bluuebunny/crossref_metadata_2025 huggingface dataset:
A Note on Groups of Ree Type
Authors: Lorimer Peter
Published: December 2010

The nonsolvable R-groups as defined by Walter [3] are groups of orders (q3+l)q3(q — 1), q = 32n+1, n ≥ 0. These are the groups of Ree type discussed by Ward [4] together with the Ree group R(3) of order 28.27.2. The R-group with parameter q has a doubly transitive representation of degree q3+1 but in this note we will prove that it cannot contain a sharply doubly transitive subset.
Presentations of the Trefoil Group
Authors: Dunwoody M. J. , Pietrowski A.
Published: December 2010

A presentation of a group G is an exact sequence of groups

where F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.
On the Diophantine Equation x(x + d)(x + 2d) +y(y + d)(y + 2d) = z(z + d)(z + 2d)
Authors: Bernstein Leon
Published: December 2010

A previous result of the author concerning the parametric representation of infinitely many solutions of the title equation is strongly improved. New classes each containing infinitely many solutions of the equation for specified values of d are stated explicitly. The method of solution hinges heavily on solving the generalized Pell’s equation x2—Dy2=c.
Orbital Lattices of SL+(2, Z)
Authors: Rankin S. A.
Published: December 2010

The results described in this paper were obtained in the study of analytic maps between flat two-tori. It is felt that analogous results will be obtained from a similar study of the analytic maps between higher-dimensional flat tori.

It is well-known that the multiplicative group GL+(2, Q) acts on the upper halfplane as a group of linear fractional transformations, i.e. for each element
A Partition Theorem of Subbarao
Authors: Gupta Hansraj
Published: December 2010

Let C(n) be the number of partitions of n such that all even multiplicities of the parts are less than 2m, m>1; and all odd multiplicities are at least (2r+1) and at most 2(m+r)—1, r≥0. Let D(n) be the number of partitions ofn into parts which are either odd multiples of (2r+1) or are even and not divisible by 2m.
A Maximality Criterion for Nilpotent Commutative Matrix Algebras
Authors: Handelman D. , Selick P.
Published: December 2010

Let A be a commutative algebra contained in Mn(F), F a field. Then A is nilpotent if there exists v such that Av=(0), and is said to have nilpotency class k (denoted Cl(A)=k) if Ak=(0), but Ak-1≠(0). A well known result asserts that matrix algebras are nilpotent if and only if every element is nilpotent. Let N = {A | A is a nilpotent commutative subalgebra of Mn(F)}.
A Short Proof of an Interpolation Theorem
Authors: Hughes Edward
Published: December 2010

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].

Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .
Self-Complementary Generalized Orbits of a Permutation Group
Authors: Frucht Roberto , Harary Frank
Published: December 2010

A permutation groupAof degree n acting on a setXhas a certain number of orbits, each a subset ofX. More generally, A also induces an equivalence relation onX(k)the set of allksubsets ofX, and the resulting equivalence classes are calledkorbits ofA, orgeneralized orbits. Aself-complementary k-orbitis one in which for everyk-subsetSin it,X—Sis also in it. Our main results are two formulas for the numbers(A)of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.
Linear Functionals and Summability Invariants
Authors: Macphail M. S. , Wilansky A.
Published: December 2010

The purpose of this paper is to continue the study of certain “distinguished” subsets of the convergence domain of a matrix, as developed by A. Wilansky [6] and G. Bennett [1], We also consider continuous linear functionals on the domain, and the extent to which their representation is unique; this turns out to be connected with the behaviour of the subsets.
Some Fixed and Common Fixed Point Theorems in Metric Spaces
Authors: Sehgal V. M.
Published: December 2010

Let (X, d) be a metric space and Ti(i=l, 2) be self mappings of X. The purpose of this paper is to investigate the fixed and common fixed points of Ti, when the pair Ti(i=l, 2) satisfies a condition of the following type:

(1)
On Von Neumann Regular Rings
Authors: Koh Kwangil
Published: December 2010

Recently, in the Research Problems of Canadian Mathematical Bulletin, Vol. 14, No. 4, 1971, there appeared a problem which asks “Is a prime Von Neumann regular ring pimitive?” While we are not able to settle this question one way or the other, we prove that in a Von Neumann regular ring, there is a maximal annihilator right ideal if and only if there is a minimal right ideal.
The Local Growth of Power Series: A Survey of the Wiman-Valiron Method
Authors: Hayman W. K.
Published: December 2010

Suppose that

1.1

is a transcendental integral function. In this article we develop the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kövari [10, 11], which describes the local behaviour off(z), near a point where |f(z) | is large, in terms of the power seriesf off(z).
Adjoints of a Geometry
Authors: Cheung Alan L. C.
Published: December 2010

To give a geometric interpretation to the inverted incidence relation between the flats of a geometry has for years been a tempting idea in combinatorial geometries [1]. If G is a geometric lattice, the inverted lattice G' is not necessarily geometric.
On Adjacency Preserving Maps
Authors: Westwick R.
Published: December 2010

In his paper [1] on homogeneous spaces W. L. Chow states that “ Any one-toone adjacency preserving transformation of the Grassmann space of all the [r] of Sn(0 < r < n— 1) onto itself is a transformation of the basic group of the space.” In the proof both the transformation and its inverse are assumed to be adjacency preserving. See also Dieudonne [2] p. 81.
On the Minimum Order of Graphs with Given Group
Authors: Babai László
Published: December 2010

ForGa, finite group letα(G) denote the minimum number of vertices of the graphsXthe automorphism groupA(X) of which is isomorphic toG.

G. Sabidussi proved [1], thatα(G)=0(nlogd) wheren=\G\anddis the minimum number of generators ofG.As0(log n) is the best possible upper bound ford,the result established in [1] implies thatα(G)=0(nlog logn).
Model Theory of Epimorphisms
Authors: Bacsich Paul D.
Published: December 2010

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .
A General Tychonoff Theorem for Multifunction
Authors: Fox Geoffrey , Morales Pedro
Published: December 2010

Let {Xa}a∈A be a non-empty family of compact spaces and let F be a subset of the m-product P{Xa:a∈A}. The Tychonoff problem is the determination of the sets F which are pointwise compact. It is known that P{Xa: a ∈ A} is pointwise compact [5, p. 400]. Also the set F of all point-closed multifunctions of P{Xa:a∈ A) is pointwise compact.
A Lower Bound for the Permanent on a Special Class of Matrices
Authors: Hartfiel D. J.
Published: December 2010

Let Un(f) denote the class of all n × n (0, 1)-matrices with precisely r-ones, r≥3, in each row and column. Then
On the Derived Cuboid
Authors: Spohn W. G.
Published: December 2010

We showed in [1] that the Eulerian family of cuboids with integral edges and face diagonals did not have integral inner diagonals. We now show that the derived family does not have integral inner diagonals except possibly when the generators are divisible by 705180. In this case there appears to be no inherent reason why the diagonals cannot be integral.
Radical Classes need not have a Unique Maximal R0-Closed Subclass
Authors: Mura Roberta Botto
Published: December 2010

It was shown in [1] that certain classes of groups which are closed under quotients, and extensions contain a unique maximal inclosed subclass. These results prompted the question whether there exists a class of groups which is closed under quotients and extensions and yet does not have a unique maximal R0-closed subclass. This note provides an example of such a class.
Some Properties of Inflated Binomial Distribution
Authors: Sobich Lucja , Szynal Dominik
Published: December 2010

The role of binomial distribution in the probability theory and its applications is well known. Recently M. P. Singh [4] has been discussing the socalled inflated binomial distribution. This distribution was introduced for situations which are described by simple binomial except for zero cell which is inflated, that is, there more observations than can be expected on the basis of simple binomial. The investigation of this inflated binomial distribution seems to be useful.
On Cayley-Dickson Rings
Authors: Britten Daniel J.
Published: December 2010

M. Slater has shown that a prime alternative (not associative) ring R such that 3R≠0 is a Cayley-Dickson ring, [7], That is, if F is the field of quotients of the center, Z, of R then F ⊗Z R is a Cayley-Dickson algebra.
Selfadjoint Metrics on Almost Tangent Manifolds Whose Riemannian Connection is Almost Tangent
Authors: Goel D. S.
Published: December 2010

Let M be a differentiable manifold of class C∞, with a given (1, 1) tensor field J of constant rank such that J2=λI (for some real constant λ). J defines a class of conjugate (G-structures on M. For λ>0, one particular representative structure is an almost product structure. Almost complex structure arises when λ<0. If the rank of J is maximum and λ=0, then we obtain an almost tangent structure. In the last two cases the dimension of the manifold is necessarily even. A Riemannian metric S on M is said to be related if one of the conjugate structures defined by S has a common subordinate structure with the G-structure defined by S. It is said to be J-metric if the orthogonal structure defined by S has a common subordinate structure.
Hausdorff Compactifications as Epireflections
Authors: Hunsaker W. N. , Naimpally S. A.
Published: December 2010

We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.
Some Remarks on Regular and Strongly Regular Rings
Authors: Raphael R.
Published: December 2010

This article presents some new algebraic and module theoretic characterizations of strongly regular rings. The latter uses Lambek’s notion of symmetry. Strongly regular rings are shown to admit an involution and form an equational category. An example due to Paré shows that the category of regular rings and ring homomorphisms between them is not equational. Remarks on quasiinverses and the generalized inverse of a matrix are included. The author acknowledges support from the NRC (A7752) and improvements from W. Blair received after announcement of the results.
On Product Decompositions
Authors: Varadarajan K.
Published: December 2010

Borsuk has asked whether there exists for each compact metric absolute neighbourhood retract X an integer l (depending only on X) with the property that if X is homotopy equivalent to a cartesian product of more than l spaces than at least one of these spaces is contractible. The answer to this question is still not known. The following theorem has been proved by Ganea and Hilton [Theorem 1.3 of [1]].
On Ramsey Graph Numbers for Stars and Stripes
Authors: Cockayne E. J. , Lorimer P. J.
Published: December 2010

Any term or symbol undefined in this paper is defined in [5]. For graphs F and G, G > F means G contains a subgraph isomorphic to F and E(G) denotes the edge set of G. If E ⊆ E(G), 〈E〉 is the subgraph of G whose edge set is E and whose vertex set is that subset of vertices of G which are incident with edges in E.
Absolute Convexity in Spaces of Strongly Summable Sequences
Authors: Maddox I. J. , Roles J. W.
Published: December 2010

The space wp of strongly Cesàro summable sequences of index p > 0 has been investigated by several authors. In [2], Kuttner proved that no Toeplitz matrix could sum all sequences in wP, a result which was extended to coregular matrices by Maddox [5]. In [1], Borwein considered the continuous dual space of wp. The more general space w(p) has also been considered [3, 4], where p = (pk) is a strictly positive sequence. The r-convexity of the spaces w∞(p) and w0(p) was dealt with in a partial way in [8]. In the present note we establish criteria for the rconvexity of some general classes of [A, p]0 and [A, p]∞ spaces (see [6] and [7] for definitions), and in particular we give the necessary and sufficient conditions for the r-convexity of w∞(p) and w0(p). For most of the relevant definitions and notation we refer to [8].
On Semiperfect Modules
Authors: Nicholson W. K.
Published: December 2010

Sandomierski (Proc. A.M.S. 21 (1969), 205–207) has proved that a ring is semiperfect if and only if every simple module has a projective cover. This is generalized to semiperfect modules as follows: If P is a projective module then P is semiperfect if and only if every simple homomorphic image of P has a projective cover and every proper submodule of P is contained in a maximal submodule.
On the Extension of Uniformly Continuous Functions(1)
Authors: Gardner L. T. , Milnes P.
Published: December 2010

A theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.
On Certain Problems in the Theory of Sequences
Authors: Higgins Rada
Published: December 2010

We are well-acquainted with the theorem about sequences which states that, the existence of

(1)

is sufficient to imply limk→∞ak = 0. Partially out of a growing interest in the theory of regularly varying sequences ([1]), and probably as an interesting problem, in and of itself, some mathematicians have tried to find conditions weaker than (1) that would guarantee limk→∞ak = 0.
Relations Between Boundary Value Functions for a Nonlinear Differential Equation and its Variational Equations*
Authors: Spencer James D.
Published: December 2010

We consider here the nonlinear differential equation

(1.1)

where x ∊l= [a, ∞). We will make the following assumptions

(A)

f is continuous on [a, ∞) × Rn,

(B)

solutions of initial value problems (I.V.P.'s) are unique and extend throughout [a, ∞).

(C)
Extension of a Semigroup Embedding Theorem to Semirings
Authors: Karyellas Paul H.
Published: December 2010

It is well known [1,3] that a commutative semigroup (S, +) can be embedded in a semigroup which is a union of groups if and only if S is separative (2a = a + b = 2b implies a = b). We extend this result to additively commutative semirings.

A semiring (S, +, ⋅) is a set S with associative addition (+) and multiplication (⋅), the latter distributing over addition from left and right. In what follows (S, +, ⋅) will denote a semiring in which the additive semigroup (S, +) is commutative. An element 0 can be adjoined, where s = s + 0, 0 = 0 ⋅ s = s ⋅ 0 for all s in S, to form S°.
A Comparison Between thin Sets and Generalized Azarin Sets
Authors: Essén M. , Jackson H. L.
Published: December 2010

Let Rp(p≥2) denote p-dimensional Euclidean space, D the half space defined by {P = (x1, x2, …, xp) ∊ Rp: xp > 0} and ∂D the frontier of D in Rp. The Martin boundary (see [2]) of D can be identified with ∂D∪{∞}.
Functions Belonging to a Dirichlet Subalgebra of the Disk Algebra
Authors: Lund Bruce
Published: December 2010

Browder and Wermer in [2] give a method for constructing Dirichlet subalgebras of the disk algebra. In this note we show that these Dirichlet algebras do not contain any non-constant functions which satisfy a Lipschitz-one condition on a subinterval of the unit circle.
Inner Functions and Isometries
Authors: Muhly Paul S.
Published: December 2010

Our objective in this note is to prove a theorem about the spectral behavior of certain multiplication operators. It was motivated by and yields an extension of part of a theorem of Lumer [3] concerning the possibility of imbedding the classical Hardy spaces in abstract ones.
On Second-Order Differential Operators With Bohr-Neugebauer Type Property
Authors: Rao Aribindi Satyanarayan
Published: December 2010

Let B be a bounded linear operator having domain and range in a Banach space. If the second-order differential operator d2/dt2–B has a Bohr-Neugebauer type property for Bochner almost periodic functions, then any Stepanov-bounded solution of the differential equation (d2/dt2)u(t) – Bu(t) = g(t) is Bochner almost periodic, with g(t) being a Stepanov-almost periodic continuous function.
La 7e Conférence Jeffery-Williams Présentée le 7 Juin 1974 à L'université Laval à L'occasion de la 28e Réunion Annuelle de la Société Mathématique
Authors: Zassenhaus Hans J.
Published: December 2010

Il me fait un très grand plaisir de vous présenter aujourd'hui la conférence Jeffrey-Williams nommée ainsi en l'honeur des distingués fondateurs de la Société Mathématique du Canada.

Il y a beaucoup d'évidences indirectes d'un vif échange d'idées entre Herman Minkowski et David Hilbert, pendant la période de temps entre la soutenance de thèse de Hilbert (décembre 1884) et la mort prématurée de Minkowski (le 12 janvier 1909).
Contractions with Fixed Points and Conditional Expectation
Authors: Al-Hussaini A. N.
Published: December 2010

Let (Ω, α, μ) be a σ-finite measure space. By Lp(Ω, α, μ) or Lp for short we denote the usual Banach space of pth power μ-integrable functions on Ω if 1≤p<+ ∞ and μ-essentially bounded functions on Ω, if p= +∞. In section (2) we characterize conditional expectation, by a method different than those used previously. Modulus of a given contraction is discussed in section (3). If the given contraction has a fixed point, then its modulus has a simple form (theorem 3.2).
An N-Parameter Chebyshev Set which is not a Sun
Authors: Braess Dietrich
Published: December 2010

Recently, Dunham has given examples for 1-parameter and 2-parameter Chebyshev sets which are not suns. In this note 2-parameter sets with these properties are described.

When studying the old problem whether Chebyshev sets are always convex, Klee [10] introduced certain sets which were called suns by Efimov and Stechkin [7]. Recently, in two shorts notes Dunham [4, 5] has given examples of 1-parameter- and 2-parameter-sets which are Chebyshev sets but not suns (cf. also [3]). The examples refer to Chebyshev sets in containing an isolated point.
Some Results on Spiral-Like Functions
Authors: Bagşöze Türkân
Published: December 2010

We begin with the following definitions.

Let f(z) be regular near z=0 and let f(0)=0, f'(0)≠0. Let α and λ be two real numbers such that |α|<π/2 and 0≤λ1. Then Rα,λ is the largest value of r such that the following conditions are satisfied for |z|<r
Another Possible Abnormality of Compact Space-Time
Authors: Lee K. K.
Published: December 2010

It is shown that the Lorentz structure of a compact prespace- time M can be so chosen such that M can not satisfy the strong energy condition. Thus, combining both the causal and the strong energy conditions, a stronger case against the compact space-times as proper arenas of physics can be made.
Packing and Covering of the Complete Graph with 4-Cycles*
Authors: Schönheim J. , Bialostocki A.
Published: December 2010

The maximal number of pairwise edge disjoint 4-cycles in the complete graph Kn and the minimal number of 4-cycles whose union is Kn are determined.
A Proof of Jacobson’s Theorem
Authors: Dolan S. W.
Published: December 2010

Let R be a ring such that, for each element a of R, there exists a positive integer n(a) > 1, depending on a, such that an(a)=a. Jacobson proved that such a ring R is necessarily commutative and the purpose of this paper is to give a proof of Jacobson’s Theorem that does not involve the use of the axiom of choice.

We take this opportunity to point out that all rings are assumed to be associative and that nothing beyond elementary ring theory is assumed in the proof to follow. Such ring theory can be found, for example, in [1].
A Characterization of the Commutator Subgroup of a Group
Authors: Thierrin G.
Published: December 2010

An element a of a semigroup S is n-potent if there exist a1, a2,..., ak∈S such that a = a1a2...ak and If S is a group, the set of n-potent elements is a normal subgroup of S and the set of 1-potent elements is the commutator subgroup of S.
Some Recent Results on Invariant Subspaces
Authors: Rosenthal Peter
Published: December 2010

This expository paper surveys work on invariant subspaces and related topics which has been done in the past few years. We recommend, naturally, that the reader consult [52] for work done prior to 1973 and [54] for a discussion of some of the consequences of Lomonosov’s Lemma; (Lomonosov’s paper has now also appeared in English ([36])).
Full Ideals and Ring Groups in Zn[x]
Authors: Suvak John A.
Published: December 2010

If we add the operation of composition to the polynomial ring R[X], where R is a commutative ring with identity, we get a tri-operational algebra . A full ideal or tri-operational ideal of is the kernel of a tri-operational homomorphism on . This is equivalent [4, pp. 73–74] to the following: A full ideal of is a ring ideal A of R[x] such that f°g ∈ A for every f ∈ A and g ∈ R[x].
1-Cohomology and Splitting of Group Extensions
Authors: Pandya G. N. , Bercov R. D.
Published: December 2010

The object of this note is to give simpler proofs of a splitting theorem of Gaschütz [1] and a related theorem for groups with operators by using cross-homomorphisms (1-cocycles) instead of 2-cohomology.

We recall that a cross-homomorphism or 1-cocycle from a group E to an abelian normal subgroup N of E is a map f from E to N such that f(ab)=(f(a))bf(b) for all a, b ∈ E where superscript denotes conjugation. Cocycles f and h are equivalent if for some n ∈ N have h(e)=nef(e)n-1 for all e ∈ E.
The Set of Julia Points for Functions Omitting Two Values
Authors: Gauthier P. M. , Hwang J. S.
Published: December 2010

Let f be a function denned in the unit disk D(|z| < 1). For each point eiθ on the unit circle C(|z| = 1) and each subset S of D, we denote by Cs(f, eiθ) the cluster set of f at eiθ relative to s, i.e.

where N(eiθ, j) = {z∈D:|z-eiθ| <1/j}.
A Map of a Polyhedron onto a Disk
Authors: Strube Richard F. E.
Published: December 2010

A map f: X → Y is said to be universal if for every map g:X → Y there exists an x ∈ X such that f(x) = g(x). In [2] W. Holsztynski observed that if B is a Boltyanskiĭ continuum (see [1]), then there exists a universal map f:B→ I2 such that the product map fxf:BxB→I2×I2 is not universal. Using this he showed that B can be replaced by a two-dimensional polyhedron. He did not, however, give a concrete example. We exhibit explicitly a two-dimensional polyhedron K and a universal map f:K→ I2 such that f×f:K×K→ I2×I2 is not universal.
A Note on Locally Quasi-Uniform Spaces
Authors: Hicks Troy L. , Huffman Shirley M.
Published: December 2010

Locally quasi-uniform spaces are studied, and it is shown that a topological space (X, t) admits exactly one compatible locally quasi-uniform structure if and only if t is finite.
A Note on the Two-Square Theorem
Authors: Fine Benjamin
Published: December 2010

One of the nicest theorems of elementary number theory is the two-square theorem of Fermat.

A prime p is expressible as the sum of 2 squares if and only if -1 is a quadratic residue mod p.
Note on Burde's Rational Biquadratic Reciprocity Law
Authors: Williams Kenneth S.
Published: December 2010

A short proof is given of a biquadratic reciprocity law proved by Burde in 1969.

Let p and q be primes ≡1 (mod 4) such that (p | q) = (q | p) = 1. Then there are integers a, b, c, d with

Set
Invertible Operators on Certain Banach Spaces
Authors: Belley J.-M.
Published: December 2010

It has long been the practice in the theory of Hilbert spaces to use the Fourier series expansion (i.e. the Levy inversion formula) for the resolution of the identity associated with a unitary operator to obtain results for this operator, and hence for any power bounded invertible operator on such spaces since they are necessarily isomorphic to unitary operators [5, p. 1945]. Though many important power bounded operators on Banach spaces are not spectral [6, p. 1045-1051] the approach of this paper permits us to deduce for such operators results similar to those known for spectral operators.
Tauberian Theorems for Strong and Absolute Borel-Type Methods of Summability(1)
Authors: Borwein D. , Smet E.
Published: December 2010

Suppose throughout that s, an (n = 0,1,2,…) are arbitrary complex numbers, that α > 0 and β is real and that N is a non-negative integer such that αN + β≥1. Let
On Spanning and Dominating Circuits in Graphs
Authors: Lesniak-Foster L. , Williamson James E.
Published: December 2010

A set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.
Modules with Finite Spanning Dimension
Authors: Rangaswamy K. M.
Published: December 2010

Modules with finite spanning dimension were defined by P. Fleury [3] in an attempt to dualize the concept of Goldie dimension. In this note we study these modules in some detail, obtain an improved structure theorem for them and also extend the work done in [2] and [3]. Projective modules with finite spanning dimension turn out to be local or artinian.
Some Pointwise Convergence Results in Lp(μ), 1 < p < ∞
Authors: Duncan Richard
Published: December 2010

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.
Some Results in the Theory of Fibrations
Authors: Piccinini Renzo A.
Published: December 2010

I wish to present here some of the results of a research in the Theory of Fibrations initiated some time ago by Peter Booth, Philip Heath, and myself. The philosophy behind the work is to approach certain aspects of the Theory of Fibrations in a unified way through the systematic use of the sections of suitable fibrations; this yields general theorems, of which some well-known results are eventually particular cases.
Infinite Geometric Products
Authors: Vagholkar M. K. , Staff P. J.
Published: December 2010

This paper is concerned with the infinite geometric products

and their generalizations to higher dimensions. Some new expressions and identities are derived for these products by using stochastic theory. The function is tabulated for p = 0(0.01)1.
Extensions of Orderable Groups
Authors: Mura R. Botto , Rhemtulla A. H.
Published: December 2010

The purpose of this note is to unify Theorem 4 of G. Baumslag [2] and a result of D. M. Smirnov in [6] in a more general setting. We prove the following result.
Images in Topoi
Authors: Rowe K. A.
Published: December 2010

The construction of images of morphisms in an elementary topos E has hithero required the use of colimits. For example, in [1], Freyd constructs the image of a morphism by taking the equalizer of the cokernel pair of the morphism. In particular, the construction of the direct image functor, or, as it is sometimes referred to, existential quantification along a morphism, has required the use of images, and hence colimits. However, Mikkelsen has defined existential quantification using only limits.
Some Generalizations of an Identity of Subhankulov
Authors: Suryanarayana D. , Walker David T.
Published: December 2010

In 1957, M. A. Subhankulov established the following identity

where ; μ is the Môbius function and J2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).

In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.
On Sheaf Representation of a Biregular Near-Ring
Authors: Szeto George
Published: December 2010

It is shown that R is a biregular near-ring if and only if it is isomorphic with the near-ring of sections of a sheaf of reduced near-rings over a Boolean space. Also, some ideal properties of a biregular near-ring are proved. These are considered as generalizations of some works of R. Pierce on biregular rings.
A Helly type Theorem for Convex Sets
Authors: Katchalski Meir
Published: December 2010

A ray in Euclidean n-dimensional space Rn is a set of the form {a + λb: λ≥ 0 } where a and b are fixed points in Rn and b≠0.

The subject of this paper is a Helly type theorem for convex sets in Rn.

If is a finite family of at least 2n convex sets in Rn and if theintersection of any 2n members of contains a ray then containsa ray.
Moulton Affine Hjelmslev Planes
Authors: Baker Catharine
Published: December 2010

Desarguesian affine Hjelmslev planes (D.A.H. planes) were introduced by Klingenberg in [1] and generalized by Lane and Lorimer in [2]. D.A.H. planes are coordinatized by affine Hjelmslev rings (A.H. rings) which are local rings whose radicals are equal to their sets of two-sided zero divisors and whose principal right ideals are totally ordered. In [5], ordered D.A.H. planes were defined and the induced orderings of their A.H. rings were discussed. In this note an ordered non-Desarguesian A.H. plane is constructed from an arbitrary ordered D.A.H. plane. The existence of such planes ensures that the discussion of ordered non-Desarguesian A.H. planes by J. Laxton in [3] is meaningful. The basic idea employed is essentially the same as the one used in the construction of the classical Moulton plane from the real affine plane (cf. [4])
The Representation of (C, k) Summable Series in Fourier Form
Authors: Cross G. E.
Published: December 2010

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series

1.1
On the Input-Output Stability of Linear Controllable Systems
Authors: Megan Mihail
Published: December 2010

The extension of bounded input, bounded output criteria of Perron [1] for the case of a linear control system

A, B

has been widely discussed (cf. e.g., references [2] to [10]). As is well known ([2]) the equation

A
How Fields Can have a Product
Authors: Zorzitto Frank
Published: December 2010

Let k be a field. Two field extensions E, F of k are said to have a product- in the category of field extensions of k (see e.g. [1, p. 30]) if and only if there exist a field extension P of k and two k -isomorphisms P→ E, P→ F satisfying the following universal property. For any field extension K of k and any pair of k-isomorphisms K→E, K→F, there exists a unique k-isomorphism K→P such that the diagrams below commute.
On Almost Contingent Manifolds of Second Class with Applications in Relativity
Authors: Duggal K. L.
Published: December 2010

D. E. Blair [1] has introduced the notion of K-manifolds as an analogue of the even dimensional Kähler manifolds and of the odd dimensional quasi-Sasakian manifolds. These manifolds have been studied with respect to a positive definite metric. In this paper, we study a more general case of if-manifolds carrying an arbitrary non-degenerate metric, in particular, a metric of Lorentz signature. This theory is then applied within the frame-work of general relativity. Using the Ruse-Synge classification [8, 9] of non-null electromagnetic fields with source, we develop a geometric proof for the existence of either two space like or one space like and one time like Killing vector fields on the space-time manifold.
Order Characterization of the Complex Field
Authors: Novoa Lino Gutierrez
Published: December 2010

It is well known that the real number field can be characterized as an ordered field satisfied the “least upper bound” property.

Using the idea of n -ordered set, introduced in [3], and generalizing the notion of l.u.b. in a suitable way, it is possible to give a similar categorical definition of the complex field.

With these extended meanings, the main theorem of this paper (Theorem 7 in the text) is stated almost identically to the one for the real field. Any directly two-ordered field, in which the "supremum property" holds, is isomorphic to the complex field.
On Weak Vitali Covering Properties
Authors: Thomson B. S.
Published: December 2010

There are now a number of Vitali covering properties which have been defined to handle problems arising in differentiation theory. Although some of these have received a unified treatment, as for example in the setting of Orlicz spaces in [1, p. 168], the underlying simplicity can be lost and the intimate connection with the original weak Vitali covering property of de Possel obscured. In this note we present an exposition of a family of covering properties and show how the original methods of de Possel in [4] can be pushed to provide an exact solution of the problem of determining necessary and sufficient covering properties for a basis which is known to differentiate a given class of integrals.
Applications of Variants of the Hölder Inequality and its Inverses: Extensions of Barnes, Marshall-Olkin, and Nehari Inequalities.
Authors: Wang Chung-Lie
Published: December 2010

The primary aim of this paper is to extend Barnes [1], Marshall-Olkin [6], and Nehari [8] inequalities as applications of some results introduced in [10] by the author.

Since several results from various sources are adopted here, a unified notation is required in order to simplify our subsequent arguments. To this end, let Lp = Lp(S, ∑, μ), p>0 (unless otherwise stated), be the space of all pth power non-negative integrable functions over a given finite measure space (S, ∑, μ) (where S may be regarded as a bounded subset of real numbers).
Translates of Sequences in Sets of Positive Measure
Authors: Borwein D. , Ditor S. Z.
Published: December 2010

Given a measurable set A of real numbers with measure mA > 0, and a sequence {dn} of real numbers converging to zero, is there always an x such that x + dn ∈ A for all n sufficiently large?

The answer to this question, which was posed to the authors by P. Erdös, is NO.
A Note on Reductive Operators
Authors: Fillmore P. A.
Published: December 2010

For a bounded linear operator T on a Hilbert space , denote by Lat0T the lattice of all linear submanifolds of such that , and by (resp. Lat T) the sublattice consisting of operator ranges (resp. closed subspaces).
On a Theorem of Niven
Authors: Rao R. Sita Rama Chandra , Murty G. Sri Rama Chandra
Published: December 2010

In [4], Niven proved that the set A of integers for all s ≥ l and all n ≥ 1 has density zero, being the sum of the sth powers of all positive divisors of n. However his argument contains a mistake (see Remark 1). In this paper we give a proof of Niven's result and establish several related results, one of which generalizes a result of Dressier (See Theorem 3 and Remark 2).
Rings all of Whose Pierce Stalks are Local
Authors: Burgess W. D. , Stephenson W.
Published: December 2010

The aim of this paper is to give a number of characterizations of the rings of the title. In particular, these turn out to be precisely those exchange rings whose idempotents are all central. They are also those rings in which every element is the sum of a unit and a central idempotent.
Some Extensions of Hardy's Inequality
Authors: Chan Ling-Yau
Published: December 2010

This note is concerned with some new integral inequalities which are extensions of the results in [2]. The method by which these results are obtained is due to D. C. Benson [1]. Throughout the present note we shall assume 1<p<∞ and f(x) a non-negative measurable function.
Square-Reduced Residue Systems (Mod r) and Related Arithmetical Functions
Authors: Sivaramakrishnan R.
Published: December 2010

We define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that

(1)

(2)

In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined by

where the summation is over a square-reduced residue system (mod r).

B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and

(3)

As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function
A Proof of an Identity for Multiplicative Functions
Authors: Krishna K.
Published: December 2010

An arithmetic function f is said to be multiplicative if f(mn) = f(m)f(n), whenever (m, n) = 1 and f(1) = 1. The Dirichlet convolution of two arithmetic functions f and g, denoted by f • g, is defined by f • g(n) = Σd|nf(d)g(n/d). Let w(n) denote the product of the distinct prime factors of n, with w(l) = 1. R.
Generalized Markov Projections and Matrix Summability
Authors: Atalla Robert E.
Published: December 2010

In [A1] is defined a class of Markov operators on C(X) (X compact T2), called Generalized Averaging Operators (g.a.o.) which yield an easy solution to the following problem: given a fixed Markov operator T, find necessary and sufficient conditions on any other Markov operator R for the relation ker T ⊂ker R to hold. The main application of this is to inclusion relations between matrix summability methods.
A Note on Bernstein's Bivariate Inequality
Authors: Mullen K.
Published: December 2010

An upper bound for P[Σ Xi≥tσ, Σ Yi ≥ tσ], where (Xi, Yi), i = 1, 2, …, n are bounded independent random variables, was given by Mullen (1973). An improvement to the bound is possible without further assumptions.
Local Unique Factorization in the Semigroup of Paths in ℝn
Authors: Putcha Mohan S.
Published: December 2010

Let S denote the semigroup of all rectifiable, piecewise continuously difïerentiable paths in ℝn under concatenation. We prove a theorem to the effect that every finite collection of paths is contained in a subsemigroup of S which has the unique factorization property with respect to certain primes and straight lines. We also determine an abstract necessary sufficient condition for a subsemigroup of S to have this unique factorization property.
Complete Diagonals of Latin Squares
Authors: Chang Gerard J.
Published: December 2010

J. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.
Smoothness Properties of Bounded Solutions of Dirichlet's Problem for Elliptic Equations in Regions with Corners on the Boundary
Authors: Azzam A.
Published: December 2010

We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.
On r*-Invariant Measure on a Locally Compact Semigroup with Recurrence
Authors: Bourne Samuel
Published: December 2010

A regular Borel measure μ is said to be r*-invariant on a locally compact semigroup if μ(Ba-1) = μ(B) for all Borel sets B and points a of S, where Ba-1 ={xϵS, xaϵB}. In [1] Argabright conjectured that the support of an r*-invariant measure on a locally compact semigroup is a left group, Mukherjea and Tserpes [4] proved this conjecture in the case that the measure is finite; however their method of proof fails when the measure is infinite.
A Commutativity Theorem for Division Rings
Authors: Richoux Anthony
Published: December 2010

Let D be a division ring with center Z. Suppose for all xϵD, there exists a monic polynomial, fx(t), with integer coefficients such that fx(x)ϵZ. Then D is commutative.
A Function which Transforms Certain Graphs into Straight Lines for Simultaneous Solution
Authors: Maher Laurence P.
Published: December 2010

A function M is defined which maps the plane onto a square region in such a way that the planar graphs In, exp, X, —X, 1/X, and all compositions formed from them are transformed into straight lines. One can then solve for their intersections. It also provides a natural definition for the repeated composition of In with itself t times, where t can be a non-integer.
Nilpotent Ideals in Alternative Rings
Authors: Rich Michael
Published: December 2010

It is well known and immediate that in an associative ring a nilpotent one-sided ideal generates a nilpotent two-sided ideal. The corresponding open question for alternative rings was raised by M. Slater [6, p. 476]. Hitherto the question has been answered only in the case of a trivial one-sided ideal J (i.e., in case J2 = 0) [5]. In this note we solve the question in its entirety by showing that a nilpotent one-sided ideal K of an alternative ring generates a nilpotent two-sided ideal. In the process we find an upper bound for the index of nilpotency of the ideal generated. The main theorem provides another proof of the fact that a semiprime alternative ring contains no nilpotent one-sided ideals. Finally we note the analogous result for locally nilpotent one-sided ideals.
A Comment on Finite Nilpotent Groups of Deficiency Zero
Authors: Robertson Edmund F.
Published: December 2010

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.
A Note on the Jensen-Gould Convolutions
Authors: Cohen M. E. , Sun H. S.
Published: December 2010

With the aid of a recent result obtained by the first author, an expression is derived which unifies the well-known Jensen and Gould formulas.
Complements of Minimal Ideals in Solvable Lie Rings
Authors: Stitzinger Ernest L.
Published: December 2010

Conditions for the existence and conjugacy of complements of certain minimal ideals of solvable Lie algebras over a Noetherian ring R are considered. Let L be a solvable Lie algebra and A be a minimal ideal of L. If L/A is nilpotent and L is not nilpotent then A has a complement in L, all such complements are conjugate and self-normalizing and if C is a complement then there exists an x∈L such that C = {y∈L; yadnx = 0 for some n = 1, 2,…}. A similar result holds if A is self-centralizing and a finitely generated R-module.
A Note on Whitney Maps
Authors: Ward L. E.
Published: December 2010

In his recent book [3] Nadler observes that the property of admitting a Whitney map is of fundamental importance in studying the internal structure of hyperspaces, especially their arc structure. Nadler presents three distinct methods of constructing a Whitney map on the hyperspace 2X of nonempty closed subsets of a continuum.

A partially ordered space is a topological space X endowed with a partial order ≤ whose graph is a closed subset of X×X. It is well-known (see, for example, [2], page 167) that if X is a regular Hausdorff space then 2X is a partially ordered space with respect to inclusion.
Sur Les M-Ideaux Dans Certains Espaces D′Operateurs Et L′Approximation Par Des Operateurs Compacts
Authors: Fakhoury H.
Published: December 2010

Sommaire

It is shown that if V=C(X) or V = L1(μ) then the subspace of compact (resp. weakly compact) operators from V into itself is not an M-ideal in the space of bounded operators. This is the contrary to what happens when V= Co(ℕ) or lp(ℕ). The main result is proved via the best approximation properties of M-ideals and some results concerning norm one projections in C(X) and L1(μ) are deduced from this fact.
Generalized n-Like Rings and Commutativity
Authors: Moore H. G.
Published: December 2010

This note continues the investigation of those rings R with unity which also satisfy the polynomial identity B(x, y) = (xy)n -xyn -xny +xy = 0, for some integer n > l. It is shown that when n is an even integer, or when n = 3, such rings are commutative. It is otherwise possible, as is shown by example, for such rings to fail to be commutative, although they are subdirect sums of local rings satisfying the polynomial identity. Each such ring has nilpotent commutator ideal.
A Theorem on the Denseness of Orbits in Metric Spaces
Authors: Boyarsky Abraham , Scarowksy Manny
Published: December 2010

Let τ be a transformation from a compact metric space X into itself. Conditions are presented which ensure that there exists an orbit which is dense in X. An example is given.