On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001     

A New Class of Infinite Products Generalizing Viète's Product Formula for π

We show how functions F(z) which satisfy an identity of the form F(α z) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π. Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained. 2000 Mathematics Subject Classification: Primary—40A20; Secondary—33E05     

Decomposition of involutions on inertially split division algebras

Let F be a Henselian valued field with , and let S be an inertially splitF}-central division algebra with involution $\sigma ^{\ast }$ that is trivial on an inertial lift in S of the field . We prove necessary and sufficient conditions for S to contain a -stable quaternion {\it F}-subalgebra, and for to decompose into a tensor product of quaternion algebras. These conditions are in terms of decomposability of an associated residue central simple algebra that arises from a Brauer group decomposition of S. Received February 1, 1999; in final form August 26, 1999 / Published online July 3, 2000     

Products and factors of Banach function spaces

Given two Banach function spaces we study the pointwise product space E · F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E · M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F.     

Integrality of L2L^2-Betti numbers

The Atiyah conjecture predicts that the -Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that is a normal subgroup, for amalgamated free products . Here F is a free group and is an arbitrary semi-direct product. This includes free products G*F and semi-direct products . We also show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new (bigger) class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with torsion-free elementary amenable quotient or with free quotient, and under certain direct and inverse limits. Received: 22 August 1998/ Revised: 10 Jannary 2000 / Published online: 28 June 2000     

The local root number of elliptic curves with wild ramification

Let E be an elliptic curve over a number field F. The root number is conjecturally the sign of the functional equation of L-function of E/F. It is defined as the product of local signs over all places of F. The purpose of this paper is to describe this local sign by the coefficients of a Weierstra? equation of E. Received: 31 March 2000 / Revised version: 22 November 2001 / Published online: 23 May 2002     

The Dimension of Subsets of Moran Sets Determined by the Success Run Behaviour of Their Codings

 Let be a Moran set associated with the set . Let Γ be a non-empty subset of with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space is a decomposition of F such that

Characterization of the sets of discontinuity points of separately continuous functions of many variables on the products of metrizable spaces

We show that a subset of the product ofn metrizable spaces is the set of discontinuity points of some separately continuous function if and only if this subset can be represented in the form of the union of a sequence ofF _σ-sets each, of which is locally projectively a set of the first category. Chernovsty University, Chernovtsy. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 740–747, June, 2000.     

Essential dimension of finite <Emphasis Type="Italic">p</Emphasis>-groups

We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F. Mathematics Subject Classification (2000) 20G15; 14C35     

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category F≀FIn

We give a new proof for the Littlewood-Richardson rule for the wreath product F?S_n where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F?S_n to F?S_n+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F?FI_n where FI_n is the category of all injective functions between subsets of an n-element set.     

Complementing a finite subgroup of a hyperbolic group by a free factor

We give necessary and sufficient conditions for a finite subgroup H of a hyperbolic group G to contain a free subgroup F of rank two in G such that F and H generate a free product F ∗ H. A verification algorithm for these conditions is pointed out.     

Realizations for direct constructions of resolvable Steiner 2‐designs with block size 5

We consider direct constructions due to R. J. R. Abel and M. Greig, and to M. Buratti, for ({ν},5,1) balanced incomplete block designs. These designs are defined using the prime fields F_p for certain primes p, are 1‐rotational over G ⊕ F_p where G is a group of order 4, and are also resolvable under certain conditions. We introduce specifications to the constructions and, by means of character sum arguments, show that the constructions yield resolvable designs whenever p is sufficiently large. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:207–217, 2000     

Frames and bases in tensor products of Hilbert spaces and Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

In this article, we study tensor product of Hilbert C*-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F, and we get more results. For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K.     

Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals

We consider parametric variational double integrals with elliptic Lagrangians F depending on the surface normal and prove a compactness theorem for -critical immersions. As a key ingredient for the relevant a priori estimates we use F. Sauvigny's F-conformal parameters adapted to the parametric integrand F. As a by-product of our analysis we obtain an isoperimetric inequality for -critical immersions generalizing the classical isoperimetric inequality for minimal surfaces. Received November 19, 1999 / Accepted February 4, 2000 / Published online July 20, 2000     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

Products of involutory matrices. I

It is shown that, for every integer ?1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n?4 or F has prime order ?5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

Determination of conductors from Galois module structure

Let E denote an unramified extension of , and set for an odd prime p and . We determine the conductors of the Kummer extensions of F by those elements such that is Galois. This follows from a comparison of the Galois module structure of with the unit filtration of F. Received: 28 August 2000; in final form: 11 October 2001 / Published online: 4 April 2002     

Quasi-isometries between groups with infinitely many ends

Let G, F be finitely generated groups with infinitely many ends and let? be graph of groups decompositions of F, G such that all edge groups are finite and all vertex groups have at most one end. We show that G, F are quasi-isometric if and only if every one-ended vertex group of is quasi-isometric to some one-ended vertex group of and every one-ended vertex group of is quasi-isometric to some one-ended vertex group of?. From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: and are all quasi-isometric. Received: April 7, 2000; revised version: October 6, 2000     

Writing Commutators of Commutators as Products of Cubes in Groups

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(γ) of an element γ of a group G, we mean the least natural number k such that γ is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = F?C _∞ is the wreath product of a free group F by the infinite cyclic group, we show that every element of W″ is a product of at most three cubes in W.     

Finite Group Actions with Fixed Points on Homology 3-Spheres

 If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds. Received 25 September 1997; in revised form 2 June 1998