bluuebunny/crossref_metadata_2025_split huggingface dataset:
Unitary Perfect Numbers
Authors: Subbarao M.V. , Warren L. J. 
Published: December 2010

Let σ*(N) denote the sum of the unitary divisors of N, that is,It is easily seen that σ*(N) is multiplicative. In fact σ*(l) = 1 if N > 1 has the prime decomposition . Let us define a positive integer to be unitary perfect whenever σ*(N) = 2N. The first four such numbers are 6, 60, 90 and 87, 360. In a recent abstract [ l ] published by one of us, the last of these numbers was overlooked. No other unitary perfect numbers are known to the authors.
Ring Theoretic Properties of Matrix Rings
Authors: Kaye S.M. 
Published: December 2010

K. Morita has shown that, given two rings R and S, there is an isomorphism between the category of left R-modules and the category of left S-modules if and only if there exists an R-S bimodule U such that(1) U is a progenerator in the category of left R-modules, and(2) S ≅ (EndR U)opp as rings.
Pure Simple and Indecomposable Rings
Authors: Fieldhouse David J. 
Published: December 2010

P. M. Cohn [7] calls a submodule P of the left A-module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E. This concept has been studied in [11] and [12]. We will call a non-zero module pure simple iff its only pure submodules are 0 and itself, and the ring A left pure simple iff it is pure simple as a left A-module. We relate these concepts to the PP and PF rings of Hattori [13], and give several new characterizations of these rings. In order to establish these, we use the following known result: the Jacobson radical of any module is the sum of all its small submodules.Parts of this paper are contained in the author's doctoral thesis [10] at McGill University.
A Simple Solution to the Word Problem for
Lattices
Authors: Day Alan 
Published: December 2010

Whitman [2] solved the word problem for lattices by giving an explicit
construction of the free lattice, FL(X),
on a given set of generators X.The solution is the following:For x, y ∊ X, and
a, b, c,
d ∊ FL(X),(W1)(W2)(W3)(W4)where [p, q] = {x;
p ≤ x ≤ q}.The purpose of this note is to give a simple nonconstructive proof that the
condition (W4) must hold in every projective (hence every free) lattice.
Jonsson [1] has shown that in every equational class of lattices (Wl), (W2),
and (W3) hold. Therefore the combination of these results gives a complete
nonconstructive solution to the word problem for lattices.
Left Cauchy Integral Bases in Linear Topological Spaces
Authors: Dyer James A. 
Published: December 2010

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.
Generalized de la Vallée Poussin Disconjugacy Tests for Linear Differential Equations(1)
Authors: Willett D. 
Published: December 2010

In this paper, we study the oscillatory behavior of the solutions of the linear differential equation(1.1)where(1.2)and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.
Some Theorems on Convex Polygons
Authors: Altman E. 
Published: December 2010

AbstractIn this paper diagonals of various orders in a (strict) convex polygon Pn are considered. The sums of lengths of diagonals of the same order are studied. A relationship between the number of consecutive diagonals which do not intersect a given maximal diagonal and lie on one side of it and the order of the smallest diagonal among them is established. Finally a new proof of a conjecture of P. Erdos, considered already in [1], is given.
A Cure for the Telephone Disease
Authors: Hajnal A. , Milner E. C. , Szemerédi E. 
Published: December 2010

The following problem due to A. Boyd, has enjoyed a certain popularity in recent months with several mathematicians. A different solution to the one given here has been given independently by R. T. Bumby and J. Spencer.The Problem, There are n ladies, and each one of them knows an item of scandal which is not known to any of the others. They communicate by telephone, and whenever two ladies make a call, they pass on to each other, as much scandal as they know at that time. How many calls are needed before all the ladies know all the scandal?
Pullbacks in Regular Categories
Authors: Meisen Jeanne 
Published: December 2010

Given a pair of mapsin a category, we would like to know whether they form part of a pullback diagram as follows:I am indebted to Basil Rattray for mentioning the solution of this problem for the category of sets. Here we shall solve it for any regular category in the sense of Barr [1].It will be useful to make the following definition.
Frieze Patterns in the Hyperbolic Plane
Authors: Garner C. W. L. 
Published: December 2010

AbstractIt is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and
The Existence of Injective Effacements
Authors: Barr Michael 
Published: December 2010

The main result of this paper is little more than a juxtaposition of a remark in [Leroux] (§5, part A), with a theorem of [Banaschewski]. However, the results are individually little known and their juxtaposition not at all. More explicitly, Leroux remarks, in somewhat different form,Let the category X be complete and cocomplete, and have a set of cogenerators. Then a necessary and sufficient condition that there exist infective effacements (with respect to the class of regular monomorphisms) is that X be coregular and satisfy, in addition, condition COEX 4*.
The Fifth Unitary Perfect Number
Authors: Wall Charles R. 
Published: December 2010

A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] thatis also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.
On the Existence and Uniqueness of Solutions of the Equation
Authors: Cleménts John C. 
Published: December 2010

AbstractThe existence and uniqueness of strong global solutions of initial-boundary value problems for the quasilinear equation utt—∂σi(uxi)/∂xi—ΔNut= f is established for functions σi(ξ), i = 1, …, N, satisfying: σi,(ξ) ∊ C1(-∞, ∞), σi(0) = 0 and
for some constant K0.
Independence in Combinatorial Geometries of Rank Three
Authors: Hall Japheth 
Published: December 2010

The class of allcombinatorial geometries of rank threeshall coincide with the class of all pairs (V, S) such thatVis a set andSis a collection of non-empty subsets ofVsuch that each pair of distinct elements ofVbelong to exactly one member ofS. (See [3].)Consider a combinatorial geometry (V, S) of rank three.
Solution of the ‘Cube’ Functional Equation in Terms of ‘Trilinear Coefficients’
Authors: Hemdan H. T. 
Published: December 2010

We consider the following three functional equations123where f:R3→R.Considering their geometric meaning, equations (1) and (2) are known as ‘Cube’ and ‘Octahedron’ functional equations, respectively. Under the assumption of continuity, Haruki [2] has proved that (1) and (2) are equivalent. Etigson [3], has proved the equivalence of (1) and (2) under no regularity assumption. We will give here another proof. Also, under the assumption of continuity, Haruki has solved the ‘Cube’ functional equation. He gave the solution as a certain polynomial of fifth degree in x, y, z individually whose terms are the partial derivatives of a given polynomial.
Amenability and Substantial Semigroups
Authors: Wong James C. S. 
Published: December 2010

AbstractIn this paper, we introduce the concept of topological left substantial subsemigroup of a locally compact semigroup S and prove that if T is such a subsemigroup in S and M(T) is the measure algebra of T, then M(T)* is topological left amenable iff M(S)* is topological left amenable, an extension of a similar result for discrete semigroups.
A Kneser Theorem for Higher order Elliptic Equations
Authors: Allegretto W. 
Published: December 2010

The problem of establishing oscillation and non-oscillation criteria for elliptic equations has recently been considered by several authors. Extensive bibliographies may be found in the books by C. A. Swanson, [7], and by K. Kreith, [3].Most of the interest has so far centered on equations of second order with some results also established for fourth order equations. Non-oscillation theorems for higher order equations have recently been established by the author, [1], and by Noussair and Yoshida, [5]. In particular, both in [1] and [5], Kneser-type theorems were established for classes of higher order elliptic equations.
On the Positive Definiteness of a Functional
Authors: Pandey J. N. 
Published: December 2010

AbstractIn this note, we have shown that the set of conditions given by Gel'fand and Vilenkin for the positive definiteness of a functional L(ϕ) [1, Theorem 7, p. 285] though sufficient is not necessary, thereby providing an answer to an open problem posed by them [1, Theorem 7, p. 285].
Commutators in Free Groups
Authors: Lyndon R. C. , Wicks M. J. 
Published: December 2010

We study the representations of an element of a free group as a commutator. For a given element g of a free group F, we are interested in the set of all pairs (x, y) of elements of F such that(1)where [x, y] = xyx-1 y-1. If g = 1, the problem is trivial. We assume henceforth that g ≠ 1.
Girth and Independence Ratio
Authors: Hopkins Glenn , Staton William 
Published: December 2010

AbstractLower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.
Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations
Authors: Littlejohn Lance L. , Shore Samuel D. 
Published: December 2010

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.
On the Ideal Equation I(B ∩ C) = IB∩IC
Authors: Anderson D. D. 
Published: December 2010

AbstractLet R be an integral domain with quotient field K and let I be a nonzero ideal of R. We show (1) that I is invertible if and only if for every nonempty collection {Bα} of ideals of R and (2) that I is flat if and only if I(B ∩ C) = IB∩IC for each pair of ideals B and C of R.
On Uniform Convergence of Continuous Functions and Topological Convergence of Sets
Authors: Beer Gerald 
Published: December 2010

AbstractLet X and Y be metric spaces. This paper considers the relationship between uniform convergence in C(X, Y) and topological convergence of functions in C(X, Y), viewed as subsets of X×Y. In general, uniform convergence in C(X, Y) implies Hausdorff metric convergence which, in turn, implies topological convergence, but if X and Y are compact, then all three notions are equivalent. If C([0, 1], Y) is nontrivial arid topological convergence in C(X, Y) implies uniform converger ce then X is compact. Theorem: Let X be compact and Y be loyally compact but noncompact. Then topological convergence in C(X, Y) implies uniform convergence if and only if X has finitely many components. We also sharpen a related result of Naimpally.
Regular Digraphs Containing a Given Digraph
Authors: Harary Frank , Karabed Razmik 
Published: December 2010

AbstractLet the maximum degree d of a digraph D be the maximum of the set of all outdegrees and indegrees of the points of D. We prove that every digraph D of order P and maximum degree d has a d-regular superdigraph H with at most d + 1 more points, and that this bound, which is independent of p, is best possible.
Zero-Free Regions for Polynomials and Some Generalizations of Eneström-Kakeya Theorem
Authors: Aziz Abdul , Mohammad Q. G. 
Published: December 2010

AbstractIn this paper we shall use matrix methods to obtain several generalizations of a well known result of Eneström and Kakeya about the location of the zeros of polynomials. We shall also obtain zero-free regions of polynomials having complex coefficients. Finally we prove some results concerning the zeros of a class of polynomials.
Modular Parametrizations of Elliptic Curves
Authors: Zagier D. 
Published: December 2010

AbstractMany — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.
On Radicals of Submodules of Finitely Generated Modules
Authors: McCasland Roy L. , Moore Marion E. 
Published: December 2010

AbstractThe concept of the M-radical of a submoduleBof anR-moduleAis discussed (Ris a commutative ring with identity and A is a unitary fl-module). The M-radical ofBis defined as the intersection of all prime submodules of A containingB.The main result of the paper is that ifdenotes the ideal radical of (B:A), then M-radB =provided that A is a finitely generated multiplication module. Additionally, it is shown that if A is an arbitrary module,wherefor some
Connectedness Properties of Lattices
Authors: Vainio R. 
Published: December 2010

AbstractLetLbe a lattice andqa convergence structure (or a topology) finer than the interval topology ofL.In case of compact maximal chains and continuous lattice translations, the connected components of the space(L,q)are characterized using lattice conditions only. Moreover, lattice conditions ofLare related to connectedness conditions of the order convergence space (L,o).Throughout this note, maximal chain conditions and maximal chain techniques play an important role.
Embedding Partial Graph Designs, Block Designs, and Triple Systems with λ > 1
Authors: Colbournt C. J. , Hamm R. C. , Lindner C. C. , Lindner C. C. , Rodger C. A. 
Published: December 2010

AbstractA general embedding technique for graph designs and block designs is developed, which transforms the embedding problem for partial designs with ƛ > 1 into the embedding problem for partial designs with ƛ = 1. Given an embedding technique for n-element partial block designs with ƛ = 1 into block designs with f(n) elements, the transformation produces a technique which embeds an «-element partial design with ƛ > 1 and block size k into a design with at most /(3k-1ƛn2) elements. For graph designs and block designs with k > 3, a finite embedding method results. For triple systems, a quadratic embedding technique is obtained immediately; the best previous result here was exponential. Finally, for partial triple systems, Mendelsohn triple systems, and directed triple systems, these quadratic embeddings are improved to linear using a colouring technique.
A General Formula on the Conjugate of the Difference of Functions
Authors: Hiriart-Urruty J.-B. 
Published: December 2010

AbstractGiven an arbitrary function g :X → (-∞, +∞] and a lowersemicontinuous convex function h:X → (-∞, +∞], we give the general expression of the conjugate (g — h)* of g - h in terms of g* and h*. As a consequence, we get Toland's duality theorem:
ON H-Sets and Open Filter Adherences(1)
Authors: Krystock Robert L. 
Published: December 2010

AbstractThe relationship between H-sets and open filter adhérences is considered. The open filter adhérences of an H-closed space are shown to be H-sets; and, a necessary and sufficient condition is given for an H-set S, of a Hausdorff space X, to be an open filter adherence. A necessary condition is determined for the existence of a minimal adherent set which contains S; and, in the case that X is H-closed, sufficient conditions are determined. As a related result, an H-closed space X is shown to be seminormal if every H-set of X possesses a neighborhood base consisting of regular open sets.
The Solution to a Problem of Grünbaum
Authors: Salamon Peter , Erdös Paul 
Published: December 2010

AbstractThe paper characterizes the set of all possible values for the number of lines determined by n points for n sufficiently large. For the lower bound of Kelly and Moser for the number of lines in a configuration with n — k collinear points is shown to be sharp and it is shown that all values between Mmin(k) and Mmax(k) are assumed with the exception of Mmax — 1 and Mmax — 3. Exact expressions are obtained for the lower end of the continuum of values leading down from In particular, the best value of c = 1 is obtained in Erdös’ previous expression for this lower end of the continuum.
Blowing-Up and Whitney (a)-Regularity
Authors: Kuo Tzee-Char , Trotman David J. A. , Xin Li Pei 
Published: December 2010

AbstractWhen a pair of adjacent C1 strata (X, Y) is Whitney (a)- regular at a point , there is, up to homeomorphism, just one germ at 0 of intersections S ∩ X where S is a Lipschitz transversal to Y at 0 (S is the graph of a Lipschitz map from The proof uses a blowing-up construction and a strong form of Verdier's (w)-regularity.
On a Result of Bruns
Authors: Kummer Martin , Churchill Richard C. , Rod David L. 
Published: December 2010

AbstractBruns' Theorem states that the classical integrals of the gravitational three-body problem generate all algebraic integrals. We show that the first step in his proof, together with Ziglin's non-integrability criterion for complex systems, can be used to prove the non-existence of energy independent algebraic integrals in certain real analytic systems. We also show that this aspect of Bruns' argument is purely algebraic: We offer a proof based on elementary differential algebraic methods.
Interpolation and Spectra of Regular LP-Space Operators
Authors: Saxe Karen 
Published: December 2010

AbstractWe consider the Banach algebra consisting of linear operators T which are defined on the simple functions and have bounded extensions Tp on LP for all values of p ∊ [1, ∞]. We show that the 'integral' operators in this algebra form a right ideal, and that each Tp associated to an integral T is regular. When the underlying measure is finite or special discrete we show further that every Tp is regular for every T in the algebra. Algebraic techniques together with interpolation results are then used to get relationships between the spectrum and the order spectrum of the associated Tp's.
Rings of Invariants and p-Sylow Subgroups
Authors: Campbell H. E. A. , Hughes I. , Pollack R. D. 
Published: December 2010

AbstractLet V be a vector space of dimension n over a field k of characteristic p. Let G ⊆ Gl(V) be a finite group with p-Sylow subgroup P. G and P act on the symmetric algebra R of V. Denote the respective rings of invariants by RG and Rp. We show that if Rp is Cohen-Macaulay (CM) so also is RG, generalizing a result of M. Hochster and J. A. Eagon. If P is normal in G and G is generated by P and pseudo-reflections, we show that if RG is CM so also is Rp. However, in general, RG may even be polynomial with Rp not CM. Finally, we give a procedure for determining a set of generators for RG given a set of generators for Rp.
Some Applications of a Theorem of Marcinkiewicz
Authors: Bullen P. S. , Vyborny R. 
Published: December 2010

AbstractA classical theorem of Marcinkiewicz states that a function is Perron integrable iff it has one continuous major and one continuous minor function. Using an elaboration of this remarkable theorem three applications are made; to obtain a new proof of a recent characterization of the Perron integral, to proofs of some theorems on interchange of limits and integration and to extend classical existence theorems for ordinary differential equations.
Vertically of (-1)-Lines in Scrolls Over Smooth Surfaces
Authors: Lanteri Antonio 
Published: December 2010

AbstractLet S be a smooth surface contained as an ample divisor in a smooth complex projective threefoldX, which is aP1-bundle, and assume thatinduces OP1(1) on the fibres ofX. The following fact is proven. The restriction toSof the bundle projection of X is exactly the reduction morphism of the pairprovided that this one is not a conic bundle. The proof is very simple and does not involve any consideration on the nefness of the adjoint bundleSome applications of the proof are given.
A Probabilistic Approach to Gradient Estimates
Authors: Cranston M. 
Published: December 2010

AbstractSuppose u is a harmonic function on a domain D and x, x' are in D. We estimate |u(x) — u(x')| using two Brownian motions started at x and x' and killed on exiting a cube Q ⊂ D. By selecting appropriate versions of the two Brownian motions, a classical gradient estimate for u is easily derived.
On the Complete Invariance Property in Some Uncountable Products
Authors: Koszmider Piotr 
Published: December 2010

AbstractWe consider uncountable products of nontrivial compact, convex subsets of normed linear spaces. We show that these products do not have the complete invariance property i.e. they include a nonempty, closed subset which is not a fixed point set (i.e. the set of all fixed points) for any continuous mapping from the product into itself. In particular we give an answer to W.Weiss' question whether uncountable powers of the unit interval have the complete invariance property.
Harmonic Spinors on Hyperbolic Space
Authors: Gaillard Pierre-Yves 
Published: December 2010

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.
Cyclic Subgroup Separability of Generalized Free Products
Authors: Kim Goansu 
Published: December 2010

AbstractWe derive a criterion for a generalized free product of groups to be cyclic subgroup separable. We see that most of the known results for cyclic subgroup separability are covered by this criterion, and we apply the criterion to polygonal products of groups. We show that a polygonal product of finitely generated abelian groups, amalgamating cyclic subgroups, is cyclic subgroup separable.