bluuebunny/crossref_metadata_2025 huggingface dataset:
Two exponential Diophantine equations
Authors: Dofs Erik
Published: May 2009

In [3], two open problems were whether either of the diophantine equations

with n ∈ Z and f a prime number, is solvable if ω > 3 and 3 √ ω, but in this paper we allow f to be any (rational) integer and also 3 | ω. Equations of this form and more general ones can effectively be solved [5] with an advanced method based on analytical results, but the search limits are usually of enormous size. Here both equations (1) are norm equations in K (√–3): N(a + bp) = fw with p = (√–1 + –3)/2 which makes it possible to treat them arithmetically.
A note on the theorem of Jarník-Besicovitch
Authors: Dickinson H.
Published: May 2009

This note draws together and extends two recent results on Diophantine approximation and Hausdorff dimension. The first, by Hinokuma and Shiga [12], considers the oscillating error function | sinq|/qτ rather than the strictly decreasing function qτ of Jarnik's theorem. The second is Rynne's extension [17] to systems of linear forms of Borosh and Fraenkel's paper [3] on restricted Diophantine approximation with real numbers. Rynne's result will be extended to a class of general error functions and applied to obtain a more general form of [12] in which the error function is any positive function.
On commutative Noetherian rings which satisfy the radical formula
Authors: Leung Ka Hin , Man Shing Hing
Published: May 2009

In this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.
On Hankel convolutors on certain Hankel transformable function spaces
Authors: Betancor Jorge J. , Rodíguez-Mesa Lourdes
Published: May 2009

In this paper we introduce the spaces of Hankel convolutors. We characterize the dual spaces of certain Hankel transformable function spaces as spaces of Hankel convolutors. Here the Hankel convolution and the Hankel transformation play an important role.
Bilinear forms on vector Hardy spaces
Authors: Blower Gordon
Published: May 2009

Let φ: ℋ → be a bilinear form on vector Hardy space. Introduce the symbol φ of Φ by (φ (Z1, Z2), a ⊗ b) = Φ (K21 ⊗ a, K22 ⊗ b ), where Kw is the reproducing kernel for w ∈ D. We show that Φ extends to a bounded bilinear form on provided that the gradient defines a Carleson measure in the bidisc D2. We obtain a sufficient condition for Φ to extend to a Hilbert space. For vectorial bilinear Hankel forms we obtain an analogue of Nehari's Theorem.
Almost-Bieberbach groups with (in)finite outer automorphism group
Authors: Malfait Wim , Szczepański Andrzej
Published: May 2009

If we investigate symmetry of an infra-nilmanifoldM, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groupsEwith finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms onM, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group ofE.
Commutator length of abelian-by-nilpotent groups
Authors: Akhavan-Malayeri Mehri , Rhemtulla Akbar
Published: May 2009

Let G be a group and C = [G, G] be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. The exact values of c{G) are computed when G is a free nilpotent group or a free abelian-by-nilpotent group. If G is a free nilpotent group of rank n>2 and class c>2, c(G) = [n/2] if c = 2 and c(G) = n if c>2. If G is a free abelian-by-nilpotent group of rank n > 2 then c(G) = n.
Existence de sous-espaces hyper-invariants
Authors: Kellay K.
Published: May 2009

Soient B un espace de Banach et ℒ(B) l'algèbre des opérateurs bornés sur B. On dit qu'un sous-espace fermé E de B est invariant pour l'opérateur T ∈ ℒ(B) lorsque TE ⊂ E et qu'il est non trivial si {0} EB. Le sous-espace E est dit hyper-invariant pour T s'il est invariant pour tout opérateur qui commute avec T.
On the C-Projectivity of ideals in Banach algebras
Authors: Pugach L. I.
Published: May 2009

The notion of projective Banach module was defined by Helemskii in [1]—the paper which properly founded the homological theory of Banach algebras. The same author introduced the definition of the (relatively) flat Banach module in [2]. Recently M. C. White [3] modified both of those definitions, introducing so called C-projective and C-flat Banach modules.
On Prefrattini residuals
Authors: Ballester-Bolinches A. , Bechtell H. , Ezquerro L. M.
Published: May 2009

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among the unique maximal normal subgroups that result from these products, the solid set of maximal subgroups X, X-prefrattini subgroups, and the residuals of formations are explored. This leads to a well-defined collected of formations, the partially nonsaturated formations, with properties analogous to those which are totally non-saturated. In the development, attention is given to a set of maximal subgroups which is the image of a solid function defined on all groups, a weaker condition than that of a solid set. A result of particular interest answers affirmatively the long-standing conjecture that a non-trivial nC-group G is solvable if and only if each G-chief factor is complemented by a maximal subgroup. This will force a critical re-examination of the classification problem for nC-groups. Since the article continues the investigations on finite groups initiated in [2], a familiarity with that article is assumed. All other notation and terminology is from [6]. If M is a maximal subgroup of a group G and G/C or e G(M) is a monolithic primitive group, i.e. a group with a unique minimal normal subgroup, then M is called a monolithic maximal subgroupof G.
The normal curvature of totally real submanifolds ofS6(1)
Authors: De Smet P. J. , Dillen F. , Verstraelen L. , Vrancken L.
Published: May 2009

We prove the pointwise inequality 0 ≥ρ+ρ⊥– 1 involving the normalized scalar curvatureρand normal scalar curvatureρ⊥of a totally real 3-dimensional submanifold of the nearly Kaehler 6-sphere. Further we classify submanifolds realizing the equality in this inequality.
Almost continuity implies closure continuity
Authors: Saleh Mohammad
Published: May 2009

The purpose of this note is to answer in the affirmative a long standing open question raised by Singal and Singal — whether every almost continuous function is closure continuous (θ-continuous).
On M-harmonic Bloch functions and their Carleson measures
Authors: Choe Boo Rim , Lee Young Joo
Published: May 2009

On the setting of the unit ball of the complex n-space, some characterizations of M-harmonic Bloch functions are obtained. As an application, Carleson measures are characterized by means of Berezin type integrals of M-harmonic Bloch functions. As one may expect, these results carry over to M-harmonic little Bloch functions and vanishing Carleson measures.
Inequalities for the Maximal Eigenvalue of a Nonnegative Matrix
Authors: Yeh Lina
Published: May 2009

We gave an elementary proof of Theorem 4 in the paper, published in the Glasgow Mathematical Journal 39(1997), 275–284. The result provides an algorithm for approximating the maximal eigenvalue of a nonnegative matrix. Recently the author has learnt that the result can be proved immediately from Theorem 6.8 in [1]. Indeed, the paper [1] determines necessary and sufficient conditions for the convergence of an iterative sequence to the maximal eigenvalue. Their proof needs knowledge of graph theoretical concepts.
On totally free crossed modules
Authors: R.-Grandjean A. , Ladra M.
Published: May 2009

In [10] we associate to a crossed module (T, G, მ) an invariant abelian crossed module H2(T, G, მ). The construction uses presentations by Set-free crossed modules. Now, Set-free crossed modules are special cases of totally free crossed modules, which are algebraic models of 2-dimensional CW complexes used by several authors (see [1] and [6]). The aim of this paper is to show that H2(T, G, მ) can also be constructed from presentations by arbitrary totally free crossed modules.
Convolution operators associated with vector measures
Authors: Robdera Mangatiana A. , Saab Paulette
Published: May 2009

Inthis note, we present a thorough investigation of convolution operators that are naturally associated to vector measures. We characterize those convolution operators that are weakly compact and compact on Ll(G) and C(G) as well as those that are p summing, (1 ≤ p ≤ ∞) and nuclear on C(G).
The spectral theorem in Banach algebras
Authors: Plafker Stephen
Published: May 2009

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.
STEPHEN GREENBLATT, NEW HISTORICISM, AND CULTURAL HISTORY, OR, WHAT WE TALK ABOUT WHEN WE TALK ABOUT INTERDISCIPLINARITY
Authors: MAZA SARAH
Published: May 2009

Michael Warner, a literary critic with a keen sense of history, wrote in 1987 that “New Historicism is a label that historians don't like very much because they understand something different by historicism. But nobody's asking historians….” This essay is an answer to questions nobody asked me, questions about interdisciplinarity and the differences between literary critical and historical practices. A return to historically informed literary criticism, which many critics still consider a dominant trend in the profession, emerged in the early 1980s following the publication of Stephen Greenblatt's acclaimed Renaissance Self-Fashioning (1980). Reacting as it did against the decontextualized abstractions of New Criticism and Deconstruction, the movement soon labeled New Historicism sought to breathe new life into canonical texts by relating them to non-literary texts and social practices of their day. This historicist inclination should have formed the basis for a coming together of the movement's practitioners with historians interested in literary representations. But no such merger has occurred: New Historicists evince little interest in the systematic, archivally based study of history, and historians have at best shown indifference to the work of Greenblatt and his followers.
Potential accuracy of object localization with multilateration systems
Authors: Chernyak Victor
Published: May 2009

Multilateration (MLAT) systems and wide area MLAT (WAM) systems are particular cases of multisite (multistatic) radar systems (MSRSs): passive MSRSs (PMSRSs) with known expected signal waveforms. One of the most stringent requirements on an MLAT system is a very high accuracy of target (emitter) localization. In view of this, the potential accuracy of emitter localization (PAEL) based on Cramer–Rao inequality is important. Its dependence on system geometry and time of arrival (TOA) measurement accuracy allows choosing reasonable system geometry and requirements on TOA measurements. PAEL for MLAT and WAM systems with different geometry is considered, including systems proposed for the Marco Polo airport in Venice, Italy. The possibility of velocity determination using PAEL for landing and taking off aircrafts is also discussed. The concept of PAEL permits one to analyze joint measurements of different signal parameters and target coordinates. The effect of additional elevation angle measurements on PAEL in the WAM system for the Marco Polo airport is shown.
On Certain Types of Solution of the Equation of Heat Conduction
Authors: Paterson Stewart
Published: May 2009

Let f(x, y, z, t) satisfy the equation

For certain purposes, particularly in connection with the propagation of a boundary of fusion etc., it is of interest to discover solutions of (1) which permit the equation:

to be solved explicitly in the form:

This suggests the examination of solutions of the type

where

and f, ϕ, ψ are functions to be determined. To save repetition, Roman capitals denote arbitrary constants throughout.
Integrals Involving E-Functions and Modified Bessel Functions of the Second Kind
Authors: Ragab F. M.
Published: May 2009

The two following formulae are to be established.

If R (m ± n) > 0, | amp z | < π,

If p ≧ q + 1, R(k ± n + 2αr) > 0, r = 1, 2, …, p, | amp z | < π,

For other values of p and q the formula holds if the integral is convergent.
On Infinite Series of Sets
Authors: Ellis David
Published: May 2009

We continue our studies (2, 3, 4, 5) of the algebraic, geometric, and analytical similarities and contrasts between Boolean algebras and the real field. In this note we contrast the convergence of series in set algebras with that in the real field.
Note on Three-Dimensional Lie Groups
Authors: Patterson E. M.
Published: May 2009

The object of this note is to construct a set of real three-dimensional Lie groups such that every real three-dimensional Lie group is locally isomorphic with some group in the set. The construction is effected by first finding canonical forms for the constants of structure of real three-dimensional Lie algebras; these canonical forms give rise to certain bilinear forms, and the Lie groups are obtained as linear groups isomorphic with groups of automorphisms which leave these bilinear forms invariant.
Integrals Involving Products of Modified Bessel Functions of the Second Kind
Authors: Macrobert T. M.
Published: May 2009

The formula to be proved is
Some Results involving Hypergeometric and E-Functions
Authors: Rathie C. B.
Published: May 2009

Recently R. S. Varma [11] gave the generalisation

for the Laplace transform

Since

(1) gives (2) when k = − m + ½.

We shall represent (1) by

and as usual, (2) will be denoted by
On Packings of Spheres in Hilbert Space
Authors: Rankin R. A.
Published: May 2009

A point x in real Hilbert space is represented by an infinite sequence (x1, x2, x3, …) of real numbers such that

is convergent. The unit “sphere“ S consists of all points × for which ‖x‖ ≤ 1. The sphere of radius a and centre y is denoted by Sa(y) and consists of all points × for which ‖x−y‖ ≤ a.
A Generalization of the Inequality of the Arithmetic-Geometric Means
Authors: Hunter John
Published: May 2009

The main result in this paper, contained in Theorem 1, is a generalisation of the inequality of the arithmetic-geometric means. A result of a similar character has been proved by Siegel (2). The present result gives an improvement in the inequality in the case when the variables involved are not all distinct, whereas Siegel's result does not. The theorem is used in § 3 to obtain a result in connection with totally real and positive algebraic integers.
A General Theory of One-dimensional Local Rings
Authors: Northcott D. G.
Published: May 2009

The development of the theory of local rings has been greatly stimulated by the importance of the applications to algebraic geometry, but it is none the less true that this stimulus has produced a theory which, on aesthetic grounds, is somewhat unsatisfactory. In the first place, if a local ring Q arises in the ordinary way from a geometric problem, then Qwill have the same characteristic as its residue field. It is partly for this reason that our knowledge of equicharacteristic local rings is much more extensive than it is of those local rings which present the case of unequal characteristics. Again, in the geometric case, the integral closure of Q in its quotient field will be a finite Q-module. Here, once more, we have a special situation which it would be desirable to abandon from the point of view of a general abstract theory.
A Theorem in Operational Calculus and some Integrals involving Legendre, Bessel and E-Functions
Authors: Rathie C. B.
Published: May 2009

In this paper we prove a theorem in Operational Calculus and use it to evaluate a few infinite integrals involving Legendre, Bessel and E-functions. We write

when

and

when

(2) is a generalisation of (1) as given by Meijer [2] and it reduces to (1) when v = ±½ by virtue of the relation
Integrals involving Products of Bessel Functions
Authors: Ragab F. M.
Published: May 2009

The first formula to be proved is

where R(z)>0.
Some Bessel Function Integrals
Authors: MacRobert T. M.
Published: May 2009

The basic formula to be proved is

where p≧q + 1, z ≠0; | amp z | < π, R(n)>0, r = 1, 2,…,p. For other values of pand qthe result holds if the integral converges. From this formula some results, involving Bessel functions and Confluent Hypergeometric functions, will be deduced.
Matrices Associated with Fractional Hankel and Fourier Transformations
Authors: Guinand A. P.
Published: May 2009

Several writers (4), (6), (7), (9) have used orthogonal expansions in discussing properties of Fourier transformations, and Kober (3) has used such expansions to derive fractional Fourier and Hankel transformations. In 1950 Barrucand (1) noted a reciprocity holding between the coefficients in the expansions in Laguerre polynomials of pairs of functions which are transforms with respect to the kernel J0(2x½).
On a Conjecture of Hanna Neumann
Authors: Neumann B. H.
Published: May 2009

In what follows, groups are written additively, and commutators are denoted by brackets:

A group is metabelian if it satisfies the law

It is conceivable, though not plausible, that this law is equivalent to a law, or a set of laws, in only three variables, or even two. The present note shows that this is not the case.
On Recurrence Formulae
Authors: Macrobert T. M.
Published: May 2009

The recurrence formulae for the Bessel, Legendre, hypergeometric and other such functions can all be related to each other by means of the E-functions. In this paper it will be shown how, starting from known recurrence formulae for the hypergeometric function, others can be derived. The E-function formulae are deduced in § 2, and the others in § 3.
Asymptotic formulae for linear oscillations
Authors: Atkinson F. V.
Published: May 2009

A number of formulae are known which exhibit the asymptotic behaviour as t→∞ of the solutions of

The aim of thisnote is to unify a group of such formulae, relating to the case in which F(t) iS on the whole positive, and suitably continuous though not necessarily analytic.
Certain expansions involving E-functions
Authors: Singh V. N.
Published: May 2009

The E-functions were defined by MacRobert [3] in 1937; they are denoted by E (p; αr: q; ps. z).

In § 3 of this paper, I prove a new expansion for E(p; αr: q; ps: z) which is similar to an expansion due to MacRobert [2], viz.,

where
On strong Rieszian summability
Authors: Glatfeld Martin
Published: May 2009

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.
On certain relations between products of bilateral hypergeometric series
Authors: Shukla Harishanker
Published: May 2009

Darling [3] in 1932 and Bailey [2] in 1933 gave certain theorems on products of hypergeometric series. Again in 1948 Sears [4] used the relation which expresses the series in terms of M other series of the same type to derive transformations between products of both basic and ordinary hypergeometric series. In this paper I give certain general theorems on products of bilateral hypergeometric series together with some of their interesting special cases.
The dynamical stresses produced in a thick plate by the action of surface forces
Authors: Fulton J. , Sneddon I. N.
Published: May 2009

The first discussion of the propagation of elastic waves in a thick plate was given by Lamb [1] for the two-dimensional problem of a harmonic wave travelling in a direction parallel to the medial plane of the plate. Lamb derived equations relating the thickness of the plate to the phase velocities of two types of wave, one symmetric with respect to the medial plane and the other antisymmetric. The symmetric modes of propagation introduced by Lamb have been studied by Holden [2] and the antisymmetric modes have been studied by Osborne and Hart [3]. More recently Pursey [4] has shown how the amplitude of the disturbance is related to a given distribution of stress, varying harmonically with time, applied to the free surfaces of the plate; two types of source are considered by Pursey, one producing a two-dimensional field of the Lamb type, and the other having circular symmetry about an axis normal to the surface of the plate.
On stratifications of Mittag-Leffler's transcendents
Authors: Wintner Aurel
Published: May 2009

Guided by an observation of Hausdorff ([4]; reproduced by Whittaker and Robinson [6, pp. 177–178]), I pointed out a long time ago [7] that his “Fourier“ treatment of certain products can be systematized so as to apply to an inclusive class of infinite convolutions. Recently I noticed [8] that an appropriate application of this method supplies the following curious result on gamma-quotients:

Corresponding to every index θ on the range 0 < θ < 1, there exists on the line - ∞ < t < ∞ a monotone function μ = μθ = μθ(t) in terms of which the identity

holds on the half-plane Re z > -1 and so, in particular, on the half-line z ≧0.
Expansion of an E-function in a series of products of E-functions
Authors: Ragab F. M.
Published: May 2009

The formula to be established is

where | amp z < π, z ≠ 0, R(α + β) > R(ς) > R(alpha;) > 0, R(ς - β) > 0.

In proving (1) use will be m ade of the two following formulae.
On Square Roots and Logarithms of Self-Adjoint Operators
Authors: Putnam C. R.
Published: May 2009

All operators considered in this paper are bounded and linear (everywhere defined) on a Hilbert space. An operator A will be called a square root of an operator B if

A simple sufficient condition guaranteeing that any solution A of (1) be normal whenever B is normal was obtained in [1], namely: If B is normal and if there exists some real angle θ for which Re(Aeιθ)≥0, then (1) implies that A is normal. Here, Re (C) denotes the real part ½(C + C*) of an operator C.
A Note on the Angles in an n-Dimensional Simplex
Authors: Guinand A. P.
Published: May 2009

Three different sets of equations connecting the sums of angles in an n-dimensional simplex have been given by Sommerville [7], Höhn [5], and Peschl [6]. The equivalence of the first two sets of equations has been proved by Sprott [7].

In the present note it is shown that results are simplified if we consider averages instead of sums, and that the averages form a sequence which is self-reciprocal with respect to the transformation

The equivalence of the sets of equations is then easily proved by symbolic methods.
Abstract theory of packings and coverings. II
Authors: Świerczkowski S.
Published: May 2009

The present paper is closely related to a paper with the same title by A. M. Macbeath [3]. We use many notions which are defined there for a measure-space; nevertheless we define them once more because we consider the slightly different case of a measure-ring.
Lattice-ordered groups having at most two disjoint elements
Authors: Conrad P. F. , Clifford A. H.
Published: May 2009

Let L = L( +, v, ＾) be a lattice-ordered group, or l-group (Birkhoff [1, p. 214]). Two elements a and b of L will be called disjoint if a > 0, b > 0, and a ＾; b = 0. It is easily seen that if L does not contain two disjoint elements, then it is linearly ordered (and, of course, conversely). What can we say about Z-groups containing two but not more than two mutually disjoint elements?

Let Aand B be linearly ordered groups (o-groups), and let A ⋏ B be the cardinal sum of A and B. That is, A ⋏ B is the direct sum of A and B, and (a, b) is positive in A + B if and only if a is positive in A and b is positive in B. An l-group L containing A ⋏ B as a convex normal subgroup (or Z-ideal) is called a lexico-extension of A ⋏ B if every positive element of L not in A ⋏ B exceeds every element of A ⋏ B. It then follows (subsection 2.9 below) that L/(A ⋏ B) is an o-group. Such an l-group L is easily seen to satisfy the following condition: (D)

There exists a pair of disjoint elements in L, but no triple of pairwise disjoint elements exists in L.
Applications of the multiplication formula for the gamma function to E-function series
Authors: Macrobert T. M.
Published: May 2009

In two recent papers [1, 2] the Barnes integral for the E-functions was employed to sum a number of infinite series of E-functions. In §2 of this paper, by making use of the multiplication formula for the gamma function, the method is extended to series of E-functions of a different type.
Some recurrence relations and series for the generalised laplace transform
Authors: Bhonsle B. R.
Published: May 2009

The Laplace transform

has been generalised by Varma [4] by the relation

which reduces to (1.1) when k = -m + ½ by virtue of the identity

We shall define πk, m, λ (p) by the relation

The object of this paper is to obtain some recurrence formulae and series for πk, m, λ (p) and to use them to obtain recurrence formulae and series for MacRobert's E-function.
On Strong and Absolute Summability
Authors: Borwein D.
Published: May 2009

Suppose throughout that λ > 0, κ > - l. γ is real and that

The series is said to be

(i) summable (C, k) to s if

(ii) strongly summable (C, k + 1) with index λ, or summable |C, k + 1|λ, to s if

(iii) absolutely summable (C, k) with indices γ, λ, or summable |C, k + 1|λ, if
An embedding theorem for groups
Authors: Chehata C. G.
Published: May 2009

Let G by any given group. A homomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, where A and B need not be distinct, is called a partial endomorphism of G. When μ is defined on the whole of G, that is when A = G, we call μ a total endomorphism of G; or simply an endomorphism of G.

A partial (or total) endomorphism μ* of a supergroup G* of G is said to extend (or continue) μ if μ* is defined on a supergroup A* of A, that is, μ* is defined for at least the elements for which μ. is defined, and moreover μ* coincides with μ on A.
On the power of a prime dividing the order of the automorphism group of a finite group
Authors: Howarth J. C.
Published: May 2009

The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequality

Later Green [2] improved this estimate to

In the Present paper this will be revised, for sufficiently large h, to
A further note on dual trigonometrical series
Authors: Tranter C. J.
Published: May 2009

This note discusses the determination of the coefficients an in the dual trigonometrical series

where p = ± 1 and F(x), G(x) are prescribed functions of X. It is shown that this problem and the corresponding one in which the sines in equations (1) are replaced by cosines are easily reduced to a form in which the results I have recently given in this journal [1] may be applied.

As with my previous paper on this subject, the analysis is purely formal and no attempt is made to give precise conditions for which the solution is valid.
Some triple integral equations
Authors: Tranter C. J.
Published: May 2009

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the form

where f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.
Some automorphisms of finite nilpotent groups
Authors: Howarth J. C.
Published: May 2009

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.
Jacobians for Measures in Coset Spaces
Authors: Świerczkowski S.
Published: May 2009

Let G be a locally compact topological group, let H be a closed subgroup and let G/H be the space of left cosets = xH with the natural topology. We denote by μ a non-negative measure in G/Hdefined on the ring of Baire sets. G acts by left multiplication as a transitive group of homeomorphisms on G/H: Every t ∈ G defines the homeomorphism We write, for E ⊂ G/H, tE = . The measure μ is called stable (cf. [3], [4]) if from t ∈ G, E ⊂ G/H and μ(E) = 0 follows μ(tE) = 0. We say that μ is locally finite [3], [5] if every set of positive measure contains a subset of positive finite measure.
Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem
Authors: Cohen Eckford
Published: May 2009

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).
Two Notes on Matrices
Authors: Aitken A. C.
Published: May 2009

1. The properties of the circulant determinant or the circulant matrix are familiar. The circulant matrix C of order 4 x 4, with elements in the complex field, will serve for illustration.

The four matrix coefficients of c0, c1 c2, c3 form a reducible matrix representation of the cyclic group ℐ4, so that C is a group matrix for this. Let ω be a primitive 4th root of 1. Then Ω as below, its columns being normalized latent vectors of C,

is unitary and symmetric, and reduces Cto diagonal form thus,

where the μr, the latent roots of C, are given by

All of the above extends naturally to the n x n case.
Torsion of beams of L-cross-section
Authors: Deutsch E.
Published: May 2009

The torsion of beams of L-cross-section was studied for the first time, from a mathematical standpoint, by Kotter [1]. He solved the problem in the case of an L-section both arms of which are infinite. Some time later, Trefftz [2], in his work on the torsion of beams of polygonal cross-section, applied his method also to an infinite L-section. In 1934, Seth [3] solved the case of a beam of an L-section with only one infinite arm. In 1949, Arutyanyan [4] solved the torsion problem of an L-section that has both arms finite, but of equal length, reducing the problem to that of solving an infinite system of equations.
On Riesz summability factors
Authors: Borwein D. , Shawyer B. L. R.
Published: May 2009

1. Suppose throughout that a, k are positive numbers and that p is the integer such that k—l≦p<k. Suppose also that φ(w), ψ(w) are functions with absolutely continuous (p+1)th derivatives in every interval [a, W] and that φ(w) is positive and unboundedly increasing. Let λ ={λn} be an unboundedly increasing sequence with λ1 > 0.

Given a series and a number m ≧0, we write

otherwise,

and A(w) = AO(w).
Extension of partial endomorphisms of abelian groups
Authors: Chehata C. G.
Published: May 2009

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroups

such that L1 ƞA is the kernel of μ and

for ι = 1, 2,…, m–1.

The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.
A generalization of Sonine's first finite integral
Authors: Tranter C. J.
Published: May 2009

In this note I show that

where Jdenotes the Bessel function of the first kind of the orders and arguments indicated, n = 0, 1, 2, 3, … and the real parts of both μand v exceed — 1. This is a generalization of Sonine's first finite integral [1, p. 373] to which it reduces in the special case n = 0.
A note on the associated Legendre polynomials
Authors: Khandekar P. R.
Published: May 2009

This paper is paper gives what appears to be a new Rodrigues’ formula for the Associated Legendre Polynomials defined by [5, p. 122]

with the restriction that m is an even positive integer, which helps to evaluate some integrals. Putting m = 2k in (1.1) and replacing Pn(x) by the Gegenbauer Polynomial and using [3, p. 176]

We obtain

Putting a =v–½ in the relation [4, p. 283]

We get
Another proof of the theorems on the eigenvalues of a square quaternion matrix
Authors: Yeung Yik-Hoi Au
Published: May 2009

The nature of the eigenvalues of a square quaternion matrix had been considered by Lee [1] and Brenner [2]. In this paper the author gives another elementary proof of the theorems on the eigenvalues of a square quaternion matrix by considering the equation Gy = μȳ, where G is an n x n complex matrix, y is a non-zero vector in Cn, μ is a complex number, and ȳ is the conjugate of y. The author wishes to thank Professor Y. C. Wong for his supervision during the preparation of this paper.
Notes on two lemmas concerning the Epstein zeta-function
Authors: Diananda P. H.
Published: May 2009

1. We use Cassels's notation and define h (m, n), Q (m, n), Zh (s), Zh (1) – ZQ (1) and G (x, y) as in [1]. Rankin [5] proved that the Epstein zeta-function Zh (s) satisfies, for s ≧ 1·035, the

THEOREM. For s > 0, Zh (s) — ZQ (s) ≧ 0 with equality if and only ifh is equivalent to Q. Rankin then asked whether the theorem is true for all s > 1. Cassels [1] answered this question in the affirmative and proved further that the theorem is true for all s > 0.
On the commutativity of the radical of a group algebra
Authors: Wallace D. A. R.
Published: May 2009

Over a field of prime characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative.
Proximate order (R) of entire functions represented by Dirichlet series (IV)
Authors: Kamthan Pawan Kumar
Published: May 2009

where be an entire

function represented by a Dirichlet series whose order (R) and proximate order (R) are respectively ρ (0 < ρ < ∞) and ρ(σ). For proximate order (R) and its properties, see the paper of Balaguer [4, p. 28].
The spectra of compact operators in Hilbert spaces
Authors: West T. T.
Published: May 2009

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.
⋏-distributive Boolean matrices
Authors: Blyth T. S.
Published: May 2009

In this paper we shall be concerned with the set Mn(B) of n x n matrices whose elements belong to a given Boolean algebra B(≤, ∪,∩,').

It is well known that Mn(B) also forms a Boolean algebra with respect to the partial ordering ≼ defined by

in which union (⋎), intersection (⋏) and complementation (*) are given by
Some integrability theorems
Authors: Chen Yung-Ming
Published: May 2009

1. P. Heywood [3] proved the following theorems:

Theorem A. If 0 ≦ λ < 2, if xy−1g(x) ε L(0, π), and if

for n = 1, 2, 3,…, then the seriesis convergent.

Theorem B. If 0 ≦ λ < 1, if xy−1f(x)ε L(0, π), and if

for n = 1, 2, 3,…, then the seriesn−yanis convergent.
On integral equations involving Whittaker's function
Authors: Srivastava K. N.
Published: May 2009

Recently some inversion integrals for integral equations involving Legendre, Chebyshev, Gegenbauer and Laguerre polynomials in the kernel have been obtained [1, 2, 3, 5, 6]. In this note, two inversion integrals for integral equations involving Whittaker's function in the kernel are obtained. We shall make use of the following known integral [4, p. 402]

The results of this note are based on the following two integrals, which are derived from (1) by writing u – t = (v – t)x.

for m + 1 > 2v > – 1;

for m + 1 > 2v > – 1.
The maxit and minit of a ring
Authors: Peinado R. E. , Leavitt W. G.
Published: May 2009

In a recent paper [2] one of the authors has introduced the concept of module type of a ring, for rings with unit. The object of this paper is to generalize this concept to arbitrary rings, without assuming the existence of a unit. This is easily accomplished for rings with one-sided unit, and we shall define the type of such a ring. Theorem 2.5 gives a relation between this type and the module type of [2], and permits the immediate extension of all results in [2] to rings with one-sided unit.
The Quadratic Law of Reciprocity and the Theory of Galois Fields
Authors: Zassenhaus Hans J.
Published: May 2009

The theory of quadratic congruences modulo an integer is dominated by the Quadratic Law of Reciprocity (see § 1), which makes it possible to decide in a very short time whether a quadratic congruence

is solvable or not. The law was first proved by Gauss.* It took him over a year to obtain his first proof, which depends on a tedious lemma in elementary number theory. He subsequently obtained seven further proofs, and today more than fifty proofs are known, most of them based on the ideas of Gauss. The object of the present paper is to present a proof which is a modernised version of Gauss's seventh proof, applying the ideas of that proof to a finite set of objects, the elements of a finite or Galois field.
The Equivalence of Asymptotic Distributions under Randomisation and Normal Theories
Authors: Silvey Samuel D.
Published: May 2009

§ 1. Introduction. A problem of some interest in mathematical statistics is that of determining conditions under which the randomisation distribution of a statistic and its normal theory distribution are asymptotically equivalent, as these two distributions are used in alternative approaches to the same inference problem.
Commutative Extension of Partial Automorphisms of Groups
Authors: Chehata C. G.
Published: May 2009

Let μ be an isomorphism which maps a subgroup A of the group G onto a second subgroup B (not necessarily distinct from A) of G; then μ is called a partial automorphism of G. If A coincides with G, that is if the isomorphism is defined on the whole of G, we speak of a total automorphism; this is what is usually called an automorphism of G. A partial (or total) automorphism μ,* extends or continues a partial automorphism μ if μ* is defined for, at least, all those elements for which μ is defined, and moreover μ* coincides with μ where μ is defined.
Some Integrals involving E-Functions
Authors: Macrobert T. M.
Published: May 2009

This paper describes a simple method of evaluating integrals of a type some cases of which have been discussed by Dr. Ragab.

The following formulae are required in the proofs

If p≧q + 1, z ≠ 0,

If. p = q + 1, | z |>1 in this formula.
Distinct silicon and germanium pathways in the soil‐plant system: Evidence from banana and horsetail
Authors: Delvigne C. , Opfergelt S. , Cardinal D. , Delvaux B. , André L.
Published: May 2009

Plants strongly impact the continental silicon cycle by taking up Si and precipitating opal phytoliths which are recycled into the soil. Studying Ge incorporation, a chemical analog of Si, relative to Si may provide a useful tracer of Si pathways. However, Ge uptake and transport through plants and the impact on Ge/Si of phytoliths remain poorly understood. Here, we report Ge uptake and accumulation and Ge/Si fractionation in all plant parts and solutions from: (1) hydroponic banana, (2) in situ sampled banana, and (3) horsetails. We further combine these data with δ29Si from banana plants. Our data reconcile opposite conclusions drawn from previous studies on Ge uptake and pathways. No discrimination of Ge occurred at the root‐solution interface. Banana and horsetails were shown to accumulate Ge in roots: a previous study provided evidence of low Ge/Si ratios in root phytoliths which contrasts with high bulk Ge/Si ratios in roots we report here. This suggests that Ge is organically trapped in roots. Consequently, shoots display lower Ge/Si ratios, without fractionation between shoot parts since Ge would follow transpiration stream as silicon, and is not discriminated between shoot parts. This contrasts with the two‐step discrimination against heavy Si isotopes, at the root‐solution interface, and then within the shoots. The soil composition (clays versus Fe oxides) has a leading role on the Ge/Si signatures of plants which may in turn impact on the Si and Ge fluxes to the global ocean.
Effect of self-orderly textured back reflectors on light trapping in thin-film microcrystalline silicon solar cells
Authors: Sai Hitoshi , Kondo Michio
Published: May 2009

Light trapping is a key issue to boost the efficiency of thin-film Si solar cells including μc-Si:H. In this work, effect of textured back reflectors on light trapping in μc-Si:H cells has been investigated with self-orderly patterned Al substrates obtained by anodic oxidation. By increasing the period of the patterned substrates from 0 to 1.1 μm, the short circuit current densities of 1-μm-thick μc-Si:H cells on the patterned substrates significantly increase from 18 to over 24 mA/cm2, which is attributed to the improved light trapping in the near infrared region. The optimum period for light trapping is determined to be around 1 μm for 1-μm-thick μc-Si:H cells. In addition, it has been demonstrated that the optimized patterned substrate enhances light trapping effect in a wide range of cell thicknesses of 0.5–4 μm. It is also confirmed that natural surface texturing during μc-Si:H film growth improves light scattering for flat substrates but deteriorates the superior light scattering property of the well-designed substrates.
Calculating the responsivity of a resonant-cavity-enhanced Si1−xGex/Si multiple quantum well photodetector
Authors: Das Mukul K. , Das N. R.
Published: May 2009

In this paper, the responsivity of a resonant-cavity-enhanced SiGe/Si multiple quantum well photodetector has been theoretically investigated. The present model considers the effects of material parameters of Si1−xGex and recombination of carriers confined by the potential barriers at the heterointerfaces of Si1−xGex/Si quantum wells on the detector responsivity for different Ge-contents (x). The effect of electric field due to applied bias on the emission of confined carriers from quantized subbands has also been included in the analysis. Results show that though high Ge-content is required to enhance the responsivity of the SiGe/Si photodetector for long-haul communication (at 1.3 and 1.55 μm), the confinement of holes at the heterointerfaces may be significant to affect the responsivity adversely. Studies at 1.3 and 1.55 μm also show that the responsivity of the photodetector for a particular Ge-content in the active layer can be increased by suitable choice of well parameters and applied bias.
Effects of nitrogen incorporation on a direction of a surface band bending investigated by polarity of terahertz electromagnetic waves in GaAs1−xNx epitaxial layers
Authors: Takeuchi Hideo , Yanagisawa Junichi , Hashimoto Jun , Nakayama Masaaki
Published: May 2009

We have investigated the polarity of the terahertz electromagnetic wave from GaAs1−xNx epitaxial layers with x=0.43% and 1.53% in order to clarify the effects of nitrogen incorporation on the direction of the surface band bending using an undoped GaAs/n-type GaAs(i-GaAs/n-GaAs) epitaxial layer structure and a semi-insulating GaAs (SI-GaAs) crystal as reference samples. A numerical calculation on the basis of the Boltzmann–Poisson model clarifies that the conduction band of the i-GaAs/n-GaAs sample bends upward at the surface region, which indicates that photogenerated electrons flow into the inside. In the measurement of the terahertz wave, it is observed that the polarity of the terahertz wave from the SI-GaAs crystal is the same as that of the i-GaAs/n-GaAs sample, namely, the present SI-GaAs crystal has an upward band bending. In contrast, the terahertz-wave polarity is reversed in the GaAs1−xNx samples in spite of the relatively small nitrogen mole fraction; namely, the GaAs1−xNx samples have a downward band bending. The above-mentioned terahertz-wave polarity reversal is attributed to the phenomenon that conduction-band bottom is considerably lowered by the band anticrossing peculiar to GaAs1−xNx, which results in approaching the conduction-band bottom to the surface Fermi level. This modifies the direction of the surface band bending connected with the polarity of the terahertz wave.