bluuebunny/crossref_metadata_2025 huggingface dataset:
The Fixed Point Set of Real Multi-Valued Contraction Mappings
Authors: Finbow Arthur S.
Published: December 2010

Let (X, d1) and (Y, d2) be metric spaces. A mapping f:X→Y is said to be a Lipschitz mapping if there exists a real number λ such that

for each x,y∊X. We call λ a Lipschitz constant for f. If λ∊[0, 1), f is called a contraction mapping. Throughout this note CB(Y) denotes the set of closed and bounded subsets of Y equipped with the Hausdorff metric induced by d2.
Approximation of Functions by a Bernstein-Type Operator
Authors: Pethe S. P. , Jain G. C.
Published: December 2010

Various generalizations of the Bernstein operator, defined on C[0, 1] by the relation

1.1

where

have been given. Note that bnk(x) is the well-known binomial distribution.
Uniqueness of Compatible Quasi-Uniformities
Authors: Votaw Charles I.
Published: December 2010

It is shown that a topological space has a unique compatible quasi-uniformity if its topology is finite. Examples are given to show the converse is false for T1 and for normal second countable spaces. Two sufficient conditions are given for a topological space to have a compatible quasi-uniformity strictly finer than the associated Császár-Pervin quasi-uniformity. These conditions are used to show that a Hausdorff, semi-regular or first countable T1 space has a unique compatible quasi-uniformity if and only if its topology is finite. Császár and Pervin described, in quite different ways, quasi-uniformities which induce a given topology. It is shown that, for a given topological space, Császár and Pervin described the same quasi-uniformity.
On Nagumo's Condition
Authors: Rogers Thomas
Published: December 2010

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.

Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and if

then there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.
Oscillation Criteria for Matrix Differential Inequalities(1)
Authors: Allegretto W. , Erbe L.
Published: December 2010

Several authors have recently considered the problem of obtaining sufficient conditions for the oscillation of the quasilinear matrix differential equation

(1)

and the associated inequality VTLV ≤ 0 (as a form). Here A, B, and V are m x m matrix functions, A(x) is symmetric, positive semidefinite and continuous in an interval [a, ∞) and B(x,V, V') is symmetric and continuous in a≤ x < ∞ for all V and V'.
Subdirectly Irreducible Semirings and Semigroups Without Nonzero Nilpotents
Authors: Cornish William H.
Published: December 2010

It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors.
On Selecting a Spurious Observation
Authors: Mount K. S. , Kale B. K.
Published: December 2010

Consider a life testing experiment in which (Xl, X2,…,Xn) are such that (n—1) of them are distributed as f(x, σ)=(l/σ)e-x/σ, x ≥ σ, > 0 and one of them is distributed as f(x, σ/α), 0 < α< l. A priori each Xi has probability 1/n of being a spurious observation distributed as f (x,σ/α). For such an experiment Kale and Sinha [2] showed that if ur denotes the probability that X(r), the rth component of the order statistic, corresponds to the spurious observation, then u1 < u2 < … < un.
A Divergence-Free Antisymmetric Tensor
Authors: Tupper B. O. J.
Published: December 2010

Consider a Vn with metric tensor gαβ. It is well known that the only tensor containing derivatives of gαβno higher than the second order, linear in the second order derivatives, and having vanishing covariant divergence is the Einstein tensor

(1)

Suppose that a system of ennuple vectors hiαand their inverses hαi is set up at each point of the Vn.
A Proof of the Calderon Extension Theorem
Authors: Marsden J.
Published: December 2010

In this note we outline a proof of the Calderon extension theorem by a technique similar to that for the Whitney extension theorem. For classical proofs, see Calderon [2] and Morrey [4]. See also Palais [6, p. 170]. Our purpose is thus to give a more unified proof of the theorem in the various cases. In addition, the proof applies to the Holder spacesCk+α, which was used in [3], and applies to regions satisfying the "cone condition" of Calderon.
On Characterizations of Conditional Expectation
Authors: Al-Hussaini A. N.
Published: December 2010

In the following (Ω, α, μ) is a totally σ-finite measure space except where noted. For a sub-σ-algebra β ⊂ α, the conditional expectation E{f|β} off given β is a function measurable relative to β, such that

In [5] R.G.Douglas proved, among other things the following, in the finite case:

Suppose μ(Ω)=l. Then a linear operator T on L1(Ω, α,μ) is a conditional expect ion if and only if

1.1

1.2

1.3

The point of this note is to characterize conditional expectation in the σ-finite case (Theorems 2, 3).
Theoreme de Müntz Pour les Fonctions de Plusieurs Variables
Authors: Aupetit Bernard H.
Published: December 2010

Le théorème de Weierstrass affirme que toute fonction réelle continue sur l'intervalle [0, 1] est limite uniforme d'une suite de polynômes (voir [5]). Dans [4], Ch. Müntz généralise ce résultat de la façon suivante.

Soit α1, α2, α3,… une suite strictement croissante et non bornée de nombres réels.
An Epi-Reflector for Universal Theories
Authors: Bacsich Paul D.
Published: December 2010

A construction of an epi-reflector by injective hull techniques is given which applies to the class of models of any universal theory with the Amalgamation Property and there yields a weak but functorial type of algebraic closure. Various completions such as the boolean envelope and quotient field constructions are identified as such injective hulls over epimorphic injections. Forms of the Amalgamation Property are also shown to eliminate various pathologies of epimorphisms and equalizers.
Existence of Solutions of Abstract Differential Equations in a Local Space
Authors: Malik M. A.
Published: December 2010

Let H be a Hilbert space; ( , ) and | | represent the scalar product and the norm respectively in H. Let A be a closed linear operator with domain DA dense in H and A* be its adjoint with domain DA*. DA and DA*are also Hilbert spaces under their respective graph scalar product. R(λ; A*) denotes the resolvent of A*; complex plane. We write L = D — A, L* = D — A*; D = (l/i)(d/dt).
Multiple Scalar Transport
Authors: Melzak Z. A.
Published: December 2010

We shall consider scalar transport phenomena in which the mass distribution of a large number of interacting mass particles varies as a result of multiple coalescence mechanisms. Four models will be considered; all the particles are of the same kind in the first three. In the first model an integer n (n≥2) is given and the coalescence mechanism is such that under suitable conditions n particles of masses xl…, xn combine to form one particle of mass we refer to this as the n/1 transport.
Images Of Class- Spaces
Authors: Rishel Thomas W.
Published: December 2010

In this paper the author characterizes images of class- spaces (as defined by Ishii, Tsuda and Kunugi, Proc. Japan Acad. 44, (1968), 897-903) under almost-open maps, bi-quotient maps, pseudoopen maps and quotient maps.
Sectorial Covers for Curves of Constant Length
Authors: Wetzel John E.
Published: December 2010

In answer to a question raised by Leo Moser, A. Meir proved some years ago that every plane arc of unit length lies in some closed semidisk of radius ½. His elegant, unpublished argument is reproduced here with his kind permission.
A New Method in Arithmetical Functions and Contour Integration
Authors: Berndt Bruce C.
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,

Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration

(1.1)

where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.
A Ring of Quotients for Group Rings which is Easy to Describe
Authors: Burgess W. D.
Published: December 2010

Recently Luedeman studied certain idempotent topologizing families of left ideals in semi-group rings AS which arise from such families of left ideals of A. Let ∑ be an idempotent topologizing family of left ideals in A and G a group, let ∑G denote the family of left ideals of AG which contain left ideals of the form LG, L ∈ ∑.
Partially Self-Injective Regular Rings
Authors: Carson Andrew B.
Published: December 2010

It is proved, for any uncountable cardinal λ, that a λ-complete Boolean ring is λ-self-injective. An example shows that the converse need not hold.
Uniquely Line Colorable Graphs
Authors: Greenwell D. L. , Kronk H. V.
Published: December 2010

A line-coloring of a graph G is an assignment of colors to the lines of G so that adjacent lines are colored differently; an n-line coloring uses n colors. The line-chromatic number χ'(G) is the smallest n for which G admits an n-line coloring.
Equalizing the Coefficients in a Product of Polynomials
Authors: Macleod R. A. , Roberts F. D. K.
Published: December 2010

In 1959, Moser [4] posed the following problem: how should a pair of n-sided dice be loaded (identically) so that, on throwing the dice, the frequency of the most frequently occurring sum is as small as possible? This can be recast in the following form: determine for each n(≥1), the polynomial Pn(x) which minimizes the maximum coefficient in the polynomial subject to the conditions that the coefficients of Pn(x) are nonnegative and sum to unity.
On the Equivalence of Modes of Convergence(1)
Authors: Tomkins R. J.
Published: December 2010

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .
Note on "Paracompactness in Small Products"
Authors: Chew James
Published: December 2010

In [1], Willard proves the following

If a regular paracompact space X has a dense Lindelöfsubspace, then X is Lindelöf.

Willard notes that the above is a generalization of the standard theorem: A separable paracompact space is Lindelöf. Actually, it is a standard fact ([2, p. 24]) that a separable metacompact space is Lindelöf. Moreover, one discovers that if a separable space X is such that each open cover of X has a point-countable open refinement, then X is Lindelöf.
A Note on the Implicational Class Generated by a Class of Structures
Authors: Grätzer G. , Lakser H.
Published: December 2010

We use the notations of [2], particularly for operators on classes of structures; in addition, P*r(K) (respectively, Pr(K)) denotes the class of all reduced products of families of structures in K (those with respect to a proper dual ideal, respectively).
On the Levitzki Radical
Authors: Anderson Tim
Published: December 2010

The Levitzki radical, which is fundamental in the study of algebras satisfying a polynomial identity, has been shown to exist in the varieties of alternative and Jordan algebras (see Zhevlakov [8], Zwier [9], and Tsai [7]— for an important application of this radical to alternative algebras satisfying a polynomial identity, see Slater [6]). In fact, Hartley [4] even investigated local nilpotence for Lie algebras, though this property can not be radical in the sense of Kurosh-Amitsur [3] for these algebras.
Inclusion Relations for General Riesz Typical Means
Authors: Jakimovski A. , Tzimbalario J.
Published: December 2010

Let α be a non-negative real number, λ≡{λ,n}(n≥0) a strictly increasing unbounded sequence with λ0≥0 and let be an arbitrary series with partial sums s≡{sn}. Write

where s(t)=sn for λn<t≤λn+1, s(t)=0 for 0≤t≤λ0. The series ∑ an or the sequence of partial sums s={sn} is summable to ṡ by the Riesz method (R, λ, α) if

as ω→∞.
A Non-Abelian Near Ring in Which (-1)r=r Implies r=0
Authors: McQuarrie Bruce
Published: December 2010

In this Bulletin Ligh [2] generalized to finite near rings with identity a theorem Zassenhaus [5] used to prove every finite near field has abelian addition. B. H. Neumann [4] extended Zassenhaus’ result, using similar techniques and showed that all near fields are abelian. It has been an open question whether Ligh’s generalization could be carried out to infinite near rings with identity. The purpose of this paper is to show that Ligh’s theorem cannot be so extended. In particular, it cannot be extended even to distributively generated near rings, a type of near ring which has been useful in studying endomorphism rings of non-abelian groups [1,3].
A Singular Perturbation Problem and A Neutral Differential-Difference Equation
Authors: Moore Edward
Published: December 2010

Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients

(1.1)

where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients

(1.2)

(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.
Lattices with Doubly Irreducible Elements
Authors: Rival Ivan
Published: December 2010

An element x in a lattice L is join-reducible (meet-reducible) in L if there exist y, z∈L both distinct from x such that x=y⋁z (x=y⋀z); x is join-irreducible (meet-irreducible) in L if it is not join-reducible (meet-reducible) in L; x is doubly irreducible in L if it is both join- and meet-irreducible in L. Let J(L), M(L), and Irr(L) denote the set of all join-irreducible elements in L, meet-irreducible elements in L, and doubly irreducible elements in L, respectively, and ℓ(L) the length of L, that is, the order of a maximum-sized chain in L minus one.
On A Problem of Purdy Related to Sperner Systems
Authors: Schonheim J.
Published: December 2010

Purdy asked whether the following conjecture is true

Conjecture. Let E be a set of 2n elements. If S={Sl, S2, …, St} is a Sperner system of E, i.e. for i≠j, i, j, =1, 2, …, t; and if

(1)

then

The proof of the conjecture will be obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337):
The Influence on a Finite Group of its Permutable Subgroups
Authors: Agrawal Ram K.
Published: December 2010

Huppert, Janko and Mann have proved the following theorems for a finite group G.

(Huppert [4]). If each second maximal subgroup of G is normal in G, then G is supersolvable. If the order of G is divisible by at least three different primes, then G is nilpotent.

(Huppert [4]). Let each third maximal subgroup of G be normal in G. Then: (i) G′ is nilpotent; (ii) the rank of G=r(G)≤2; (iii) if |G| is divisible by at least three different primes, then G is supersolvable.
Modules Et Anneaux Quasi-Continus
Authors: Jeremy Louis
Published: December 2010

Le but de ce travail est de prolonger les travaux de Y. Utumi sur les anneaux continus.

Nous appelons module quasi-continu un module stable par tout projecteur de son enveloppe injective. Un anneau unitaire A est dit quasi-continu à gauche (resp. à droite) si le A-module à gauche (resp. à droite ) est quasi-continu. Cette définition est plus générale que celle d'anneau continu à gauche donnée par Yuzo Utumi, et quand l'anneau est régulier, elles coïncident (§111).
On the Irreducibility of a Class of Euler Frobenius Polynomials
Authors: Sharma A. , Straus E. G.
Published: December 2010

In [1,2] the sequence of polynomials

which we shall call Euler-Frobenius polynomials were considered and it was conjectured that these polynomials are irreducible in Q[x] for all odd values of n.
Steepest Descent and Least Squares Solvability
Authors: Groetsch C. W.
Published: December 2010

Let T be a bounded linear operator defined on a Hilbert space H. An element z∈H is called a least squares solution of the equation

if . It is easily shown that z is a least squares solution of (1) if and only if z satisfies the normal equation
On the Matrix With Elements 1/(r+s-1)
Authors: Sawyer W. W.
Published: December 2010

A standard example of a matrix for which the computation of eigenvectors and eigenvalues is very awkward is the matrixAwithars=1/(r+s—1), 1≤r≤n, 1 ≤s≤n. It is therefore of interest thatAcommutes with a tridiagonal matrix.
A Note on the Rate of Convergence of Hermite-Fejér Interpolation Polynomials*
Authors: Saxena R. B.
Published: December 2010

The Hermite-Fejér interpolation polynomial Hn[f] of degree ≤2n—1 is defined by

(1)

Where

(2)

are the zeroes of Chebyshev polynomial of first kind Tn(x)=cos n(arc cos x).
A General and Sharpened form of Opial's Inequality
Authors: Shum D. T.
Published: December 2010

Z. Opial [11] proved in 1960 the following theorem:

Theorem 1. 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.
When is a Matrix Positive
Authors: Brand Daniel
Published: December 2010

Throughout this note we shall use the following conventions and notations: All matrices have entries in the field of complex numbers. I denotes the identity matrix with compatible dimensions. A* is the conjugate transpose of a matrix A. A being self adjoint means A = A*.
A Note Concerning the Real Inflexions of Real, Plane, Algebraic Curves
Authors: Griffith Gareth J.
Published: December 2010

Theorem. “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double point.”
Generalized Sverdrup's Lemma and the Treatment of Less than Full Rank Regression Model
Authors: Kabe D. G.
Published: December 2010

Generalized Sverdrup's lemma, Kabe [5], is used here to give a more direct treatment of less than full rank regression model.
On 2-Summing Operators
Authors: Duncan Richard
Published: December 2010

In this note all Banach space are assumed to be real and separable and their norms will be denoted by || ||. The canonical bilinear form between a Banach space B and its topological dual B′ will be denoted by 〈x, y〉, x ∊ B, y ∊ B′.
On the Branching Theorem of the Symplectic Groups
Authors: Lee C. Y.
Published: December 2010

In [1], Zhelobenko introduced the concept of a Gauss decomposition ZtDZ of a topological group and gave characterizations of irreducible representations of the classical groups. In this setting, vectors of representation spaces are polynomial solutions of a system of differential equations and the problem of obtaining branching theorem with respect to a subgroup G0 is to find all polynomial solutions that are invariant under Z ∩ G0 and have dominant weight with respect to D ∩ G0
On Abundant-Like Numbers
Authors: Erdös Paul
Published: December 2010

Problem 188, [3], stated: Apart from finitely many primes p show that if npis the smallest abundant number for which p is the smallest prime divisor of np then np is not squarefree.
Une Remarque Sur L'interpolation Des Fonctions Entieres
Authors: Giroux André
Published: December 2010

Considérons une fonction entiére f(z) de type exponentiel T. Il est bien connu (voir [1]) que si le module de f(z) reste borné le long de Taxe réel, on a

pour tout x réel.
On a Class of Right Hereditary Semigroups
Authors: Feller E. H.
Published: December 2010

Throughout this paper all semigroups will have identity, and all S-systems (operands) will be right unitary S-systems. All homomorphisms will be S-homomorphisms unless specified.
Mappings Related to Contractions
Authors: Kiang Mo Tak
Published: December 2010

Some results generalizing a fixed point theorem due to R. Kannan are presented.

0. Let be a family of self-mappings of a metric space (X, d).
L’arithmétique Toposienne Des Mots Naturels
Authors: Labelle Gilbert
Published: December 2010

Cette note a pour but d’expliciter une structure mathématique spéciale: . Cette structure sera aux fonctions finies (i.e. fonctions d’un ensemble fini dans un autre) ce que l’anneau ℤ des entiers est aux ensembles finis. Nous ne nous servirons des résultats de la théorie des catégories et des topos que comme outils de base pour les constructions considérées. Le lecteur ne doit donc pas s′attendre à des résultats catégoriques fondamentaux mais bien à un nouvel anneau pouvant servir en théorie des nombres, en combinatoire et peut-être même en physique.
Boundary Points of the Numerical Range of an Operator
Authors: Lin C.-S.
Published: December 2010

The purpose of this note is to investigate boundary points of the numerical range of an operator in terms of inner and outer center points. Some applications on commutators are given.

Throughout this note, an operator will always mean a bounded linear operator on a Hilbert space X.
A Note on Klein’s Oscillation Theorem for Periodic Boundary Conditions
Authors: Faierman M.
Published: December 2010

Recently Howe [4] has considered the oscillation theory for the two-parameter eigenvalue problem

1a

1b

subjected to the boundary conditions

2a

2b

where for i = 1, 2, — ∞<ai<bi<∞, and qi are real-valued, continuous functions in [ai, bi], pi is positive in [ai, biz], and pi(ai)=pi(bi). Furthermore, it is also assumed that (A1B2—A2B1)≠0 for all values of x1 and x2 in their respective intervals.
The Exact Non-Null Distribution of Wilks’ Λ Criterion in the Bivariate Collinear Case
Authors: Mikhail N. N. , Tracy D. S.
Published: December 2010

It is well-known that Wilks’ Λ criterion is distributed as the product of p independent beta variables in the p-variable null-case [3]. In the collinear case, Λ is still distributed as the product of p independent beta variables, one of them following a non-central beta density. Thus when p=2, the exact non-null distribution of Λ in the collinear case is given by the product of two independent beta variables, one central and the other having non-centrality parameter λ.
Hamiltonian Circuits on the N-Cube
Authors: Smith D. H.
Published: December 2010

The problem of finding bounds for the number h(n) of Hamiltonian circuits on the n-cube has been studied by several authors, (1), (2), (3). The best upper bound known is due to Larman (5) who proved that .

In this paper we use a result of Nijenhuis and Wilf (4) on permanents of (0, 1)- matrices to show that for n≥5

where τ, a and c are constants.
Hamiltonian Cycles in Products of Graphs
Authors: Zaks Joseph
Published: December 2010

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.
Conditions for a Zero Sum Modulo n
Authors: Bovey J. D. , Erdös Paul , Niven Ivan
Published: December 2010

In this paper the following result is proved.

Let n > 0 and k ≥ 0 be integers with n — 2k ≥ 1. Given any n — k integers

there is a non-empty subset of indices I ⊂ {1, 2,…, n — k} such that the sum Σi∊I ≡ 0(mod n) if at most n — 2k of the integers (1) lie in the same residue class modulo n.

The result is best possible if n ≥ 3k — 2 in the sense that if "at most n — 2k" is replaced by "at most n — 2k + 1" the result becomes false. This can be seen by taking aj = 1 for 1 ≤ j ≤ n — 2k + 1 and aj = 2 for n —2k + 2 ≤ j ≤ n — k, noting that the number of 2's here is n — k — (n — 2k + 1)= k — 1 ≤ n — 2k + 1.
Most Power Series have Radius of Convergence 0 or 1*
Authors: Fournier J. J. F. , Gauthier P. M.
Published: December 2010

Consider a random power series Σ0∞ cn zn, that is, with coefficients {cn}0∞ chosen independently at random from the complex plane. What is the radius of convergence of such a series likely to be?

One approach to this question is to let the {cn}0∞ be independent random variables on some probability space. It turns out that, with probability one, the radius of convergence is constant. Moreover, if the cn are symmetric and have the same distribution, then the circle of convergence is almost surely a natural boundary for the analytic function given by the power series (See [1, Ch. IV, Section 3]). Our treatment of the question will be elementary and will not use these facts.
Regular Ultrafilters and Long Ultrapowers
Authors: Jorgensen Murray
Published: December 2010

The Ultrapower construction, which builds a new structure AI/D from a relational structure A and an ultrafilter D on a set I, is by now a familiar tool in Model Theory and many other branches of mathematics. In this note we present a result that belongs in the theory of ordered sets, i.e. where the relational structure A has just a single binary relation <, which satisfies the axioms for a strict linear order. We assume that the reader is familiar with the definition and standard notation for ultrapowers, as may be found, for example, in [1]. We differ from [1] only in our eschewing of Gothic capitals.
The Maximal Ideal Space of Subalgebras of the Disk Algebra
Authors: Lund Bruce
Published: December 2010

Let X be a compact Hausdorff space and C(X) the complexvalued continuous functions on X. We say A is a function algebra on X if A is a point separating, uniformly closed subalgebra of C(X) containing the constant functions. Equipped with the sup-norm ‖f‖ = sup{|f(x)|: x ∊ X} for f ∊ A, A is a Banach algebra. Let MA denote the maximal ideal space.

Let D be the closed unit disk in C and let U be the open unit disk. We call A(D)={f ∊ C(D):f is analytic on U} the disk algebra. Let T be the unit circle and set C1(T) = {f ∊ C(T): f'(t) ∊ C(T)}.
Note on Integers Representable by Binary Quadratic Forms
Authors: Williams Kenneth S.
Published: December 2010

Let B be the set of positive integers prime to d which are representable by some primitive, positive, integral binary quadratic form of discriminant d. It is the purpose of this note to show that the following asymptotic estimate for the number of integers in B less than or equal to x can be proved using only elementary arguments:

(1)

where c1 is the positive constant given in (17) below. (Using the deeper methods of complex analysis James [2] has proved this result with the error term ((log x)-1/2) replacing ((log log x)-1). Heupel [1] using a transcendental method as in James [2] improved this to ((log x)-1).)
Absolutely Continuous Measures on Locally Compact Semigroups(1)
Authors: Wong James C. S.
Published: December 2010

Let S be a locally compact Borel subsemigroup of a locally compact semigroup G. It is shown that the algebra of all "absolutely continuous' measures on S is isometrically order isomorphic to the algebra of all measures in M(G) which are "concentrated" and "absolutely continuous" on S.
The Jacobson Radical and Regular Modules
Authors: Fieldhouse David J.
Published: December 2010

Let A be an associative, but not necessarily commutative, ring with identity, and J = J(A) its Jacobson radical. A (unital) module is regular iff every submodule is pure (see (1)). The regular socle R(M) of a module M is the sum of all its submodules which are regular. These concepts have been introduced and studied in (2).
On a Theorem of Rav Concerning Egyptian Fractions
Authors: Webb William A.
Published: December 2010

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability of

where k is fixed.

In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.
Weakly Homogeneous Order Types
Authors: Adams M. E.
Published: December 2010

An order type α is said to be weakly homogeneous (ℵ0 homogeneous) if for any x1 < x2 and y1 < y2 there exists an order preserving bijection f on α such that f(xi)= y i for i = 1, 2. The reverse of an order type a is denoted, as usual, by α*. We say that α is order invertible if α*≤α. J. Q. Longyear [5] has asked whether for a weakly homogeneous order type α such that no (non-trivial) interval of α is order invertible we may deduce that every interval of α contains a copy of ηω1 or (ηω1)*.
Direct Product Decompositions of Twisted Wreath Products
Authors: Brown Jeffrey M.
Published: December 2010

The twisted wreath product of two groups was first defined by B. H. Neumann ([1]) who used this construction to present a group-theoretic proof of a theorem due to Auslander and Lyndon. In this paper we present a complete characterization of the direct product decompositions of a restricted twisted wreath product of two groups A and B provided this product is not simply a semi-direct product of A by B.
Semi-Simple Artinian Rings of Fixed Points
Authors: Cohen Miriam , Montgomery Susan
Published: December 2010

Let G be a finite group of automorphisms of the ring R, and let RG denote the ring of fixed points of G in R; that is, RG={x∊R|Xg = x,∀∊ G}. Let |G| denote the order of G. In this note, we prove the following:

Theorem.Assume that R has no nilpotent ideals and no |G|-torsion. Then if RG is semi-simple Artinian, R is semi-simple Artinian.
On Ring Properties of Injective Hulls
Authors: Lang N. C.
Published: December 2010

Let R be an associative ring and denote by the injective hull of the right module RR. If can be endowed with a ring multiplication which extends the existing module multiplication, we say that is a ring and the statement that R is a ring will always mean in this sense.

It is known that is a regular ring (in the sense of von Neumann) if and only if the singular ideal of R is zero.
Almost Polar-Dense Lattices
Authors: Redfield R. H.
Published: December 2010

We introduce almost polar-dense lattices and prove that the generalized interval topology of an almost polar-dense, modular lattice is equivalent to its interval topology. Furthermore, for totally ordered sets, the converse holds: if the generalized interval topology is the interval topology, then the set is almost polar-dense.
Limit Distributions for Sums of Weighted Random Variables
Authors: Beuerman D. R.
Published: December 2010

Let X1, X2, X3 … be i.i.d., Sn their nth partial sum with Sn = 0; Suppose that
Commuting k-Tuples of Self Adjoint Operators and Matrix Measures
Authors: Browne Patrick J.
Published: December 2010

In this note we give a generalization of a spectral representation theorem for self adjoint operators in a Hilbert space recently obtained by the author [1]. Our interest here is to develop corresponding results for a k-tuple (T1, …, Tk) of commuting self adjoint operators.
A Note on Upper Bounds for the Eigenvalues ofy″ + λpy=0
Authors: Gentry Rodney D.
Published: December 2010

The natural modes of a small planar transversal vibration of a fixed string of unit length and tension are determined by the eigenvalues and associated eigenfunctions of the differential equation

(1)

subject to the boundary condition

(2)

where the non-negative functionpdescribes the mass distribution of the string. That the distribution of mass on the string influences the modes of vibration, may be reflected by observing that the eigenvalues determined by the system (1–2) may be considered functions of the densityp, λn(p), where λ1(p)<λ2(p)<….
Quasi-Injective and Pseudo-Infective Modules
Authors: Jain S. K. , Singh Surjeet
Published: December 2010

Let R be a ring with identity not equal to zero. A right R-module is said to be quasi-injective (pseudo-injective) if for every submodule N of M, every R-homomorphism (R-monomorphism) of N into M can be extended to an R-endomorphism of M [7] ([13]).
Three Remarks on the Measurement of Unit Spheres
Authors: Betts J. R. , Thompson A. C.
Published: December 2010

Let X and Y be isomorphic normed linear spaces and let i ℒ denote the set of isomorphisms from X to Y (i.e. bounded linear mappings from X onto Y which have bounded inverses).
Approximation Polynomiale Generalisee Dans Certains Espaces Semi-Normes
Authors: Lavoie Mario
Published: December 2010

Dans cet article, nous nous intéressons à trouver des conditions suffisantes au problème d'approximation polynômiale généralisée dans certains espaces seminormes. Notre résultat principal est basé sur un théorème d'unicité de Malliavin et sur la théorie de la représentation de Choquet.
Morphismes et Multiplications Dans Une Algèbre
Authors: Lemaire Claude
Published: December 2010

Soient R un anneau commutatif unitaire, S une R-algèbre associative unitaire, A et B des sous-R-modules de S. Nous étudions des situations dans lesquelles HomR(A, B) peut être emboîté dans S. Les définitions et le résultat de base se trouvent dans la section 1; dans la section 2, nous examinons des applications à l'étude des structures d'algèbre que l'on peut définir sur A; la section 3 contient des exemples.
A New Family of Irreducible Representations of An
Authors: Lemire F. W.
Published: December 2010

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.
A Short Combinatorial Proof of the Vaught Conjecture
Authors: Edmunds Charles C.
Published: December 2010

In [5] R. C. Lyndon gave the first proof of the Vaught conjecture: that if a, b9 and c are elements of a free group F such thata2b2=c2, thenab=ba. Lyndon's proof has been followed by many alternative proofs and generalizations [1, 2, 3, 4, 6, 8, 9, 10, 11, 13, 14] all of which involve rather long combinatorial arguments or group theoretical arguments of a noncombinatorial nature.This note provides a short, purely combinatorial proof of the Vaught conjecture.
On a Result of Faith
Authors: Herstein I. N.
Published: December 2010

In a paper several years ago, Faith [2] proved an extension of a well-known theorem of Kaplansky [4]. His proof, even for the division ring case, was somewhat complicated. Using an old trick of Brauer [1] we show how Faith's theorem follows from Kaplansky's immediately.
Stability of Almost Periodic Solutions of an Autonomous Equation
Authors: Chang K. W.
Published: December 2010

The purpose of this paper is to extend to almost periodic (a.p.) solutions a stability result on the periodic solutions of the autonomous equation

x′ = F(x),

(cf. Coppel [1], p. 82 or Coddington and Levinson [2], p. 323.)
Degrees of Vertices in a Friendship Graph
Authors: Kotzig Anton
Published: December 2010

A friendship graph is a graph in which every two distinct vertices have exactly one common adjacent vertex (called a neighbour). Finite friendship graphs have been characterized by Erdós, Rényi and Sós [2]: Each finite friendship graph Fn which consists of n edge disjoint triangles such that all n>1 triangles have one vertex in common (F1 is a triangle i.e. the complete graph with three vertices). Thus Fn has 2n+1 vertices, 2n of them being of degree two and the remaining one (the common vertex of n triangles if n>1) being of degree 2n.
On Sequences {ξtn(mod 1)}
Authors: Strzelecki E.
Published: December 2010

J. Mycielski has conveyed to me the following problem by P. Erdös. Let {tn} be a sequence of natural numbers such that

(a) Does there exist an irrational number η such that the sequence {tnη (mod 1)} is not dense in [0,1)?
The Range of a Gap Series
Authors: Hwang J. S.
Published: December 2010

Theorem. Letbe a function holomorphic in the disk, wherep is a natural number andIfthen then f(z) assumes every complex value infinitely often in every sector.

The purpose of this note is to prove the above result. To do this, we first observe that from the condition a<∞, we can easily show that the derivative f′(z) satisfying
Pointwise Finite Families of Mappings
Authors: Roberts James W.
Published: December 2010

In [3], Montgomery proved that if h is a pointwise periodic homeomorphism of a connected manifold without boundary onto itself, then h is periodic. Kaul generalized this result in [2] by showing that if X is a connected metrizable manifold without boundary and if (X, T)is a transformation group with T countable such that T is pointwise periodic, then T is periodic.
Modules Whose Cyclic Submodules Have Finite Dimension
Authors: Berry David
Published: December 2010

R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.

A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.
On the Property (PU) for *-Regular Rank Rings
Authors: Burke John L.
Published: December 2010

In this paper we consider an irreducible *-regular ring with order k for some k≥4. If is also a Baer ring it is a rank ring. Our first result is:

Theorem 1.3. Let be an irreducible *-regular Baer ring with order k for some k≥4. The following are equivalent.
A Global Existence and Uniqueness Theorem for Ordinary Differential Equations
Authors: Derrick W. , Janos L.
Published: December 2010

As observed by A. Bielecki and others ([1], [3]) the Banach contraction principle, when applied to the theory of differential equations, provides proofs of existence and uniqueness of solutions only in a local sense. S. C. Chu and J. B. Diaz ([2]) have found that the contraction principle can be applied to operator or functional equations and even partial differential equations if the metric of the underlying function space is suitably changed.
Centralizing Automorphisms of Prime Rings
Authors: Mayne Joseph H.
Published: December 2010

Let R be a prime ring and T be a nontrivial automorphism of R. If xxT—xTx is in the center of the ring for every x in R, then R is a commutative integral domain.
A Note on Extending Locally Finite Collections
Authors: Shapiro H. L. , Smith F. A.
Published: December 2010

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.
Oscillation Criteria for Semilinear Equations in General Domains
Authors: Allegretto W.
Published: December 2010

Oscillation theory for nonlinear ordinary differential equations has been extensively developed in recent years by several authors. We refer the reader to the recent paper by Noussair and Swanson, [2], where an extensive bibliography may be found. The situation is somewhat different for the case of second order partial differential equations, an area which has recently been virtually untouched, except for the establishment of criteria which depend on a comparison with suitable linear equations and therefore are essentially linear in nature. A bibliography of such results may also be found in [2]. Of a more general nature have been the paper by the author [1], and the more recent work of Noussair and Swanson, [2]. Although the methods employed and the equations considered in [1] and [2] are different, both papers obtained results only for a class of exterior domains of Rn, of which the typical example is the complement of a bounded sphere.
Alternating Chebyshev Approximation with A Non-Continuous Weight Function
Authors: Dunham Charles B.
Published: December 2010

Let [α, β] be a closed interval and C[α, β] be the space of continuous functions on [α, β], For g a function on [α, β] define

Let s be a non-negative function on [α, β]. Let F be an approximating function with parameter space P such that F(A, .)∊ C[α, β] for all A∊P. The Chebyshev problem with weight s is given f ∊ C[α, β], to find a parameter A* ∊ P to minimize e(A) = ||s * (f - F(A, .))|| over A∊P. Such a parameter A* is called best and F(A*,.) is called a best approximation to f.
Hamiltonian Cycles in Squares of Vertex-Unicyclic Graphs
Authors: Fleischner Herbert , Hobbs Arthur M.
Published: December 2010

In this paper we determine necessary and sufficient conditions for the square of a vertex-unicyclic graph to be Hamiltonian. The conditions are simple and easily checked. Further, we show that the square of a vertex-unicyclic graph is Hamiltonian if and only if it is vertex-pancyclic.
Polynomial Modules Over Macaulay Modules
Authors: Gordon Robert
Published: December 2010

In [2] we introduced a concept of a Macaulay module over a right noetherian ring by saying that all associated primes of the module have the same codimension. That is to say that a module M over a right noetherian ring R is Macaulay if K dim R/P = K dim R/Q for all P, Q ∈ Ass M. Our main aim here is to extend Nagata’s useful result [6], that Macaulay rings are stable under polynomial adjunction, to a noncommutative setting. Specifically, we prove where x is a commuting indeterminate, that the polynomial module M[x] = M⊗RR[x] is a Macaulay R[x]-module if and only if M is a Macaulay R-module. But actually, we prove a more general result. We show that when M is any module over a right noetherian ring, the associated primes of M[x] are precisely the extensions of the associated primes of M.
Local Compactness in Set Valued Function Spaces
Authors: Kaul Saroop K.
Published: December 2010

Recently Hunsaker and Naimpally [2] have proved: The pointwise closure of an equicontinuous family of point compact relations from a compact T2-space to a locally compact uniform space is locally compact in the topology of uniform convergence. This is a generalization of the same result of Fuller [1] for single valued continuous functions.

For a range space which is locally compact normal and uniform theorem B below is an improvement on the result of Hunsaker and Naimpally quoted above [see Remark 3 at the end of this paper].
Local Rings with Elementary Abelian Units
Authors: Nicholson W. K. , Springer H. J.
Published: December 2010

In [2] the structure of all semiperfect rings with abelian group of units has been obtained in terms of commutative local rings. It follows easily that the structure of semiperfect rings with elementary abelian group of units is determined by commutative local rings whose unit groups are elementary abelian. In this note such local rings are completely characterized. It is shown that a local ring having an elementary abelian group of units has characteristic two, four or eight and is a homomorphic image of ZkG/E(ZkG) where G is some elementary 2-group and E(ZkG) is the ideal of ZkG generated by {1 - u2:u∈(ZkG)*}.
Reduction des Points Singuliers Des Feuilletages a Singularites Non Degenerees de M3
Authors: Wagneur E.
Published: December 2010

Soit M une variété réelle de classe C∞ (connexe, paracompacte) et ω une un-forme complètement intégrable sur M (ω ∧ dω = 0). On admet qu’en certains points, dits singuliers, la un-forme peut s’annuler. Désignons par Jω(A) la matrice jacobienne de ω en un tel point A. Un point singulier A de ω est non dégénéré si le rang de Jω(A) est maximum. II est clair que ces singularités sont toujours isolées dans M.
Finite Principal Ideal Rings
Authors: Fisher James L.
Published: December 2010

This paper determines the structure of finite rings whose two sided ideals are principal as left ideals, and as right ideals. Such rings will be called principal ideal rings. Although finite rings have been studied extensively [1], [5], [12], [14] and the tools necessary for describing finite principal ideal rings have been available for thirty years, these structure theorems (which are essentially given in a more general setting in [4]) seem to have been overlooked. In particular, let or be an endomorphism of a ring V.
Absolutely Free Algebras in a Topos Containing an Infinite Object
Authors: Schumacher D.
Published: December 2010

This note confirms that the existence proof for absolutely free algebras originated by Dedekind in [2] and completely developed for instance in [4] can still be carried out in a topos containing an infinite object i.e. an object N for which N ≃ N+1 if the type of the algebras considered is finite, pointed and internally projective i.e. is a finite sequence of objects, (Ij)i≤j≤k for which the functors ( )Ij preserve epimorphisms and each of which has a global section.
Representing Homology Classes on Surfaces
Authors: Schafer James A.
Published: December 2010

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.
Tensor Products and Bimorphisms
Authors: Banaschewski Bernhard , Nelson Evelyn
Published: December 2010

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.
Closure Theorem for Analytic Subgroups of Real Lie Groups
Authors: Djoković D. Ž.
Published: December 2010

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .

This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.
About the # Function
Authors: Klincsek G. T.
Published: December 2010

The use of decreasing rearrangement formulas, and particularly that of the weak N inequality, is illustrated by deriving from Eτ |f-f(τ -)|≤Eτu (where ft is some stochastic process and τ arbitrary stopping time) the estimate ||f||≤Const||u|| in the class of structureless norms with finite dual Hardy bound.

The basic estimate is
Beatty Sequences, Continued Fractions, and Certain Shift Operators
Authors: Stolarsky Kenneth B.
Published: December 2010

Letθ=θ(k) be the positive root ofθ2+ (k-2)θ-k= 0. Letf(n) = [(n+ l)θ]-[nθ] for positive integersn, where [x] denotes the greatest integer inx. Then the elements of the infinite sequence (f(l),f(2),f(3),…) can be rapidly generated from the finite sequence (f(l),f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.