bluuebunny/crossref_metadata_2025_split huggingface dataset:
Eta-products which are simultaneous eigenforms of Hecke operators
Authors: Biagioli Anthony J. F. 
Published: May 2009

The Dedekind eta-function is defined for any τ in the upper half-plane bywhere x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a functionwhere N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ bywhen a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier seriesthen we define a Hecke operator Tp bywhereand
Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections
Authors: Drewnowski Lech , Florencio Miguel , Paúl Pedro J. 
Published: May 2009

The research presented in this paper started by extending a theorem of Swetits [18]about barrelledness of subspaces of metrizable AK-spaces to general AK-spaces of scalar sequences. The extension reads as follows.(1) A subspace λ0 of a barrelled AK-space λ such that λ0 ⊃ φ is barrelled if and only if its dualis weak* sequentially complete. If in addition λ0 is monotone, then it is barrelled if and only ifequals the Köthe dualof λ0.As an easy consequence of this extension, we obtained the following result of Elstrodt and Roelcke [8, Corollary 3.4].(2) If λ is a barrelled monotone AK-space, then also its subspace ℒ(λ), consisting of all sequences in λ with zero-density support, is barrelled.
On an arithemtical inequality
Authors: Srinivasan S. 
Published: May 2009

Here we extend an arithmetical inequality about multiplicative functions obtained by K. Alladi, P. Erdős and J. D. Vaaler, to include also the case of submultiplicative functions. Also an alternative proof of an extension of a result used for this purpose is given.Let Uk, for integral k, denote the set {1,2,…, k}, and Vk denote the collection of all subsets of Uk. In the following, all unspecified sets like A,…, are assumed to be subsets of Uk. Let σ = {Si} and τ = {Tj} be two given collections of subsets of Uk. SetandLet ′ denote complementation in Uk (but for in the proof of (3) where it denotes complementation in C). For any collection p of subsets of Uk, let p′ denote the collection of the complements of members of p.
Symmetries of surfaces: an extension of Kulkarni's theorem
Authors: Jones Gareth A. 
Published: May 2009

In [6] Kulkarni considered the set of values of g for which a given finite group G acts faithfully as a group of orientation-preserving self-homeomorphisms of a compact, connected, orientable surface σg of genus g. Let us denote this set by (G). Then Kulkarni showed that there exists a positive integer Kdepending only on the order d = |G| of G, the exponent e= exp G of G and the structure of a Sylow 2-subgroup G2 of G, satisfying:Theorem 1. (Kulkarni [6]) (G) consists of all but finitely many non-negative integers g ≡ 1 mod K.
On 𝓕-subnormal subgroups and Frattini-like subgroups of a finite group
Authors: Ballester-Bolinches A. , Pérez-Ramos M. D. 
Published: May 2009

Throughout the paper we consider only finite groups.J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.
On dilation equations and the Hölder continuity of the de Rham functions
Authors: Pan Yibiao 
Published: May 2009

AbstractWe use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.1. Consider the following dilatation equationwhere |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfywhere is the Fourier transform of f, andwhich immediately leads to
Representations and positive definite functions on topological semigroups
Authors: Baker J. W. , Lashkarizadeh-Bami M. 
Published: May 2009

AbstractA number of theorems are established about positive definite functions and representations of certain topological semigroups. In particular we establish theorems which show that measurable positive definite functions and measurable representations can each be decomposed into the sum of two parts one of which is continuous and the other of which is “small”.
A note on comonotonic additivity
Authors: Parker June M. 
Published: May 2009

AbstractThe axiom of comonotonic independence for a preference ordering was introduced by Schmeidler [9]. It leads to the comonotonic additivity for the functional representing the preference ordering, which is necessarily a Choquet integral.The aim of this paper is to illuminate the concepts of comonotonicity, comonotonic independence and comonotonic additivity. For example the seemingly weaker condition of weak comonotonic independence used by Chateauneuf in [2] is seen to be equivalent to comonotonic independence. Comonotonic additivity is characterized as additivity on chains of sets. From this the characterization of Choquet integrals in [4], [1], [8] follows easily.
Substitutional lemma for G-spaces of 1-dimensional groups
Authors: Pérez Juan Antonio 
Published: May 2009

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of Xwhere EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extensionwhere G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order
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 inequalityHere
Semigroup identities on units of integral group rings
Authors: Dokuchaev Michael A. , Gonçalves Jairo Z. 
Published: May 2009

AbstractLet 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 diophantine equation x2 + 7 = yn
Authors: Le Maohua 
Published: May 2009

AbstractIn 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

AbstractWe 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].
Multiplicity of Boardman strata and deformations of map germs
Authors: Ballesteros J. J. Nuño , Saia M. J. 
Published: May 2009

AbstractWe 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

AbstractWe 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

AbstractWe 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.
Extension operators for Sobolev spaces commuting with a given transform
Authors: Burenkov Viktor , Schulze Bert-Wolfgang , Tarkhanov Nikolai N. 
Published: May 2009

AbstractWe 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 chainwhere B-i is the trivial class, B0 is the class of Boolean algebras, and B1 is the class of Stone algebras.
Generalisations of Some Hypergeometric Function Transformations
Authors: Macrobert T. M. 
Published: May 2009

The formulaewhere γ = ½ (α + β + l), andwhere 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 expansionwhereand 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]).
A Reduction Formula
Authors: Watson G. N. 
Published: May 2009

In this paper I construct a reduction formula for the integralthe 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.
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,
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 seriessuch 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 equationdis 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 byand, 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

Letbe the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved thatand 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 nand Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),
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 offor 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, αmfor all large such that R(t) > C, then, as ⃗ ∞ in a suitable sector in the z-plane, we havewhere Z is an appropriate value of z1/z.
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 letWith Flett [4], we say that the series is summable , q real, ifwhere 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 sumsandwhere 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 letwhere 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. Ifdefine , 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 thatIt 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 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 identityab = 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)}.
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 equationprovided that the integral is convergent and R(p)> 0.As usual, we shall denote (1) by the symbolic expressionφ(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.