bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Non-Existence of Injective Near-Ring Modules
Authors: Banaschewski Bernhard , Nelson Evelyn
Published: December 2010

Associated with any near-ring R, and any set S of distributive elements of R, one has a natural notion of S-distributive R-modules, analogous to that of modules over rings and including the latter as special case (Frohlich [6]; we recall the details in Section 1). Since near-rings can be viewed as slightly deficient rings, it makes sense to enquire whether such near-ring modules share with modules over rings the familiar and important property of having injective hulls.
A Note on the Hopficity of Finitely Generated Free-by-Nilpotent Groups
Authors: Boler James
Published: December 2010

Our purpose is to deduce from a theorem of P. Hall the following observation.

Let G be a finitely generated group and F a free normal subgroup of G with G/F nilpotent. Then G is hopfian.

Here a group G is hopfian if every epiendomorphism G→G is an automorphism.
A Proof of the Equivalence of Helly's and Krasnoselski's Theorems
Authors: Borwein J.
Published: December 2010

We show that Krasnoselski's Theorem, which is usually derived from Helly's Theorem, is in fact equivalent to it.
Multiple Series Manipulations and Generating Functions
Authors: Grimson R. C.
Published: December 2010

Let γ be an increasing function on the real numbers such that γ(0) = 0 (which, by translation of axes, is no restriction) and suppose that γ(n) is a positive integer if n is a positive integer. Let γ- denote the inverse function of γ. Furthermore, let L(x) be the least integer ≥ x; let [x] be the greatest integer ≤x, and suppose that c0, c1 … is an arbitrary sequence of numbers.
A Note on the Category of the Telescope
Authors: Hardie K. A.
Published: December 2010

Let X be an infinite connected CW-complex which is the union of an increasing sequence of subcomplexes Xr. Let cat X denote the Lusternik-Schnirelmann category of X, normalized to take the value 0 on contractible spaces. Suppose that cat Xr ≤ K (r ≥ 1). In his problem list [1], T. Ganea proved that cat Xr ≤ 2k +1 and asked (Problem 5) whether this is the best possible upper bound. The purpose of this note is to prove that cat X ≤ 2k.
An Asymptotic Estimate for the Bernoulli and Euler Numbers
Authors: Leeming David J.
Published: December 2010

We derive here simple asymptotic estimates for both the Euler and Bernoulli numbers. The derivations follow easily from known results, but I am unable to find them elsewhere in the literature. C. Jordan [1, p. 245 and p. 303] gives some related inequalities. Other properties of these two classical sets of numbers may be found in [1], [3] and [4].
Numerical-Valued Fourier Transforms
Authors: Struble Raimond A.
Published: December 2010

It is shown that the classical Fourier transform can be extended as an algebra isomorphism onto the algebra of all complex-valued functions, which are measurable and finite a.e., under pointwise addition and multiplication. The extended Fourier transform agrees with the distributional Fourier transform on the space of all distributions which have regular transforms. It is defined on an algebra of Mikusiński-type operators in which multiplication is convolution in the subspace of integrable distributions.
Completions of Rank Rings
Authors: Handelman David
Published: December 2010

In this note, we prove three results on regular rings possessing a rank function: (a) the completion of a *-regular rank ring is a regular Baer *-ring; (b) (a) is used to construct regular Baer * factors of type IIf with centre any complex subfield closed under conjugation; (c) the units of a unit-regular rank ring form a dense topological subgroup of the units of the completion.
On Integral Abel-Type and Logarithmic Methods of Summability
Authors: Heagy E. C. , Shawyer B. L. R.
Published: December 2010

In this paper, we define an integral logarithmic method of summability, extending the integral Abel-type methods defined by Jakimovski [6]. We examine the behaviour of the product of this method with integral Hausdorff methods. A full scale of strict inclusions for integral Abel-type methods is obtained and the integral logarithmic method is placed in this scale.
Boundary-Value Problems of a Degenerate Sobolev-Type Differential Equation
Authors: Pao C. V.
Published: December 2010

The purpose of this paper is to study a degenerate Sobolev type partial differential equation in the form of Mut + Lu = f, where M and L are second order partial differential operators defined in a domain (0, T]×Ω in Rn+1. The degenerate property of the equation is in the sense that both M and L are not necessarily strongly elliptic and their coefficients may vanish or be negative in some part of the domain (0, T]×Ω. Two types of boundary conditions are investigated.
Lecture—Recent Results on Fermat's Last Theorem
Authors: Ribenboim P.
Published: December 2010

This text is an elaboration of a lecture delivered at the First Winter Meeting of the Canadian Mathematical Congress, December 1975. Subsequently, this same lecture was presented at various occasions, (Colloquium at Université Pierre et Marie Curie, in Paris; Seventh Iranian Mathematical Congress in Tabriz; Université de Paris-Sud à Orsay, Illinois Institute of Technology in Chicago, Colloquium at the University of Toronto, and at the University of Waterloo).
A Note on Convergence of Fourier Series of a Function on Wiener's Class Vp
Authors: Siddiqi Rafat N.
Published: December 2010

Let f be a 2π-periodic function defined on [0, 2π]. We set

where suprema have been taken with respect to all partitions P:a = t0<t1<t2<…<tn<=b of any segment [a, b] contained in [0, 2π]. We call the pth total variation of f on [a, b]. If we denote pth total variaiton of f on [0, 2π] by Vp(f), then we can define Wiener's class simply by
A Note on Injective Modules Over a d.g. Near-Ring
Authors: Oswald A.
Published: December 2010

In [3] an attempt was made at proving the following result:

(A) An N-module M over a d.g. near-ring is injective if and only if for each right ideal u of N and each N-homomorphism f:u→M there exists an element mϵM with f(a) = ma for all aϵu.

In this note we present two examples. The first is a counterexample to (A) and the second illustrates one point at which the attempt made in [3] fails.
Dilations of Positive Contractions on Lp Spaces*
Authors: Akcoglu M. A. , Sucheston L.
Published: December 2010

Throughout this article p denotes a fixed number such that 1 ≤ p < ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.
A Characterization of the Lebesgue Integral
Authors: Thomson B. S.
Published: December 2010

The first general integrability criterion is due to Riemann. He observed that a necessary and sufficient condition for a function f(x) on [a, b] to be integrable in his sense is that for each ε > 0 there exists a δ > 0 so that

where {x0, x1, …, xn} is a partition of [a, b] with diameter less than δ (and ω(f, I) denotes the oscillation of f on the interval I).
On the Order of the Error Function of the (k, r)-Integers—II
Authors: Subbarao M. V. , Suryanarayana D.
Published: December 2010

In the first part of this note we give a very simple and elegant proof of the theorem on the order of the error function of the (k, r)-integers, which the authors proved earlier using elaborate calculations. We also obtain an improvement of an earlier result on the order of the same error function on the basis of the Riemann hypothesis.
A Note on (r, λ)-Systems
Authors: Gardner B.
Published: December 2010

An (r, λ)-system is an arrangement of v objects (or varieties) into subsets (or blocks) such that each variety appears in exactly r blocks and each pair of distinct varieties appears in exactly λ blocks. To avoid trivial designs, we assume that 1≤λ<r.
Coset Enumeration in a Finitely Presented Semigroup
Authors: Jura Andrzej
Published: December 2010

The enumeration method for finite groups, the so-called Todd-Coxeter process, has been described in [2], [3]. Leech [4] and Trotter [5] carried out the process of coset enumeration for groups on a computer. However Mendelsohn [1] was the first to present a formal proof of the fact that this process ends after a finite number of steps and that it actually enumerates cosets in a group. Dietze and Schaps [7] used Todd-Coxeter′s method to find all subgroups of a given finite index in a finitely presented group. B. H. Neumann [8] modified Todd-Coxeter′s method to enumerate cosets in a semigroup, giving however no proofs of the effectiveness of this method nor that it actually enumerates cosets in a semigroup.
A Construction of the Pair Completion of a Quasi-Uniform Space
Authors: Lindgren William F. , Fletcher Peter
Published: December 2010

Although L. Nachbin introduced the concept of a quasiuniform space in connection with the study of uniform ordered spaces [7], it was A. Császár who first developed a theory of completion for these spaces. In the review of Császár′s work [3], J. Isbell wrote: “A topogenic space is already complete according to the present definition; an unfortunate side effect is that not every convergent filter is Cauchy.” This aspect of Császár′s theory of completeness seems to have stalled the investigation of its applications, so that even in the study of uniform ordered spaces Császár′s important work has often been overlooked. While it may have seemed that Isbell had pointed out the Achilles′ heel of Császár′s theory, in this paper we outline a construction of Császár′s completion for quasi-uniform spaces, which differs from Császár′s original construction and the construction given by S. Salbany [8], in which Isbell′s objection does not obtain.
Isomorphisms of Function Algebras and Algebras of Analytic Functions
Authors: Lund Bruce
Published: December 2010

Let R be a finite open Riemann surface with analytic boundary Γ. Set and define is analytic on R}. Conditions are given on a function algebra A on a compact Hausdorff space X which imply that A is isomorphic to a subalgebra of A(R) of finite codimension.
Uniform Contractification
Authors: Tan Kok-Keong
Published: December 2010

Let (X, τ) be a metrizable space and {fn: n = 1,2,…} be a commuting family of continuous mappings on X with a common fixed point ξ ∈ X such that (I) for each k = l, 2, … and (II) Then for each c ∈ (0,1), there exists a metric d on X inducing the topology τ such that , for all x, y∈X and n = l, 2, …. The above result is also generalized to Tychonoff spaces.
First Countable Spaces that Have Special Pseudo-Bases
Authors: White H. E.
Published: December 2010

Two types of pseudo-bases, σ-disjoint and σ-discrète, are utilized in this note. In the next section, we show that a first countable Hausdorff space has a σ-disjoint pseudo-base if and only if it has a dense metrizable subspace. This result implies that many first countable spaces have dense metrizable subspaces. In section 3, we show that if X is a Hausdorff space that either is quasi-developable or has a base of countable order, then X has a dense metrizable subspace if and only if it has a dense metrizable Gδ subspace. We give an example to show that the conclusion of this theorem is false for semi-metrizable spaces. Finally, in the last section, we investigate when a quasi-developable (resp. semi-metrizable) space can be embedded as a dense subspace of a quasi-developable (resp. semi-metrizable) Baire space.
A Characterization of Biregular Group Rings
Authors: Burgess W. D.
Published: December 2010

In this note biregular group rings are characterized and an example is given to show that Renault′s conjecture is false.

A ring A with 1 is biregular if for all a∈A, AaA is generated by a central idempotent Equivalently, A is biregular iff all the stalks of its Pierce sheaf are simple.

In [1] Bovdi and Mihovski showed that for a ring A, if the group ring AG is biregular then: (*) A is biregular and G is locally normal with the order of each finite normal sub-group of G invertible in A. A proof is found in Renault [7]. In [6] Renault showed that (*) is necessary and sufficient in case A is a finitely generated module over its centre or if A is right self-injective. He conjectured that (*) is necessary and sufficient in general. In fact (*) is not sufficient as the example below shows. Some familiarity with Pierce sheaf techniques is assumed (see [5] or [2]).
Strong Extensions vs. Weak Extensions of C*-Algebras
Authors: Cho S. J.
Published: December 2010

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such that

for all a in A.
On the Weak Global Dimension of Pseudovaluation Domains
Authors: Dobbs David E.
Published: December 2010

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.
Some Smoothness Properties of Measures on Topological Spaces
Authors: Hegde Shankar
Published: December 2010

Abstrct

V. S. Varadarajan has classified the bounded linear functional on the algebra C(X) of bounded continuous functions on a topological space X, according to the properties of their smoothness and related this classification to the corresponding natural classification of finitely additive regular measures on the zero sets of X. In this paper, some of these results are extended to the linear functionals on an arbitrary uniformly closed algebra A of bounded functions on a set X.
Zeros of Nonlinear Monotone Operators in Hilbert Space*
Authors: Schöneberg R.
Published: December 2010

Around 1960, the Russian mathematician Kachurovski [1] introduced the notion of monotone operators in Hilbert spaces: Let E be a Hilbert space and X ⊂ E. An operator T:X→E is said to be monotone, iff

.
S-Barrelled Topological Vector Spaces*
Authors: Snipes Ray F.
Published: December 2010

N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called “espaces tonnelés” or “barrelled spaces.” These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8].
Russian Dolls
Authors: Wilker J. B.
Published: December 2010

In an earlier number of this Bulletin, P. Erdös [1] posed the following problem. “For each line ℓ of the plane, Aℓ is a segment of ℓ. Show that the set ∪ℓAℓ contains the sides of a triangle.” One objective of this paper is to prove a strengthened version of this result in iV-dimensions. As usual denotes the cardinality of the natural numbers and c, the cardinality of the real numbers.
On a Result of Levitzki
Authors: Felzenszwalb B.
Published: December 2010

A well known result of Levitzki [2, Lemma 1.1] is the following:

Theorem. Let R be a ring and U a non-zero one-sided ideal of R. Suppose that given a∈U, an = 0 for a fixed integer n ≥ 1; then R has a non-zero nilpotent ideal.

The purpose of this note is to observe some additional results which are related to the above.

Theorem 1. Let R be a ring with no non-zero nil ideals and U an ideal of R. Suppose that a∊R is such that for every x∊U, axn(x) = 0 where n(x) ≥ 1 depends on x; then aU= Ua = 0.
Unipotent Matrix Groups Over Division Rings
Authors: Mochizuki H. Y.
Published: December 2010

If G is a unipotent group of n×n matrices over a division ring of characteristic 0 or prime p greater than (n- 1) (n-[n/2]) where [n/2] is the greatest integer less than or equal to n/2, then it is proved that G can be simultaneously triangularized.
A Global Existence and Uniqueness Theorem for Ordinary Differential Equations of Generalized Order
Authors: Al-Abedeen Ahmed Z. , Arora H. L.
Published: December 2010

We extend the Picard's theorem to ordinary differential equation of generalized order α, 0 ≤ α ≤ l, and prove a global existence and uniqueness theorem by using the Banach contraction principle.
Certain nth Order Differential Inequalities in the Complex Plane
Authors: Al-Amiri H. S.
Published: December 2010

Let w(z) be regular in the unit disc U:|z|<l, with w(0) = 0 and let h(r, s, t) be a complex function defined in a domain D of C3. The author determines conditions on h such that if

z∈U, then |w(z)|< 1 for z ∈ U and n= 0, 1, 2, …. Here Dnw(z) = (z/(l-z)n+1*w(z), where * stands for the Hadamard product (convolution). Some applications of the results to certain differential equations are given.
Continuous Rings and Rings of Quotients
Authors: Page S. S.
Published: December 2010

Throughout R will denote an associative ring with identity. Let Zℓ(R) be the left singular ideal of R. It is well known that Zℓ(R) = 0 if and only if the left maximal ring of quotients of R, Q(R), is Von Neumann regular. When Zℓ(R) = 0, q(R) is also a left self injective ring and is, in fact, the injective hull of R. A natural generalization of the notion of injective is the concept of left continuous as studied by Utumi [4]. One of the major obstacles to studying the relationships between Q(R) and R is a description of J(Q(R)), the Jacobson radical of Q(R). When a ring is left continuous, then its left singular ideal is its Jacobson radical. This facilitates the study of the cases when either Q(R) is continuous or R is continuous.
Parallel Lines
Authors: Coxeter H. S. M.
Published: December 2010

About a hundred years ago, the author of Through the Looking-Glass wrote another book called Euclid and his Modern Rivals. Were these rivals Lobachevsky and Bolyai, Riemann and Schlafli? No, they were merely the authors of dull school textbooks that would soon be forgotten. The sad truth is that, in 1873, hardly anyone in England knew of the breakthrough that had occurred on the Continent some fifty years before: only Cayley in Cambridge, Clifford in London, and a few students. Even if Cayley or Clifford had visited Oxford and given a lecture there, it is doubtful that he would have succeeded in convincing the conservative Dodgson that Euclid's postulates could be modified to yield two new worlds, surpassing in strangeness the worlds of the two Alice books and yet just as logically consistent as Euclid.
An Asymptotic Formula for Reciprocals of Logarithms of Certain Multiplicative Functions
Authors: Koninck Jean-Marie de , Ivić Aleksandar
Published: December 2010

Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)>0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.
An Asymptotic Formula for a Sum of Products of Powers
Authors: Evans Ronald J.
Published: December 2010

Fix an integer r ≥ 2 and positive numbers b1, …, br. Write σ = b1+ …+br Let . In this note we evaluate the constant A (when it exists) for which

1

where the sum is over all vectors

2

We also obtain upper and lower bounds for the sum in (1).
Determination of [nθ] by its Sequence of*Differences
Authors: Fraenkel A. S. , Mushkin M. , Tassa U.
Published: December 2010

For any real numberθletwhere [x] denotes the greatest integer not exceedingx.A method is given for computingfθfrom its first few terms. A similar method is given for computing the characteristic functiongθ(n)of[nθ]. The given methods converge rapidly, and generalize previous results of Bernoulli, Markorf, and Stolarsky. Note that either of the sequencesfθand gθdetermines the sequence[nθ] (n = 1, 2,…).
Properties of Hereditary Hypergraphs and Middle Graphs
Authors: Cockayne E. J. , Hedetniemi S. T. , Miller D. J.
Published: December 2010

The middle graph of a graph G=( V, E) is the graph M(G) = (V∪E, E′), in which two vertices u, v are adjacent if either M is a vertex in V and v is an edge in E containing u, or u and v are edges in E having a vertex in common. Middle graphs have been characterized in terms of line graphs by Hamada and Yoshimura [7], who also investigated their traversability and connectivity properties. In this paper another characterization of middle graphs is presented, in which they are viewed as a class of intersection (representative) graphs of hereditary hypergraphs. Graph theoretic parameters associated with the concepts of vertex independence, dominance, and irredundance for middle graphs are discussed, and equalities relating the chromatic number of a graph to these parameters are obtained.
Principal Irreducible Lie-Algebra Modules
Authors: Servedio Frank J.
Published: December 2010

Let V be a finite dimensional vector space over k, a field of characteristic 0, L be an algebraic Lie-subalgebra of Endk(V), with the latter a Lie algebra in the canonical way, and let V be an L-module in the canonical way. For X ∈ V, let LX = {AX | A ∈ L{. Call V a principal L-module if ∃ X ∈ V such that LX= V; X will be called a principal generator of the L-module V.
Sums of Powers in Arithmetic Progressions
Authors: Small Charles
Published: December 2010

The papers [2] and [3] study the function g(k, n), defined for integers k > 1 and n > 1 as the smallest r with the property that every integer is a sum of r kth powers mod n. This note identifies g′(k), defined as the maximum over all n of g(k, n), with the function Γ(k) studied by Hardy and Littlewood [1] fifty years ago in connection with Waring's problem.
Sur la Structure Extremale de la Somme de Deux Convexes
Authors: Bair J.
Published: December 2010

Ces dernières années, de nombreuses recherches ont été consacrées à l’étude de la structure extrémale de la somme de deux convexes; signalons, par exemple, les travaux de Bair-Fourneau-Jongmans [3], Edelstein-Fesmire [5], Husain-Tweddle [6], Jongmans [7], Klee [8] et Roy [9]. Dans cette direction, Klee [8] a obtenu ce résultat très intéressant, qui a d’ailleurs été généralisé plus tard par Jongmans [7]: dans un espace euclidien ℝd, tout point extrême z de la somme d’un convexe fermé A et d’un convexe compact B s’écrit de façon unique comme la somme d’un point extrême xz de A et d’un point extrême yz de B; de plus, tout point extrême x de A livre un point extrême y de B tel que z = x + y soit un point extrême de A + B. Par contre, il peut exister des points extrêmes de B dont la somme avec un point extrême de A n’est jamais un point extrême de A + B; cette constatation conduit au problème: est-il possible de caractériser l’ensemble des yz?
Geometric Inequalities for Plane Convex Bodies
Authors: Chakerian G. D.
Published: December 2010

In what follows we shall mean by a plane convex body K a compact convex subset of the Euclidean plane having nonempty interior. We shall denote by h (K, θ) the supporting function of K restricted to the unit circle. This measures the signed distances from the origin to the supporting line of K with outward normal (cos θ, sin θ). The right hand and left hand derivatives of h (K, θ) exist everywhere and are equal except on a countable set.
A Lower Bound for the Number of Negative Zeros of Power Series
Authors: Gawronski W.
Published: December 2010

In this paper we are concerned with power series of the type

1

which admit unique analytic extension onto a domain containing the negative real axis. Our primary object is to establish a general theorem giving a lower estimate for the number of different zeros of (1) on the negative real axis. W. Jurkat and A. Peyerimhoff showed that for a certain class of coefficient functions a(z) the number of negative zeros of (1) is closely related to the behaviour of a(z) at z = 0. In particular they proved the following theorem [4, p. 219, Theorem 4].
Sommes de Fonctions Additives Restreintes à une Class de Congruence
Authors: Mercier Armel
Published: December 2010

Une fonction arithmétique f est une fonction définie sur l'ensemble des entiers naturels et à valeurs dans ℂ. On dit qu'une telle fonction f est additive si f(mn)=f(m) + f(n) si(m, n) = l et par ailleurs qu'elle est multiplicative si f(mn)=f(m)f(n) si (m, n) = l. Les sommes de la forme Σn≤xf(n) ont été largement étudiées.
Some Progression-Free Partitions Constructed using Folkman's Method
Authors: Rabung John R.
Published: December 2010

Almost from the day that B. L. van der Waerden [10] proved his now famous theorem on arithmetic progressions, mathematicians have been working to find a new or an improved constructive proof of that result, but without much success.
Bounds on the Coarseness of the n-Cube
Authors: Hartman Jehuda
Published: December 2010

The coarseness, c(G), of a graph G is the maximum number of edge disjoint nonplanar subgraphs contained in G For the n-dimensional cube Qn we obtain the inequalities
On Berndt's Method in Arithmetical Functions and Contour Integration
Authors: Krishnaiah P. V. , Rao R. Sita Rama Chandra
Published: December 2010

If f is a suitable meromorphic function then, by a classical technique in the calculus of residues, one can evaluate in closed form series of the form

1.1
Internal Hom Functors for Polarities
Authors: Nelson Evelyn
Published: December 2010

The notion of functionality for an internal horn functor H in a concrete category K was introduced in Banaschewski and Nelson [1], formalizing the condition that the structure on the H(A, B) is "pointwise" structure on sets of functions.
Coherent Overrings
Authors: Papick Ira J.
Published: December 2010

In the study of particular categories of integral domains, it has proved useful to know which overrings of the domains of interest lie within the category, and indeed whether all such overrings do. (Recall: an overring of R is a ring T with R ⊆ T ⊆ quotient field of R.) Two classes of domains classically studied in this setting are Prüfer domains and one-dimensional Noetherian domains. Since both of these classes are contained in the category of coherent domains, it is natural to investigate this category in this setting.
A Criterion for Taylor Summability of Fourier Series
Authors: Holland A. S. B. , Sahney B. N. , Tzimbalario J.
Published: December 2010

Let {ank} be a matrix defined by

1

and n taking only non-negative integer values.

Let f(x) ∈ L [0, 2π] and be periodic with period 2π outside this interval. Let the Fourier series associated with the function f(x) be given by

and let

where s is a constant.
Functional Dependence and Analytic Functions
Authors: Ramankutty P.
Published: December 2010

Without appealing to the Cauchy theorem or its corollaries, it is proved that the real and imaginary parts of a non-constant complex-valued analytic function of several complex variables are functionally independent. This unifies and generalizes some results sporadically treated in standard treatises on function theory.
Approximation on Boundary Sets
Authors: Wang James Li-Ming
Published: December 2010

Let U be a bounded open subset of the complex plane. By a well known result of A. M. Davie, C(bU) is the uniformly-closed linear span of A(U) and the powers (z-zi)-n, n = 1, 2, 3, … with zi a point in each component of U. We show that if A(U) is a Dirichlet algebra and bU is of infinite length, then one power of (z - zi) is superfluous.
Strongly Monotone Solutions of Retarded Differential Equations
Authors: Sficas Y. G.
Published: December 2010

Let us consider the retarded differential equation

1

for which the following assumptions are made:

(i) p: [t0, ∞) → [0, ∞) is continuous and not identically zero for all large t.

(ii) σ [t0, ∞)→ ℝ is continuous, strictly increasing,

(iii) φ: ℝ → ℝ is continuous, non-decreasing and
A Commutattvity Theorem for Rings and Groups
Authors: Nicholson W. K. , Yaqub Adil
Published: December 2010

The following theorem is proved: Suppose R is a ring with identity which satisfies the identities xkyk = ykxk and xlyl = ylxl, where k and l are positive relatively prime integers. Then R is commutative. This theorem also holds for a group G. Furthermore, examples are given which show that neither R nor G need be commutative if either of the above identities is dropped. The proof of the commutativity of R uses the fact that G is commutative, where G is taken to be the group R* of units in R.
A Note on a Theorem of Ky Fan
Authors: Lin Tzu-Chu
Published: December 2010

Fan ([2, Theorem 2]) has proved the following theorem:

Let K be a nonempty compact convex set in a normed linear space X. For any continuous map f from K into X, there exists a point u∈K such that

In this note, we prove that the above theorem is true for a continuous condensing map defined on a closed ball in a Banach space. We also prove that it is true for a continuous condensing map defined on a closed convex bounded subset of a Hilbert space.
Uniform Spaces as Nice Images of Nice Uniform and Metric Spaces(1)
Authors: Willmott Richard
Published: December 2010

The classical theorem that a complete separable metric space is the image under a one-to-one continuous function of a closed subset of the irrational numbers has been extended in two directions, the first leading to various characterizations in descriptive set theory of Borel and analytic sets or generalizations of them as continuous images of certain subsets of the irrationals, or generalizations of them (see, e.g. [3] and references cited there; [4]; [6]). The second direction originates in the observation that a closed subset of the irrationals is a complete 0-dimensional metric space (under a suitable metric), and leads to the general question asked by Alexandroff [1], "Which spaces can be represented as images of 'nice' (e.g. metric, 0-dimensional) spaces under 'nice' [e.g. one-to-one, open, closed, perfect] continuous mappings?" (See, e.g. [7], [9] and the survey articles [1], [2] and [11].)
Notes on Frattini Subgroups of Generalized Free Products with Cyclic Amalgamation
Authors: Allenby R. B. J. T. , Tang C. Y. , Tang S. Y.
Published: December 2010

The problem of the exact location of the Frattini subgroup 4>(G) of a generalized free product G = (A*B)H was first raised by Higman and Neumann [5]. Solutions to special cases of the problem can be found in [1], [2], [8], [9] and [10]. The purpose of this note is to extend the results of [2], [8], and to simplify the proof of Whittemore's theorem [10]. We also apply our result to give simple proofs of certain classes of knot groups that have trivial Frattini subgroups. The proof that every knot group has trivial Frattini subgroup hard and long (footnote 2, p. 56).
On Some Generalization of Inequalities of Opial, Yang and Shum
Authors: Lee Cheng-Shyong
Published: December 2010

In 1960, Z. Opial [20] proved the following interesting integral inequality:

Theorem A. If u is a continuously differentiable function on [0, b], and if u(0) = u(b) = 0, and u(x)>0 for x∊(0, b), then

1

where the constant b/4 is the best possible.
Second-Order Evolution Equations Associated with Convex Hamiltonians(1)
Authors: Aubin J. P. , Ekeland I.
Published: December 2010

Many problems in mathematical physics can be formulated as differential equations of second order in time:

with V a convex functional. This is the Euler equation for the Lagrangian

which is convex with respect to x, and concave with respect to x.
A Note on the Realization of Types
Authors: Adamson Alan
Published: December 2010

Let L be a countable first-order language and T a fixed complete theory in L. If is a model of T, is an n-sequence of variables, and ā=〈a1,…, an〉 is an n-sequence of elements of M, the universe of , we let where ranges over formulas of L containing freely at most the variables υ1,…υn. ā is said to realize in We let be where is the sequence of the first n variables of L.
Function Theoretic Integral Operator Methods for Partial Differential Equations(1)
Authors: Kreyszig Erwin
Published: December 2010

It is well known that complex analytic functions and harmonic functions of two real variables are closely related, so that from methods and results in complex function theory one can easily obtain theorems on those harmonic functions. This is the prototype of a relation between complex analysis and a partial differential equation (Laplace's equation in two variables). In the case of more general linear partial differential equations, one can establish similar relations.
On the Structure of Locally Solid Topologies
Authors: Aliprantis C. D. , Burkinshaw O.
Published: December 2010

This paper considers what conditions on the order structure of a Riesz space will insure that one locally solid topology is finer than another, or when does one topology induce a finer topology than another on the order bounded subsets. The basic tool employed for the comparisons will be the carrier of a locally solid topology. We shall deal mainly with topologies whose carriers are order dense; a locally solid topology with order dense carrier will be called entire.
Fixed Point Theorems for Multi-Valued Mappings
Authors: Hu Thakyin
Published: December 2010

Two fixed point theorems for multi-valued mappings in a complete, ε-chainable metric space are proved. The theorems, thus established, extend result of M. Edelstein, Peter K. F. Kuhfittig, Hwei-mei Ko and Yueh-hsia Tsai, S. B. Nadler, Jr. and S. Reich.
Irréductibilité des Polynömes Sur un Corps Fini
Authors: Agou S.
Published: December 2010

Dans ce qui suit f(X) désigne un polynöme irréductible de degré n de et un polynöme de tel que a≠0 et r désigne un entier non nul.

On montre dans cet article que les polynÔmes ne sont pas irréductibles sur pour m≥3.
Decomposition of Kn into Dragons
Authors: Huang C. , Schonheim J.
Published: December 2010

It is shown that if 1<n ≡ 0 or 1 (mod 2 m), then the edges of Kn may be partitioned into isomorphic copies of a graph D3(m) and also of a graph D4(m), graphs consisting respectively of a triangle with an attached path of m - 3 edges or a quadrilateral with an attached path of m - 4 edges. If m is a power of 2 then the above condition is shown to be necessary and sufficient for the existence of such a partition.
On the Boundedness and Range of the Extended Hankel Transformation
Authors: Rooney P. G.
Published: December 2010

For l≤p<∞ μ∈ℝ, let ℒμ.p denote the collection of functions f, measurable on (0, ∞) and such that

Let C0 be the collection of functions continuous and compactly supported on (0, ∞); it is known that C0 is dense in ℒμ.p—see [2; Lemma 2.2]. If X and Y are Banach spaces, denote by [X, Y] the collection of bounded linear operators from X into Y, abbreviating [X, X] to [X].
Gaps between Spheres in Normed Linear Spaces
Authors: Lohman Robert H.
Published: December 2010

The geometric notions of a gap and gap points between “concentric” spheres in a normed linear space are introduced and studied. The existence of gap points characterizes finitedimensional spaces. General conditions are given under which an infinite-dimensional normed linear space admits concentric spheres such that both these spheres and their dual spheres fail to have gap points.
First Countable Lindelöf Extensions of Uncountable Discrete Spaces
Authors: Bell Murray , Ginsburg John
Published: December 2010

The existence of a first countable Lindelöf extension L of an uncountable discrete space D for which L-D is countable is considered. Assuming CH, such extensions exist; however it is also consistent that no such spaces exist, as follows from MA + ¬ CH
Towards a Classification of Convolution-Type Operators From l1 to l∞
Authors: Crombez G. , Govaerts W.
Published: December 2010

Let Z be the additive group of integer numbers with discrete topology, the space of complex-valued integrable functions on Z with respect to normalized Haar measure, the space of bounded functions on Z.
Infinitesimal Isometries on Compact Manifolds
Authors: Liang Chao-Chu
Published: December 2010

Let X denote a non-vanishing infinitesimal isometry on a compact Riemannian manifold Mn. Let denote the deRham complex of M. We write i(X) for the operator of interior product, and L(X) the Lie derivative on the elements of A(M). We define E(M) = {u ∈ A(M)| i(X)u = 0, L(X)u= 0}.
Derived Subspaces of Metric Spaces
Authors: Martin Harold W.
Published: December 2010

It is shown that the boundary of the set of accumulation points of a metrizable space X is compact iff X has a compatible metric d such that d(A, B)>0 whenever A and B are disjoint closed subsets of X, each of which is disjoint from the set of accumulation points.
Arithmetic Progressions Contained in Sequences with Bounded Gaps
Authors: Nathanson Melvyn B.
Published: December 2010

Van der Waerden [1, 4, 5] proved that if the nonnegative integers are partitioned into a finite number of sets, then at least one set in the partition contains arbitrarily long finite arithmetic progressions. This is equivalent to the result that a strictly increasing sequence of integers with bounded gaps contains arbitrarily long finite arithmetic progressions. Szemerèdi [3] proved the much deeper result that a sequence of integers of positive density contains arbitrarily long finite arithmetic progressions. The purpose of this note is a quantitative comparison of van der Waerden's theorem and sequences with bounded gaps.
Complete Quasi-Uniform Spaces
Authors: Huffman Shirley M. , Hicks Troy L. , Carlson John W.
Published: December 2010

An example is given of a topological space that does not admit a strongly complete quasi-uniform structure.
An Ultrafilter Completion of a Nearness Space
Authors: Carlson John W.
Published: December 2010

An ultrafilter completion is constructed for a nearness space. It is shown to preserve the Tl separation axiom. Characterizing conditions are given for it to be topological or for its topology to be compact. It is shown to have the simple extension topology and for a given Hausdorff space a compatible nearness structure is found for which its ultrafilter completion is homeomorphic to the Katetov H-closed extension.
The Average Number of Divisors in an Arithmetic Progression
Authors: Smith R. A. , Subbarao M. V.
Published: December 2010

Let l and k be positive integers. Then for each integer n ≥ 1, define d(n; l, k) to be the number of (positive) divisors of n which lie in the arithmetic progression I mod k. Note that d(n;1,1) = d(n), the ordinary divisor function.
The Zeroes of Functions Related to Dirichlet L-Functions
Authors: Weinstein Lenrd
Published: December 2010

Hecke, [3], has shown for x a real Dirichlet character modulo q, the associated Dirichlet L-function L(s, x) has infinitely many zeroes on the line

Here, using a method of Polya, [5], we show that both the real and imaginary parts of a function associated to L(s, x) through the functional equation, have infinitely many zeroes on any line
A Remark on Nonexpansive Mappings
Authors: Goebel Kazimierz , Koter Małgorzata
Published: December 2010

Let X be a closed convex subset of a Banach space and let T: X → X be a nonexpansive mapping, i.e.
The Phragmén-Lindelöf Principle
Authors: Humphries Michael D.
Published: December 2010

The theorem below is one version of the Phragmén-Lindelöf principle [4], which extends the maximum modulus theorem. The theorem has many applications, including the proof of a better-known but less general result [3], which is sometimes attributed to Phragmén and Lindelöf.
Continuous Preimages of Spaces with Finite Compactifications
Authors: Cain George L.
Published: December 2010

A compactification αX of the space X is called an n -point compactification if the remainder αX — X consists of exactly n points. K. D. Magill [5] showed that if Y has an n-point compactification and if f:X→ f(x) = Y is a compact continuous mapping of the space X onto Y, then X also has an n-point compactification.
Operator Topologies and Invariant Operator Ranges
Authors: Ong Sing-Cheong
Published: December 2010

The invariant operator range lattices of a wide class of uniformly closed algebras (including C*-algebras) are stable under weak closures. There is an algebra whose invariant operator range lattice contains properly the corresponding lattice of its norm closure. An operator range transitive algebra is operator range n -transitive for all n. A normal operator is algebraic if and only if each of its invariant operator ranges is the range of some operator commuting with it.
Sommes de la Forme
Authors: Mercier Armel
Published: December 2010

Dans [8], nous avons étudié l'ordre de grandeur des sommes de la forme , pour ou k, où g(n) et f(n) appartiennent respectivement à une classe de fonctions multiplicatives et additives.
On the Total Variation of a Function
Authors: Thomson B. S.
Published: December 2010

There are a number of theories which assign to a function defined on the real line a measure that reflects somehow the variation of that function. The most familiar of these is, of course, the Lebesgue-Stieltjes measure associated with any monotonie function. The problem in general is to provide a construction of a measure from a completely arbitrary function in such a way that the values of this measure provide information about the total variation of the function over sets of real numbers and from which useful inferences can be drawn.
Outer Measures and Total Variation
Authors: Thomson B. S.
Published: December 2010

In this note we collect some observations on the outer measures ψf and ψf that have been introduced in [4] and which describe the total variation of the function f. These properties have direct applications to the study of the derivative and the relative derivative. For definitions and notation the reader is referred to [4].
Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem
Authors: Schirmer Helga
Published: December 2010

Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.
Summability of Matrix Transforms of Subsequences
Authors: Keagy Thomas A.
Published: December 2010

D. F. Dawson has considered several questions of the following nature. Suppose T is a regular matrix summability method. If A is a regular matrix and x is a sequence having a finite limit point, then there exists a subsequence y of x such that each finite limit point of x is a T-limit point of Ay. In the present paper, we show the regularity condition for A may be replaced by the requirement that A be a limit preserving bv to c map. This leads to summability characterizations for several classes of sequences.
On the Cohomological Dimension of Soluble Groups
Authors: Gildenhuys D. , Strebel R.
Published: December 2010

It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities

We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.
Semiprime Rings with Nilpotent Derivatives
Authors: Chung L. O. , Luh Jiang
Published: December 2010

There has been a great deal of work recently concerning the relationship between the commutativity of a ring JR and the existence of certain specified derivations of R. Bell, Herstein, Procesei, Schacher, Ligh, Martindale, Putcha, Wilson, and Yaqub [1, 2, 6, 8, 9, 10, 11, 12, 14] have studied conditions on commutators which imply the commutativity of rings.
Reflexive Topological Semilattices
Authors: Hong S. S. , Nel L. D.
Published: December 2010

The duality between compact 0-dimensional semilattices and discrete semilattices studied by K. H. Hofmann et al. [2] is here extended to larger categories of topological semilattices.

We regard topological semilattices as objects in the category CvSl of convergence semilattices, believing CvSl to be the appropriate setting for this study. The referee objected to this at first, which prompted us to set forth more carefully our reasons for doing so, as follows.
Bivectors over a Finite Field
Authors: MacDougall J. A.
Published: December 2010

Let U be an n -dimensional vector space over a finite field of q elements. The number of elements of Λ2U of each irreducible length is found using the isomorphism of Λ2U with Hn, the space of n x n skew-symmetric matrices, and results due to Carlitz and MacWilliams on the number of skew-symmetric matrices of any given rank.
Remark on Irreducible Factorizations of Polynomials in Several Variables*
Authors: Rodman Leiba
Published: December 2010

It is proved that a polynomial on several complex variables, whose coefficients depend analytically on a parameter ε, admits a factorization which is irreducible for every value of the parameter, with the possible exception of an analytic set of points. Moreover, the coefficients of the irreducible factors can be chosen to depend analytically on ε in a neighborhood of every point not belonging to this analytic set.
Remarks on Two Weak Forms of Continuity
Authors: Espelie M. Solveig , Joseph James E.
Published: December 2010

New characterizations of weakly-continuous andθ-continuous functions are presented, andθ-continuity is applied to characterizeH(i) spaces; a recent characterization of closed graph functions is utilized to characterizeH-closed spaces. Noiri has shown that a function λ which is almost-continuous in the sense of Husain is weakly-continuous if cl(λ−1(W)) ⊂ λ−1(cl(W)) for all openW. It is established here that almost-continuity is superfluous in this statement.
Remarques sur les Solutions Presquepériodiques de L’Équation [(d/dt) − A] x = 0
Authors: N’Guérékata Gaston Mandata
Published: December 2010

L’auteur démontre la presque-périodicité des solutions à trajectoires relativement compactes de l’Équation différentielle abstraite dans un espace de Hilbert: [(d/dt)−A]x = 0 où A =A+ + A, avec A+ un opérateur linéaire symétrique et A_antisymétrique.
Integral Representation by Boundary Vector Measures
Authors: Saab Paulette
Published: December 2010

In this paper we show that if X is a compact Hausdorff space, A is an arbitrary linear subspace of C(X, C), and if E is a Banach space, then each element L of (A ⊗ E)* can be represented by a boundary E*-valued vector measure of the same norm as L.
Systems of Briot-Bouquet Equations with Analytic Solutions
Authors: Westreich David , Podolak Ester
Published: December 2010

In this note we use functional analysis arguments to prove the existence of families of analytic solutions for the singular system of complex ordinary differential equations zW′ = h(z, W).