Towards a Characterization of Self-Similar Tilings in Terms of Derived Voronoï Tessellations

In this paper, a technique for analyzing levels of hierarchy in a tiling of Euclidean space is presented. Fixing a central configuration P of tiles in , a `derived Voronoï' tessellation _P is constructed based on the locations of copies of P in . A family of derived Voronoï tilings is formed by allowing the central configurations to vary through an infinite number of possibilities. The family will normally be an infinite one, but we show that for a self-similar tiling it is finite up to similarity. In addition, we show that if the family is finite up to similarity, then is pseudo-self-similar. The relationship between self-similarity and pseudo-self-similarity is not well understood, and this is the obstruction to a complete characterization of self-similarity via our method. A discussion and conjecture on the connection between the two forms of hierarchy for tilings is provided.     

Discussing Knarr's 2-Surface of \mathbb{R} 4 which Generates the First Single Shift Plane

The generating line of the first single shift plane (cf. [11, p. 435]) is a 2-surface of ⁴ which we call the the affine part of Knarr's surface. We compute all affinities leaving invariant. After embedding ⁴ into PG(4, ) we calculate the uniquely determined projective closure _Kn of . Using a suitable projection we transform questions on Knarr's surface to questions on Cayley's surface in PG(3, ). In this way we determine all planes carrying 1-dimensional algebraic varieties of _Kn . We exhibit all automorphic collineations of _Kn .     

Foliations with One-Sided Branching

We generalize the main results from the author's paper in Geom. Topol. 4 (2000), 457–515 and from Thurston's eprint math.GT/9712268 to taut foliations with one-sided branching. First constructed by Meigniez, these foliations occupy an intermediate position between -covered foliations and arbitrary taut foliations of 3-manifolds. We show that for a taut foliation with one-sided branching of an atoroidal 3-manifold M, one can construct a pair of genuine laminations ^± of M transverse to with solid torus complementary regions which bind every leaf of in a geodesic lamination. These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston (unpublished), which maps monotonely and ₁(M)-equivariantly to each of the circles at infinity of the leaves of , and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of . In particular, let denote the pulled-back foliation of the universal cover, and co-orient so that the leaf space branches in the negative direction. Then for any pair of leaves of with , the leaf is asymptotic to in a dense set of directions at infinity. This is a macroscopic version of an infinitesimal result from Thurston and gives much more drastic control over the topology and geometry of , than is achieved by him. The pair of laminations ^± can be used to produce a pseudo-Anosov flow transverse to which is regulating in the nonbranching direction. Rigidity results for ^± in the -covered case are extended to the case of one-sided branching. In particular, an -covered foliation can only be deformed to a foliation with one-sided branching along one of the two laminations canonically associated to the -coveredfoliation constructed in Geom. Topol. 4 (2000), 457–515, and these laminations become exactly the laminations ^± for the new branched foliation. Other corollaries include that the ambient manifold is -hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.     

Convexes Hyperboliques et Quasiisométries

For any m ≥ 3, we construct properly convex open sets Ω in the real projective space whose Hilbert metric is Gromov hyperbolic but is not quasiisometric to the hyperbolic space . We show that such examples cannot exist for m = 2. Some of our examples are divisible, i.e. there exists a discrete group Г of projective transformations preserving Ω with a compact quotient Г\Ω. The open set Ω is strictly convex but the group Г is not isomorphic to any cocompact lattice in the isometry group of .     

Bounds for the lattice point enumerator

For the lattice point enumerator of a lattice and a convex body K we give bounds in terms of the intrinsic volumes of K and of minimal determinants of . The intrinsic volumes are the normalized Minkowski quermassintegrals and the minimal determinants are analogous functionals of .     

Stochastic integrals on general topological measurable spaces

Summary A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M ². For each M ², a real valued measure on the predictable -algebra is constructed. The stochastic integral of a random function with respect to is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I is an isometry from L ²() to a stable subspace of M ², its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.     

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.     

Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p

The aim of this paper is to give a geometric interpretation of the continued fraction expansion in the field of formal Laurent series in X ^–1 over , in terms of the action of the modular group on the Bruhat–Tits tree of , and to deduce from it some corollaries for the diophantine approximation of formal Laurent series in X ^–1 by rational fractions in X.     

A note on threefolds with a hyperplane section of general type

This paper deals with polarized pairs , where is a nonsingular projective threefold and is a very ample line bundle on it, such that for one smooth member  | |, one has (Â)=2. A large class of pairs whose adjoint line bundle is nef and big was indirectedly studied by Beltrametti and co-workers. We add some more information, both in this general case and also when the adjoint line bundle fails to be nef and big.     

Isometries of Quaternionic Lie Groups

We study connected Lie groups whose Lie algebra is obtained as the tensor product of a real associative algebra and the algebra of quaternions. It is proved that they carry a natural integrable -structure. We endow such quaternionic Lie groups with a left-invariant Hermitian metric and study the identity connected component of their isometry groups. The determination of such identity connected component is illustrated with a family of examples.     

Designs and partial geometries over finite fields

Let be a finite field, and let (, B) be a nontrivial 2-(n, k, 1)-design over . Then each point induces a (k–1)-spread S on /. (, B) is said to be a geometric design if S is a geometric spread on / for each . In this paper, we prove that there are no geometric designs over any finite field .Research partially supported by NSF grant DMS-8703229.     

Four-dimensional compact projective planes with doubly transitive action on the fixed pencil

We consider a four-dimensional compact projective plane =( , ) whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag pW, acts transitively on _p \{W}, and fixes no point in the set W{p}. We study the actions of and N on and on the pencil _p \{W}, in the case that does not contain a three-dimensional elation group. In the special situation that acts doubly transitively on _p {W}, we will determine all possible planes . There are exactly two series of such planes.     

Sub-weakly Embedded Singular and Degenerate Polar Spaces

We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG , a commutative field, is the projection of a full embedding in some subspace PG of PG , where PG contains PG and is a subfield of . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space contains exactly two points of . This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field , where in the characteristic 2 case is perfect if the polar space is symplectic and the degree of the embedding is 2.     

Tubes of even order and flat π.C 2 geometries

Atube of even orderq=2 ^d is a setT={L, } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q ₃ ⁺ (q) in PG(3,q); letL be an exterior line, and let consist of the polar line ofL together with a regulus onQ.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PL(4,q) fixing a tube {L, } cannot act transitively on . As pointed out by a construction due to Pasini, this implies new results for the existence of flat .C ₂ geometries whoseC ₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing hyperoval by conic in the definition), and consider briefly a related extremal problem.Dedicated to luigi antonio rosati on the occasion of his 70th birthday     

A structure theorem for weak buildings of spherical type

Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building , there is canonically associated a thick building whose Weyl group W( ) can be considered as a reflection subgroup of the Weyl group W() of . One can reconstruct from together with the embedding W( ) W(). Conversely, if is any thick building and W any reflection group containing W( ) as a reflection subgroup, there exists a weak building with Weyl group W and associated thick building .     

A New Cover of the 3-Local Geometry of Co1

The 3-local geometry of the sporadic simple group Co₁ has been known to have a cover with a flag-transitive automorphism group which is a nonsplit extension of an elementary Abelian 2-group of rank 24 (the Leech lattice modulo 2) by Co₁. It was conjectured that was simply connected. We disprove this conjecture by constructing a double cover of . The automorphism group of is of the shape . However, it is not isomorphic to the involution centralizer of the Monster sporadic simple group.     

On the Admissibility of Some Linear and Projective Groups in Odd Characteristic

A bijective mapping defined on a finite group G is complete if the mapping defined by , , is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL and PGL(2,q),q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n,q GL , admit a complete mapping.     

Beispiele starrer,topologischer Faserungen des reellen projektiven 3-Raums

A spread of a projective 3-space is said to be rigid (German: starr) if the only collineation of leaving invariant is the identity; it is called nearly rigid if there are only finitely many collineations of this kind. A spread of real projective 3-space is called topological if the associated translation plane in the sense of André (or Bruck and Bose) is a topological plane; it is then a 4-dimensional translation plane (abbreviated: 4-dtp) in the terminology of Betten. is rigid if and only if every collineation of the associated 4-dtp fixes the translation line pointwise. In 1977 D. Betten asked for such 4-dtps and termed them rigid. If is nearly rigid, the collineation group of the associated 4-dtp is 5-dimensional.In the present paper, examples of rigid and nearly rigid 4-dtps are constructed. The central tool is the method of crosswise tacking together two topological spreads of along a common regulus, which yields two further topological spreads. In a first step, this method when applied to known spreads produces nearly rigid spreads. Rigid spreads are then obtained by iteration of the method; the simplest example is composed of parts of four elliptic linear line congruences. The rigidness of a spread of is proved by arguments from projective differential geometry applied to the image ( ) under Klein's correspondence from line geometry.     

Eine Variante des Fano-Axioms fü angeordnete Ebenen

An ordered plane is an incidence structure ( ) with an order function , which satisfies the axioms (G), (V) and (S), but no continuation--axiom is required. Points a, b E are said to be in distinct sides of a line iff and in the same side if , respectively. For any lines , and we prove that if b,c are in the same side of line A and a,c are in the same side of B , then a and b are in distinct sides of C. As conclusions we deduce that is harmonic and that in each complete quadrangle the intersection points of the diagonals are never collinear, which is known as the axiom of Fano. So the Fano-axiom holds in each ordered plane, and also in those with boundary points.