bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Metanilpotent Varieties of Groups
Authors: Gupta Narain 
Published: December 2010

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.
On Baer Involutions of Finite Projective Planes
Authors: Cofman Judita 
Published: December 2010

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.
Fixed Points of Automorphisms of Compact Riemann Surfaces
Authors: Moore M. J. 
Published: December 2010

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.
On a Generalization of Alternative Rings
Authors: Morgan Raymond V. 
Published: December 2010

Bruck and Kleinfeld [3] proved that any alternative ring with characteristic prime to 2 must satisfy the identitywhere the associator (x,y,z) is defined by (x, y, z) = (xy)z – x(yz) and . Linearization of the identity (x2, y, z) = 2x · (x, y, z) yields for characteristic prime to 2 an equivalent identity(1)Using the right alternative law (x, y, z) = –(y, x, z) and the flexible law (x, y, z) = –(z, y, x) which is satisfied in any alternative ring we obtain(2)and(3)
Weak Normality and Related Properties
Authors: Ball Eugene S. 
Published: December 2010

In [5], Zenor stated the definition of weakly normal. In the main, since weak normality does not imply either normality or regularity, various properties related to either normality or regularity will be considered in the context of weak normality.Throughout this paper the word “space” will mean topological space. The closure of a point set M will be denoted by cl(M). The closure of a point set M with respect to the subspace K will be denoted by cl(M, K).Definition 1. A space S is weakly normal provided that if is a monotonically decreasing sequence of closed sets in S with no common part and H is a closed set in S not intersecting H1, then there is a positive integer N and an open set D such that HN ⊂ D and cl(D) does not intersect H.
A Note on a Generic Hyperplane Section of an Algebraic Variety
Authors: Kuan Wei-Eihn 
Published: December 2010

Let V be an irreducible algebraic variety of dimension > 1 defined over a field k in an affine n-space over k, and let H be the generic hyperplane defined by u0 + u1X1 + … + unXn = 0, where u0, u1, …, un are indeterminates over k. It is well known that:(1) if V is normal over k, then V ∩ H is normal over k(u0, …, un) (see [6]), and(2) if P is in the intersection V ∩ H, then P is absolutely simple on V ∩ H over k(u0, …, un) if and only if P is absolutely simple on V over k (see [2; 5]).In this paper we prove:(1′) if V is factorial over k, then V ∩ H is also factorial over k(u0, …, un) (Theorem 3), and(2′) if P is in V ∩ H, then P is normal on V ∩ H over k(u0, …, un) if and only if P is normal on V over k (Theorem 2).
Gentle Perturbations of with Application to
Authors: Derzko N. A. 
Published: December 2010

The theory of gentle perturbations was introduced by Friedrichs [3] as a tool to study the perturbation theory of the absolutely continuous spectrum of a self-adjoint operator H0 and developed in an abstract form by Rejto [7; 8]. Two examples of gentle structures are well knowTn. In the first of these, the gentle operators have Hölder continuous complex or operator-valued kernels, and in the second, the kernels are Fourier transforms of L1 functions [4].The gentle structure has traditionally been verified in the case when H0 is in its spectral representation, that is, when H0 is the simple differentiation operator. This is not the natural setting for the second example mentioned above where one should consider the simple differentiation operator in a suitable L2-space and perturbations with L1 kernels.
Finite Subsghemes of Group Schemes
Authors: Shatz Stephen S. 
Published: December 2010

If G is an ordinary group and H is a non-empty subset of G, then there are two elementary criteria for H to be a subgroup of G. The first and more general is that the mapping H × H → G × G → G, via 〈x, y〉 ⟼ xy–1 factor through H. The second is that H be finite and closed under multiplication.In the category of group schemes, if one writes down the hypotheses for the first criterion in diagram form, one can supply the proof by a suitable translation of the classical arguments. The only point that causes any difficulty whatsoever is that one must assume that the structure morphism πH: H → S (S is the base scheme) is an epimorphism in order to factor the identity section through H. The second criterion is also true for group schemes under a mild finite presentation hypothesis. It is our aim to provide a simple proof for the following theorem.
Suspension of the Lusternik-Schnirelmann
Category
Authors: Gilbert William J. 
Published: December 2010

Let cat be the Lusternik-Schnirelmann category structure as defined by
Whitehead [6] and let be the
category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for
connected X. Dually, if X is simply
connected,1. We work in the category of based
topological spaces with the based homotopy type of CW-complexes and based
homotopy classes of maps. We do not distinguish between a map and its
homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5;
1] which unify the definitions of the numerical homotopy
invariants akin to the Lusternik-Schnirelmann category.
Full-Completeness in Weighted Spaces
Authors: Summers W. H. 
Published: December 2010

That the class of fully complete (B-complete) locally
convex spaces has an essential role has been well-established (see, e.g.,
Collins [2], Husain [12], and Pták
[16]). However, it is still the case that very little is
known concerning full-completeness in function spaces except when implied by
formally stronger properties (e.g., Fréchet), and our aim in this article is
to shed some light in this direction. Our approach is to use the unifying
notion of the weighted spaces CV0(X) discussed in [19; 20], since
such spaces provide a general setting for the study of virtually all
continuous function spaces encountered in analysis.The results to be presented here divide naturally into three sections with §
2 being consigned the preliminary definitions and theorems. In § 3 we obtain
a necessary condition for full-completeness in a class of function spaces,
and consider certain consequences of this result.
Bicommutators of Cofaithful, Fully Divisible Modules
Authors: Beachy John A. 
Published: December 2010

We define below a notion for modules which is dual to that of faithful, and a notion of “fully divisible” which generalizes that of injectivity. We show that the bicommutator of a cofaithful, fully divisible left R-module is isomorphic to a subring of Qmax(R), the complete ring of left quotients of R.In recent papers, Goldman [2] and Lambek [3] investigated rings of left quotients of a ring R constructed with respect to torsion radicals. It is known that every ring of left quotients of R is isomorphic to the bicommutator of an appropriate injective left R-module. We investigate below subrings of rings of quotients which are determined by radicals rather than torsion radicals, and show that any such ring can be constructed as the bicommutator of a fully divisible left R-module.
Solution to a Problem of Spector
Authors: Lachlan A. H. 
Published: December 2010

In [6, p. 586] Spector asked whether given a number e there exists a unary partial function from the natural numbers into {0, 1} with coinfinite domain such that for any function ƒ into {0, 1} extending it is the case thatWe answer this question affirmatively in Corollary 1 below and show that can be made partial recursive (p.r.) with recursive domain. The reader who is familiar with Spector's paper [6] will find the new trick that is required in the first paragraph of the proof of Lemma 2 below.From one point of view, this is a theorem about trees which branch twice at every node. We shall formulate a generalization which applies to trees which branch n times at every node.
On Modules of Singular Submodule Zero
Authors: Cateforis Vasily C. , Sandomierski Francis L. 
Published: December 2010

In this paper we generalize to modules of singular submodule zero over a ring of singular ideal zero some of the results, which are well known for torsion-free modules over a commutative integral domain, e.g. [2, Chapter VII, p. 127], or over a ring, which possesses a classical right quotient ring, e.g. [13, § 5].Let R be an associative ring with 1 and let M be a unitary right R-module, the latter fact denoted by MR. A submodule NR of MR is large in MR (MR is an essential extension of NR) if NR intersects non-trivially every non-zero submodule of MR; the notation NR ⊆′ MR is used for the statement “NR is large in MR” The singular submodule of MR, denoted Z(MR), is then defined to be the set {m ∈ M| r(m) ⊆’ RR}, where
The Topological Degree of A-Proper Maps
Authors: Wong H. Ship Fah 
Published: December 2010

Recently several fixed-point theorems have been proved for new classes of non-compact maps between Banach spaces. First, Petryshyn [15] generalized the concept of compact and quasi-compact maps when he introduced the P-compact maps and proved a fixed-point theorem for this class of maps. Then in [6], de Figueiredo defined the notion of G-operator to unify his own work on fixed-point theorems and that of Petryshyn. He also proved that the class of G-operators was a fairly large one.We notice the following facts: (i) The essential idea in the above cases is that if certain finite-dimensional “approximations” of the map have fixed points, then the map has a fixed point; (ii) One of the tools used in proving fixed-point theorems in the finite-dimensional case is the Brouwer degree and its generalization to maps of the type Identity + Compact in [8].
Some Theorems on the Structure of Nearly Equicontinuous Transformation Groups
Authors: Roberson Fred A. 
Published: December 2010

The purpose of this paper is to extend the theorems in [3; 7] to uniform spaces and to prove some additional theorems. These results are related to [4; 5]. Notation and definitions are as in the book [2]. For a general reference on nets see [6]. All topological spaces are assumed to be Hausdorff.THEOREM 1. Let (X, T, Π) be a transformation group, where X is a locally compact, locally connected, uniform space. Let E denote the set of all points at which T is equicontinuous and N = X – E. Let N be closed totally disconnected and each orbit closure in N be compact and let E be connected. Then N contains at most two minimal sets. (Note: We will assume that N ≠ ∅ so that N will contain at least one minimal set.)
On the “Third Definition” of the Topology on the Spectrum of a C*-Algebra
Authors: Gardner L. Terrell 
Published: December 2010

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the formwhere the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.
The Arithmetic of the Quasi-Uniserial Semigroups without Zero
Authors: Behrens Ernst August 
Published: December 2010

An element a in a partially ordered semigroup T is called integral ifis valid. The integral elements form a subsemigroup S of T if they exist. Two different integral idempotents e and f in T generate different one-sided ideals, because eT = fT, say, implies e = fe ⊆ f and f = ef ⊆ e.Let M be a completely simple semigroup. M is the disjoint union of its maximal subgroups [4]. Their identity elements generate the minimal one-sided ideals in M. The previous paragraph suggests the introduction of the following hypothesis on M.Hypothesis 1. Every minimal one-sided ideal in M is generated by an integral idempotent.
On 3-Manifolds with Sufficiently Large Decompositions
Authors: Heil Wolfgang 
Published: December 2010

In [6] it is shown that two (compact) orientable 3-manifolds which are irreducible, boundary irreducible and sufficiently large are homeomorphic if and only if there exists an isomorphism between the fundamental groups which respects the peripheral structure. In this note we extend this theorem to reducible 3-manifolds.Any compact 3-manifold M has a decomposition into prime manifolds [1; 4].1Here the connected sum of two bounded manifolds N1, N2 is denned by removing 3-balls B1 B2 in int N1, int N2, respectively, and glueing the resulting boundary spheres together. The M1's which occur in the decomposition (1) are either irreducible or handles (i.e., a fibre bundle over S1 with fibre S2). If (1) contains a fake 3-sphere, we assume it to be Mn.
On a Theorem of Bovdi
Authors: Parmenter M, M. 
Published: December 2010

If p is a prime, we call an element x ≠ 1 of a group G a generalized p-element if, for every n ≧ 1, there exists r ≧ 0 such that xpr ∈ Gn, where Gn is the nth term of the lower central series of G. Bovdi [1] proved that if
G is a finitely generated group having a generalized p-element, and if ∩nΔn(Z(G) = 0 where Δ(Z(G)) is the augmentation ideal, then G is residually a finite p-group.We recall that if R is a ring, then the nth dimension subgroup of G over R, denoted by Dn(R(G)), is defined to be {g | g – 1 ∈ Δn(R(G))}. In this note, we show that if G is finitely generated, then ∩nDn(Zp∧(G)) = 1 ⇔ ∩nΔn(Zp∧ (G)) = 0 ⇔ G is residually a finite p-group.
Mercerian Conditions for the Method (F, dn)
Authors: Skerry H. B. 
Published: December 2010

This paper sets forth conditions sufficient that the generalized Lototsky method (F, dn) be regular and Mercerian. If the dn's are real and of constant sign, then the conditions are also necessary. Moreover, it follows that if f is a polynomial, then under the same conditions the method (f,dn) is equivalent to the Sonnenschein method generated by f . Various related results are also given.Definition 2.1. Let f be a nonconstant function holomorphic on the closed unit disk and let be a complex sequence with f (l) + dn ≠ 0. Suppose
Singular Integrals on Ultraspherical Series
Authors: Dunkl Charles F. 
Published: December 2010

One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G.
Baer Subplanes in Finite Projective and Affine Planes
Authors: Cofman Judita 
Published: December 2010

Let π be a projective or an affine plane ; a configuration C of π is a subset of points and a subset of lines in π such that a point P of C is incident with a line I of C if and only if P is incident with I in π. A configuration of a projective plane π which is a projective plane itself is called a projective subplane of π, and a configuration of an affine plane π’ which is an affine plane with the improper line of π‘ is an affine subplane of π‘.Let π be a finite projective (respectively, an affine) plane of order n and π0 a projective (respectively, an affine) subplane of π of order n0 different from π; then n0 ≦ . If n0 = , then π0 is called a Baer subplane of π. Thus, Baer subplanes are the “biggest” possible proper subplanes of finite planes.
The Augmentation Terminals of Certain Locally Finite Groups
Authors: Gruenberg K. W. , Roseblade J. E. 
Published: December 2010

Let G be a group and ZG be the integral group ring of G. We shall write 𝔤 for the augmentation ideal of G; that is to say, the kernel of the homomorphism of ZG onto Z which sends each group element to 1. The powers gλ of 𝔤 are defined inductively for ordinals λ by 𝔤λ = 𝔤μ𝔤, if λ = μ + 1, and otherwise. The first ordinal λ for which gλ = 𝔤λ+1 is called the augmentation terminal or simply the terminal of G. For example, if G is either a cyclic group of prime order or else isomorphic with the additive group of rational numbers then gn > 𝔤ω = 0 for all finite n, so that these groups have terminal ω.The groups with finite terminal are well-known and easily described. If G is one such, then every homomorphic image of G must also have finite terminal.
Familles De Valuations A Caractere Compact
Authors: Wong Cécile M. 
Published: December 2010

Soit Ω une famille de valuations d'un corps commutatif K. Dénotons par , l'anneau de la valuation v, et Mv =, l'idéal maximal de l'anneau Rv. Soit l'anneau défini par la famille Ω; nous supposons toujours Ω tel que K soit le corps des fractions de R, et tel que Rv ≠ Rw dès que v ≠ w (v et w ∈ Ω).Pour tout x ∈ K, posons et munissons Ω de la topologie la plus faible pour laquelle les VΩ(X) sont des ouverts. On dit que la famille Ω est à caractère compact (respectivement, fini) si pour chaque élément non nul x de K, VΩ(X) est quasi compact (respectivement, est un ensemble fini). S'il n'y a pas de confusion possible quant à la famille Ω, on écrit V(x) plutôt que VΩ(X).
Extended Chromatic Polynomials
Authors: Sobczyk Andrew , Gettys James O. 
Published: December 2010

Let G be a finite graph with non-empty vertex set (G) and edge set (G) (see [2]). Let λ be a positive integer. Tutte [5] defines a λ-colouring of G as a mapping of (G) into the set Iλ = {1, 2, 3, …, λ} with the property that two ends of any edge are mapped onto distinct integers. The elements of Iλ are commonly called “colours.” If P(G, λ) represents the number of λ-colourings of G, it is well known that P(G, λ) can be expressed as a polynomial in λ. For this reason P(G, λ) is usually referred to as the chromatic polynomial of G.The chromatic polynomial P(G, λ) was first suggested as an approach to the four-colour conjecture. To quote Tutte [5]: ”… many people are specially interested in the value λ = 4.
On the Plethysm of S-Functions
Authors: Plunkett S. P. O. 
Published: December 2010

Many authors have studied the theory and calculation of the plethysms of S-functions. The significance of S-functions lies in their relationship [9] to the characters of the continuous groups, and plethysms play a crucial role in the determination of branching rules associated with the decomposition of a continuous group into its subgroups [2 ; 14 ; 16]. Tables have been published for the plethysm {λ{ ⊗ {μ{, where (λ) and (μ) are any partitions of l and m, respectively, with Im ≦ 18. These tables have been drawn up both with [1] and without [5] the aid of computers and some results are also known for Im > 18 [3; 4; 7].The method given here deals with the notion of q-quotients and is based on a theorem of Littlewood's relating these to plethysms of S-functions with symmetric power sums.
A Generalization of Commutative and Alternative Rings
II
Authors: Kleinfeld Erwin 
Published: December 2010

We shall call a linear function on the elements of a ring R
skew-symmetric if it vanishes whenever at least two of the variables are
equal. Here we shall study rings R of characteristic not 2
which satisfy the following two identities:1(2)is skew-symmetric.Both of these identities hold in alternative rings. The fact that
F(w, x, y, z) is skew-symmetric in alternative rings is
an important tool in the study of such rings. It is also obvious that both
identities hold in commutative rings. But unlike other recent
generalizations of commutative and alternative rings it turns out that there
exist simple, finite dimensional algebras of degree two which are neither
alternative nor commutative and satisfy (1) and (2).
Polynomes a Valeurs Entieres
Authors: Cahen Paul-Jean 
Published: December 2010

Soit A un anneau intègre de corps des fractions
K, on s'interesse ici aux polynômes de
K[X] qui prennent sur A leurs valeurs
dans A. Ils forment bien évidemment un sous anneau de
K[X] qui contient A[X], nous le notons
As.Cet article reprend, dans un langage moderne, les idées de deux publications
assez anciennes, dues à Polya [3] et Ostrowsky
[4], en 1919. Les auteurs montraient que si A
est principal, As est un A-module libre qui admet sur
A une base formée de polynômes de degré croissant ; si
A est plus généralement l'anneau des entiers d'un corps
de nombres ils mettaient en évidence une suite d'idéaux fractionnaires de
A. En exposant le cas de l'anneau A
principal, pour un anneau de valisation discrète dans le premier paragraphe,
puis en généralisant aux anneaux de Dedekind dans notre second paragraphe,
nous pouvons énoncer de façon plus explicite, que As est un A-module projectif, que l'on peut
décomposer canoniquement en somme d'idéaux fractionnaires de
A.