bluuebunny/crossref_metadata_2025 huggingface dataset:
Equivalence Locale des Structures de Contact de Codimension
Un
Authors: Kumpera A.
Published: December 2010

Une structure de contact sur une variété M est un champ
d'éléments de contact tangents Σ dont le système
caractéristique Δ est régulier. Lorsque codim Σ = 1 une
telle structure est définie localement par une forme de Pfaff
ω ou encore par l'équation ω = 0. Le
théorème classique de Darboux nous donne un critère d'équivalence de deux
Pfaffiens ω et ω′ à l'aide de l'invariant
classe ω. Cet invariant est cependant trop fort pour
déterminer l'équivalence des structures de contact Σ et
Σ′ définies par ω = 0 et
ω′ = 0 respectivement. D'une façon précise, les
structures Σ et Σ′ peuvent être
équivalentes sans que les Pfaffiens ω et
ω′ le soient. Dans cette note nous considérons un
invariant plus faible, classe Σ, attaché à une structure de
contact Σ et nous montrons qu'il détermine l'équivalence en
codimension 1 (cf. corollaire 3).
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 that

and

It 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.
Characteristically Nilpotent Algebras
Authors: Ravisankar T. S.
Published: December 2010

Our aim in this paper is to extend (Theorem 1.7) to general algebras a classical result of Lie algebras due to Léger and Togo [6]. This extension requires, in turn, extension to general algebras of the concept of characteristically nilpotent algebras introduced by Dixmier and Lister [3] for Lie algebras. Based on this extended concept, we introduce in § 2 a new concept of radical (and semisimplicity) for general algebras and Lie triple systems. We study in some detail the consequences of the newly introduced concepts, furnishing necessary examples. With a stronger notion of characteristically nilpotent Mal'cev algebra arising out of these concepts, we obtain (Proposition 3.6) for such an algebra the parallel to the Leger-Tôgô result mentioned at the outset. In § 4, we deal with a further generalization of the concept of characteristic nilpotency leading to extension of very recent results of Chao [1] and Tôgô [12].
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 that

We 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 form

where 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.
On Torsion-Free Discrete Subgroups of PSL(2, C) with Compact Orbit Space
Authors: Best L. A.
Published: December 2010

In 1883 Poincaré [13] recognized that the discrete subgroups of PSL(2, C) could be extended from their natural action on the complex plane to acting on hyperbolic 3-space and he attempted to analyze these groups in an analogous manner to his classical treatment of Fuchsian groups, with fundamental polyhedra playing the role of the fundamental polygons for Fuchsian groups. This approach, however, did not lead very far, perhaps not surprisingly when one appreciates the close connection between the geometry of these groups and the topology of 3-manifolds. Since that time the state of knowledge remained essentially unchanged until 1964 when work by Ahlfors [1] and soon afterwards by Bers [3] revitalized the subject of Kleinian groups. The modern approach tends to use analytic methods, although recently Marden [11] has had considerable success in carrying forward Poincaré's geometric approach.
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 if

is 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 the η Function of Brown and Pearcy and the Numerical Function of an Operator
Authors: Salinas Norberto
Published: December 2010

Throughout this paper will denote an infinite dimensional, separable complex Hilbert space, and will denote the unit sphere of (i.e. ). Also will represent the algebra of all bounded linear operators on , and will represent the ideal of all compact operators on . Furthermore will denote the set of all (orthogonal) projections on and will denote the sublattice of consisting of all finite rank projections. In most of the cases (especially when limits are involved) will be regarded as a directed set with the usual order relation inherited from .
An Inclusion Theorem for Bohr-Hardy Summability Factors
Authors: Thorpe B.
Published: December 2010

1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank)(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B-l) and that B ⇒ A if and only if F = AB-1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an† is summable A if its sequence of partial sums is A-limitable.
On the Mackey Borel Structure
Authors: Gardner L. Terrell
Published: December 2010

Let A be a C*-algebra and a Hilbert space which is infinite dimensional and of Hilbert dimension ≧ dim π for all π ∈ Â. Suppose that the set Irr of all non-null *-representations π of A on , irreducible on the essential space , is given the relative strong topology as a subspace of Rep [2; 4; 6]. That is, the topology is that of simple convergence in with the strong topology. Finally, let ∼ denote equivalence of representations in Irr implemented by partial isometries in if and only if there exists a partial isometry with vv* ⊃ H(π1) and v*v ⊃ H(π2) satisfying π2(a) = v*π1(a)v for all a ∈ A.
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].

1

Here 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.
Orthogonal Matrices with Zero Diagonal. II
Authors: Delsarte P. , Goethals J. M. , Seidel J. J.
Published: December 2010

C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v ≡ 2 (mod 4), and the skew C-matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [24] and by Reid and Brown [21].
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
Pseudo Harmonic Measures and the Dirichlet Problem
Authors: Arsove Maynard , Leutwiler Heinz
Published: December 2010

For the case of plane regions bounded by finitely many disjoint Jordan curves, the solution of the Dirichlet problem can be expressed in terms of the classical harmonic measure of boundary arcs. At an appropriate stage in the development it is, in fact, useful to observe that the existence of such harmonic measures is equivalent to solvability of the Dirichlet problem (although one subsequently proves that all such regions are Dirichlet regions). We propose here to carry over this order of ideas to a quite general setting, in which arbitrary regions and ideal boundary structures are allowed. The counterparts of the classical harmonic measures of arcs are then harmonic functions with analogous boundary properties, but they no longer appear as measures in the boundary sets, in general. We shall refer to them as “pseudo harmonic measures”. Our main result shows how pseudo harmonic measures can be used to solve the Dirichlet problem.
Actions That Fiber and Vector Semigroups
Authors: Hanson T. H. McH.
Published: December 2010

From [2], we can derive a criterion for determining when an action of a Lie group on a locally compact space leads to a fiber bundle. Here, we present an equivalent criterion which can be stated purely in the language of actions of groups on spaces. This is Theorem I. Using this result, we are able to give a version of a result of Home [1] for dimensions greater than one. This is done in Theorem IV and Corollary IVA. In Theorem II, we show that if a vector semigroup acts on a space X, then whenever the map t ↦ tx is 1 — 1 from onto x, it is in fact a homeomorphism. Also, is a closed subset of X. This is also a version of a result in [1].
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.
The Cone = Hyperspace Property
Authors: Rogers James T.
Published: December 2010

The author has recently shown [11] that the hyperspace of subcontinua of a solenoid is homeomorphic to the cone over that solenoid. This is an interesting result, for it is the first time that the hyperspace of subcontinua of a complicated space has been recognized. This homeomorphism, moreover, is the expected map; it maps the singletons onto the base of the cone and the point corresponding to the whole space onto the vertex of the cone. We say that spaces for which such natural homeomorphisms exist have the cone = hyperspace property. In the first section we prove the following theorem.
Equidistant Loci and the Minkowskian Geometries
Authors: Phadke B. B.
Published: December 2010

The spaced of this paper is a metrization, with a not necessarily symmetric distance xy, of an open convex set D in the n-dimensional affine space An such that xy + yz = xz if and only if x, y, z lie on an affine line with y between x and z and such that all the balls px ≦ p are compact. These spaces are called straight desarguesian G-spaces or sometimes open projective metric spaces. The hyperbolic geometry is an example; a large variety of other examples is studied by contributors to Hilbert's problem IV. When D = An and all the affine translations are isometries for the metric xy, the space is called a Minkowskian space or sometimes a finite dimensional Banach space, the (not necessarily symmetric) distance of a Minkowskian space being a (positive homogeneous) norm. In this paper geometric conditions in terms of equidistant loci are given for the space R to be a Minkowskian space.
Ideals and Higher Derivations in Commutative Rings
Authors: Brown William C. , Kuan Wei-Eihn
Published: December 2010

In this paper, we wish to generalize the following lemma first proven by O. Zariski [5, Lemma 4]. Let O be a complete local ring containing the rational numbers and let m denote the maximal ideal of O. Assume there exists a derivation δ of O such that δ(x) is a unit in O for some x in m. Then O contains a ring O1 of representatives of the (complete local) ring O/Ox having the following properties: (a) δ is zero O1; (b) x is analytically independent over O1; (c) O is the power series ring O1[[x]]. In [4], A. Seidenberg used Zariski's lemma extensively to study conditions under which an affine algebraic variety V over a base field of characteristic zero is analytically a product along a given subvariety W of V. We should like to generalize Zariski's lemma by removing the condition that O contain the rationals. We could then get some conditions under which an arbitrary affine variety V would be analytically a product along a subvariety W.
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.
A Universal Property of the Takahashi Quasi-Dual
Authors: Poguntke Detlev
Published: December 2010

Topological group always means Hausdorff topological group, homomorphism (isomorphism) between topological groups always means continuous homomorphism (homeomorphic isomorphism). For a topological group G, the topological commutator subgroup (the closure of the algebraic commutator subgroup) is denoted by G’. For each locally compact group G, Takahashi has constructed a locally compact group GT (called the Takahashi quasi-dual) and a homomorphism G → GT such that GT is maximally almost periodic, and GT’ is compact. The category of all locally compact groups with these two properties is denoted by [TAK]. Takahashi's duality theorem states that G → GT is an isomorphism if G ∊ [TAK].
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.
Approximation of Piecewise Continuous Functions by Quotients of
Bounded Analytic Functions
Authors: Sarason Donald
Published: December 2010

This paper concerns a certain subalgebra of the Banach algebra of complex
valued, essentially bounded, Lebesgue measurable functions on the unit
circle in the complex plane (denoted here by L∞). My interest in
this subalgebra was prompted by a question of R. G. Douglas. Let
H∞ denote the space of functions in L∞ whose Fourier
coefficients with negative indices vanish (equivalently, the space of
boundary functions for bounded analytic functions in the unit disk). Douglas
[5] has asked whether every closed subalgebra of
L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra
generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known
for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3].
At the time of this writing, no algebra is known for which the above
question can be answered negatively.
Separability in an Algebra with Semi-Linear
Homomorphism
Authors: Winter David J.
Published: December 2010

The purpose of this paper is to outline a simple theory of separability for
a non-associative algebra A with semi-linear homomorphism
σ. Taking A to be a finite dimensional
abelian Lie p-algebra L and
σ to be the pth power operation in
L, this separability is the separability of
[2]. Taking A to be an algebraic field
extension K over k and σ
to be the Frobenius (pth power) homomorphism in
K, this separability is the usual separability of
K over k. The theory also applies to
any unital non-associative algebra A over a field
k and any unital homomorphism σ from
A to A such that σ(ke) ⊂ ke,
e being the identity element of A.
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.
Taming Wild Simple Closed Curves with Monotone
Maps
Authors: Boyd W. S. , Wright A. H.
Published: December 2010

Hempel [6, Theorem 2] proved that if S is a
tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and
f(E3 - S) =
E3- f(S), then f(S) is
tame. Boyd [4] has shown that the converse is false; in fact,
if S is any 2-sphere in E3, then there is a monotone map f of
E3 onto itself such that f |S is a homeomorphism,
f(E3 — S) = E3 — f(S), and
f(S) is tame.

It is the purpose of this paper to prove that the corresponding converse for
simple closed curves in E3 is also false. We show in Theorem 4 that if J is
any simple closed curve in a closed orientable 3-manifold
M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism,
f(J) is tame and unknotted, and
f(M3 - J) = S3 -
f(J).

In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a
simple closed curve in an orientable 3-manifold.
On Conformally Flat Spaces with Commuting Curvature and Ricci
Transformations
Authors: Bishop R. L. , Goldberg S.I.
Published: December 2010

Let (M, g) be a C∞ Riemannian manifold and
A be the field of symmetric endomorphisms corresponding
to the Ricci tensor S; that is,

We consider a condition weaker than the requirement that A
be parallel (▽ A = 0), namely, that the “second exterior
covariant derivative” vanish ( ▽x▽YA — ▽Y ▽XA — ▽[X,Y]A
= 0), which by the classical interchange formula reduces to the
property

where R(X, Y) is the curvature transformation determined by
the vector fields X and Y.

The property (P) is equivalent to

To see this we observe first that a skew symmetric and a symmetric
endomorphism commute if and only if their product is skew symmetric.
On Dirac's Generalization of Brooks' Theorem
Authors: Kronk Hudson V. , Mitchem John
Published: December 2010

It is easy to verify that any connected graph G with
maximum degree s has chromatic number χ(G)
≦ 1 + s. In [1], R. L. Brooks proved that
χ(G) ≦ s, unless s = 2 and
G is an odd cycle or s > 2 and
G is the complete graph Ks+1. This was the first significant theorem connecting the structure of
a graph with its chromatic number. For s ≦ 4, Brooks' theorem says that
every connected s-chromatic graph other than Ks contains a vertex of degree > s — 1. An
equivalent formulation can be given in terms of s-critical graphs. A graph
G is said to be s-critical if
χ(G) = s, but every proper subgraph has chromatic number
less than s. Each scritical graph has minimum degree ≦ s —
1. We can now restate Brooks' theorem: if an s-critical graph, s ≦ 4, is not
Ks and has p vertices and q edges,
then 2q ≦ (s — l)p + 1. Dirac [2]
significantly generalized the theorem of Brooks by showing that
2q ≦ (s — 1)£ + s — 3 and that this result is best
possible.
Certain Subsets of Products of Metacompact Spaces and
Subparacompact Spaces are Realcompact
Authors: Zenor Phillip
Published: December 2010

We will say that a space X has property (*) if and only if
each discrete subset of X is realcompact; i.e., the
cardinality of each discrete subset of X is nonmeasurable.
In [8], Shirota shows that a completely regular
T1-space X is realcompact if and only if
X has property (*) and X is complete
with respect to some uniformity. In [7], Moran, using measure
theoretic techniques, shows that any normal metacompact T1-space with property (*) is realcompact.
The Number of Generators of a Linear
p-Group
Authors: Isaacs I. M.
Published: December 2010

Let G be a finite p-group, having a faithful
character χ of degree f. The object of this paper is to bound the number,
d(G), of generators in a minimal
generating set for G in terms of χ and in particular in
terms of f. This problem was raised by D. M. Goldschmidt,
and solved by him in the case that G has nilpotence class
2.
The Strict Topology in a Completely Regular Setting: Relations
to Topological Measure Theory
Authors: Mosiman Steven E. , Wheeler Robert F.
Published: December 2010

Let X be a locally compact Hausdorff space, and let
C*(X) denote the space of real-valued bounded
continuous functions on X. An interesting and important
property of the strict topology β on C*(X) was proved by Buck [2]: the dual
space of (C*(X), β) has a natural
representation as the space of bounded regular Borel measures on
X.

Now suppose that X is completely regular (all topological
spaces are assumed to be Hausdorff in this paper). Again it seems natural to
seek locally convex topologies on the space C*(X) whose dual spaces are (via
the integration pairing) significant classes of measures.
Graphical Regular Representations of Non-Abelian Groups, I
Authors: Nowitz Lewis A. , Watkins Mark E.
Published: December 2010

In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). IfXis such a graph with vertex setV(X)and automorphism groupA(X),we say thatXis agraphical regular representation(GRR) of a given abstract groupGif

(I) G ≅ A(X) , and

(II)A(X)acts onV(X) as a regular permutation group; that is, givenu, v∈V(X), there exists a uniqueφ∈A(X)for whichφ(u) =v.

That for any abstract groupGthere exists a graphXsatisfying (I) is well-known (cf. [3]).
Products of Locally Compact Groups with Zero and Their Actions
Authors: Hanson T. H. McH.
Published: December 2010

In [4], Hofmann defines a locally compact group with zero as a Hausdorff locally compact topological semigroup, S, with a non-isolated point, 0, such that G = S — {0} is a group. He shows there that 0 is indeed a zero for 5, G is a locally compact topological group, and the identity of G is the identity of S. The author has investigated actions of such semigroups on locally compact spaces in [1; 2]. In this paper, we are investigating direct products of semigroups of the above type and actions of these products; for a special case of this, the reader is referred to [3].
Modular Hadamard Matrices and Related Designs, II
Authors: Marrero O. , Butson A. T.
Published: December 2010

Anhbyhmatrix with entries ±1 is called amodular Hadamard matrixif the inner product of any two distinct row vectors is a multiple of a fixed (positive) integern;such a matrix is also referred to as an“H(n, h) matrix” with parametersnandh.Modular Hadamard matrices and the related combinatorial designs were introduced in [2]; that paper was concerned mainly with two of the related designs, the “pseudo (ν, k, λ)- designs” and the “ (m, v, k1,λ1,k2,λ2,f, λ3)-designs” (the reader is referred to [2] for the definition of these designs).
Extension Closed and Cluster Closed Subspaces
Authors: Harris Douglas
Published: December 2010

One of the most useful properties of a compact Hausdorff space is that such a space is closed whenever embedded into a Hausdorff space. This property does not extend to compact spaces with respect to embeddings into arbitrary spaces. Thus, an interesting topological problem is to characterize the types of absolute “closure” properties that are possessed by compact spaces. This is the problem that is solved in the present paper.

The following notation and terminology will be used below. We shall consider a fixed space X and subspace A, representing arbitrary nonempty open subsets of X (respectively A ) by W (respectively V).
Series Expansions for Dual Laguerre Temperatures
Authors: Haimo Deborah Tepper
Published: December 2010

In a recent paper [2], the author, with F. M. Cholewinski, derived criteria for the series expansions of solutions u(x, t) of the Laguerre differential heat equation xuxx + (α + 1 - x)ux = ut in terms of the Laguerre heat polynomials and of their temperature transforms. Our present goal is the characterization of those solutions which are representable in a Maclaurin double series in xe-t and in 1 — e-t Some of the results are analogous to those derived by D. V. Widder in [4] for the classical heat equation and by the author in [1] for the generalized heat equation.
On the Ranges of Certain Fractional Integrals
Authors: Rooney P. G.
Published: December 2010

Suppose 1 ≦ P < ∞, μ is real, and denote by Lμ,p the collection of functions f, measurable on (0, ∞ ), and which satisfy

1.1

Also denote by [X] the collection of bounded operators from a Banach space X to itself. For v > 0, Re α > 0, Re β > 0, let

1.2

and

1.3

where ξ and η are complex numbers. Iv,α,ξ and Jv,β,η, are generalizations of the Riemann-Liouville and Weyl fractional integrals respectively, and consequently we shall refer to them as fractional integrals.
Axioms for an n-metric Structure
Authors: Grant Kerry E.
Published: December 2010

From Euclid to Hilbert, and beyond, the primitive terms of geometry have been taken as “point,” “line,” etc., while “distance” plays a secondary role. The reversal of this situation is a modern development. Frechet [4], in 1906 first considered the properties of distance which should be formalized. The most significant contributions to the geometric properties of metric spaces have been by Menger [10] and Blumenthal [2; 3].
ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3
Authors: Cannon J. W.
Published: December 2010

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).
Uniqueness Theorems for a Singular Partial Differential Equation
Authors: Ramankutty P.
Published: December 2010

A singular partial differential equation which occurs frequently in mathematical physics is given by

where is the Laplacian operator on Rn of which the generic point is denoted by x = (x1, … , xn) and s and k are real numbers. The study of solutions of this equation for the case k = 0 was initiated by A. Weinstein [5], who named it ‘Generalized Axially Symmetric Potential Theory'. Numerous references to the literature on this equation can be found in [1; 3; 6]. The analytic theory of equations of the type mentioned above has extensively been treated in [2].
On Unsymmetric Dirichlet Forms
Authors: Elliott Joanne
Published: December 2010

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.
Angle Bisection and Orthoautomorphisms in Hilbert Lattices
Authors: Morash Ronald P.
Published: December 2010

The lattices of all closed subspaces of separable, infinitedimensional Hilbert space (real, complex, and quaternionic) share the following purely lattice-theoretic properties. Each is complete, orthocomplemented, atomistic, irreducible, separable, M-symmetric, and orthomodular [2]. We will call any lattice possessing these seven properties a Hilbert lattice. The general situation which motivates the investigations of this paper concerns infinite-dimensional Hilbert lattices (the dimension of a Hilbert lattice being the cardinality of any maximal family of orthogonal atoms). There are several lattice theoretic properties, possessed by the three canonical lattices, whose only known proofs involve the analytic properties of the underlying Hilbert space, that is, there is no known purely lattice-theoretic proof of these properties.
Some Almost Simple Rings
Authors: Hill Paul
Published: December 2010

Herein, a ring is not required to have an identity. All rings are associative but not necessarily commutative. However, we specialize to the commutative case for some of our results. The paper is concerned primarily with rings having the property that all unbounded ideals or all unbounded homomorphic images are isomorphic to the ring. We say that a ring R is bounded if nR = 0 for some positive integer n; alternately, R, with or without 1, is said to have finite characteristic. Unbounded rings having the property that all proper subrings are bounded were characterized in [8].
Idempotents in Noetherian Group Rings
Authors: Formanek Edward
Published: December 2010

If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.
On the Closure of the Linear Span of a Weighted Sequence in Lp(0, ∞)
Authors: Siddiqi Jamil A.
Published: December 2010

Let {λn} be an increasing sequence of positive numbers. The question of the closure in Lp(0, ∞ ) (1 ≦ p ≦ ∞ ) of the linear span of the sequence has been considered by several authors, notably by Boas (1) and Fuchs [3;4].
Some Extremal Properties of Bipartite Subgraphs
Authors: Edwards C. S.
Published: December 2010

Gp = (V, X) is a graph on p vertices, with vertex set V and edge set X. Occasionally, to avoid ambiguity, V and X will be written V(GP) and X(GP), respectively.
Solutions Asymptotiques au Problème des Secrétaires
Authors: Dubuc Serge
Published: December 2010

Nous reprenons le problème des secrétaires tel que considéré en particulier par Chow, Moriguti, Robbins et Samuels [2]. Nous commençons par dégager une formule sur le rendement de certains temps d'arrêt dans le problème des secrétaires. Nous faisons voir comment une certaine fonction d'une infinité de variables est étroitement reliée aux temps d'arrêt optimaux. Enfin, nous étudions le comportement asymptotique du rendement de certains temps d'arrêt pour le problème des n secrétaires lorsque n tend vers l'infini.
The Frechet Differential of a Primary Matrix Function
Authors: Powers David L.
Published: December 2010

LetXbe a given complex matrix of ordern.Iff(z)is analytic at the eigenvalues ofX,one may define the primary matrix functionf(X)with stem functionf(z)by using any of several well-known methods: for instance, canonical forms, power series, or interpolating polynomials [9].
The Dual of Hp(R+n+1) for p < 1
Authors: Walsh T.
Published: December 2010

The dual of Hp of the unit disk for 0 < p < 1 has been characterized by Duren, Romberg and Shields (see [3]). The present paper is concerned with the analogous result for Hp(R+n+1) in the sense of Stein and Weiss (see [11]). In this connection it may be recalled that the dual of H1 has been characterized by Fefferman (see [4]).
Edge Partition Properties of Graphs
Authors: Henson C. Ward
Published: December 2010

Erdös and Hajnal [1] have introduced an edge partition relation for graphs

(1)

which means that whenever the edges of G are separated into two sets, E1 and E2, there exists a subgraph G’ of G such that G’ is isomorphic to Hi and the edges of G’ are all in Ei. for i = 1 or 2. A class of graphs has the G-R (Galvin-Ramsey) property [2] if for each H in there exists a G in which satisfies G→(H,H).
Integral Extensions of Commutative Banach Algebras
Authors: Lindberg John A.
Published: December 2010

In this paper, we continue the study of integral extensions begun in [7]. Whereas in the previous paper, we dealt exclusively with the extension A[x]/(α(x)), α(x) a monic polynomial over A, we now deal with arbitrary integral extensions. Applications of the results presented herein will be made in subsequent papers.

To simplify our presentation, we make the following conventions. By an algebra, we will always mean a commutative complex algebra with an identity element, usually denoted by e
Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula
Authors: Rummler Hansklaus
Published: December 2010

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).
Boundary Behavior and Quasi-Normality of Finitely Valent Holomorphic Functions
Authors: Haddad David C.
Published: December 2010

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.
On Certain Sublattices of the Lattice of Subgroups Generated by the Prefrattini Subgroups, the Injectors and the Formation Subgroups
Authors: Makan A. R.
Published: December 2010

Various characteristic conjugacy classes of subgroups having covering/avoidance properties with respect to chief factors have recently played a major role in the study of finite soluble groups. Apart from the subgroups which are now called Hall subgroups, P. Hall [7] also considered the system normalizers of a finite soluble group and showed that these form a characteristic conjugacy class, cover the central chief factors and avoid the rest. The system normalizers were later shown by Carter and Hawkes [1] to be the simplest example of a wealth of characteristic conjugacy classes of subgroups of finite soluble groups which arise naturally as a consequence of the theory of formations.
Subgroups of Central Separable Algebras
Authors: Elgethun E. D.
Published: December 2010

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.
Subgroups of One-Relator Fuchsian Groups
Authors: Best L. A.
Published: December 2010

Let Γ be the Fuchsian group with canonical presentation:

Generators:

Relations:

where . The genus g, the period partition {m1, m2, … , mr} and the number s are invariants of Γ. The symbol (g|m1, m2, … , mr|s) is called the signature of Γ and it determines Γ up to isomorphism.
The Cohomology of (Λ2X, ΔX)
Authors: Larmore Lawrence L.
Published: December 2010

Let X be any connected C.W. complex. Let Λ2X be the 2-fold symmetric product of X, the set of unordered pairs (not necessarily distinct) of elements of X. Let ΔX ⊂ X2 be the diagonal, and also (by a slight abuse of notation) let ΔX ⊂ Λ2X also denote the set of unordered pairs {x, x}.