Journal of the London Mathematical Society - recent issues

Two and a Half Remarks on the Marica-Schonheim Inequality

Aharoni, R., Holzman, R. — 1993-12-01

The Marica-Schönheim inequality states that the number of distinct differences of the form A\B, with A, B taken from a given finite family A of sets is at least |A|. We prove that equality occurs essentially if and only if A is the product of an ideal and a filter. We also prove an infinite version of the theorem, conjectured (in weaker form) by Daykin and Lovasz. Finally, we note that a generalization (due to Ahlswede and Daykin) of the inequality which considers two families A and B holds under a weaker assumption on the relation between A and B.

Submodules of the Deficiency Module

Migliore, J. C. — 1993-12-01

Proof of the Evans-Root Conjectures for Selberg Character Sums

Van Wamelen, P. B. — 1993-12-01

From a Large Sieve to the Orthogonality of Operators

Elliott, P. D. T. A. — 1993-12-01

On the Number of Characters in a Block and the k(GV) PROBLEM FOR SELF-DUAL V

Gow, R. — 1993-12-01

Sobolev Inequalities and Harmonic Functions of Polynomial Growth

Alexopoulos, G., Lohoue, N. — 1993-12-01

We prove a Sobolev inequality for functions not necessarily with compact support, on a connected Lie group G of polynomial volume growth. To prove this inequality we have to characterise the harmonic functions of polynomial growth on G.

Integral Operators on the Cone of Monotone Functions

Stepanov, V. D. — 1993-12-01

Necessary and sufficient conditions for the boundedness of linear integral operators from $${L}_{U}^{P}$$(R⁺) to $${L}_{Q}^{W}$$(R⁺) restricted to the cones of monotone functions are given. In addition a general approach to a number of classical operators is explicitly described. In particular, we determine when the Hardy-Littlewood maximal operator is bounded in the classical Lorentz space _p(v) consisting of those measurable functions on Rⁿ such that $${\left({\int }_{0}^{\infty }f**{\left(t\right)}^{p}\upsilon \left(t\right)dt\right)}^{1/p}\hbox{ \hspace{0.17em} } < \infty .$$

The Boundary Behaviour of Bloch Functions

Rohde, S. — 1993-12-01

Proof of a Conjecture of Hayman Concerning f and f''

Langley, J. K. — 1993-12-01

We prove the following, which confirms a conjecture of W. K. Hayman from 1959. If f is meromorphic in the plane such that f and f'' have only finitely many zeros, then f(Z) = R(z) exp (P(Z)), where R is rational and P is a polynomial. The theorem is related to earlier results of Frank, Mues and others.

Uniqueness and Extension Theorems for Subharmonic Functions

Gardiner, S. J. — 1993-12-01

Automatic Well-Posedness with the Abstract Cauchy Problem on a Frechet Space

Delaubenfels, R. — 1993-12-01

For an arbitrary closed linear operator, A, on a Fréchet space, we introduce what we shall call its solution space. This is a Fréchet space that contains all initial data for which the corresponding abstract Cauchy problem has a unique global mild solution. We show that A, restricted to this space, generates a locally equicontinuous strongly continuous semigroup.

Corollaries include an almost immediate proof of the fundamental relationship between generating a strongly continuous semigroup and having a unique mild solution, for all initial data. More generally, we show how the solution space may be used to present a simplified and unified approach to different types of semigroups and their relationships to each other and the abstract Cauchy problem.

Quantum Flows with Unbounded Structure Maps and Finite Degrees of Freedom

Fagnola, F., Sinha, K. B. — 1993-12-01

We prove a general existence theorem for quantum flows with finite degrees of freedom and unbounded structure maps satisfying an analyticity assumption.

The Rate of Escape for Pairs of Windings on the Riemann Sphere

Gruet, J-C., Mountford, T. S. — 1993-12-01

At a typical time t, the distribution of the continuous argument of Brownian motion on the sphere about a typical point is Cauchy like. One might expect that a pair of windings would have the same rate of escape as two independent Cauchy processes (as computed in Taylor [8]), we show however that this is not the case. We also give an exact integral test for the rate of escape.

Transplantation Et Isospectralite II

Berard, P. — 1993-12-01

The Existence of Large {omega}1-Homogeneous But Not {omega}-Homogeneous Permutation Groups is Consistent with Zfc+Gch

Shelah, S., Soukup, L. — 1993-10-01

Denote by Perm () the group of all permutations of a cardinal . A subgroup G of Perm () is called K-homogeneous if and only if for all X, Y []^K there is a g G with g''X = Y. We show that if either (i) ⁺ holds and we add ₁ Cohen reals to the ground model, or (ii) we add 2^₁ Cohen reals to the ground model, then in the generic extension for each ≥ ₂ there is an ₁-homogeneous subgroup of Perm () which is not -homogeneous.

The Small Index Property for {omega}-Stable ({omega}-Categorical Structures and for the Random Graph

Hodges, W., Hodkinson, I., Lascar, D., Shelah, S. — 1993-10-01

We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any -stable -categorical structure, and (ii) the random graph have the small index property. We also show that the automorphism group of such a structure is not the union of a countable chain of proper subgroups.

Harish-Chandra Series of Brauer Characters in a Finite Group with a Split BN-Pair

Hiss, G. — 1993-10-01

On Finitely Presented Soluble Groups with Small Abelian-by-Finite Images

Wilson, J. S. — 1993-10-01

Level Sets and the Uniqueness of Measures

Alspach, D. E. — 1993-10-01

A result of Nymann is extended to show that a positive -finite measure with range an interval is determined by its level sets. An example is given of two finite positive measures with range the same finite union of intervals but with the property that one is determined by its level sets and the other is not.

Shuffled Verma Modules and Principal Series Modules over Complex Semisimple Lie Algebras

Irving, R. S. — 1993-10-01

A Proof of Hall's Conjecture on Starlike Mappings

Balasubramanian, R., Karunakaran, V., Ponnusamy, S. — 1993-10-01

In this paper we show that $${\left|f\left(r{e}^{1\theta }\right)\right|}^{-1}{\displaystyle \underset{0}{\overset{r}{\int }}\left|{f}^{\prime }\left(u{e}^{1\theta }\right)\right|}du\le {r}^{-1}\mathrm{arcsin}r,0 < r < 1$$ where f lies in the class of starlike functions of order 1/2, that is, which are regular and univalent for |z| < 1 and such that $$\mathrm{Re}\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)} > \frac{1}{2}for\left|z < 1\right|$$

Sharp Distortion Theorems Associated with the Schwarzian Derivative

Chuaqui, M., Osgood, B. — 1993-10-01

A Discontinuous Homomorphism from C(X) without CH

Woodin, W. H. — 1993-10-01

A Characterization of Pick Bodies

Cole, B., Lewis, K., Wermer, J. — 1993-10-01

Let z = (z₁..., z_n) be an n-tuple of distinct points in the open unit disk. We define the Pick body D;(z) as the totality of points w = (w₁,...,w_n) in Cⁿ such that there exists fH with ||f|| ≤ 1 and f(z₁) = w₁, for 1 ≤j≤ n. We discuss the properties of Pick bodies and characterize them among compact subsets of Cⁿ. We also study related questions concerning certain algebras of operators on Hilbert space.

Complexes that Arise in Cohomological Dimension Theory: A Unified Approach

Dydak, J., Walsh, J. J. — 1993-10-01

Let : $$\tilde{P}$$ -> P be a combinatorial map (that is, ^-1(L) is a subcomplex of $$\tilde{P}$$ whenever L is a subcomplex of P) between CW-complexes. A map f: X -> P is said to approximately lift with respect to provided that there is a map $$\tilde{f}$$: X -> $$\tilde{P}$$ such that, for each x X, there is a cell in P containing both o$$\tilde{f}$$(x) andf(x).

A characteristic property of a compact metric space X having covering dimension dim X ≤ n is that each map from X to a CW-complex P has an approximate lift with respect to the inclusion P(ⁿ) P. An analogous characterization of compacta X having integral cohomological dimension dim_z X ≤ n emerged from work of R. D. Edwards [12] and was introduced in [19]. Complexes and maps : EWz(P,n) -> P are associated to each simplicial complex P so that a compactum X has dim_z X ≥ n if and only if every map f : X -> P to a simplicial complex P can be approximately lifted to EW_z(P,n). These complexes provide a l‘combinatorial’ approach to cohomological dimension theory that has supported many of the recent developments in the area.

Historically, cohomological dimension theory with respect to groups other than Z has provided computational machinery for determining covering dimension. Hence, it is not surprising that it has been useful to consider comparable complexes EW_G(L, n) for other groups G. The goal of this paper is to present a unified exposition of these complexes. As an application, they are used to provide an alternative construction to that of Dranishnikov [4, 6] of compact metric spaces realizing the Bockstein functions.

The Geometric Equivariant Segal Conjecture for Toral Groups

Greenlees, J. P. C. — 1993-10-01

Fibrewise Hopf Structures on Sphere-Bundles

Cook, A. L., Crabb, M. C. — 1993-10-01

Mixed Multiplicities, Joint Reductions and Quasi-Unmixed Local Rings

Swanson, I. — 1993-08-01

Embedding GCD Domains in Bezout Domains

Shamash, J., Smith, S. T. — 1993-08-01

We show that an arbitrary GCD domain can be embedded in a Bézout domain without changing the set of units. This result continues to hold if we require the domains to be discretely ordered. We can thus construct Bézout domains which are models of open induction and in which the order type of the set of positive infinite primes is arbitrary of cardinality ≤ N₁.

Weakly Finite Matrix Localizations

Malcolmson, P. — 1993-08-01

A ring is called weakly finite if all of its right invertible square matrices are invertible.Given a set of square matrices over a ring, there is the notion of the (universal) matrix localization with respect to that set, which adjoins a universal inverse for each matrix in the set. Under mild conditions on the set of matrices to be inverted (that it is a multiplicative factor-closed set of full matrices), we show that such a matrix localization of any ring is weakly finite. Related results are also given.

Embedding Arbitrary Graphs of Maximum Degree Two

Aigner, M., Brandt, S. — 1993-08-01

Let (H) be the minimum degree of the graph H. We prove that a graph H of order n with (H) ≥ (2n–1)/3 contains any graph G of order at most n and maximum degree (G) ≤ 2 as a subgraph, and this bound is best possible. Furthermore, this result settles the case (G) = 2 of the well-known conjecture of Bollobás, Catlin and Eldridge on packing two graphs with given maximum degree.

The Undecidability of the Unification Problem for Nilpotent Groups of Class >= 5

Burke, E. K. — 1993-08-01

We study the unification problem for the theories of nilpotent groups of class at least 5. This is equivalent to the problem of constructing an algorithm which will solve any equation in a free nilpotent group of class at least 5. Romankov in 1977 showed that the ‘endomorphic reducibility’ problem is undecidable for free nilpotent groups of class at least 9 and it follows that the Unification Problem for the theories of nilpotent groups of class at least 9 is undecidable. We shall improve this to 5 by reducing the problem of algorithmically solving an arbitrary diophantine equation of degree 4 to that of solving equations in a free nilpotent group of class 5 and then appealing to Matiyasevitch's Theorem.

Unipotent Finitary Linear Groups

Leinen, F., Puglisi, O. — 1993-08-01

Permutation Representations of the Symmetry Groups of Regular Hyperbolic Tessellations

Mushtaq, Q., Servatius, H. — 1993-08-01

Higman has questioned which discrete hyperbolic groups [p, q] have representations onto almost all symmetric and alternating groups. We call this property H and show that, except perhaps for finitely many values of p and q, [p, q] has property H.

A Class of Eventually Regular Semigroups Determined by Pseudo-Random Sets

Higgins, P. M. — 1993-08-01

A class of covers of the free monogenic inverse monoid is introduced in one-to-one correspondence with the irrational numbers greater than one. Each member of the class is an eventually regular semigroup with a generator a, the regular powers of which form a pseudo-random set. This then affords an example of a direct product of two eventually regular semigroups that is not eventually regular, thus answering a question of P. M. Edwards.

On Weighted Norm Inequalities with Three Weights

Oinarov, R. — 1993-08-01

A characterization is given of the weights r, p, v when the weighted norm inequality $${\left({{\displaystyle {\int }_{a}^{b}\left|rf\right|}}^{q}\right)}^{1/q}\le C{\left({{\displaystyle {\int }_{a}^{b}\left|pf\text{'}\right|}}^{q}+{{\displaystyle {\int }_{a}^{b}\left|vf\right|}}^{p}\right)}^{1/p}$$ holds. Both cases 1 ≤ p ≤ q ≤ and 1 ≤ q < p ≤ are considered and the related problems of compactness are also studied.

Suzuki Groups and Surfaces

Jones, G. A., Silver, S. A. — 1993-08-01

We show that the least genus of any compact Riemann surface S, admitting a simple Suzuki group G = Sz(q) as a group of automorphisms, is equal to 1 + |G|/40. We compute the number of such surfaces S as the number of normal subgroups of the triangle group (2,4,5) with quotient-group G, and investigate the associated regular maps of type {4, 5}.

A Characterization of Hardy Spaces on the Unit Ball of Cn

Stoll, M. — 1993-08-01

A function f holomorphic in the unit ball B of Cⁿ lies in the Hardy space H^p, 0 < p < , if and only if$${\displaystyle {\int }_{B}{\left(1-{\left|z\right|}^{2}\right)}^{n}}{\left|f\left(z\right)\right|}^{p-2}{\left|\tilde{\nabla }f\left(z\right)\right|}^{2}d\lambda \left(z\right) < \infty $$ where $$\tilde{\nabla }$$ and denote the invariant gradient and invariant measure on B, respectively. Furthermore, if fH^p, then$$lim{(1-{r}^{2})}^{n}{{\int }_{{B}_{r}}\left|f\left(z\right)\right|}^{v-2}{\left|\tilde{\nabla }f\left(z\right)\right|}^{2}d\lambda \left(z\right)=0$$. An analogous characterization is also given for invariant harmonic functions for the case 1 < p ≤ .

On the Best Constant for a Weighted Sobolev-Hardy Inequality

Chou, K. S., Chu, C. W. — 1993-08-01

On the Spectrum of L{infty}(G)

Lau, A. T., Medghalchi, A. R., Pym, J. S. — 1993-08-01

We investigate the algebraic structure of the spectrum of L(G) for a locally compact group G. In contrast to the compact and discrete cases, when G has neither of these properties, is never a semigroup. For compact G we determine exactly when the product of two elements of . is in , but we present an example which suggests that for general groups the underlying set theory may have an effect. Our principal tool, which has independent interest, is a topological structure theorem for the LB-compactification of an arbitrary locally compact group.

The Hardy Operator, L{infty} and BMO

Qinsheng, L., Pick, L. — 1993-08-01

We give necessary and sufficient conditions in order that the Hardy operator is bounded from an arbitrary weighted Banach function space (X, v) into L or weighted BMO. Moreover, we give necessary and sufficient conditions in order that the Hardy operator is compact from (X, v) into L or non-weighted BMO. Applications to the case of weighted BMO are discussed as well.

Classification of Symmetric Caustics II: Caustic Equivalence

Janeczko, S., Roberts, M. — 1993-08-01

Symplectic equivalence of Lagrangian projections is too strong to yield a useful classification of projections which commute with a symmetry group action. A weaker equivalence relation, caustic equivalence, is introduced and used to classify the caustics of Lagrangian submanifolds that are invariant under symplectic involutions.