Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

Study of a filtration expanded to include an honest time

Summary Let (,, P) be a measurable space, and { _t} be a filtration on (,). Then, given a fixed honest timeL a new filtrationG _t} is defined, the smallest containing { _t} and for whichL is a stopping time, and the martingales, semimartingales and stopping times of this new filtration are characterised.     

Singular points of a star-finite tiling

Let be a star-finite tiling of a topological vector space of dimension greater than one and let S() denote the set of singular points of . We show that S() is either uncountable or empty by investigating the density in S() of certain subcollections of geometrically interesting singular points.     

On the homogeneous Riemannian structures of type ℐ1⊕ℐ3

We study homogeneous Riemannian structures belonging to the class ₁₃ of the classification given by Tricerri and Vanhecke. The main result is the following: a connected, simply connected Riemannian manifoldM admits a homogeneous structureT of type ₁₃,T₃, if and only ifM is isometric to a hyperbolic space ⁿ .This work was partially supported by M.P.I.     

Tessellations generated by hyperplanes

Let be a locally finite system of hyperplanes in ^d with the property that the cells of the induced cell complex decomposition of ^d have uniformly bounded diameters. If is simple and the density of the vertices in exists, then the density of thek-cells in exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in exists and can be calculated. For certain nonsimple systems , corresponding inequalities are obtained.     

A gaussian central limit theorem for trimmed products on simply connected step 2-Nilpotent Lie groups

A limit theorem due to J. Kuelbs and M. Ledoux, valid for dilatation-stable laws on type 2-Banach spaces, is carried over to stable laws on simply connected step 2-nilpotent Lie groupsG. We show that for products of i.i.d. random variables in the of attraction of a nondegenerate semigroup onG, where is a one-parameter automorphism group acting contracting onG, a certain intermediate trimming procedure, together with a suitable norming, always yields a nondegenerate centered Gaussian limit.     

Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

Modules over matrix near-rings and the ℐ0-radical

This paper clears up some questions concerning type 0 modules over matrix near-rings and the ₀-radical in matrix near-rings. It is shown that, unlike in the type 2 case, type 0 modules over matrix near-rings may arise in several non-isomorphic ways. As a result, we do not always have the same nice relationship between the ₀-radicals of a near-ring and the corresponding matrix near-ring, as we do for the ₂-radical. All near-rings concerned are zero-symmetric with identity element.     

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let be a Galois ring of characteristic 4,X a cyclotomic scheme over with classD and the Terwilliger algebra ofX, then we show that most of the irreducible -modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible -modules using Jacobi sums.     

On the geometric structure of independence systems

A bouquet of matroids is a combinatorial structure that generalizes the properties of matroids. Given an independence system, there exist several bouquets of matroids having the same family of independent sets. We show that the collection of these geometries forms in general a meet semi-lattice and, in some cases, a lattice (for instance, when is the family of the stable sets in a graph). Moreover, one of the bouquets that correspond to the highest elements in the meet semi-lattice provides the smallest decomposition of into matroidal families, such that the rank functions of the different matroids have the same values for common sets. In the last section, we give sharp bounds on the performance of the greedy algorithm, using parameters of some special bouquets in the semi-lattice.     

Bourbaki spaces of topological groups

A study is presented of the relationship between the topological and uniformity properties of a group G and the spaces (G), (G) of all nonempty closed subsets and closed subgroups of G. A base for the neighborhood system of a closed subset X of G is formed by the sets S(X, U)={Y Y XU, X YU}, where U ranges over all neighborhoods of the identity in G. Criteria are obtained for the space (G) and some of its subspaces to be totally bounded and locally totally bounded. Some classes of groups with compact spaces (G) are described. It is proved that the spaces (G), (G) are complete in the case of projective metrizable groups G.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 542–549, April, 1990.     

Complements to ‘The Bruhat order on symmetric varieties’

We give several complements to the paper The Bruhat order on symmetric varieties. Our main result shows that the partial order on the set of twisted involutions in the Weyl groupW, which was introduced in the earlier paper, agrees with the partial order on induced by the usual Bruhat order onW.R. W. Richardson died on 15 June. 1993.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

A certain periodic control problem for differential equations with impulses in the space of bounded numerical sequences

One considers the differential equation dx/dt=f(t, x) with the impulse action ¦_t=t_i=H_i(t_i,x) in the space of bounded numerical sequences, where f(t, x), H_i(t, x) are T-periodic, countable-dimensional vector-valued functions, is a positive parameter. One gives conditions for the existence of a control (₁,₂) such that the solution of the equation dx/dt=f(t, x)–₁ with impulse action x¦_t=t_i=H_i(t_i,x)–₂ assuming for t= the value x=x₀, be T-periodic.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 271–275, February, 1990.     

On representing contexts in line arrangements

Acontext is defined to be a triple (G, M, J) of setsG, M and an incidence relationJ G×M.A finite set ofn oriented lines in general position in the euclidean plane induces a cell decomposition of the plane. For a givenk-element subset of cells of dimension 2, we define an incidence relationJ × as follows:t _i andl _j are incident if and only ift _i lies on the positive side with respect tol _j .We call a context (G, M, J)represented in a line arrangement if and only if there are relation preserving bijections betweenG and ,M and , respectively. We study conditions for a context to be representable in a line arrangement.Especially, we provide a non-trivial infinite class of contexts which can not be represented in a line arrangement.     

Happy families and completely Ramsey sets

Summary We use games of Kastanas to obtain a new characterization of the classC of all sets that are completely Ramsey with respect to a given happy family . We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC . We also study some cardinals that can be associated with the ideal of nowhere -Ramsey sets.Part of this research was done while the author was visiting I.V.I.C. in Caracas in September 1989. The author would like to thank Carlos Di Prisco for his hospitality.     

A lower bound for n-diameters

Let D be a subset of the s-dimensional lattice Z^S, M=M(D) the number of elements in D, _Dthe space of trigonometric polynomials on the torus T^S with spectrum concentrated in D and having unit norm in L₂(T^S). In this paper we give the following bound for the Gel'fand diameter:d ⁿ( _D,C(T^s))M/2–N/2. This bound is subsequently used for actual functional classes.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 413–419, October, 1972.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), where v = k, k = a ₁, (v – k – 1) = k(k – 1 – ) and c ₂ + 1, since a -graph is a regular graph with valency . If c ₂ = + 1 and c ₂ 1, then is a Terwilliger graph, i.e., all the -graphs of are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c ₂ 2 and obtained that there are only three such examples. In this article we consider the case c ₂ = + 2 3, i.e., the case when the -graphs of are the Cocktail Party graphs, and obtain that either = 0, = 2 or is one of the following graphs: (i) a Johnson graph J(2m, m) with m 2, (ii) a folded Johnson graph J¯(4m, 2m) with m 3, (iii) a halved m-cube with m 4, (iv) a folded halved (2m)-cube with m 5, (v) a Cocktail Party graph K _{m × 2} with m 3, (vi) the Schläfli graph, (vii) the Gosset graph.     

Homotopy types for tamely approximable Sobolev maps between manifolds

We define a class ^p (M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu ^p (M, N) has a well defined [p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in [p]-homotopy classes for some polyconvex variational problems.