bluuebunny/crossref_metadata_2025 huggingface dataset:
On commutative semigroup algebras
Authors: Munn W. D.
Published: November 2008

This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).
The group of homotopy self-equivalence classes of CW complexes
Authors: Rutter John W.
Published: November 2008

Exact sequences, which were subsequently used for calculating the group ℰ(X) of homotopy classes of self-homotopy equivalences of a space X, were given by Barcus and Barratt (§5 of (1)) in the case where X is obtained from a simply-connected (q + 1 > 1)-dimensional complex by adding one (q + l)-cell (q ≥ 3): these were later extended by Kudo and Tsuchida (theorems 2·2 and 2·8 of (6)) and by the author (theorem 3·1* of (15)), who also obtained a related sequence (theorem 2·3* of (15)). In the case of a two-cell complex, one or more of these sequences has been shown to split by Oka, Sawashita and Sugawara (theorems 3·9, 3·13 and 3·15 of (11)). The sequences have been used to calculate ℰ (X) for a number of complexes having two, three or more cells by various authors, including Oka (8), Oka, Sawashita and Sugawara (11), Rutter (17) and Sawashita (18). However the aforementioned sequences are only applicable to the addition of top-dimensional cells if the complex has no cells in its penultimate dimension. In this article I obtain sequences which are applicable without this latter restriction, show that one of them is generally split, and in special cases where there is only one top-dimensional cell obtain a further splitting: sequences are given which are valid without the assumption that A is simply connected. Also I give a new formula for calculating ℰ(X)in the case where X is not 2-connected.
On the interaction between Tollmien-Schlichting and Rayleigh-Bénard instabilities in the presence of a longitudinal magnetic field
Authors: Rudraiah N. , Shivaraya E. S.
Published: November 2008

The method used by Gage and Reid(10) to investigate hydrodynamic stability of thermally stratified fluid is extended to hydromagnetic stability to study the effect of aligned magnetic field on the stability of unstable thermal stratification under the assumption of small magnetic Reynolds number. The interaction between the Tollmien–Schlichting–Stuart mechanism of instability due to shear and magnetic field and Rayleigh–Bénard–Thompson mechanism of instability due to thermally unstable stratification and magnetic field is brought out in detail. It is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, Squire's theorem is not valid. This conclusion follows from the fact that in our analysis the Richardson number Ri ( < 0) will not be greater than the value −0·92 × 10−6. In particular, it is shown that for the values of stratification parameter n ≤ 0·6 the effect of magnetic field for small values of Stuart number N is to augment instability and impose the restriction on the validity of our numerical procedure. However, for η = 0·8 a sharp transition from unstable to stable flow takes place at N = 0·3. A physical explanation for this based on eddies is given.
Quotients of decidable objects in a topos
Authors: Johnstone Peter T.
Published: November 2008

In (5), Peter Freyd recently raised the question of whether every Grothendieck topos could be obtained from the topos of sets by means of the two constructions of taking sheaves on a locale and of taking continuous actions of a topological group (i.e. the topos-theoretic analogues of the set-theorists' techniques of forcing extensions and permutation models). He showed that these two constructions do suffice to within epsilon; provided we allow ourselves the freedom to take exponential varieties (4) (which do not change the internal logic of the topos) we can obtain every Grothendieck topos in this way.
Semifinite traces on JBW-algebras
Authors: King W. P. C.
Published: November 2008

A JB-algebra is a real Jordan algebra A which is also a Banach space and whose norm and multiplication satisfy the two following conditions

(i) ∥a2∥ = ∥a∥2,

(ii) ∥a2 − b2∥ ≤ max{∥a2∥, ∥b2∥},

for all elements a and b in A. A JB-algebra which is also a Banach dual space is called a JBW-algebra. The properties of JB-algebras and JBW-algebras can be found in (3), (4), (8) and (15).
An arithmetic characterization of algebraic number fields with a given class group
Authors: Rush David E.
Published: November 2008

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.
Generalized Bianchi identities for horizontal distributions
Authors: Crampin M.
Published: November 2008

A linear connection on a differentiable manifold M defines a horizontal distribution on the tangent bundle T(M). Horizontal distributions on tangent bundles are of some interest even when they are not generated by connections. Much of linear connection theory generalizes to arbitrary horizontal distributions. In particular, there are generalized versions of Bianchi's two identities for the torsion T and curvature R

where C denotes the cyclic sum with respect to X, Y and Z. These identities are derived below by associating with a horizontal distribution a graded derivation of degree 1 in a graded Lie algebra of vertical forms on M. This approach reveals the fundamentally algebraic origin of the Bianchi identities.
Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions
Authors: Saab Paulette
Published: November 2008

Given a compact Hausdorff spaceX, EandFtwo Banach spaces, letT: C(X, E) → Fdenote a bounded linear operator (hereC(X, E)stands for the Banach space of all continuousE-valued functions defined onXunder supremum norm). It is well known [4] that any such operatorThas a finitely additive representing measureGthat is defined on the σ–field of Borel subsets ofXand thatGtakes its values in the space of all bounded linear operators fromEinto the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance[1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that ifE*andE**have the Radon-Nikodym property then a bounded linear operatorT: C(X, E) → Fis weakly compact if and only if the measureGis continuous at Ø (also called strongly bounded), i.e. limn||G|| (Bn) = 0 for every decreasing sequenceBn↘ Ø of Borel subsets ofX(here ||G|| (B) denotes the semivariation ofGatB), and if for every Borel setBthe operatorG(B)is a weakly compact operator fromEtoF. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties thenE*andE**must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators onC(X, E). It is known [6] that ifT: C(X, E) → Fis an unconditionally converging operator, then its representing measureGis continuous at 0 and, for every Borel setB, G(B)is an unconditionally converging operator fromEtoF. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spacesEwith the Schur property in terms of properties of Dunford-Pettis operators onC(X, E)spaces.
Non-amalgamation of ordered groups
Authors: Glass A. M. W. , Saracino D. , Wood C.
Published: November 2008

An ordered group (o-group for short) is a group endowed with a linear (i.e. total) order such that for all x, y, z, xz ≤ yz and zx ≤ zy whenever x ≤ y. A group for which such an order exists is called an orderable group. A group G is said to be divisible if for each positive integer m and each g ε G, there is x ε G such that xm = g.
A Poincaré limit theorem for wrapped probability distributions on compact symmetric spaces
Authors: Jupp P. E.
Published: November 2008

In an article on the philosophy of chance, Poincaré[7] showed that the distribution of the stopping position of a needle pivoted about its centre tends to the uniform distribution on the circle as the distribution of the initial push becomes spread out along the real line. This result was formalized by Feller [2] and strengthened by Mardia [6] as follows.
Unstable families related to the image of J
Authors: Gray Brayton
Published: November 2008

The object of this paper is to describe certain families of unstable elements in the homotopy groups of spheres at an odd prime. In so doing we completely account for the image of J as possible Hopf invariants of unstable elements. The analogous result for p = 2 was obtained in [13]. In addition we will discuss other periodic phenomena. Our main results have been independently obtained by Bendersky[5] using BP*. Our methods, however, are entirely geometric, and we actually construct the elements, rather than detect them. Our basic tool is the map .All our constructions are made in BΣp and transferred over.
Semimartingales and the empirical distribution
Authors: Al-Hussaini Ata N. , Elliott Robert J.
Published: November 2008

Processes related to the empirical distribution are shown to converge to a time changed Brownian motion. In particular, the time change of the Brownian motion differs from the time change obtained by using the distribution function: this gives rise to two different representations of a related Brownian bridge.
Soluble-by-periodic skew linear groups
Authors: Wehrfritz B. A. F.
Published: November 2008

Let D be a division ring with central subfield F, n a positive integer and G a subgroup of GL(n, D) such that the F-subalgebra F[G] generated by G is the full matrix algebra Dn×n. If G is soluble then Snider [9] proves that G is abelian by locally finite. He also shows that this locally finite image of G can be any locally finite group. Of course not every abelian by locally finite group is soluble. This suggests that Snider's conclusion should apply to some wider class of groups.
Fourier restriction theorems for curves with affine and Euclidean arclengths
Authors: Drury S. W. , Marshall B. P.
Published: November 2008

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequality

for every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].
Finite quasisimple groups of 2×2 matrices over a division ring
Authors: Hartley B. , Shojaei M. A. Shahabi
Published: November 2008

In 1955 [1], Amitsur determined all the finite groupsGthat can be embedded in the multiplicative groupT*=GL(1, T)of some division ringTof characteristic zero. IfGcan be so embedded, then the rational span ofGinTis a division ring of finite dimension over ℚ,andGacts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero isSL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.
A tabular method for correcting skewness
Authors: Hall Peter
Published: November 2008

We propose a smooth correction for skewness in an asymptotically normal statistic. Unlike the case of approximation by Edgeworth expansion, our correction applies uniformly in all values of the level. The correction is based on a ' comparison statistic', and, in the case of the Studentized mean, it enables removal of all effects of skewness up to terms of order.
A note concerning the global 2 -forms on a x = 0 threefold
Authors: Wilson P. M. H.
Published: November 2008

Let V be a smooth complex algebraic threefold with k(V) = 0. Consider the space of holomorphic 2-forms on V, and let s denote the rank of the subsheaf of generated by these global forms. It was conjectured by Uenoin[8] that ≤ 3-here as always, hi denotes the dimension of the corresponding cohomology group Hi.
On realization of Whitehead torsion
Authors: Kwasik Slawomir
Published: November 2008

We prove that the Poincaré conjecture implies that the 4-dimensional version of the realization theorem for Whitehead torsion is false. We show that infinitely many examples may be constructed to demonstrate this.
A note on the cohomology of metabelian groups
Authors: Kropholler P. H.
Published: November 2008

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.
Infinitely many ribbon knots with the same fundamental group
Authors: Suciu Alexander I.
Published: November 2008

We work in the DIFF category. A knot K = (Sn+2, Sn) is a ribbon knot if Sn bounds an immersed disc Dn+1 →Sn+2 with no triple points and such that the components of the singular set are n-discs whose boundary (n – l)-spheres either lie on Sn or are disjoint from Sn. Pushing Dn+1 into Dn+3 produces a ribbon disc pair D = (Dn+3, Dn+1), with the ribbon knot (Sn+2, Sn) on its boundary. The double of a ribbon (n+1)-disc pair is an (n + l)-ribbon knot. Every (n+l)-ribbon knot is obtained in this manner.
Analytic sets from the point of view of compact sets
Authors: Becker Howard
Published: November 2008

A set A ⊂ ωω is called compactly if, for every compact K ⊂ ωω, A ∩ K is . Consider the proposition that every compactly set is . (AD implies that it is true, ZFC + CH implies that it is false.) We are concerned here with whether this is consistent with ZFC, particularly when n = 1. In the case of sets (that is, analytic sets), this consistency question is due to Fremlin (see [7], page 483, problem 18). Kunen and Miller [3] have proved the following two theorems.
Form rings and projective equivalence
Authors: Katz Daniel , Ratliff L. J.
Published: November 2008

If I and J are ideals in a Noetherian ring R, then I and J are projectively equivalent in case (Ii)a = (Jj)a for some positive integers i, j (where Ka denotes the integral closure in R of the ideal K) and the form ring F(R, I) of R with respect to I is the graded ring R/I ⊕ I/I2 ⊕ I2/I3 ⊕ …. These two concepts have played an important role in many research problems in commutative algebra, so they have been deeply studied and many of their properties have been discovered. In a recent paper [13] they were combined to show that a semi-local ring R is unmixed if and only if for every ideal J in R there exists a projectively equivalent ideal J in R such that every prime divisor of zero in F(R, J) has the same depth. It seems to us that results similar to this are interesting and potentially quite useful, so in this paper we prove several additional such theorems. Namely, it is shown that all ideals in all local rings have a projectively equivalent ideal whose form ring is fairly nice. Also, a characterization similar to the just mentioned result in [13] is given for the class of local rings whose completions have no embedded prime divisors of zero, and several analogous new characterizations are given for locally unmixed Noetherian rings. In particular, it is shown that if I is an ideal in an unmixed local ring R such that height(I) = l(I) (where l(I) denotes the analytic spread of I), then there exists a projectively equivalent ideal J in R such that Ass (F(R, J)) has exactly m elements, all minimal, where m is the number of minimal prime divisors of I (so if I is open, then F(R, J) has exactly one prime divisor of zero and is a locally unmixed Noetherian ring).
Groups with two projective characters
Authors: Higgs R. J.
Published: November 2008

All representations and characters studied in this paper are taken over the field of complex numbers, and all groups considered are finite. The reader unfamiliar with projective representations is referred to [8] for basic definitions and terminology.
A problem in additive number theory
Authors: Brüdern Jörg
Published: November 2008

The determination of the minimal s such that all large natural numbers n admit a representation as

is an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.
Additive arithmetic functions on intervals
Authors: Elliott P. D. T. A.
Published: November 2008

§1. A real-valued arithmetic function f is additive if it satisfies the relation f(ab) = f(a) + f(b) for all mutually prime positive integers a, b. In the present paper I establish three theorems concerning the value distribution of such functions on intervals.
The dual J* of the James space has cotype 2 and the Gordon-Lewis property
Authors: Pisier Gilles
Published: November 2008

We prove the result in the title. More generally we consider the Banach space υp, of all sequences (xn) of scalars such that

where the supremum runs over all increasing sequences n1 ≤ n2 ≤ …. We show that is of cotype 2 if p ≽ 2 and of cotype p′, where 1/p′ + 1/p = 1, if p ≤ 2. Similar results are obtained for the analogous function spaces.
Corrigendum
Authors: Kleidman Peter B. , Wilson Robert A.
Published: November 2008

Volume 102 (1987), 17–23

‘The maximal subgroups of Fi22’

We reported on some computer calculations which we used to complete the enumeration of the maximal subgroups of the sporadic simple group Fi22 and of its automorphism group Fi22:2. Unfortunately there was an error in these calculations. We have therefore repeated all the calculations, incorporating much more thorough checking routines.
Multiplicative functions at consecutive integers. II
Authors: Hildebrand Adolf
Published: November 2008

The global behaviour of multiplicative arithmetic functions has been extensively studied and is now well understood for a large class of multiplicative functions. In particular, Halász [5] completely determined the asymptotic behaviour of the means

for multiplicative functions g satisfying |g| ≤ 1, and gave necessary and sufficient conditions for the existence of the ‘mean value’
The tensor product of Archimedean ordered vector spaces
Authors: Grobler J. J. , Labuschagne C. C. A.
Published: November 2008

A Riesz space tensor product of Archimedean Riesz spaces was introduced by D. H. Fremlin[2, 3]. His construction as well as a subsequent simplified version by H. H. Schaefer[10] depended on representation techniques and it is our aim to find a more direct way to prove the existence of the tensor product and to derive its properties. This tensor product proved to be extremely useful in the theory of positive operators on Banach lattices (see [3] and [10]) and should be considered as one of the basic constructions in the theory of Riesz spaces. It is therefore of interest to construct it in an intrinsic way. The problem to do this was already posed by Fremlin in [2]. In this paper we shall present two different approaches, the first of which is analogous to the formation of a free lattice generated by a given partially ordered set. (See [5], p. 41.) In the second one we first assume the Riesz spaces involved to have the principal projection property. In this case a simple method of construction by step-elements is available and the tensor product of arbitrary Archimedean Riesz spaces can then be obtained by embedding the spaces into their Dedekind completions. To complete the latter step we need results on the extension of Riesz bimorphisms which will be proved in §1. Both our approaches hinge on results about the tensor product of ordered vector spaces. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the Riesz space tensor product. This is investigated in §2.
Quasi-free stochastic integral representation theorems over the CCR
Authors: Wilde Ivan F.
Published: November 2008

It is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.
Boolean algebras in a localic topos
Authors: Banaschewski B. , Bhutani K. R.
Published: November 2008

When a familiar notion is modelled in a certain topos E, the natural problem arises to what extent theorems concerning its models in usual set theory remain valid for its models in E, or how suitable properties of E affect the validity of certain of these theorems. Problems of this type have in particular been studied by Banaschewski[2], Bhutani[5], and Ebrahimi[6, 7], dealing with abelian groups in a localic topos and universal algebra in an arbitrary Grothendieck topos. This paper is concerned with Boolean algebras, specifically with injectivity and related topics for the category of Boolean algebras in the topos of sheaves on a locale and with properties of the initial Boolean algebra in .
On even-dimensional fibred knots obtained by plumbing
Authors: Lines Daniel
Published: November 2008

In this article we extend the study of ‘plumbing’ initiated in [10] to the case of fibred even-dimensional knots. Plumbing is a geometric operation on the fibre-surfaces of two fibred knots of the same dimension that produces another such knot.
Counterexamples to a conjecture about spherical diagrams
Authors: Gersten S. M. , Howie James
Published: November 2008

A spherical diagram over a 2-complex X consists of a tessellation T of the 2-sphere, together with a combinatorial map (that is, one which maps each cell homeomorphically on to a cell). In [2] and [6] a number of conjectures were made concerning spherical diagrams over a 2-complex X with H2(X) = 0. There are also related conjectures in [7]. The motivation for all of these conjectures is that they imply the Kervaire Conjecture: every non-singular system of equations over a group can be solved in some overgroup (see [1, 2, 4, 6, 7] for details and discussion).
Matrix summability of Fourier series based on inclusion theorems
Authors: Rhoades B. E.
Published: November 2008

Letdenote the Fourier series expansion of a function. Féver's celebrated theorem states that the series is summable (C, 1) at each point of continuity off, where (C, 1) denotes the Cesàro method of summability of order 1. Riesz extended this result to (C, α) for each α > 0. Since then many authors have established sufficient conditions on various methods of summability to guarantee similar results. Over the years the pattern has been to strive for weaker conditions on the matrix, and to replace the condition of continuity on the function by a less stringent one. Theorems have been proved not only for the summability off, but the summability of the derived series, and other series, related tof. Beginning with the work of Hille and Tamarkin[13], many of the theorems have been extended to absolute summability.
On the equivalence of the properties of quasi-translation invariance and absolute continuity of measures on a locally compact abelian group
Authors: Bell Denis
Published: November 2008

We give a simple proof that a measure on a locally compact abelian group G is quasi-translation invariant with continuous translation densities if and only if it is equivalent to Haar measure on G and has a continuous positive density.
A reduction theory of second order meromorphic differential equations, I
Authors: Jurkat W. B. , Zwiesler H. J.
Published: November 2008

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:

(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )

(ii) A(z) has trace tr for some c ∈ ℂ,

(iii) Ar−1 has different eigenvalues,

(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,

(v) if A−1 = μI, then Ar−1 is diagonal,

(vi) Ar−1 and A−1 are triangular in opposite ways,

(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.
Further results on the relative entropy
Authors: Donald Matthew J.
Published: November 2008

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.
Free quotients of congruence subgroups of SL2 over a Dedekind ring of arithmetic type contained in a function field
Authors: Mason A. W.
Published: November 2008

Let R be a commutative ring with identity and let q be an ideal in R. For each n ≽ 2, let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. We put SLn(R, q) = Ker(SLn(R)→SLn(R/q)), the principal congruence subgroup of GLn(R) of level q. (By definition En(R, R) = En(R) and SLn(R, R) = SLn(R).)
Graphs which are vertex-critical with respect to the edge-chromatic number
Authors: Hilton A. J. W. , Johnson P. D.
Published: November 2008

In this paper, multigraphs will have no loops. For a multigraph G, the least number of colours needed to colour the edges of G in such a way that no two edges on the same vertex of G have the same colour, is called the edge-chromatic number, or the chromatic index, of G, and denoted χ′(G). It is clear that if Δ(G) denotes the maximum degree of G, then Δ(G) ≤ χ′(G). If Δ(G) = χ′(G), then G is Class 1, and otherwise G is Class 2.
Relative semicharacteristic classes
Authors: Davis James F.
Published: November 2008

In 1973 Ronnie Lee introduced the notion of semicharacteristic classes, which are invariants of the bordism group ℜ*(Bπ) of closed manifolds equipped with a free action of a finite group π. In this paper we relativize his theory. Associated to a homomorphism G → π of finite groups, there is the relative bordism group ℜ*(BG → Bπ), which is the bordism group of compact manifolds M with a free π-action, so that the action on ∂M is induced from a free G-action, i.e. ∂M = π xGN for some manifold N with a free G-action. The invariants defined here are invariants of this relative group.
Steenrod squares of polynomials and the Peterson conjecture
Authors: Wood R. M. W.
Published: November 2008

We are concerned in this paper with polynomials in n variables xi over the field 2 of two elements. The Steenrod squares Sqk are linear operators which act on the variables by the rules

and on the product uv of homogeneous polynomials u and v by the Cartan formula
Natural constructions of the Mathieu groups
Authors: Curtis R. T.
Published: November 2008

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.
On complete d-sequences and the defining ideals of Rees algebras
Authors: Huckaba Sam
Published: November 2008

If R is a Noetherian local ring and I = (x1, …, xn)R is an ideal of R then the Rees algebra R[It] can be represented as a homomorphic image of the polynomial ring R[Z1, …, Zn]. The kernel is a homogeneous ideal, and the smallest of the degree bounds among all generating sets, called the relation type of I, is independent of the representation. We derive formulae connecting the relation type of I with the reduction number of I when the analytic spread of I exceeds height(I) by one. In the process we define complete d-sequences with respect to I and use them to help achieve our results. In addition some results on the behaviour of the relation type modulo an element are proved, and examples where the relation type is explicitly computed are presented.
Crossed complexes and chain complexes with operators
Authors: Brown Ronald , Higgins Philip J.
Published: November 2008

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex of cellular chains of the universal cover of a reduced CW-complex X. The group of operators here is the fundamental group of X.
On ideals and sublattices in linear lattices and F-lattices
Authors: Abramovich Y. A. , Lipecki Z.
Published: November 2008

The main aim of this paper is to study the role of the assumptions in the following

Theorem 0. Every ideal I of codimension 1 in an F-lattice X is closed†.

For X a Banach lattice this result appears in [11], ii.5·3, corollary 3. The proof carries over to the case where X is an. F-lattice, i.e. a topological linear lattice the topology of which is metrizable and complete, with the help of a result due to Klee ([10], theorem v.5·5; see also [3], theorem 16·6). We first note that the assumption of metric completeness is essential (Example 2). We then extend Theorem 0 to the case of ideals of finite codimension (Theorem 1), and show that non-closed ideals exist whenever X is infinite-dimensional (Corollary 2). Our main result commenting on Theorem 0 is, however, the existence of dense sublattices of arbitrary codimension between 1 and in every infinite-dimensional F-lattice (Theorem 6).
Scattering operator and Eisenstein integral for Kleinian groups
Authors: Mandouvalos N.
Published: November 2008

In this paper we study certain aspects of the problem of analytic continuation of the scattering operator and Eisenstein integral which we introduced in [7, 8, 10], for Kleinian groups Γ with exponent of convergence δ(Γ) ≥ 1. The corresponding problem for groups with δ(Γ) < 1 was examined and solved in [9].
Some strong limit-2 and Dirichiet criteria for fourth order differential expressions
Authors: Race David
Published: November 2008

Conditions are given on the real coefficients p, q and r and the weight w, for the fourth order formally symmetric differential expression

to have the properties of being strong limit-2 and Dirichlet at ∞, when considered in the weighted Hilbert space, . These extend existing results due to both W. N. Everitt and V. Krishna Kumar and cover an expression which is important in the study of certain orthogonal polynomials.
Some recent developments concerning asymptotic distributions of pontograms
Authors: Eastwood Vera R.
Published: November 2008

The data analytical problem of testing whether an empirical set of n given points in the plane could be considered to contain too many straight line configurations in a situation where the generating mechanism of the n points is unknown, was recently reintroduced to the literature by D. G. Kendall and W. S. Kendall[7]. Taking the Land's end data problem (cf. also Broadbent [1] and further references given there) as the anchor point and motivation for their discussion, D. G. and W. S. Kendall developed a new testing device, called the pontogram. The pontogram is a one- parameter stochastic process defined pointwise by

where N denotes a Poisson process in [0, 1] with N(0) = 0 and unknown intensity parameter μ > 0 and where R = N(1) gives the number of Poisson events in the entire [0, 1] section.
Hopf algebras and regular isotopy invariants for link diagrams
Authors: Hennings M. A.
Published: November 2008

In this paper, we shall consider the following method for obtaining regular isotopy invariants of link diagrams. Given any link diagramL, equip it with a Morse functionh, so that the diagram consists entirely of crossings, maxima, minima and vertical arcs. Introduce 2-valent graphical vertices to separate the various segments of the diagram. Given a finite index setI, a state σ forLhis an assignation of one element ofIto each graphical vertex. Each segment of the diagram now has a weight

associated with it, given in terms of tensor coordinates indexed by the setIby the pictures

and, for any state σ, [Li|σ] denotes the product of the various weights. We then define 〈Lh〉 to be the sum of [Lh|σ] over all possible states σ,
On Waring's problem for cubes
Authors: Brüdern Jörg
Published: November 2008

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shape

where (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.
Reidemeister's theorem for 3-manifolds
Authors: Sundheim Paul A.
Published: November 2008

The Reidemeister theorem describes the equivalence of links in terms of diagrams (this theorem was proven in detail by Alexander [1]). A diagram for a link in S3 can be found by projecting the link to any disc recording the over or under crossings. It was shown that two links are equivalent if and only if their diagrams are related by a sequence of so called Reidemeister moves and isotopy in the disc.
Compact groups and products of the unit interval
Authors: Cleary Joan , Morris Sidney A.
Published: November 2008

It is proved that if G is a compact connected Hausdorff group of uncountable weight, w(G), then G contains a homeomorphic copy of [0, 1]w(G). From this it is deduced that such a group, G, contains a homeomorphic copy of every compact Hausdorff group with weight w(G) or less. It is also deduced that every infinite compact Hausdorff group G contains a Cantor cube of weight w(G), and hence has [0, 1]w(G) as a quotient space.
Bounds on the derivatives of polynomials on Banach spaces
Authors: Sarantopoulos Yannis
Published: November 2008

We generalize the classical Bernstein's and Markov's Inequalities for polynomials on any real Banach space. We also give estimates for the derivatives of homogeneous polynomials on real Banach spaces.
Word problems related to derivatives of the displacement map
Authors: Devlin J.
Published: November 2008

In [6], we considered the equation

where z ∈ ℂ and the pi are real-valued functions; abstract word-problem concepts and techniques were applied to the local problem of the bifurcation of periodic solutions out of the solution Z ≡ 0. This paper is a sequel to [6]; we present an extension of certain concepts given in that paper, and give a global version of some of our word-problem results.
ℤ2-equivariant hierarchies of irreducible 3-manifolds containing two-sided projective planes
Authors: Kalliongis John
Published: November 2008

For a compact irreducible 3-manifold containing 2-sided projective planes and admitting a ℤ2-action, we consider the problem of when there exists a ℤ2-equivariant hierarchy. We show that a ℤ2-equivariant hierarchy always exists if either every irreducible summand of the orientable double cover has infinite first homology or every 2-sided projective plane is boundary parallel.
The preimages of submanifolds
Authors: Rong Yongwu , Wang Shicheng
Published: November 2008

All manifolds in this paper are piecewise linear (or smooth if one wishes).
A centred norm inequality for singular integral operators
Authors: Bass Richard F.
Published: November 2008

Let K be a standard singular integral kernel on ℝ satisfying the usual Hölder continuity condition of order δ, and define (where c is chosen so that the integral of w is 1), the mean of g with respect to the measure w(x) dx, and ‖·‖p the Lp norm with respect to w(x) dx. Although the inequality is not true in general, the centred norm inequality does hold for 1 < p < ∞ if α < δ.
On the existence of minimal Seifert manifolds
Authors: Silver Daniel S.
Published: November 2008

For n ≥ 3, an n-knot K has a minimal Seifert manifold if and only if its group is isomorphic to an HNN-extension with finitely presented base. In this case, any Seifert manifold for K can be converted to a minimal Seifert manifold for K by some finite sequence of ambient 0- and 1-surgeries.
Mod p K-theory of Ω∞Σ∞X revisited
Authors: Kashiwabara Takuji
Published: November 2008

In this note we present a new proof of a theorem of McClure on K*(Ω∞Σ∞X, Z/p) [11], in the special case when X is a finite complex with K1(X; Z/p) = 0. Although our method does not work in the full generality covered by his work, our argument requires neither a geometric interpretation of complex k-theory nor all the delicate coherence properties of its multiplication. Since BP-theory is not likely to possess such coherence properties [9], the possibility of generalizing his approach to the case of higher Morava K-theory does not seem feasible. On the contrary, the main ingredient of our approach is the rank formula for the Morava K-theory of the Borel construction [5], which works for any K(n); thus our approach is better adapted to the potential generalization [8]. Throughout the paper we assume that p > 2 so that mod p K-theory possesses a commutative multiplication, and denote by K*(−) the mod p K-theory. Since it is simpler to state our results in terms of CX, the combinatorial model for QX, rather than QX itself, we shall do so. This is sufficient, as when X is connected CX is homotopy equivalent to QX, and when not, K*(QX) can be easily recovered from K*(CX) (see e.g. [11]).
The general theory of 3-transposition groups
Authors: Hall J. I.
Published: November 2008

A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.
The Gehring–Hayman inequality for quasihyperbolic geodesics
Authors: Heinonen Juha , Rohde Steffen
Published: November 2008

The quasihyperbolic metric in a proper subdomain D of Rn is defined by

where the infimum is taken over all rectifiable arcs γ in D joining x and y. There always exists an arc, called a quasihyperbolic geodesic in D, for which the infimum above is attained. We refer to [3], [4], [16], and [17] for the motivation and basic properties of the quasihyperbolic metric.
QuantumSU(2)-invariants dominate Casson'sSU(2)-invariant
Authors: Murakami Hitoshi
Published: November 2008

In 1988, E. Witten proposed an array of invariants (quantumG-invariants) for a 3-manifold associated with a compact simple Lie groupGbased on the quantum field theory [22]. N. Reshetikhin and V. Turaev [20] gave a mathematical approach to these invariants forG=SU(2) by using representations of the quantum groupUq(sl(2, ℂ)). See also [2, 11, 12, 14, 15, 16] for other approaches to these invariants: [16] may be the simplest way (for knot theorists) to understand the existence of these invariants.
Effective resistances for harmonic structures on p.c.f. self-similar sets
Authors: Kigami Jun
Published: November 2008

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].
Optimal choice between parametric and non-parametric bootstrap estimates
Authors: Lee Stephen Man Sing
Published: November 2008

A parametric bootstrap estimate (PB) may be more accurate than its non-parametric version (NB) if the parametric model upon which it is based is, at least approximately, correct. Construction of an optimal estimator based on both PB and NB is pursued with the aim of minimizing the mean squared error. Our approach is to pick an empirical estimate of the optimal tuning parameter ε∈[0, 1] which minimizes the mean square error of εNB+(1−ε) PB. The resulting hybrid estimator is shown to be more reliable than either PB or NB uniformly over a rich class of distributions. Theoretical asymptotic results show that the asymptotic error of this hybrid estimator is quite close in distribution to the smaller of the errors of PB and NB. All these errors typically have the same convergence rate of order. A particular example is also presented to illustrate the fact that this hybrid estimate can indeed be strictly better than either of the pure bootstrap estimates in terms of minimizing mean squared error. Two simulation studies were conducted to verify the theoretical results and demonstrate the good practical performance of the hybrid method.
On Schur algebras and related algebras III: integral representations
Authors: Donkin Stephen
Published: November 2008

Let G be a reductive group over an algebraically closed field K. In [8] and [9] we defined and studied certain finite dimensional K-algebras SK(π), associated to G via a finite saturated set π of dominant weights. The algebras are defined over ℤ, i.e. SK(π) = K ⊗ℤSℤ(π) for an order Sℤ(π) of Sℚ(π), and if G is a general linear group or a Chevalley group then the order Sℤ(π) arises naturally from the corresponding group scheme G over ℤ (or Kostant ℤ-form Uℤ). These algebras may be regarded as (and were obtained as) direct generalizations of the Schur algebras S(n, r) studied by Green in [10].