Isometric Shifts and Metric Spaces

 Let M be a complete metric space. If admits an isometric shift, then M is separable. (Received 2 February 2001; in revised form 9 April 2001)     

On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties

Let X be a complete toric variety and Y a smooth projective variety with . We prove that, if is a surjective morphism then . Received: 15 May 2001; in final form: 22 October 2001/ Published online: 4 April 2002     

A Characterization of Astral (n 4 ) Configurations

A conjecture of Branko Grünbaum concerning what astral (n ₄ ) configurations exist is shown to be true. Received July 8, 2000, and in revised form February 14, 2001. Online publication August 28, 2001.     

Automorphism groups of Weyl-type algebras

Let F be a field of characteristic 0, be n commuting variables over F, and be the field of all rational functions. Let . We have the simple Weyl type algebra . In this paper, the automorphism group of the associative algebra and the automorphism group of the Lie algebra are determined, and it is proved that . Received: 4 October 2001 / Revised version: 5 November 2001     

Proper holomorphic discs avoiding closed convex sets

Denote by the open unit disc in . Let C be a closed convex subset of . We prove that for each there is a proper holomorphic map such that and if and only if either C is a complex line or C does not contain any complex line. Received: 17 July 2001; in final form: 22 November 2001 / Published online: 5 September 2002     

Disk-Like Self-Affine Tiles in R 2

We give simple necessary and sufficient conditions for self-affine tiles in R ² to be homeomorphic to a disk. Received October 10, 2000, and in revised form February 16, 2001, and April 25, 2001. Online publication July 25, 2001.     

Defining Integrality at Prime Sets of High Density over Function Fields

 Let F be a function field of characteristic over a finite field . Then for any finite set of F-primes S and any , there exists a set of F-primes W of density greater than such that and has a Diophantine definition over . (Here and is defined analogously.) Received 5 March 2001; in revised form 4 October 2001     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002     

Linear embeddings of semialgebraic G-spaces

Let G be a compact semialgebraic linear group. We prove that every regular semialgebraic G-space admits a semialgebraic G-embedding into some semialgebraic orthogonal representation space of G. Received: 9 January 2001; in final form: 17 August 2001 / Published online: 28 February 2002     

Smoothness of solutions of nonlinear ODE

Smoothness of a -function f is measured by (Carleman) sequence we say if with As an extension of our results in [2] we prove the following type statements: Let u be -function on [0,1] such that where is an entire function of order 1, finite type, on z and of a nonanalytic Carleman class on t. Then u is in the same Carleman class. Received: 16 January 2001 / Revised version: 10 August 2001 / Published online: 6 August 2002     

Hard Tiling Problems with Simple Tiles

It is well known that the question of whether a given finite region can be tiled with a given set of tiles is \NP -complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process we show that Monotone 1-in-3 Satisfiability is \NP -complete for planar cubic graphs. In higher dimensions we show \NP -completeness for the domino and straight tromino for general regions on the cubic lattice, and for simply connected regions on the four-dimensional hypercubic lattice. Received March 8, 2000, and in revised form May 14, 2001, and June 18, 2001. Online publication October 12, 2001.     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14]. Received 30 September 2001; in revised form 10 December 2001     

The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

We prove the Kato conjecture for elliptic operators and N×N-systems in divergence form of arbitrary order 2m on . More precisely, we assume the coefficients to be bounded measurable and the ellipticity is taken in the sense of a G?rding inequality. We identify the domain of their square roots as the natural Sobolev space . We also make some remarks on the relation between various ellipticity conditions and G?rding inequality. Received May 4, 2001; accepted September 6, 2001.     

Adelic version of Margulis arithmeticity theorem

We generalize Margulis's S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one and set . Let S be any subset of R. For each , let be a connected semisimple adjoint -group and be a compact open subgroup for each finite prime . Let denote the restricted topological product of 's, with respect to 's. Note that if S is finite, . We show that if , any irreducible lattice in is a rational lattice. We also present a criterion on the collections and for to admit an irreducible lattice. In addition, we describe discrete subgroups of generated by lattices in a pair of opposite horospherical subgroups. Received: 30 November 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Let G be a connected reductive linear algebraic group over , and X a compact connected Riemann surface. Let be a Levi factor of some parabolic subgroup of G, with its maximal abelian quotient. We prove that a holomorphic G-bundle over X admits a flat connection if and only if for every such L and every reduction of the structure group of to L, the -bundle obtained by extending the structure group of is topologically trivial. For , this is a well-known result of A. Weil. Received: 1 December 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

Asymptotique de la norme $L^2$ d'un cycle géodésique dans les revêtements de congruence d'une variété hyperbolique arithmétique

Let M be an arithmetic hyperbolic manifold and be a codimension 1 geodesic cycle. In this paper, we study the asymptotic growth of the -norm of the lifts of F in the congruence tower above M. We obtain an explicit value for the growth rate of this norm. In particular, we provide a new proof of a celebrated result of Millson [Mi] on the homology of the arithmetic hyperbolic manifolds. The method is quite general and gives a new way of getting non zero homology classes in certain locally symmetric spaces. Received: 20 April 2001; in final form: 26 September 2001 / Published online: 28 February 2002     

The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and an integer. Let be irreducible supercuspidal representations of GL with . One knows that there exists an irreducible supercuspidal representation of GL, with , such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) are distinct. In this paper, we show that, when is an unramified twist of , one may here takem dividingn and , for a prime divisor ofn depending on and the order of : in particular, , where is the least prime divisor of . This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations. Received: 11 November 2000 / Accepted: 15 January 2001 / Published online: 23 July 2001     

About 4-Isosceles Planar Sets

A finite planar set is k -isosceles for k \geq 3 , if every k -point subset of the set contains a point equidistant from the other two. This paper gives affirmative answers to a few open problems about 4-isosceles sets. Received August 14, 2001, and in revised form September 10, 2001. Online publication December 21, 2001.     

Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes. Received August 5, 2000, and in revised form March 29, 2001, and May 3, 2001. Online publication October 12, 2001.     

Bouquet and join theorems for disentanglements

The disentanglement of certain augmentations is shown to be the topological join of a disentanglement and a Milnor fibre. The kth disentanglement of a finite map is defined and for corank 1 maps from ℂ ⁿ to ℂ ^{n +1} it is shown that they are homotopically equivalent to a wedge of spheres. Applications to the Mond conjecture are given. Oblatum 24-VII-2000 & 5-VII-2001?Published online: 12 October 2001