On Weighted Distortion in Conformal Mapping II

Some years ago, Blatter [1] gave a result of the form for any function f regular and univalentin D: |z| < 1, where is the hyperbolic distance betweenz₁ and z₂. Kim and Minda [5] pointed out that the multiplieron the right is incorrect. They say that Blatter's proof givesthe correct multiplier, but Blatter's formula for in termsof z₁, z₂ is wrong. Kim and Minda proved the generalized formula where D₁(f) = f'(z) (1 – |z|²),valid for p P with some P, . In each case there was an appropriate equality statement. Kimand Minda made the important and easily verified remark thatthese problems are linearly invariant in the sense that if theresult is proved for f, then it follows for , where U is a linear transformation of the planeonto itself and T is a linear transformation of D onto itself.This means that we need to prove such results only in an appropriatelynormalized context. 1991 Mathematics Subject Classification30C75, 30F30.     

Hopf-Galois Structures on Field Extensions with Simple Galois Groups

Let L/K be a finite Galois extension of fields whose Galoisgroup is a nonabelian simple group. It is shown that L/K admitsexactly two Hopf–Galois structures. 2000 Mathematics SubjectClassification 12F10, 16W30.     

Left Cotorsion Rings

It is proved that if R is an associative ring that is cotorsionas a left module over itself, and J is the Jacobson radicalof R, then the quotient ring R/J is a left self-injective vonNeumann regular ring and idempotents lift modulo J. In particular,if R is indecomposable, then it is a local ring. 2000 MathematicsSubject Classification 16D50, 16D90, 16L30.     

On Pointed Hopf Algebras of Dimension 2n

We give a structure theorem for pointed Hopf algebras of dimension2ⁿ, having coradical kC₂, where k is an algebraically closedfield of characteristic zero. 1991 Mathematics Subject Classification16W30.     

Tilting Cotorsion Pairs

Let R be a ring and T a 1-tilting right R-module. Then T isof countable type. Moreover, T is of finite type in the casewhere R is a Prüfer domain. 2000 Mathematics Subject Classification16D90, 16D30, 13G05 (primary), 03E75, 20K40, 16G99 (secondary).     

The Grunsky Norm and Some Coefficient Estimates for Bounded Functions

We estimate the growth order in n of the Taylor coefficientsof bounded univalent functions in the unit disk. Our estimatedepends on the norm of the Grunsky operator. In particular,we improve the known results for coefficients of quasiconformallyextendible univalent functions. 1991 Mathematics Subject Classification30C50, 30C75, 30C55.     

The Ok Corral and the Power of the Law (A Curious Poisson-Kernel Formula for a Parabolic Equation)

Two lines of gunmen face each other, there being initially mon one side, n on the other. Each person involved is a hopelessshot, but keeps firing at the enemy until either he himselfis killed or there is no one left on the other side. Let µ(m,n) be the expected number of survivors. Clearly, we have boundaryconditions: We also have theequation This is because theprobability that the first successful shot is made by the sidewith m gunmen is m/(m + n). On using the recurrence relation(1.2) together with the boundary condition (1.1), the computerproduces Table 1 below, in which 1991 Mathematics Subject Classification 60F05.     

Zeros of Composite Meromorphic Functions and Ahlfors's Theory of Covering Surfaces

This paper improves a theorem of Bergweiler on the zeros offunctions f g — R with f meromorphic, g entire and Rrational. The proof yields a simple inequality for the numberof zeros and T(r, g). 2000 Mathematics Subject Classification30D35.     

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

It is proved here that if is an elliptic function and q is the maximal multiplicity ofall poles of f, then the Hausdorff dimension of the Julia setof f is greater than 2 q/(q + 1), and the Hausdorff dimensionof the set of points that escape to infinity is less than orequal to 2q/(q + 1). In particular, the area of this latterset is equal to 0. 2000 Mathematics Subject Classification 37F35(primary); 37F10, 30D30 (secondary).     

On the Bargmann Transform and the Wigner Transform

We observe that the Bargmann transform arises from the polardecomposition of a simple restriction operator. By virtue ofthe Bargmann transform, the Wigner transform is imbedded inthe geometric symmetry group of phase space. From this imbedding,we easily obtain that the spectrum of the Wigner transform consistsof {e^ik/12:k=0, 1, ..., 23}. 1991 Mathematics Subject Classification81R30, 47B38, 81S30.     

Dimensions of Julia Sets of Hyperbolic Meromorphic Functions

It is known that, if f is a hyperbolic rational function, thenthe Hausdorff, packing and box dimensions of the Julia set,J(f), are equal. In this paper it is shown that, for a hyperbolictranscendental meromorphic function f, the packing and upperbox dimensions of J(f) are equal, but can be strictly greaterthan the Hausdorff dimension of J(f). 2000 Mathematics SubjectClassification 30D05, 37F10, 37F15, 37F35, 37F50.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

Asymptotically Sharp Bounds in the Hardy-Littlewood Inequalities on Mean Values of Analytic Functions

Let f be analytic in the unit disc, and let it belong to theHardy space H^p, equipped with the usual norm ||f||_p. It is knownfrom the work of Hardy and Littlewood that for q > _p, theconstants [formula] with the usual extension to the case where q = , have C(p,q)< . The authors prove that [formula] and [formula] 2000 Mathematics Subject Classification 30D55, 30A10.     

Estimates for the Norm of the nth Indefinite Integral

Let T be the Volterra operator on L²[0, 1] where f L²[0, 1], 0 x 1. It is well known that||n!Tⁿ|| = O(1/n!). In a recent paper [1], D. Kershaw has provedthat a result which wasfirst conjectured by Lao and Whitley in [2]. It is easy to provethat For completeness, wegive the proof using the familiar Schmidt norm estimate forthe norm of an integral operator (see Section 2 below). Kershawproves that by analysingthe special positivity preserving properties of T*T. He usesone of the many abstract theorems on eigenvalues and eigenfunctionsof compact operators which preserve a cone. In this paper weshall reprove (1), giving a short and direct proof of (2). 1991Mathematics Subject Classification 47G10, 45-04.     

Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

We study finite centralizing extensions AH of noetherian Hopfalgebras. Our main results provide necessary and sufficientconditions for the fibres of the surjection spec Hspec A tocoincide with the X-orbits in spec H, where X denotes the finitegroup of characters of H that restrict to the counit of A. Inparticular, all of the fibres are X-orbits if and only if thefibre over the augmentation ideal of A is an X-orbit. An applicationto the representation theory of quantum function algebras, atroots of unity, is presented. 1991 Mathematics Subject Classification16D30, 16S20, 16P40, 16W30, 81R50.     

Steepest Descent on Real Flag Manifolds

Real flag manifolds are the isotropy orbits of noncompact symmetricspaces G/K. Any such manifold M is acted on transitively bythe (noncompact) Lie group G, and it is embedded in euclideanspace as a taut submanifold. The aim of this paper is to showthat the gradient flow of any height function is a one-parametersubgroup of G, where the gradient is defined with respect toa suitable homogeneous metric s on M; this generalizes the Kählermetric on adjoint orbits (the so-called complex flag manifolds).2000 Mathematics Subject Classification 53C30, 53C35.     

Ordinary Meeting

Held on Friday 23 May and Saturday 24 May 1997 at the Universityof Liverpool, Professor J. M. Ball, FRS, President, in the Chair.There were present about 30 members and visitors for all orpart of the meeting.     

A Proof of a Generalized Nakayama Conjecture

In a recent paper, Külshammer, Olsson, and Robinson proveda deep generalization of the Nakayama conjecture for symmetricgroups. We provide a similar but a shorter and relatively elementaryproof of their result. Our method enables us to obtain a moregeneral H-analogue of the Nakayama conjecture where H is a setof positive integers. 2000 Mathematics Subject Classification20C30 (primary), 20C15, 20C20 (secondary).     

Infinite Simple (2, 3, n)-Groups and Congruence Hulls in the Modular Group

We prove that if n > 66 and (n, 30) = 1, then there existuncountably many infinite simple (2, 3, n)- groups, that is,groups generated by a pair of elements x, y, say, where theorders of x, y and xy are 2, 3 and n, respectively. This extendsprevious results of Schupp and the authors. These results are used to prove the existence of subgroups ofthe modular group with special arithmetic properties. 1991 MathematicsSubject Classification 20F06.     

On Sets Where Iterates of a Meromorphic Function Zip Towards Infinity

For a transcendental meromorphic function f, various propertiesof the set [formula] were obtained in [8] and [9]. Here we establish analogous propertiesfor the smaller sets [formula] introduced in [5], and [formula] We deduce a symmetry result for Julia sets J(f), and also indicatesome techniques for showing that certain invariant curves liein I'(f), Z(f) and J(f). 2000 Mathematics Subject Classification30D05, 37F10, 37F50.