bluuebunny/crossref_metadata_2025 huggingface dataset:
A problem of Hooley in Diophantine approximation
Authors: Harman Glyn
Published: May 2009

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).

Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequality

Here
Spectra of conjugated ideals in group algebras of abelian groups of finite rank and control theorems
Authors: Tushev Anatolii V.
Published: May 2009

Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have finite torsion-free rank if it has a finite series in which each factoris either infinite cyclic or locally finite; its torsion-free rank r0(Γ) is then defined to be the number of infinite cyclic factors in such a series.
Primary decompositions over domains
Authors: Fuchs Laszlo , Lee Sang Bum
Published: May 2009

Throughout, R denotes a commutative domain with 1, and Q (≠R) its field of quotients, which is viewed here as an R-module. The symbol K will stand for the R-module Q/R, while R denotes the multiplicative monoid R/0.
The kernel relation for a strict extension of certain regular semigroups
Authors: Petrich Mario
Published: May 2009

Let R be a regular semigroup and denote by (R) its congruence lattice. For , the kernel of pis the set ker . The relation K on (R) defined by λKp if ker λ = ker p is the kernel relation on (R). In general, K is a complete ∩-congruence but it is not a v-congruence. In view of the importance of the kernel-trace approach to the study of congruences on a regular semigroup (the trace of p is its restriction to idempotents of R), it is of considerable interest to determine necessary and sufficient conditions on R in order for K to be a congruence. This being in general a difficult task, one restricts attention to special classes of regular semigroups. For a background on this subject, consult [1].
Semigroup identities on units of integral group rings
Authors: Dokuchaev Michael A. , Gonçalves Jairo Z.
Published: May 2009

Let U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.
A note on the representation theory of the Hecke algebra of type F4
Authors: Pallikaros C. A.
Published: May 2009

In [4] Dipper and James investigated the representation theory of Hecke algebras of type Bn, H(Bn). Using the results in [4] and exploiting the fact that the Hecke algebra of type F4, denoted by H(W), contains two copies of H(B3) certain right ideals of H(W) will be constructed in this paper. These right ideals will be proved to be irreducible on the assumption that H(W) is semisimple.
A note on the diophantine equation x2 + 7 = yn
Authors: Le Maohua
Published: May 2009

In this note we prove that the equation x2 + 1 = yn, x, y, n ɛ ℕ, n>2, has no solutions (x, y, n)with 2 × y. Moreover, all solutions (x, y, n)of the equation with 2| y satisfy n < 5. 106 and y < exp exp exp 30.
Moufang affine buildings have Moufang spherical buildings at infinity
Authors: van Maldeghem H. , van Steen K.
Published: May 2009

We show in a direct and elementary way that the spherical building at infinity of every rank 3 affine building which satisfies Tits' Moufang condition, is itself a Moufang building. This result is also true for higher rank affine buildings by Tits' classification [4].
The efficiency of PSL(2, p)3 and other direct products of groups
Authors: Campbell C. M. , Miyamoto I. , Robertson E. F. , Williams P. D.
Published: May 2009

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15].
Inequalities for the maximal eigenvalue of a nonnegative matrix
Authors: Yeh Lina
Published: May 2009

Two-sided bounds are obtained for the maximal eigenvalue of a positive matrix by iterating computations of row sums. The result provides an algorithm for approximating the maximal eigenvalue of a nonnegative matrix.
Square integrable highest weight representations
Authors: Neeb Karl-Hermann
Published: May 2009

If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.
Quadratic algebras with few relations
Authors: Zhang James J.
Published: May 2009

Throughout V will be a finite dimensional vector space over a field k and T(V) will denote the tensor algebra over V. For simplicity the symbol ⊗ will be omitted in the writing of the elements of T(V). Let be a basis of V ordered by Xi<Xi+1 for all i. Then we order the non-commutative monomials and 1 ≤ is ≤ n for s = 1,…, l} lexicographically from the left. D. Anick [1, p. 652] defines the high term of an element b in T(V) to be the highest monomial appearing in b. As a consequence of [1,3.2], if the set of the high terms of homogeneous relations is combinatorically free in the sense of no overlap ambiguities, then the connected algebra has global dimension 2. The purpose of this note is to prove this result and more for quadratic algebras under other hypotheses on the relations.
Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components
Authors: Pogorzały Zygmunt
Published: May 2009

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.
Multiplicity of Boardman strata and deformations of map germs
Authors: Ballesteros J. J. Nuño , Saia M. J.
Published: May 2009

We define algebraically for each map germ f:Kn,0→Kp, 0 and for each Boardman symbol i=(i1,…,ik) a number ci(f) which is -invariant. If f is finitely determined, this number is the generalization of the Milnor number of f when p = 1, the number of cusps of f when n = p = 2, or the number of cross caps when n = 2, p = 3. We study some properties of this number and prove that, in some particular cases, this number can be interpreted geometrically as the number of Σi points that appear in a generic deformation of f. In the last part, we compute this number in the case that the map germ is a projection and give some applications to catastrophe map germs.
Dehn functions and complexes of groups
Authors: Brick Stephen G. , Corson Jon M.
Published: May 2009

We study the Dehn functions of the fundamental groups of complexes of groups. We study a function known asthe Howie function, which has a natural geometric formulation. We make use of the Howie function to obtain an upper bound for the Dehn function of the complex of groups. And we show a connection between the Howie function and actions on complexes.
Bounded mean curvature isometric immersions of a compact Riemannian manifold with images contained in a tube
Authors: Carreras Francisco J. , Giménez Fernando , Miquel Vicente
Published: May 2009

We characterize some isometric immersions of a compact Riemannian manifold into a tube of Sn(λ) or CPn(λ) (in fact, in some more general spaces in the real case) around a totally geodesic Sq(λ) or CPq(λ) respectively, with the norm of the mean curvature of the immersion bounded from above. This bound depends on the radius of the tube, and is related with the mean curvature of its boundary.
On the spectrum of n-tuples of p-hyponormal operators
Authors: Duggal B. P.
Published: May 2009

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε ℋU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the ℋW(p + 1/2) operator Â by A = |A|1/2U |A|l/2 Let Â = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.
Extension operators for Sobolev spaces commuting with a given transform
Authors: Burenkov Viktor , Schulze Bert-Wolfgang , Tarkhanov Nikolai N.
Published: May 2009

We consider a real-valued function r = M(t) on the real axis, such that M(t) < 0 for t < 0. Under appropriate assumptions on M, the pull-back operator M* gives rise to a transform of Sobolev spaces Ws.p (-∞, 0) that restricts to a transform of Ws.p(-∞, ∞). We construct a bounded linear extension operator Ws.p(-∞, 0) → Ws.p(−∞, ∞), commuting with this transform.
Free distributive P-algebras: a new approach
Authors: Katriňák Tibor
Published: May 2009

It is well known (Lee [13]) that the class of all distributive p-algebras B = Bω is a variety and that the class of all subvarieties of B forms a chain

where B-i is the trivial class, B0 is the class of Boolean algebras, and B1 is the class of Stone algebras.
An operator satisfying Dunford's condition (C) but without bishop's property (β)
Authors: Miller T. L. , Miller V. G.
Published: May 2009

ForXa complex Banach space andUan open subset of the complex plane С, letO (U, X)denote the space of analyticX-valued functions defined onU. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the spaceXmay be viewed as simply the constants inO(U, X). Every bounded operatorTonXinduces a continuous mappingTUonO(U, X)given by(Tuf)(λ)=(λ – T)f(λ)for every f eO(U, X)and λe U. Corresponding to each closed F ⊂ С there is also an associated analytic subspaceXT(F) = X ∩ran(7c//F). For an arbitraryT e L(X), the spacesXT(F)are T-invariant, generally non-closed linear manifolds inX.
A lower bound on the homological bidimension of a non-unital C*-algebra
Authors: Ermert Olaf
Published: May 2009

Let A be a C*-algebra. For each Banach A-bimodule X, the second continuous Hochschild cohomology group H2(A, X) of A with coefficients in X is defined (see [6]); there is a natural correspondence between the elements of this group and equivalence classes of singular, admissible extensions of A by X. Specifically this means that H2(A, X) ≠ {0} for some X if and only if there exists a Banach algebra B with Jacobson radical R such that R2 = {0}, R is complemented as a Banach space, and B/R ≅ A, but B has no strong Wedderburn decomposition; i.e., there is no closed subalgebra C of B such that B ≅ C © R. In turn this is equivalent to db A ≥ 2, where db A is the homological bidimension of A; i.e., the homological dimension of A#, the unitization of A, as an,A-bimodule [6, III. 5.15]. This paper is concerned with the following basic question, which was posed in [7].
Cone length of the exterior join
Authors: Marcum Howard J.
Published: May 2009

The cone length Cl(f) of a map f: X → Y is defined to be the least number of attaching maps possible in a conic (or iterated mapping cone) structure for f. Cone length is a homotopy invariant in the sense that if φ: X → X and ρ: Y → Y are homotopy equivalences then Cl (ρ°f°φ) = Cl(f). Furthermore Cl(f) depends only on the homotopy class of f. It was shown by Ganea [8] that the cone length of the map * → X coincides with the strong Lusternik-Schnirelmann category of X as a space (see Proposition 1.6 below). Recent work of Cornea ([3]–[6]) is much concerned with cone length and its role in critical point theory. For example, let f be a smooth real valued function on a manifold triad (M; V0, V1) with V0 ≠ θ. Under certain conditions, if f has only “reasonable” critical points then it must have at least Cl(V0↪M) of them (see [6]).
Bifidogenic effect and stimulation of short chain fatty acid production in human faecal slurry cultures by oligosaccharides derived from lactose and lactulose
Authors: Cardelle-Cobas Alejandra , Fernández María , Salazar Nuria , Martínez-Villaluenga Cristina , Villamiel Mar , Ruas-Madiedo Patricia , de los Reyes-Gavilán Clara G
Published: May 2009

Bifidogenic effect and stimulation of short chain fatty acid (SCFA) production by fractions of oligosaccharides with a DP⩾3 and Gal β(1-6) linkages synthesised from lactose or lactulose by Pectinex Ultra SP-L and Lactozym 3000 L HP G were evaluated in human faecal slurries. Results were compared with those obtained for the commercial oligosaccharide mixture Vivinal®-GOS. Quantitative real-time PCR showed that all galacto-oligosaccharide (GOS) fractions stimulated slightly higher bifidobacteria growth than lactose, lactulose and Vivinal®-GOS. GOS fractions promoted the production of total SCFA and acetic acid in a similar way to Vivinal®-GOS and more than glucose, lactose and lactulose. In conclusion, oligosaccharides derived from lactose and lactulose may represent a suitable alternative to lactulose for the design of new functional food ingredients.
Prediction of total quarter milk somatic cell counts based on foremilk sampling
Authors: Wellnitz Olga , Doherr Marcus G , Woloszyn Marta , Bruckmaier Rupert M
Published: May 2009

Determination of somatic cell count (SCC) is used worldwide in dairy practice to describe the hygienic status of the milk and the udder health of cows. When SCC is tested on a quarter level to detect single quarters with high SCC levels of cows for practical reasons, mostly foremilk samples after prestimulation (i.e. cleaning of the udder) are used. However, SCC is usually different in different milk fractions. Therefore, the goal of this study was the investigation of the use of foremilk samples for the estimation of total quarter SCC. A total of 378 milkings in 19 dairy cows were performed with a special milking device to drain quarter milk separately. Foremilk samples were taken after udder stimulation and before cluster attachment. SCC was measured in foremilk samples and in total quarter milk. Total quarter milk SCC could not be predicted precisely from foremilk SCC measurements. At relatively high foremilk SCC levels (>300×103cells/ml) foremilk SCC were higher than total quarter milk. At around (50–300)×103cells/ml foremilk and total quarter SCC did not differ considerably. Most interestingly, if foremilk SCC was lower than 50×103cells/ml the total quarter SCC was higher than foremilk SCC. In addition, individual cows showed dramatic variations in foremilk SCC that were not very well related to total quarter milk SCC. In conclusion, foremilk samples are useful to detect high quarter milk SCC to recognize possibly infected quarters, only if precise cell counts are not required. However, foremilk samples can be deceptive if very low cell numbers are to be detected.
Piezoelectric materials parameters for piezoelectric thin films in GHz applications
Authors: Muralt P. , Conde J. , Artieda A. , Martin F. , Cantoni M.
Published: May 2009

Piezoelectric thin films have existing and promising new applications in microwave filter technologies. The final performance depends on many parameters, and very specifically on the materials properties of each involved material. In this article, materials and properties for thin-film bulk acoustic wave resonators are discussed on some selected issues: the piezoelectric coefficients and acoustic losses of AlN, the relation of the first one with microstructural parameters, the inclusion of parasitic elements, and the merits of and problems with ferroelectric materials.
Solutions of the Diffusion Equation for a Medium Generating Heat
Authors: Sneddon Ian N.
Published: May 2009

The problem of solving the equation of thermal conduction for cases in which heat is generated in the interior of the medium under consideration arises frequently in physics and engineering. It occurs, for instance, when we consider the diffusion of heat in a solid undergoing radioactive decay (1) or which is absorbing radiation (2). Complications of a similar nature arise when there is a generation or absorption of heat in the solid as a result of a chemical change-for example, the hydration of cement (3). The particular case in which the rate of generation of heat is independent of the temperature arises in the theory of the ripening of apples and has been discussed by Awberry (4).
Generalisations of Some Hypergeometric Function Transformations
Authors: Macrobert T. M.
Published: May 2009

The formulae

where γ = ½ (α + β + l), and

where γ = 1 + β, were given by Gauss (Ges. Werke, iii, pp. 225, 226). It is here proposed to find the corresponding expressions for the hypergeometric function when γ has general values (not zero or negative integral). These will be derived in section 2 by applying Lagrange's expansion

where

and that root of equation (4) in x is taken which is equal to λ when w = 0. Two generalisations of Whipple's Transformation will be obtained in section 3.
The Representations of Lie Algebras of Prime Characteristic
Authors: Zassenhaus Hans
Published: May 2009

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are

(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.

(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf.[4]).

(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf.[4]).

(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf.[6], [2], [4]).
The Asymptotic Distributions of Statistics arising in Certain Non-Parametric Tests
Authors: Silvey Samuel D.
Published: May 2009

In a previous paper [5] the equivalence of randomisation and normal theory distributions of linear combinations was discussed. In the present paper we discuss the asymptotic randomisation distributions of statistics used in analysis of variance and in a closely related problem which includes, in particular, the “problem of m-rankings“. Kruskal [4] has studied the first of these questions in the case where observations are replaced by ranks.
A Reduction Formula
Authors: Watson G. N.
Published: May 2009

In this paper I construct a reduction formula for the integral

the formula connects any three consecutive members of a set I0, I1, I2, …, Im. We regard m as given, and, in order to avoid wasting time over trivialities, we postulate that the constants a, α, β, γ δ (which are not restricted to be real) have real parts large enough to ensure (i) that the integrals In under consideration and the integrals related to them which will be introduced subsequently are all absolutely convergent, and (ii) that, in all the partial integrations which will be effected, the integrated parts vanish at both limits.
A Problem associated with a Particular Markov Chain
Authors: Silvey Samuel. D.
Published: May 2009

The problem discussed in this paper is a rather specialised problem associated with a particular random walk with reflecting barriers. Such processes are discussed in particular by Feller (2). It happens that probabilities at any time associated with the system here discussed are relatively easy to determine and it is this fact which makes the given solution possible. This problem arose in certain investigations concerned with computing machines carried out by Mr. A. E. Roy of the Astronomy Department in this University and I wish to thank Mr. Roy for bringing it to my notice.
General Products of Two Finite Cyclic Groups
Authors: Yacoub K. R.
Published: May 2009

Groups that can be represented as the product of two proper subgroups have been studied extensively; one of the latest contributions is a paper by Wielandt (8), in which references to previous work can be found. In the case where the two proper subgroups have only the unit element in common, we adopt the term ‘general product’introduced by Neumann (1).
Integrals Involving E-Functions and Associated Legendre Functions
Authors: Macrobert T. M.
Published: May 2009

The formulae to be proved are as follows.

If p ≧ q + 1, R(l + m) > 0, R(αr − l − m + n)> −1, R(αr − l − m − n)> 0, r = 1,2,…p,

If p ≧ q + 1, R(l >0, R(l+m >0, R(αr − l − m + n)> −1,
Prime Ideals in Matrix Rings
Authors: Sands Arthur D.
Published: May 2009

Let R be a ring and let Rn be the complete ring of n × n matrices with coefficients from R.
The Structure of Normal Inverse Semigroups
Authors: Preston G. B.
Published: May 2009

In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.
On The Closed Graph Theorem
Authors: Robertson Alex. P. , Robertson Wendy
Published: May 2009

The closed graph theorem is one of the deeper results in the theory of Banach spaces and one of the richest in its applications to functional analysis. This note contains an extension of the theorem to certain classes of topological vector spaces. For the most part, we use the terminology and notation of N. Bourbaki [1], contracting “locally convex topological vector space over the real or complex field” to “convex space”; here we confine ourselves to convex spaces.
On Semi-Special Permutations I
Authors: Yacoub K. B.
Published: May 2009

In an earlier paper [1] on groups which are the products of two finite cyclic groups with trivial intersection, certain permutations, called “semi-special”, played a certain role. The permutation π of the numbers 1, 2,…, n is semi-special if† πn=n, and if, for every y ε [n],

is again a permutation, namely a power (depending on y) of π.
Remarks on the Upper Centralc Series of a Group
Authors: McLain D. H.
Published: May 2009

Following, for example, Kurošs [8], we define the (transfinite) upper central series of a group G to be the series

such that Zα + 1/Za is the centre of G/Zα, and if β is a limit ordinal, then If α is the least ordinal for which Zα =Zα+1=…, then we say that the upper central series has length α, and that Zα= His the hypercentre of G. As usual, we call G nilpotent if Zn= Gfor some finite n.
A Note on Integer Solutions of the Diophantine Equation x2-dy2=1
Authors: Hunter John
Published: May 2009

In the equation

dis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given by

and, in particular, all positive integer solutions are given by
On the growth of the cyclotomic polynomial in the interval (0, 1)
Authors: Erdös P.
Published: May 2009

Let

be the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved that

and Emma Lehmer [5] showed that An>cn1/3 for infinitely many n; in fact she proved that n can be chosen as the product of three distinct primes. I proved [3] that there exists a positive constant q such that, for infinitely many n

and Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),
The construction of branched covering Riemann surfaces
Authors: Rankin R. A.
Published: May 2009

In some recent work on uniformization [2], I found it necessary to consider a regular branched covering Riemann surface Ȓ of a given Riemann surface Rf, where Rf is an unlimited branched, but not necessarily regular, covering surface of a portion Rz of the extended complex z-plane Z(2-sphere). The branching of Ȓ over Rf had to be chosen so that Ȓ was regular over Rz, since the uniformization of the functions on Rf is then simpler; in particular, the Schwarzian derivative is then a single-valued function of z.
On a Theorem on Ordered Groups
Authors: Chehata C. G.
Published: May 2009

The following work establishes a new proof of the theorem: Every archimedean ordered group is abelian. This theorem has been proved differently by many authors. It was first proved by O. Hölder [2]. A second proof has been given by H. Cartan [1]: he uses the topology which is naturally introduced in the group by its order.
A recursion formula for the coefficients in an asymptotic expansion
Authors: Wright E. M.
Published: May 2009

Many authors have proved results deducing an asymptotic expansion of

for large from the behaviour of f(t), when f(t) is regular in an appropriate part of the complex t-plane. For example, if, for some k > 0 and some Am, αm

for all large such that R(t) > C, then, as ⃗ ∞ in a suitable sector in the z-plane, we have

where Z is an appropriate value of z1/z.
On coefficient fields
Authors: Geddes A.
Published: May 2009

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.
An extension of a theorem on the equivalence between absolute Rieszian and absolute Cesàro summability
Authors: Borwein D.
Published: May 2009

be a given series and let

With Flett [4], we say that the series is summable , q real, if

where Summability |C, k, 0|1 is identical with absolute Cesàro summability (C, k), or summability |C, k|, as defined by Fekete [3].
Integration of E-functions with respect to their parameters
Authors: Macrobert T. M.
Published: May 2009

A number of integrals of E-functions with respect to their parameters have been given by Ragab [1, 5]. Two further integrals of this type are given in § 2 and § 3.

In § 4 it is shown that these can be employed to sum series of products of E-functions.
On some infinite series involving the zeros of Bessel functions of the first kind
Authors: Sneddon Ian N.
Published: May 2009

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sums

and

where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.
Measures equivalent to the Haar measure
Authors: Świerczkowski S.
Published: May 2009

We call two measures equivalent if each is absolutely continuous with respect to the other (cf. [1]). Let G be a locally compact topological group and let μ be a non-negative Baire measure on G (i.e. μ is denned on all Baire sets, finite on compact sets and positive on open sets). We say that μ is stable if μ (E)=0 implies μ(tE)=0 for each t ∈ G. A. M. Macbeath made the conjecture that every stable non-trivial Baire measure is equivalent to the Haar measure. In this paper we prove the following slightly stronger result:

Theorem. Every stable non-trivial measure defined on Baire sets and finite on some open set is equivalent to the Haar measure.
Integrals involving E-functions
Authors: Rathie C. B.
Published: May 2009

In this paper three integrals involving E-functions are evaluated in terms of E-functions. The formulae to be established are:

where n is a positive integer, | args z < π, R(γ ± m ÷ ½) > 0, αρ+ν = (2γ + ν)/2n (ν = 1, 2, …, 2n), αρ+2n+i = (γ + m - ½ + i)/n, αρ+3n+i = (γ - m - ½ + i)/n, βα+i = (γ + κ + i)/n, βα+ν+i = (γ - κ + i)/n(i = 1, 2, …, n).

where n is a positive integer, |arg z| < π, R(λ±μ±ν) > 0, αp+i+1 = (λ + μ + ν + i)/n, αp+n+i+1 = (λ - μ + ν + i)/n, αp+2n+i+1 = (λ + μ - ν + i)/n, αp+3n+i+1 = (λ - μ - ν + i)/n (i = 0, 1, 2, …, n - 1), βa+i+1 = (2λ + j)/2n (j = 0, 1, 2, …, 2n - 1).

where n is a positive integer, R(λ) > ½, |arg z| < π, αp+i+1 = (2λ - 1 + i)/2n (i = 0, 1, 2, …, 2n-1), βq+j+1 = (λ + μ j)/n, βq+n+j+1 = (λ - μ + j)/n (j = 0, 1, 2, …, n-1).
Evaluation of an E-Function when two of the upper Parameters differ by an Integer
Authors: Macrobert T. M.
Published: May 2009

If p ≧ q + 1, [1, p. 353]

where, if p = q + 1, | z | < 1. The dash in the product sign indicates that the factor for which t = αr – αr + 1 is omitted, while the asterisk indicates that the parameter αr−αr + 1 is omitted.
The Divisibility of Divisor Functions
Authors: Rankin R. A.
Published: May 2009

For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].
A Note on Regular Coverings of Closed Orientable Surfaces
Authors: Mennicke Jens
Published: May 2009

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.

Let the closed orientable surfaceFhof genushbe a covering ofF2and letand f be the fundamental groups respectively. Thenis a subgroup of f of indexn = h − 1. A covering is called regular ifis normal in f.

Conversely, letbe a normal subgroup of f of finite index. Then there is a uniquely determined regular coveringFhsuch thatis the fundamental group ofFh. The coveringFhis an orientable surface. Since the indexnofin f is supposed to be finite,Fhis closed, and its genus is given byn=h − 1.

The fundamental group f can be defined by

.
Factorization of Matrices and Minkowski's Conjecture
Authors: Macbeath A. M.
Published: May 2009

Let Rn denote real Euclidean space of n dimensions. If

define , and (as usual) so that, by the inequality of arithmetic and geometric means,

Let Λ0 be the integer lattice, consisting of those points in Rn whose co-ordinates are integers. A non-singular n × n matrix M will be called a Minkowski matrix, if, for any point a ∈ Rn, there exists a point x ∈ Λ0 such that

It was shown by Minkowski that, when n = 2, every non-singular matrix is a Minkowski matrix, and that, for general n, every rational non-singular matrix is a Minkowski matrix. Minkowski is also said to have conjectured that every non-singular matrix is a Minkowski matrix, whatever the value of n. For n = 3, this was proved by Remak [5], and a much simpler proof was given later by Davenport [2]. For n = 4, it was proved by Dyson [3], who used a method similar to that of Remak and Davenport, but required the methods of algebraic topology to deal with some of the complications which arise in the higher dimension. Since this method depends also on the reduction of quadratic forms, it is quite likely that it might fail for higher values of n even if the topological difficulties could be overcome. Therefore, an alternative proof for n = 3, due to Birch and Swinnerton-Dyer [1], is of some interest, though in this dimension it is more complicated than Davenport's proof. A good deal of their analysis applies to general n, and they showed that for all n there is a neighbourhood of the unit matrix I that consists entirely of Minkowski matrices.
Some Series and Recurrence Relations for MacRobert's E-Function
Authors: Bhonsle B. R.
Published: May 2009

Since [3]

where , a result involving Wk, m, (z) can be transformed into a result involving MacRobert's E-function. Further this result can be generalised with the help of the known integrals for E-functions.

The object of this paper is to use this method to obtain some recurrence relations and series for MacRobert's E-functions.
On the natural representation of the symmetric groups
Authors: Farahat H. K.
Published: May 2009

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,
On a set of normal subgroups
Authors: Macdonald I. D.
Published: May 2009

The commutator [a, b] of two elements a and b in a group G satisfies the identity

ab = ba[a, b].

The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.

The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation

(A) [g, x]Y = [g, y]−1[g, xy]

holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that

{g}G = {g, K(g)}.
Group algebras with radicals of square zero
Authors: Wallace D. A. R.
Published: May 2009

Over a field of 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. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself.
Some infinite integrals involving E-functions
Authors: Saxena R. K.
Published: May 2009

A function φ(p) is operationally related to h(t) when they satisfy the integral equation

provided that the integral is convergent and R(p)> 0.

As usual, we shall denote (1) by the symbolic expression

φ(p) ≑ h(t).

The object of this paper is to evaluate some infinite integrals involving E-functions by the methods of the operational calculus. Most of the results obtained are believed to be new.
An identity in combinatorial analysis
Authors: Mordell L. J.
Published: May 2009

In a very recent paper [1], Basil Gordon discusses generalizations of Jacobi's identity

where x and z are complex numbers and |x| <1. He notes that some of its consequences, inter alia Euler's formula

are of interest in number theory and combinatory analysis. He proves the apparently new and striking result

where |s|<1, and also considers the possibility of generalizations. His methods are algebraic and quite simple, but perhaps do not make obvious what underlies such formulae. It may be worth while to do so, especially since the details become simpler and the presentation more perspicuous. The method given here assumes no more knowledge than his does, although the new proof is expressed in terms of theta-functions, in simple properties of which, formulae such as (3) have their origin. Further, (3) appears in a slightly more symmetrical form.
Positive matrices and eigenvectors
Authors: Putnam C. R.
Published: May 2009

For i, j = 1, 2, …, let aij be real. A matrix A = (aij) will be called positive (A>0) or non-negative (A≧0) according as, for all i and j, aij>0 or aij≧0 respectively. Correspondingly, a real vector x = (x1, x2, …) will be called positive (x>0) or non-negative (x≧0) according as, for all i, xi>0 or x≧0. A matrix A is said to be bounded if ∥ Ax ∥ ≦M ∥ x ∥ holds for some constant M, 0 ≦ M < ∞, and all x in the Hilbert space H of real vectors x = (x1, x2, …) satisfying . The least such constant M is denoted by ∥ A ∥. If x and y belong to H, then (x, y) will denote as usual the scalar product Σxiyi. Whether or not x is in H, or A is bounded, y = Ax will be considered as defined by

whenever each of the series of (1) is convergent.
Some further properties of a q-analogue of MacRobert's E-function
Authors: Agarwal R. P.
Published: May 2009

1. Introduction. Recently, I gave an analogue [1] of the MacRobert's E-function [4[ in the form

where the symbol denotes that a similar expression with a and β interchanged is to be added to the expression following it. It has since then been generalized by N. Agarwal [2], who defined and studied the q-analogue of the generalized E-function. In this paper I give some further properties of the Eq-function.
On generating points of a lattice in the region
Authors: Foster D. M. E.
Published: May 2009

A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), where

for some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.

If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the form

and Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.
Residuation theory and Boolean matrices
Authors: Blyth T. S.
Published: May 2009

We begin this paper by considering a Boolean algebra as a lattice which is relatively pseudo-complemented (i.e., residuated with respect to intersection) and give, in this case, certain properties of the equivalences of types A, B and F(as introduced by Molinaro [1]). We then show how these results carry over to the case of Boolean matrices, which form a Boolean algebra residuated also with respect to matrix multiplication. Other properties of matrix residuals are established and we conclude with three algebraic characterisations of invertible Boolean matrices.
Note on a paper of Tsuzuku
Authors: Farahat H. K.
Published: May 2009

In [2], Tosiro Tsuzzuku gave a proof of the following:

THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.

For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.
The Stress on the boundary of an elastic half-plane in which body forces are acting
Authors: Sneddon Ian N.
Published: May 2009

In this note we consider the problem of determining the stress on the boundary y = 0 of the elastic half-plane y ≡ 0 when there are prescribed body forces acting in the interior and the boundary is free from applied stress. Expressions for the components of stress at a general point of the half-plane when the imposed body force is concentrated at a single point have been derived by Melan [1], Sneddon [2] and Green [3], each author making use of a different method.
Schreier systems in free products
Authors: Dey I. M. S.
Published: May 2009

In 1927 Schreier [8] proved the Nielsen-Schreier Theorem that a subgroup H of a free group F is a free group by selecting a left transversal for H in F possessing a certain cancellation property. Hall and Rado [5] call a subset T of a free group F a Schreier system in F if it possesses this cancellation property, and consider the existence of a subgroup H of F such that a given Schreier system T is a left transversal for H in F.
Generalized Cesaro means of order −1
Authors: Maddox I. J.
Published: May 2009

A series ∑an is said to be summable (C, — 1) to s if it converges to s and nan = o(1) [8]. It is well known that this definition is equivalent to tn→s (n→∞), where tn = sn + nan, sn = ao + … + an. The series is summable | C, — 1 | to s if the sequence t = {tn} is of bounded variation (t ∈ B.V.), i.e. ∑ |; ▲tn |; = ∑ | tn - tn-1 | < ∞, and ∑ ▲tn = lim tn = s. An equivalent condition is ∑ | an |; < ∞, ∑an = s and ∑ | ▲(nan) | < ∞. For, suppose that ∑an = s | C, - 1 |. Since {sn} is the sequence of (C, 1)-means of {tn} and since | C, 0 | ⊂ | C, 1 |, we have ∑ | an | < ∞ and ∑an = s whence ∑ | ▲(nan) | < ∞. Conversely, ∑ | an | < ∞, ∑an = s and ∑ | ▲(nan) | < ∞ imply t ∈ B.V. and ∑▲n = s + lim nan. But lim nan = 0, since ∑ | an | < ∞.
Certain fundamental congruences on a regular semigroup
Authors: Howie J. M. , Lallement G.
Published: May 2009

In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.
Isomorphic congruence groups and Hecke operators
Authors: Rankin R. A.
Published: May 2009

Let G, H, K be groups such that G is normal in K and G ⊆ H ⊆ K. Let I(H, K) be the set of inner automorphisms of K restricted to H; thus α ∊ I(H, K) if and only if, for some κ∊ K, α(h) = k-1hk for all h ∊ H. Let φ be an isomorphism of H/G onto a subgroup Hφ/G of K/G. An isomorphism Φ of H onto H(φ) is called an extension of ø if

Φ(h)G = φ(hG) for all h∊H.
The structure of cancellative power-free groups
Authors: Geddes A. , Walker R. G.
Published: May 2009

The definition of a power-free group will be found in [1]. It is a partial algebraic system which, roughly speaking, may be thought of as a group in which (with the exception of the identity) squares and higher powers of an element are not defined.

It has been shown [1, Theorem 3.3] that the usual cancellation laws need not hold in a power-free group. When these laws do hold, the power-free group is called cancellative. In this paper we shall be solely concerned with cancellative power-free groups and the term ‘power-free group’ is to be understood to mean ‘cancellative power-free group’.
Non-cancellative power-free groups
Authors: Walker R. G.
Published: May 2009

A. Geddes [1, Theorem 3.3] has shown that the partial algebraic system which he has called a power-free group need not be cancellative. In other words, there exist power-free groups containing at least one element a with the property that ab can equal ac when b ≠ c. In the present paper we propose to study the structure of such non-cancellative power-free groups, and we shall in fact obtain a complete solution to this problem.
Fourier Expansions
Authors: Stewart C. A.
Published: May 2009

The theory of operational solutions of differential equations in applied mathematics suggests a method of developing the theory of Fourier and allied series that is simpler for ordinary applications than the classical development. It may be useful to those whose interests lie in such applications rather than in the deeper analytical processes associated with this subject.