bluuebunny/crossref_metadata_2025 huggingface dataset:
Reduction of modules
Authors: Rees D.
Published: November 2008

The notion of a reduction of an ideal I of a noetherian ring R and the related notions of the integral closure I* of I in R and the integral dependence of an element x of R on I have proved useful in many situations in commutative algebra. The purpose of this paper is to extend these notions to modules. To motivate the definitions which follow, it will be useful to consider these notions for ideals before extending them to modules.
The total Steenrod operation is induced by an A∞ ring homomorphism
Authors: Kozlowski A.
Published: November 2008

Let X be a (based) space of the homotopy type of a CW-complex. Let H(X) denote the classical (ungraded) cohomology ring Πi≥0Hi (X;Z/2). In [1] Atiyah and Hirzebruch described the group of natural ring automorphisms of H(X) (‘cohomology automorphisms’) with group operation given by composition. They showed that is isomorphic to the group of formal power series of the form with group operation given by ‘substitution’ of one power series into another. In particular the most famous ‘cohomology automorphism’, the total Steenrod Square, corresponds to x + x2.
The Artin–Rees equations
Authors: Caruth A.
Published: November 2008

Let R denote a commutative Noetherian ring with an identity element and N a finitely generated R -module. When K is a submodule of N and A an ideal of R the Artin–Rees lemma states that there is an integer q ≥ 0 such that AnN ∩ K = An−q(AqN ∩ K) for all n ≥ q (Rees[4]; Northcott [3], theorem 20, p. 210; Atiyah and Macdonald [1], proposition 10·9, p. 107; Nagata [2], theorem (3·7), p. 9). The above equation belongs to the family of module equations involving A and K which is considered below. We characterize, in terms of A and K, the set of submodules X of N for which there is an integer q = q(X) ≥ 0 satisfying the equation

Equation (1), which we call the Artin–Rees equation related to A and K, gets its maximal force when X is largest and we determine the best possible solution in this sense. Notice that for any submodule X satisfying (1), X ⊆ K:NAn for all n ≥ q(X). Since N is a Noetherian R-module ([3], proposition 1 (corollary), p. 177), there is an integer t ≥ 1 such that K:NAt = K:NAt+n for all n ≥ 0. We define M = K:NAt and prove, in Theorem 1, that X = Q satisfies equation (1), for a suitable integer q(Q) ≥ 0, if and only if K ⊆ Q:NAυ ⊆ M for some integer υ ≥ 0. In topological terms, the A-adic topology of K coincides with the topology induced by the A-adic topology of N on the subspace Q if the inequality K ⊆ Q:NAυ ⊆ M is satisfied. It follows that the solution set of equation (1) includes every submodule of N of the form An−rK when n ≥ r = q(K) as well as every submodule lying between K and M. Hence, X = M is the strongest solution, in the sense that M is the largest such submodule contained in An−s (AsN ∩ K): NAn for all n ≥ s = q(M). Recall that M is strictly larger than K if and only if A is contained in at least one prime ideal of R belonging to K ([3], theorem 14 (corollary 1), p. 193). Thus, equation (1) has a unique solution (necessarily X = K) if and only if A is not contained in any prime ideal of R belonging to any solution.
Double multipliers andA*-algebras of the first kind
Authors: Kassem M. S. , Rowlands K.
Published: November 2008

LetAbe anA*-algebra and letdenote its auxiliary norm closure. The multiplier algebras of dualA*-algebras of the first kind have been studed by Tomiuk [12], [13] and Wong[15]. In this paper we study the double multiplier algebra ofA*-algebras of the first kind. In particular, we prove that, ifAis anA*-algebra of the first kind, then the double multiplier algebraM(A) ofAis *-isomorphic and (auxiliary norm) isometric to a subalgebra ofM(), extending in the process some results established by Tomiuk[12]. We also consider the embedding of the double multiplier algebra ofAin**, when the latter is regarded as an algebra with the Arens product, and, in particular, we show that, for an annihilator A*-algebra,M(A) is *-isomorphic and (auxiliary norm) isometric to**.
Records, permutations and greatest convex minorants
Authors: Goldie Charles M.
Published: November 2008

Theorems on random permutations are translated into distribution-free results about record times and greatest convex minorants, by defining them together on appropriate probability spaces. The Bernoulli random variables that appear in the standard representation of the number of sides of the greatest convex minorant of a random walk are identified.
Geometry of singular sets
Authors: Bruce J. W.
Published: November 2008

Singularity theory is concerned with the study of smooth mappings between smooth manifolds. Given two such manifolds X and Y and a pair of smooth mappings f1,f2: X→Y we say that f1 and f2 are -equivalent if there are diffeomorphisms α: X→X and β: Y→Y with βof1oα = f2. Clearly -equivalence is an equivalence relation, and one aims to classify smooth mappings f: X→Y up to this equivalence.
An application of descent to a classification theorem for toposes
Authors: Bunge Marta
Published: November 2008

The aim of this paper is to answer the following question. For a spatial groupoidG, i.e. for a groupoid in the category Sp of spaces (in the sense of [20]) in a topos, and continuous maps, the toposBG, of étaleG-spaces, is called ‘the classifying topos of G’ by Moerdijk[22]. This terminology is suggested by the case ofGa discrete group (in Sets), as thenBG, the topos ofG-sets, classifies principalG-bundles. This means that, for each topological spaceX, there is a bijection between isomorphism classes of principalG-bundles overXand isomorphism classes of geometric morphisms from Sh(X) toBG. The question is: what doesBGclassify, in terms ofG, in the general case of a spatial groupoidGin a topos?
Constructive complete distributivity. I
Authors: Fawcett Barry , Wood R. J.
Published: November 2008

The relationships, in many cases equivalences, between lattice distributivity, adjunction and continuity have been studied by many authors, for example [1, 3–8, 12, 13, 15, 17–20, 22, 23]. Very roughly, we refer to the following circle of ideas. LetLbe an ordered set, andLa class of subsets ofL, and suppose thatLhas a supremum for each element inL. We might say thatLhas-sups. The ‘distributivity’ we refer to is that of infs over-sups. The ‘adjunction’ is that given by a left adjoint to the map V:L→L. Now the latter has a left adjoint if and only if it preserves infs, and this means roughly that the-sup of an intersection is an inf of-sups. When one does succeed in identifying the-sup of an intersection as a-sup of infs, one has an instance of distributivity.
Quasi-standard C*-algebras
Authors: Archbold R. J. , Somerset D. W. B.
Published: November 2008

Abstact

A necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.
Fine limits of superharmonic functions
Authors: Gardiner Stephen J.
Published: November 2008

LetOdenote the origin of ℝn(wheren≥ 2), letB(X, r) denote the open ball in ℝnof centreXand radiusr, and define φn: (0, + ∞) → ℝ by φn(t) =t2-nifn≥ 3 andφ2(t) = log (1/t). A sequence (Xk) inB(O, 1) is said toconverge regularlytoOifXk→Oand there is a constantk> 1 such that φn(|Xk+1|) k′|Xk| for eachk(cf. [18], p. 149).
A stochastic calculus approach to the shape distribution induced by a complex normal model
Authors: Le Huiling
Published: November 2008

An approach via stochastic calculus was given by Kendall in [7] to the Mardia–Dryden shape distribution of three labelled independent -random points (j = 1, 2, 3). We give here the analogous approach for the general case discussed in [3] in which k labelled random points have a complex normal distribution.
On the group determinant
Authors: Johnson K. W.
Published: November 2008

The original motivation for the introduction by Frobenius of group characters for non-abelian groups was the problem of the factorization of the group determinant corresponding to a finite groupG. The original papers are [5] and [6] and a good historical survey of the work is given in [7] and [8]. IfGis of ordern, thegroup matrix XGis defined to be then×nmatrix {xg, h} wherexg, h=xgh∈G. Here thexg,g∈G, represent variables. Thegroup determinantΘGis defined to be det(XG), and is thus a polynomial of degreenin thexg. This determinant is the same, up to sign, as that of the matrix obtained from the unbordered multiplication table ofGby replacing each elementgbyxg. If there is no ambiguity ΘGwill be written as Θ.
Intermediate value theorems for holomorphic maps in complex Banach spaces
Authors: Włodarczyk Kazimierz
Published: November 2008

One of the most celebrated theorems of mathematical analysis is the intermediate value theorem of Bolzano which, in a simple case, states that a real-valued continuous map f of a closed interval [a, b], such that f(a) and f(b) have different signs, has a zero in (a, b). Recently, Shih in [5] observed that without loss of generality we may suppose that a 7 < 0 < b and f(a) < 0 < f(b) and, consequently, the condition f(a).f(b) < 0 becomes x.f(x) > 0 for x∈∂Ω where ∂Ω denotes the boundary of the interval Ω = (a, b); then the conclusion is that f has at least one zero in ω. It is a remarkable fact that Shih extends this form of Boizano's theorem to analytic maps in ℂ [5] and, subsequently, in ℂn [6]. He proved that if Ω is a bounded domain in ℂn containing the origin, is continuous in and analytic in Ω and Re for z∈∂Ω, then f has exactly one zero in Ω. In this paper we extend Shih's result to Banach spaces.
Smooth majorants for functions of arbitrarily rapid growth
Authors: Earl J. P. , Hayman W. K.
Published: November 2008

Valiron[3] showed that if f(z) is an entire function for which

where A(r) = log(sup{|f(z)|:|z| = r}), then there is a function x(r) = rρ(r) satisfying both
A characterization of pie limits
Authors: Power John , Robinson Edmund
Published: November 2008

It is well-known that limits in 2-categories are more complex than limits in ordinary categories. Most readers will at least be familiar with terms such as ‘lax limit’ and ‘pseudo-limit’. In the more modern treatments, these become special cases of a more general class of ‘weighted’ or ‘indexed’ limits (see Kelly [7] and Section 1 of this paper).
Weakly compact subsets of symmetric operator spaces
Authors: Dodds Peter G. , Dodds Theresa K. , Pagter Ben De
Published: November 2008

Under natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.
Conditional expectation and stochastic integrals in non-commutative Lp spaces
Authors: Goldstein Stanisław
Published: November 2008

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).
On exchanges between convective modes in a slightly tilted porous cavity
Authors: Impey M. D. , Riley D. S.
Published: November 2008

We use Liapunov–Schmidt reduction and singularity-theory methods to investigate the bifurcation structure of steady free convection in a finite two-dimensional saturated porous cavity heated from below. In particular, we develop a qualitative model describing the modal exchanges that occur as the aspect ratio of a slightly tilted cavity varies. ℤ2-symmetry breaking bifurcations are involved in these exchanges and the mechanism is significantly different from those in previously studied physical systems. The work is akin to, but distinct from, that of Schaeffer's on the Taylor–Couette problem. We also derive and clearly state conditions for the application of this model to modal exchange in other systems.
Generators and relations for metabelian Lie algebras
Authors: Wisliceny Jürgen , Zerck Rainer
Published: November 2008

The aim of this paper is to prove an inequality of Golod-Shafarevich type for metabelian Lie algebras and to show that this inequality is best possible up to a constant factor. Investigations of this kind were started in [4] in connection with the solution of the class field tower problem. It was shown that if there is a finite p-group which may be presented as a pro-p-group with d ≥ 2 generators and r relations then the inequality r > ¼(d − 1)2 holds; and Vinberg [8] improved this result by showing

The inequality (1) also holds for presentations of nilpotent Lie algebras (see [6]) (with the exception (d, r) = (2, 1)) and nilpotent associative algebras (see [4, 8, 6]). If the relations have of degree at least m ≥ 3 then more relations are needed. More precisely, Koch [5] has shown that the inequality

holds for finite p-groups, and the corresponding results hold for associative algebras and Lie algebras.
Spectral characterization of the radical in Banach and Jordan–Banach algebras
Authors: Aupetit Bernard
Published: November 2008

If a is a n × n matrix such that a + m is invertible for every invertible a + m matrix m, then a = 0, by a classical result of Perlis [8]. Unfortunately the same result is not true in general for semi-simple rings as shown by T. Laffey. In the general situation of Banach algebras, Zemánek[12] has proved that a is in the Jacobson radical of A if and only if ρ(a+x) = ρ(x), for every x in A, where ρ denotes the spectral radius. More sophisticated characterizations of the radical were given in [4] and [3], theorem 5·3·1. The arguments used in all these situations depend heavily on representation theory.
The sheaf of relative differentials of a fibred surface
Authors: Serrano Fernando
Published: November 2008

Let Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.
Links with super-additive tunnel number
Authors: Boileau Michel , Lustig Martin , Moriah Yoav
Published: November 2008

LetL⊂S3be a knot or a link. A tunnel system forLis a collection of disjoint arcst1, …tsproperly embedded inS3−N(L) so thatis a handlebody. Alternatively one can view thet1, …tsascocores of 2-handles which have to be attached to a handlebody in order to obtainS3−N(L). It follows immediately that this handlebody is of genuss+1. Thetunnel numberofL, denoted byt(L), is defined to be the minimal cardinality of all such collections.
Topological methods to compute Chern-Simons invariants
Authors: Auckly David R.
Published: November 2008

In this paper we develop a method which may be used to compute the Chern-Simons invariants of a large class of representations on a large class of manifolds. This class includes all representations on all Seifert fiber spaces, all graph manifolds, and some hyperbolic manifolds. I owe many thanks to Peter Scott, John Harer, Frank Raymond, Ron Fintushel, Paul Kirk and Eric Klassen, without whose help and support this paper could not have been written.
Ordered orbits of the shift, square roots, and the devil's staircase
Authors: Bullett Shaun , Sentenac Pierrette
Published: November 2008

An orbit of the shift σ: t ↦ 2t on the circle = ℝ/ℤ is ordered if and only if it is contained in a semi-circle Cμ = [μ, μ+½]. We investigate the ‘devil's staircase’ associating to each μ ε the rotation number ν of the unique minimal closed σ-invariant set contained in Cμ; we present algorithms for μ in terms of ν, and we prove (after Douady) that if ν is irrational then μ is transcendental. We apply some of this analysis to questions concerning the square root map, and mode-locking for families of circle maps, we generalize our algorithms to orbits of the shift having ‘sequences of rotation numbers’, and we conclude with a characterization of all orders of points around realizable by orbits of σ.
Static solutions of the spherically symmetric Vlasov–Einstein system
Authors: Rein Gerhard
Published: November 2008

We consider the Vlasov—Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular centre and have isotropic or anisotropic pressure, and solutions which have a Schwarzschild-singularity at the centre.
The even master system and generalized Kummer surfaces
Authors: Bertin Jose , Vanhaecke Pol
Published: November 2008

In this paper we study a generalized Kummer surface associated to the Jacobian of those complex algebraic curves of genus two which admit an automorphism of order three. Such a curve can always be written as y2 = x6 + 2kx3 + 1 and k2 ╪ 1 is the modular parameter. The automorphism extends linearly to an automorphism of the Jacobian and we show that this extension has a 94 invariant configuration, i.e. it has 9 fixed points and 9 invariant theta curves, each of these curves contains 4 fixed points and 4 invariant curves pass through each fixed point. The quotient of the Jacobian by this automorphism has 9 singular points of type A2 and the 94 configuration descends to a 94 configuration of points and lines, reminiscent to the well-known 166 configuration on the Kummer surface. Our ‘generalized Kummer surface’ embeds in ℙ4 and is a complete intersection of a quadric and a cubic hypersurface. Equations for these hypersurfaces are computed and take a very symmetric form in well-chosen coordinates. This computation is done by using an integrable system, the ‘even master system’.
Varieties and modules with vanishing cohomology
Authors: Carlson Jon F. , Robinson Geoffrey R.
Published: November 2008

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context ofkG-modules wherekis a field of finite characteristicpandGis a finite group whose order is divisible byp. Aside from some general considerations, the main results of [2] related the existence ofkG-modulesMwithH*(G, M) = 0 to the structure of the centralizers of thep-elements inG. Specifically it was shown that there exists a non-projective moduleMin the principal block ofkGwithH*(G, M) = 0 whenever the centralizer of somep-element ofGis notp-nilpotent. The converse was proved in the special case that the primepis an odd integer (p> 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.
Chaos map for the universal enveloping algebra ofU(N)
Authors: Hudson R. L. , Parthasarathy K. R.
Published: November 2008

It is shown that the family of representations {jt,t∈ ℝ+} of the universal enveloping algebra U of theN-dimensional unitary group which is generated by theN-dimensional number process of quantum stochastic calculus can be expressed in the form

where ψ is a bijective linear map from U onto the space S of symmetric tensors over the Lie algebra, andItis the iterated (chaotic) integral on S. The chaotic product * is defined by the formula

and satisfies

This work generalizes and completes earlier results on the centre of U.
A representation of distributions supported on smooth hypersurfaces of Rn
Authors: Guo Kanghui
Published: November 2008

Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula
Algebraic surfaces with nef and big anti-canonical divisor
Authors: Zhang D.-Q.
Published: November 2008

Let S be a normal projective algebraic surface over C with at worst quotient singularities. S is a quasi-log del Pezzo surface if the anti-canonical divisor — Ks is nef (= numerically effective) and big, i.e. — Ks. C ≥ 0 for all curves C on S and (−Ks)2 > 0. Further, if — Ks is ample we say S is a log del Pezzo surface.
QuantumSO(3)-invariants dominate theSU(2)-invariant of Casson and Walker
Authors: Murakami Hitoshi
Published: November 2008

For a compact Lie groupG, E. Witten proposed topological invariants of a threemanifold (quantumG-invariants) in 1988 by using the Chern-Simons functional and the Feynman path integral [30]. See also [2]. N. Yu. Reshetikhin and V. G. Turaev gave a mathematical proof of existence of such invariants forG=SU(2) [28]. R. Kirby and P. Melvin found that the quantumSU(2)-invariantassociated toq= exp(2π √ − 1/r) withrodd splits into the product of the quantumSO(3)-invariantand[15]. For other approaches to these invariants, see [3, 4, 5, 16, 22, 27].
Compact embeddings of some weighted Sobolev spaces on ℝN
Authors: Chiappinelli Raffaele
Published: November 2008

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces
Geometry of quartic curves
Authors: Wall C. T. C.
Published: November 2008

In recent work [5] which involved enumeration of singularity types of highly singular quintic curves, it was necessary to use rather detailed information on the geometry of quartic curves (for the case when the quintic consists of the quartic and a line). The present paper was written to supply this background. The cases of primary interest for this purpose are the rational quartics, and we concentrate on these.
Moduli spaces of compact complex submanifolds of complex fibered manifolds
Authors: Merkulov Sergey A.
Published: November 2008

In 1962 Kodaira[11] proved that ifX ↪ Y is a compact complex submanifold with normal bundle N such that H1(X, N) = 0, then X belongs to a locally complete family {Xt: t ∈ M} of complex submanifolds Xt of Y with the moduli space M being a (dimcH0(X, N))-dimensional complex manifold, and there exists a canonical isomorphism

between the tangent space of M at a point t ∈ M and the space of all global sections of the normal bundle Nt of the embedding Xt ↪ Y.
A uniform Kadec-Klee property for symmetric operator spaces
Authors: Dodds P. G. , Dodds T. K. , Dowling P. N. , Lennard C. J. , Sukochev F. A.
Published: November 2008

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q-estimate with constant 1 then the corresponding space E(M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces Lq, p and the Lorentz-Schatten classes Cg, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q-estimate.
The operator extension theory, semitransparent surface and short range potential
Authors: Popov I. Yu.
Published: November 2008

Solvable model of a quantum dot as a resonator with semitransparent boundary is constructed in the framework of the operator extensions theory. It is proved that the model operator is a limit of the short range Hamiltonians in the norm resolvent sense.
Congruence of Ankeny–Artin–Chowla type for cyclic fields of prime degree l
Authors: Jakubec Stanislav
Published: November 2008

Ankeny–Artin–Chowla obtained several congruences for the class number hk of a quadratic field K, some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and ε = t + u √p/2 is the fundamental unit of K, then
The square-full numbers in an interval
Authors: Huxley M. N. , Trifonov O.
Published: November 2008

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with
Highly singular quintic curves
Authors: Wall C. T. C.
Published: November 2008

The origin of this paper lies in a study of the unfolding space of the stratumN16of singularity theory, and the question, at which points of the stratum the versal deformation space ceases to be topologically trivial over the stratum. This question turns out to be closely related to the study of how a plane section (= binary quintic) of a quintic curve varies as we deform the curve, either rigidly (underGL3) or equisingularly.
Some decomposition numbers for Hecke algebras of general linear groups
Authors: Richards Matthew J.
Published: November 2008

The theorem which is still known as Nakayama's Conjecture shows how the modular characters of the symmetric group Sn can be divided into blocks of various weights, those with the same weight having similar properties. In fact, all blocks of weight one have essentially the same decomposition numbers and these are easy to describe. In recent work, Scopes [16, 17] has shown that there are essentially only finitely many possibilities for the decomposition numbers for blocks of any given weight, and has given a bound for the number. We develop the combinatorics implicit in her work, and so establish an improved bound.
On 2-dimensional aspherical complexes and a problem of J. H. C. Whitehead
Authors: Luft E.
Published: November 2008

In [W] J. H. C. Whitehead posed the following question: ‘Is every subcomplex K of a 2-dimensional aspherical complex L itself aspherical ?’

This problem is usually referred to as the ‘Whitehead Conjecture’ though it was only stated in the form of a question. For convenience we treat it also as a conjecture.

The Whitehead Conjecture has been proved in special cases: if the subcomplex K has only one 2-cell, and also in the case where π1(K) is either finite, abelian, of free [C] For more partial results see, for example, the introduction of [H1].
The packing dimension of projections and sections of measures
Authors: Falconer Kenneth J. , Mattila Pertti
Published: November 2008

We show that for a probability measure μ on ℝn

for almost all m–dimensional subspaces V, provided dimH μ≤m. Here projv denotes orthogonal projection onto V, and dimH and dimp denote the Hausdorff and packing dimension of a measure. In the case dimH μ > m we show that at μ-almost all points x the slices of μ by almost all (n − m)-planes Vx through x satisfy

We give examples to show that these inequalities are sharp.
Lagrangian submanifolds satisfying a basic equality
Authors: Chen Bang-Yen , Vrancken Luc
Published: November 2008

In [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following inequality:

He then introduced three families of Riemannian n-manifolds and two exceptional n-spaces Fn, Ln and proved the existence of a Lagrangian isometric immersion pa from into ℂPn(4) and the existence of Lagrangian isometric immersions f, l, ca, da from Fn, Ln, , into ℂHn(− 4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesie ones, these are the only Lagrangian submanifolds in ℂPn(4) and in ℂHn(− 4) which satisfy this basic equality. In this article, we obtain the explicit expressions of these Lagrangian immersions. As an application, we obtain new Lagrangian immersions of the topological n-sphere into ℂPn(4) and ℂHn(−4).
Ramanujan's formula for the logarithmic derivative of the gamma function
Authors: Bradley David
Published: November 2008

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.
The mod p cohomology group of extra-special p-group of order p5 and of exponent p2
Authors: Minh Pham Anh
Published: November 2008

Let p be an odd prime number, and let M2 be the extra-special p-group of order p5 and of exponent p2. For every p-group K, we denote by H*(K) the mod p cohomology of K. The purpose of this paper is to calculate the mod p cohomology groups of M2 and of Cp2* M2 (the central product of the cyclic group Cp2 of order p2 and M2).
On the geometry of homotopy invariants of links
Authors: Mio Washington
Published: November 2008

One of the central problems in higher-dimensional knot theory is the classification of links up to concordance. In 14, Le Dimet constructed a universal model for (disk) link complements, which allowed him to formulate this problem in the framework of surgery theory by applying the Cappell-Shaneson program for studying codimension two embeddings of manifolds 1. The concordance classification was reduced to questions in L-theory (-groups 1) and homotopy theory (of Vogel local spaces 14). While recent results of Cochran and Orr2 (see also 18) provide rich information on the -theoretic part of the problem (in particular, they settle the question of the existence of links not concordant to boundary links), little is known about Le Dimet's homotopy invariant of links; for example, it is not known whether it may ever be non-trivial, or phrasing it more geometrically (according to 19), whether there are links that are not concordant to sublinks of homology boundary links. This motivated us to look at simpler classes of links, for which a more direct geometric approach to the problem is also possible, in an attempt to get some insight on the geometry carried by the homotopy invariants.
Generalized maximal surfaces in Lorentz–Minkowski space L3
Authors: Estudillo Francisco J. M. , Romero Alfonso
Published: November 2008

In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L3, with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L3. In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L3, i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L3. So, the term ‘regular’ means free of branch points. As in the minimal case, a conformal structure is naturally induced on M, which becomes a Riemann surface S. The corresponding isometric immersion is then conformal on S, and it does not have any singular points on S (i.e. points on which the differential of the mapping is not one-to-one). This is the way in which generalized maximal surfaces include regular ones. Moreover, branch points are the singular points of the conformal mapping on S. Whereas branch points of generalized minimal surfaces are isolated, we shall show in Section 2 that, in addition to isolated branch points, a generalized maximal surface in L3. may have non-isolated ones, in fact they constitute a 1-dimensional submanifold in a certain open subset of S (see Section 2). So our purpose is two-fold, firstly to study and explain in detail the branch points, and secondly to state several geometric results involving prescribed behaviour of those points on the surface.
T-systems of certain finite simple groups
Authors: Evans Martin J.
Published: November 2008

Let Fn be the free group of rank n freely generated by x1, x2,…, xn and write d(G) for the minimal number of generators of the finitely generated group G.
A generalization of the Whitney rank generating function
Authors: Farr G. E.
Published: November 2008

The Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.
Upper lp-estimates in vector sequence spaces, with some applications
Authors: Castillo Jesús M. F. , Sánchez Fernando
Published: November 2008

In [11], Partington proved that if λ is a Banach sequence space with a monotone basis having the Banach-Saks property, and (Xn) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space ΣλXn has this same property. In addition, Partington gave an example showing that if λ and each Xn, have the weak Banach-Saks property, then ΣλXn need not have the weak Banach-Saks property.
Recurrence and transience of reflecting Brownian motion in the quadrant
Authors: Hobson D. G. , ROGERS L. C. G.
Published: November 2008

In this paper we obtain criteria for a reflecting Brownian motion in the first orthant of the plane to reach an arbitrary open neighbourhood of the origin in finite time, or in finite mean time. The reflecting Brownian motion (RBM) is assumed to have a constant non-zero drift, and a constant non-singular covariance, and the directions of reflection on the two sides of are constant along each side, but not necessarily normal. We explain why the criteria we find are to be expected.
Simultaneous rational approximations and related diophantine equations
Authors: Rickert John H.
Published: November 2008

In this paper we shall consider simultaneous approximations to algebraic numbers of the form for rational t1, …, tm, v. If t1, …, tm, are sufficiently close to 1 with respect to a common denominator we shall derive an effective inequality of the form max {|θ1 – p1 / q|, …, |θm – pm / q|} > cq–1–λ for all integers p1, …, pm, q with q > 0. The technique is used to derive the inequality

We use this to show that the only solution to the simultaneous equations

are the trivial ones.
On the decimal representation of integers
Authors: Drazin M. P. , Griffith J. Stanley
Published: November 2008

Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such that

where, subject to the conditions

the integers αk(r, n) are uniquely determined, and, in fact, clearly

αk(r, n) = [n/rk] − r[n/rk+1]

(square brackets denoting integral parts, according to the usual convention).
Tauberian theory for the asymptotic forms of statistical frequency functions
Authors: Hammersley J. M.
Published: November 2008

An Abelian theorem is a theorem stating that a given behaviour on the part of each of several quantities entails similar behaviour for their average. A Tauberian theorem is a converse to an Abelian theorem. As a rule, a given behaviour of an average will not entail similar behaviour of the individual quantities themselves unless there is some condition imposed to secure reasonably uniform behaviour amongst the individuals. Such a condition, known as a Tauberian condition, is usually sufficient but not necessary, and it enters into the premises of the Tauberian theorem. We interpret ‘average’ in a wide sense to include any kind of smoothing process; for example, the integral of a functionf(t) is an average of the values off(t) corresponding to individual values oft;and we may seek a sufficient Tauberian condition such that a limit-behaviour of an integral-average

entails the corresponding limit-behaviour

for individual values oft.
Statistical mechanics and the partition of numbers II. The form of crystal surfaces
Authors: Temperley H. N. V.
Published: November 2008

The classical theory of partition of numbers is applied to the problem of determining the equilibrium profile of a simple cubic crystal. It is concluded that it may be thermo-dynamically profitable for the surface to be ‘saw-toothed’ rather than flat, the extra entropy associated with such an arrangement compensating for the additional surface energy. For both a two- and a three-dimensional ‘saw-tooth’ the extra entropy varies, to a first approximation, in the same way as the surface energy, i.e. is proportional to or respectively, where N is the number of molecules in a ‘tooth’. For the simple cubic lattice, the entropy associated with the formation of a tooth containing N atoms is estimated to be 3.3 It is also possible to estimate the variation of the ‘equilibrium roughness’ of a crystal with temperature, if its surface energy is known.
Removal of an inconsistency in the theory of diffraction
Authors: Jones D. S.
Published: November 2008

It is indicated by means of a particular example how an inconsistency arises in one formulation of diffraction problems. It is shown that the inconsistency is removed if different boundary conditions are imposed during the formulation. In the proof it is necessary to determine the contributions to the field from parts of a sphere of radius R at an interior point when the field on the sphere is a plane wave. It is found that, as R → ∞, the part of the sphere near the source which subtends a solid angle of order at the centre contributes the plane wave and that the remainder of the sphere provides a contribution which vanishes in the limit. The demonstration involves the use of the principle of stationary phase for an integral of two variables.

In the final section special boundary conditions are given for an exceptional three-dimensional problem.
The generalized function (ℵ + iO)λ
Authors: Fisher B.
Published: November 2008

The generalized function (x + iO)λ is defined by

for λ ╪ -1, -2, … and (x + iO)λ is denned by

for n = 1, 2,…, see (l).
A quantum theory of Wheeler–Feynman electrodynamics
Authors: Davies P. C. W.
Published: November 2008

A quantum mechanical theory of the action-at-a-distance electrodynamics of Wheeler and Feynman is given using an S-matrix approach. The response of the universe is introduced, and a perturbation expansion leads to the usual expression for the spontaneous transition rate between atomic energy levels, an effect normally attributed to quantized field oscillators. The Feynman propagator is then recovered, leading to the familiar self-energy formulae. Finally, a comparison of the formal structure of the new theory with the conventional is shown to establish a complete mathematical equivalence to all orders in the expansion.
A ‘short’ proof of the Riesz representation theorem
Authors: Garling D. J. H.
Published: November 2008

Textbook proofs of the Riesz theorem on the representation of linear functionals on C(X) by measures tend to be self-contained, but consequently are rather long, and use ad hoc methods (see, for example (2, 4, 5)). The purpose of this note is to give a short proof by appealing to standard methods of modern analysis.
Continuity of automorphic representations
Authors: Moffat J.
Published: November 2008

Let ℛ be a von Neumann algebra, with predual ℛ*, acting on a Hilbert space ℋ; G a locally compact group with left Haar measure m, and α a representation of G on aut (ℛ), the group of all *-automorphisms of ℛ, i.e. α is a group homomorphism from G to aut (ℛ). We shall show that if ℋ is separable, then very weak measurability assumptions on the representation α produce strong continuity properties. This will be used to obtain results on the extension of representations from a C*-algebra to its weak closure, giving a much simpler proof of a result of Aarnes ((1), theorem 8, p. 31), and on continuity of tensor products of representations. The main result was suggested by the analogous theory concerning unitary representations of locally compact groups, and its proof employs ideas frequently used in that context. (See, for example, (5), theorem 22.20 (b), p. 347.)
Random growth in a tessellation
Authors: Richardson Daniel
Published: November 2008

Let S be n dimensional Euclidean space and let T be a division of S into cells. Assume that each cell must be either white or black at any time t. At time 0 the cell at the origin, α0, is black and all other cells are white. Let G be some stochastic growth process which tends to change white cells with black neighbours into black cells. Let C(t) be the black shape at time t. For a family, F, of such growth processes we prove the following theorem.
Circular inclusion in an infinite elastic medium with a circular hole
Authors: Bhargava R. D. , Kapoor O. P.
Published: November 2008

This paper deals with the two-dimensional problem of a circular inclusion undergoing spontaneous dimensional changes in an infinite elastic medium with a circular hole. This provides a more realistic model for some physical phenomena where inclusions have found applications. The effect of a concentrated force acting at an arbitrary point of an infinite medium with a circular hole has been found as an auxiliary result. After the evaluation of the effect of a concentrated force, Eshelby's. point force approach (1) has been used to get an exact solution to the inclusion problem.
Higher-order functions for parsing
Authors: Hutton Graham
Published: November 2008

In combinator parsing, the text of parsers resembles BNF notation. We present the basic method, and a number of extensions. We address the special problems presented by white-space, and parsers with separate lexical and syntactic phases. In particular, a combining form for handling the ‘offside rule’ is given. Other extensions to the basic method include an ‘into’ combining form with many useful applications, and a simple means by which combinator parsers can produce more informative error messages.
The complexity of type inference for higher-order typed lambda calculi
Authors: Henglein Fritz , Mairson Harry G.
Published: November 2008

We analyse the computational complexity of type inference for untyped λ-terms in the second-order polymorphic typed λ-calculus (F2) invented by Girard and Reynolds, as well as higher-order extensionsF3,F4, …,Fωproposed by Girard. We prove that recognising theF2-typable terms requires exponential time, and forFωthe problem is non-elementary. We show as well a sequence of lower bounds on recognising theFk-typable terms, where the bound forFk+1is exponentially larger than that forFk.

The lower bounds are based on generic simulation of Turing Machines, where computation is simulated at the expression and type level simultaneously. Non-accepting computations are mapped to non-normalising reduction sequences, and hence non-typable terms. The accepting computations are mapped to typable terms, where higher-order types encode reduction sequences, and first-order types encode the entire computation as a circuit, based on a unification simulation of Boolean logic. A primary technical tool in this reduction is the composition of polymorphic functions having different domains and ranges.

These results are the first nontrivial lower bounds on type inference for the Girard/Reynolds system as well as its higher-order extensions. We hope that the analysis provides important combinatorial insights which will prove useful in the ultimate resolution of the complexity of the type inference problem.
Safe fusion of functional expressions II: Further improvements
Authors: Chin Wei-Ngan
Published: November 2008

Large functional programs are often constructed by decomposing each big task into smaller tasks which can be performed by simpler functions. This hierarchical style of developing programs has been found to improve programmers' productivity because smaller functions are easier to construct and reuse. However, programs written in this way tend to be less efficient. Unnecessary intermediate data structures may be created. More function invocations may be required.

To reduce such performance penalties, Phil Wadler proposed a transformation algorithm, called deforestation, which could automatically fuse certain composed expressions together to eliminate intermediate tree-like data structures. However, his technique is currently safe (terminates with no loss of efficiency) for only a subset of first-order expressions.

This paper will generalise the deforestation technique to make it safe for all first-order and higher-order functional programs. Our generalisation is explained using a model for safe fusion which views each function as a producer and its parameters as consumers. Through this model, syntactic program properties are proposed to classify producers and consumers as either safe or unsafe. This classification is used to identify sub-terms that can be safely fused/eliminated. We present the generalised transformation algorithm, illustrate it with examples and provide a termination proof for the transformation algorithm of first-order programs. This paper also contains a suite of additional techniques to further improve the basic safe fusion method. These improvements could be viewed as enhancements to compensate for some inadequacies of the syntactic analyses used.
A unifying type-theoretic framework for objects
Authors: Hofmann Martin , Pierce Benjamin
Published: November 2008

We give a direct type-theoretic characterization of the basic mechanisms of object-oriented programming, including objects, methods, message passing, and subtyping, by introducing an explicit constructor for object types and suitable introduction, elimination, and equality rules. The resulting abstract framework provides a basis for justifying and comparing previous encodings of objects based on recursive record types (Cardelli, 1984; Cardelli, 1992; Bruce, 1994; Cooket al., 1990; Mitchell, 1990a) and encodings based on existential types (Pierce & Turner, 1994).