It is shown that a Carter subgroup of a non-soluble unitary group can only be the normalizer of a Sylow 2-subgroup. This result, combined with previous results, implies that no finite simple group can be a minimal counterexample to the conjugacy conjecture of Carter subgroups in a finite group.The third author was partially supported by the University of Padova, the RFBR, grants 02-01-00495 and 02-01-06226, Ministry of Education RF, E00-1.0-77, ‘Universities of Russia’ UR.04.01.031.

This paper investigates how to relate the syzygy periodicity of a self-injective algebra $A$ to its Auslander–Reiten periodicity. Moreover, a characterization is provided of the Auslander–Reiten bounded $A$–$A$-bimodules that are periodic.

The authors present a technique of deriving basic hypergeometric identities from specializations using fewer parameters, by using the classical Cauchy identity on the expansion of the power of $x$ in terms of the $q$-binomial coefficients. This method is referred to as ‘Cauchy augmentation’. Despite its simple appearance, the Cauchy identity plays a key role in parameter augmentation. For example, one can reach the $q$-Gauss summation formula from the Euler identity by using the Cauchy augmentation twice. This idea also applies to Jackson's $_2\phi_1$ to $_3\phi_1$ transformation formula. Moreover, a transformation formula analogous to Jackson's formula is obtained.

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.The author was partially supported by NSF grant #DMS 9802558.

The set of vectors tangent to complete simple geodesics at a Weierstrass point is investigated, and interpreted as an infinitesimal version of the Birman–Series set. Three new identities for the lengths of simple geodesics are deduced. Fundamental use is made of the properties of the elliptic involution.

The paper proves a result on the existence of finite flat scheme covers of Deligne–Mumford stacks. This result is used to prove that a large class of smooth Deligne–Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to a classical question on Brauer groups of schemes.Research of the first author supported in part by the Wolfgang Paul program of the Humboldt Foundation; research of the second author supported in part by the University of Bologna, funds for selected research topics.

Let $G$ be a simple simply connected complex Lie group. Some criteria are given for the nonexistence of exceptional principal $G$-bundles over a complex projective surface. As an application, it is shown that there are no exceptional $G$-bundles over a surface whose arithmetic genus is zero or one. It is also shown that there are no stable exceptional $G$-bundles over an abelian surface.

By using a more efficient way of ‘almost diagonalizing’, known results on small fractional parts of general real cubic forms in a small number of variables, are improved. In particular, for ternary forms the method gives a bound superior to that arising from simply reducing to a cubic form in one variable.

Two normalised Hecke eigenforms $f$ and $g$ are considered in this paper, of level one and weights $k'\,{>}\,k$, lying in a $p$-adic family, such that $f\,{\equiv}\,g\pmod{p^r}$ as $q$-expansions. Interpreting the congruence in terms of the associated Galois representations leads to the existence of non-trivial global $p^r$-torsion for the motive associated with the tensor product $L$-function at $s\,{=}\,k'\,{-}\,1$. (It must be assumed that the (mod $p$) Galois representation attached to $g$ is irreducible.) This contributes a factor of $p^r$ to the denominator of the Bloch–Kato conjectural formula for the $L$-value. The $p$-part of the numerator is considered, using recent work of Diamond, Flach and Guo. Using Shimura's Rankin–Selberg integral formula and the Clausen–von Staudt theorem, the ratio of $L$-values is examined at $s\,{=}\,k'-1$, for the tensor product and a quadratic twist; confirmation is given (under certain conditions) that, at $p$, this is as predicted by Bloch and Kato.

It is shown that, given continuous, absolutely irreducible representations of ${\rm Gal}(\mathbb{Q}^{\rm ac}/\mathbb{Q})$ with values in ${\rm GL}_2(\mathbb{F}_2)$ and ${\rm GL}_2(\mathbb{F}_3)$, and having cyclotomic determinant, there is a weight 2 newform of some level whose mod $2$ and mod $3$ representations are equivalent to those given.

The action of translation operators on wavelet subspaces in higher dimensions is investigated. This action defines an equivalence relation on the set of single wavelets of $L^2(\mathbb R^n)$ associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of the wavelets. Further, examples of wavelets in each of these classes are constructed. This construction shows the existence of wavelets for which the associated wavelet subspaces are invariant under various groups of translation operators.

Simply connected closed symplectic 4-manifolds with $b_{2}^+ \,{=}\,1$ and $K^2 \,{=}\,0$ are investigated. As a result, it is confirmed that most of homotopy elliptic surfaces $\{E(1)_{K} \,{|}\,K\, \mathrm{is\, a\, fibred\, knot}$ in $S^3\}$ constructed by R. Fintushel and R. Stern in Invent. Math. 134 (1998) 363–400 are simply connected closed minimal symplectic 4-manifolds that do not admit a complex structure.This work was supported by grant No. R14-2002-007-01002-0 from KOSEF.

Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$, then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

Let $H$ be a not necessarily separable Hilbert space, and let $\mathcal{B}(H)$ denote the space of all bounded linear operators on $H$. It is proved that a commutative lattice $\mathcal{D}$ of self-adjoint projections in $H$ that contains $0$ and $I$ is spatially complete if and only if it is a closed subset of $\mathcal{B}(H)$ in the strong operator topology. Some related results are obtained concerning commutative lattice-ordered cones of self-adjoint operators that contain $\mathcal{D}$.

It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$, $K \in {\mathbb N}$, such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.