Mathematical Proceedings of the Cambridge Philosophical Society - Current Issue

Volume 145 Issue 03

Sat, 01 Nov 2008 00:00:00 GMT

Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03

Mathematical Proceedings is one of the few high-quality journals publishing original research papers that cover the whole range of pure and applied mathematics, theoretical physics and statistics. All branches of pure mathematics are covered, in particular logic and foundations, number theory, algebra, geometry, algebraic and geometric topology, classical and functional analysis, dynamical systems, probability and statistics. On the applied side, mechanics, mathematical physics, relativity and cosmology are included.

The effect of twisting on the 2-Selmer group

Research Articles
PETER SWINNERTON–DYER,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 513-526

Abstract
Let b be its quadratic twist by b. Subject to a mild additional condition on b as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of has odd or even dimension.

Ubiquity and large intersections properties under digit frequencies constraints

Research Articles
JULIEN BARRAL, STÉPHANE SEURET,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 527-548

Abstract
We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n 1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.

An algebraic generalization of Kripke structures

Research Articles
SÉRGIO MARCELINO, PEDRO RESENDE,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 549-577

Abstract
The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

Bases for commutative semigroups and groups

Research Articles
NEIL HINDMAN, DONA STRAUSS,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 579-586

Abstract
A base for a commutative semigroup (S, +) is an indexed set xttA in S such that each element x S is uniquely representable as n xt. We investigate those commutative semigroups or groups which have a base. We obtain the surprising result that has a base. More generally, we show that an abelian group has a base if and only if it has no elements of odd finite order.

A criterion for a monomial in P (3) to be hit

Research Articles
A. S. JANFADA,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 587-599

Abstract
Let P(n) = [x1, . . ., xn] = d i Sqi(hi), called a hit equation, where the pre-images hi M have grading strictly less than d and the Sqi, called the Steenrod squares, generate . One of the important parts of the hit problem is to check whether a given polynomial in M is hit or not. In this article we study this problem in the 3-variable case.

A Cohen–Macaulay algebra has only finitely many semidualizing modules

Research Articles
LARS WINTHER CHRISTENSEN, SEAN SATHER-WAGSTAFF,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 601-603

Abstract
We prove the result stated in the title, which answers the equicharacteristic case of a question of Vasconcelos.

Analytic continuation of multiple Hurwitz zeta functions

Research Articles
JAMES P. KELLIHER, RIAD MASRI,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 605-617

Abstract

The threefold containing the Bordiga surface of degree ten as a hyperplane section

Research Articles
HIDETOSHI MAEDA,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 619-622

Abstract
Let be a very ample vector bundle of rank 2 on with c1() = 4 and c2() = 6. Then it is proved that is the cokernel of a bundle monomorphism , where is the tangent bundle of . This gives a new example of a threefold containing a Bordiga surface as a hyperplane section.

Hilbert polynomials and powers of ideals

Research Articles
JÜRGEN HERZOG, TONY J. PUTHENPURAKAL, JUGAL K. VERMA,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 623-642

Abstract
The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k 0 with Jk Ik for all k with the property that JkJ and IkI for all k and 1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

Research Articles
DANIELA KRAUS, OLIVER ROTH,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 643-667

Abstract
A classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation u = (z)e2u where lder continuous function. As an application a higher Ahlfors Schwarz lemma for complete conformal Riemannian metrics is obtained.

Exceptional sets for self-affine fractals

Research Articles
KENNETH FALCONER, JUN MIAO,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 669-684

Abstract
Under certain conditions the formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

Invariant measures for meromorphic Misiurewicz maps

Research Articles
JANINA KOTUS, GRZEGORZ ŚWIATEK,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 685-697

Abstract
We study the existence of finite absolutely continuous invariant measures for meromorphic Misiurewicz maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that such a measure must exist. We show that it is not so for meromorphic maps unless an additional condition on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.

Complex asymptotics of Poincaré functions and properties of Julia sets

Research Articles
GREGORY DERFEL, PETER J. GRABNER, FRITZ VOGL,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 699-718

Abstract
The asymptotic behaviour of the solutions of Poincar z) = p(f(z)) ( 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(z z)), if f(z) for z = log deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

Hausdorff dimension of hairs and ends for entire maps of finite order

Research Articles
KRZYSZTOF BARAŃSKI,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 719-737

Abstract
We study transcendental entire maps f of finite order, such that all the singularities of f 1 are contained in a compact subset of the immediate basin B of an attracting fixed point of f. Then the Julia set of f consists of disjoint curves tending to infinity (hairs), attached to the unique point accessible from B (endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class (i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

A devil's staircase from rotations and irrationality measures for Liouville numbers

Research Articles
DOYONG KWON,
Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 739-756

Abstract
From Sturmian and Christoffel words we derive a strictly increasing function ) distinguishes some irrationality measures of real numbers.

Proceedings of the meetings held during the session 2007–2008

Meeting Report

Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145 Issue 03 , pp 757-763

Abstract