f ^–1(f()·1(x)), oùf est une bijection de au groupeG ₁ isomorphe avecG et 1 est un homomorphisme deG àG ₁. On considère aussi le cas oùG forme un espace vectoriel sur le corps des nombres rationels.Si est un intervalle ayant plus qu'un point etG = R ^m avec l'addition comme l'opération on trouve des conditions pour que la fonction continueF soit de la formeF(, x ₁,, x _m ) =f ^–1(f() + c ₁ x ₁ + +c _m x _m ), oùf est une homéomorphie de àR et (c ₁,,c _m ) R ^m .

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Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

The Trace Conjecture – A Counterexample

In the paper Geometric K-Theory for Lie Groups and Foliations, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group with torsion, the trace map tr: K₀(C^*)R maps K₀(C^*) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of . We construct a counterexample to this conjecture.     

FG(2)-Expansions

Let be a finite geometry of rank n 2 with a selected type of elements, called'points'. Let m be the number of'points' of . Under some mild hypotheses on we can consider an affine expansion of to AG(m,2). We prove that the geometries obtained by applying this construction to matroids are simply connected. Then we exploit this result to study universal covers of certain geometries arising from hyperbolic quadrics and symplectic varieties over GF.     

Some remarks on embeddings of the flag geometries of projective planes in finite projective spaces

The flag geometry of a finite projective plane II of orders is the generalized hexagon of order (s, 1) obtained from II by putting equal to the set of all flags of II, by putting equal to the set of all points and lines of II and where I is the natural incidence relation (inverse containment), that is, is the dual of the double of II in the sense of [8]. Then we say that is fully (and weakly) embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of generates PG(d, q) (and if the set of points of not opposite any given point of does not generate PG(d, q)). We have classified all such embeddings in [3, 4, 5, 6]. In the present paper, we weaken the hypotheses in some special cases, and we give an alternative formulation of the classification.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Generalized quadrangles with an ovoid that is translation with respect to opposite flags

The article [6] contains the result that if a finite generalized quadrangle of order s has an ovoid that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then is self-polar and is the set of absolute points of a polarity. In particular, if is the classical generalized quadrangle Q(4, q) then is a Suzuki-Tits ovoid. In this article, we remove the need to assume that is Q(4, q) in order to conclude that is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that is Q(4, q). At the same time, we also relax the requirement that have order s.Received: 14 May 2004     

A finite element method for the nonlinear Tricomi problem

Summary We consider a finite element procedure for numerical solution of the nonlinear problem:L[u]=yu _xx +u _yy +r(x,y)u=f(x, y, u) in a simply connected regionG in thex-y plane. The boundary ofG consists of ₀, ₁, and ₂ and we impose the boundary condition . ₀ is assumed to be a piecewises smooth curve lying in the half-planey>0 with endpointsA(–1, 0) andB(0, 0). ₁ and ₂ are characteristics of the operatorL issued fromA andB which intersect at the pointC(–1/2,y _c). An error analysis of the method is also given.     

Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

Free Algebras of Automorphic Forms on the Upper Half-Plane

We find all pairs (,a) consisting of a cocompact Fuchsian group of genus zero and an automorphy factor a of for which the graded algebra of a--automorphic forms is free.     

Differentiable Connections with Poincar{e} Groups of Local Transformations. {III}

In the canonical smooth fiber bundles :ⁿ⁺¹ⁿ, we study generalized differentiable connections constructed by the author in papers [4] and [5]. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincar{e} groups (1,n) and extended Poincar{e} groups canonically acting in the given connections. We found all first-order nonholonomic affine, ₁, ₂, and _1,2-connections with the groups (1,n) and of local transformations and also constructed classes of the corresponding invariant second-order connections.     

An Arithmetic Reduction of Finite Rank 3 Geometries with Linear Spaces as Plane Residues and with Dual Linear Spaces as Point Residues

Let be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in .We prove that one of the following holds (up to a permutation of the words point and plane):A) the planes of and the dual of the point residues belong to the same family and have the same orders,B) the diagram of is in one of six families,C) the diagram of belongs to a list of seven sporadic cases.Finally, we consider the particular case where the line residues of are generalized digons.     

On some locally finite groups which are sharply triply transitive

We classify the groups satisfying the following conditions: i) is locally finite; ii) is a sharply triply transitive permutation group; iii) all elements of have fixpoints.Published in Ukraninskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1060–1065, July–August, 1991.     

Statistical Cluster Points of Sequences in Finite Dimensional Spaces

In this paper we study the set of statistical cluster points of sequences in m-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m-dimensional spaces too. We also define a notion of -statistical convergence. A sequence xis -statistically convergent to a set Cif Cis a minimal closed set such that for every > 0 the set has density zero. It is shown that every statistically bounded sequence is -statistically convergent. Moreover if a sequence is -statistically convergent then the limit set is a set of statistical cluster points.     

Die Anzahl der Nullstellen eines Polynoms in Gebieten mit stückweise rationalen Randkurven

Summary We are dealing with the problem of calculating the numberN of zeros of a polynomialw(z) in certain regions of the complex plane. If is a halfplane or a disc, there exist many well-known solutions of the problem. If is of a more general type, there is only a paper ofSherman ⁴) for the case, that is the intersection of two halfplanes. In this paper we give the solution for the case that the region is simply connected and the boundary of is composed by a finite number of pieces of rational curves. Our method generalizes the method ofWielandt ³) for the halfplane and the method ofSherman ⁴). The core of our paper is the way of computing the variation of argw, ifw runs along a piece of a rational curve. This computation is based on a lemma about generalized Sturm sequences.