Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n−1(4n−1)

A construction of a pair of strongly regular graphs ⁿ and ⁿ of type L _2n–1(4n–1) from a pair of skew-symmetric association schemes W, W of order 4n–1 is presented. Examples of graphs with the same parameters as ⁿ and ⁿ, i.e., of type L _2n–1(4n–1), were known only if 4n–1=p ³, where p is a prime. The first new graph appearing in the series has parameters (v, k, )=(225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for ⁿ and ⁿ, thus to prove that ⁿ and ⁿ are not rank three graphs if n>2.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

On the dimensions of automorphism groups of eight-dimensional ternary fields,II

LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF of is connected, then is either isomorphic to one of the compact Lie groupsG ₂ or SU₃, or the (covering) dimension of is at most 7.     

Geometries of type Cn and F4 with flag-transitive automorphism groups

Let be a finite thick geometry of type C_n (n 4) or F₄. We prove that is a building iff Aut() is flag-transitive.     

Hypergraphs and hypergroups

We associate to every hypergraph a commutative quasi-hypergroupH _qG and find a necessary and sufficient condition on so thatH is associative. For certain, any finite included, we determine a sequence= ₀, ₁,, _n of hypergraphs such that ifH ₀ ,H ₁ ,H, H _n is the sequence of the associated quasi-hypergroups,H _n is a join space.Presented by I. Rosenberg.     

ApproximatingL 2-invariants by their finite-dimensional analogues

LetX be a finite connectedCW-complex. Suppose that its fundamental group is residually finite, i.e. there is a nested sequence ... _m + 1 _m ... of in normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL ²-Betti number ofX is the limit of the sequenceb _p(X_m)/[: _m ] whereb _p(X_m) is the (ordinary)p-th Betti number of the finite covering ofX associated with _m .     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

On the null-controllability of linear discrete-time systems with restrained controls

This paper is devoted to the null-controllability problem for linear, autonomous, discrete-time systems with restrained controls, i.e., systems described by the equationx _k+1=Ax _k+u _k,u _k , where 0 , but is not necessarily an interior point of . Some criteria for local and global null controllability are given with proofs. These results can be considered as extensions to discrete-time systems of the known corresponding criteria for linear, continuous-time systems.     

On the critical set for photogrammetric reconstruction using line tokens in P 3(ℂ)

We prove that in general the critical set for photogrammetric reconstruction using lines in P ₃() is a line congruence of order 3 and class 6; has 10 singular points and no singular planes. The general hyperplane sections of (ruled surfaces formed by intersecting with linear line complexes) have genus 5. can be found in Fano's classification of congruences of order 3, and further properties of can be found in the literature.     

Bounded isometries and homogeneous Riemannian quotient manifolds

Let M be a Riemannian manifold that admits a transitive semisimple group G of isometries, G of noncompact type. Then every bounded isometry of M centralizes G and so is a Clifford translation (constant displacement). Thus a Riemannian quotient M is homogeneous if and only if consists of Clifford translations of M. The technique of proof also leads to a determination of the group of all isometries of M.IMAF, Córdoba, Argentina. Partially supported by Conicet, Argentina, and by IMPA, Rio de Janeiro, Brazil.IMAF, Córdoba, Argentina. Partially supported by Conicet, Argentina.University of California at Berkely, U.S.A. Partially supported by National Science Foundation Grant DMS-8200235.     

Isoperimetric inequalities for lattices in semisimple lie groups of rank 2

LetG be a connected semisimple Lie group with finite center. Let be an irreducible non-uniform lattice inG. We show that if the real rank ofG is 2, then the Dehn (or filling) function of is exponential.     

Zur Vollständigkeit der Induktiven Gruppoide der Partiellen Automorphismen von Algebren

For any algebraA let(A) be the set of partial automorphisms (isomorphisms between subalgebras). With the natural multiplication it is an inductive groupoid in the sense of Ehresmann.(A) is complete iff every subset of(A) which is compatible with the semi-ordering has an upper bound. The fact, whether(A) is complete or not, depends on the defining operations ofA. For every direct familyF = (, (A ) ,()_, , of algebras such that all ,, with are one-to-one functions, the direct limit is complete iff all(A) are complete. We give some theorems on the decomposition of inductive groupoids, and employ them in proving the completeness of(A) to variousA.In particular, we obtain that, in case whenG is a finite group,(G) is complete iffG is either cyclic or direct product of a noncyclic group of order 4 and a cyclic group of odd order. For finite acyclic ringsR and finite fieldsK the inductive groupoids(R) and(K) are complete.Further we deal with the question, to what extent algebras are determined by their inductive groupoids. (An algebraA of a classS is defined by(A) iff, for any algebraB of the classS, isomorphism between(A) and(B) implies isomorphism betweenA andB.) Particular attention is paid to finite groups. In general, algebras of classesS are not defined within the classS by their inductive groupoids.

     

Über die endliche Lösbarkeit des Plateau-Problems in Riemannschen Mannigfaltigkeiten

We prove finiteness results of Plateau's problem for boundary curves in a normal chart of radius R < /2 in an analytic Riemannian manifold. It is shown that for analytic there exist only finitely many minimal surfaces of the type of the disc bounded by , which represent an absolute minimum of area, and if boundary branch points are excluded, there exist only finitely many relative minima in the normal chart. If lies on the boundary of a strict convex manifold, there cannot exist any boundary branch points.

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Die vorliegende Arbeit entstand als Dissertation am Mathematischen] Institut der Universität Saarbrücken.

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Herrn Prof. Dr. F. Tomi möchte ich für die Bereitstellung des Themas und für seine Betreuung sehr herzlich danken.     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

On the Neumann-Poincaré operator

Let be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L _C ² () the space of all complex-valued functions on which are square integrable w.r. to the arc-length on . Let L ²() stand for the space of all real-valued functions in L _C ² () and put Since the Cauchy singular operator is bounded on L _C ² (), the Neumann-Poincaré operator C ₁ sending each h L ²() into , is bounded on L ²(). We show that the inclusion characterizes the circle in the class of all AD-regular Jordan curves .     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

Powers of the curvature operator of space forms and geodesics of the tangent bundle

It is well known that if is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve = _1463-01 are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k ₃, k ₆, and k ₁₀ for the real, complex, and quaternionic space forms, respectively.__________Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1231–1243, September, 2004.     

On distance-regular graphs withk i =k j,II

Let be a distance-regular graph of diameterd andi-th valencyk _i. We show that ifk ₂ = k_j for 2 +j d and 2 <j, then is a polygon (k = 2) or an antipodal 2-cover (k _d = 1). We also give a short proof of Terwilliger's inequality for bipartite distance-regular graphs and a refinement of Ivanov's argument on diameter bound.     

Classes caracteristiques de Γ(G,H)-structures et finitude de leur evaluation

LetG be a Lie group andH a closed subgroup ofG. We denote by (G,H) the groupoïd of germs of left translations ofG over the homogeneous spaceG/H.LetV be a compact manifold andx the universal characteristic class of dimensionk which belongs to the vector spaceH _cont ^k ((G, H)).The evaluation ofx over all the (G, H)-structures overV determines a subsetA _{(G, H)} (x, V) of the vector spaceH ^k (V;).We show that in some cases this set is finite.