bluuebunny/crossref_metadata_2025 huggingface dataset:
Analytic Evaluation of Certain Characteristic Classes(1)
Authors: Jeffries Clark D.
Published: December 2010

We have an alternative proof of the following result of Kervaire [2]:

Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.
On the Irregular Sets of a Transformation Group
Authors: Kaul S. K.
Published: December 2010

We assume throughout that (X, T, π) is a transformation group [2], where X is a topological space which is always assumed to be regular and Hausdorff. We call a point x ∊ X regular under T if for any open set U in X and any subset G of T such that , there exists an open set V containing x, such that VG ⊆ U [7]. Let R(X) denote the interior of the set of all the regular points of X under T, and I(X) the set of irregular points of X under T, that is the set of points which are not regular under T.
On Cycles and Connectivity in Planar Graphs
Authors: Plummer M. D. , Wilson E. L.
Published: December 2010

Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.
Profinite Modules
Authors: Cohen Gerard Elie
Published: December 2010

An inverse limit of finite groups has been called in the literature a pro-finite group and we have extensive studies of profinite groups from the cohomological point of view by J. P. Serre. The general theory of non-abelian modules has not yet been developed and therefore we consider a generalization of profinite abelian groups. We study inverse systems of discrete finite length R-modules. Profinite modules are inverse limits of discrete finite length R-modules with the inverse limit topology.
On m-Paracompact Spaces and ω-Mappings
Authors: Gantner T. E.
Published: December 2010

The purpose of this note is to present a characterization of m-paracompact normal spaces in terms of ω-mappings into m-separable metric spaces; this result is almost contained in Morita's original paper [2] and is implicitly contained in Shapiro's thesis paper [5].
Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets
Authors: Eberhart C. , Fugate J. B. , Mohler L.
Published: December 2010

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)
A Countable Nowhere First Countable Hausdorff Space
Authors: Harley P. W.
Published: December 2010

Here an example of a countable, nowhere first countable Hausdorff space is given.
On the Martingales Upcrossings Inequality
Authors: Menon M. V. , Seshadri V.
Published: December 2010

In this note we present a very simple proof of the upcrossings inequality (see [6], and the note at the end) for martingale sequences—one of the basic results in the theory of martingales—which does not make use of the notion of optional random variable, as is done in the usual proofs of the inequality.
Finite-Dimensional Extensions of Certain Symmetric Operators(1)
Authors: Michael I. M.
Published: December 2010

Let H be a Hilbert space with inner product 〈,). A well-known theorem of von Neumann states that, if S is a symmetric operator in H, then S has a selfadjoint extension in H if and only if S has equal deficiency indices. This result was extended by Naimark, who proved that, even if the deficiency indices of S are unequal, there always exists a Hilbert space H1 such that H ⊆ H1 and S has a selfadjoint extension in H1.
Systems of Congruences
Authors: Mordell L. J.
Published: December 2010

An interesting problem is to discuss the solutions of the congruences in n variables (x)=(xl,…,xn),

(1)

where
Nets of Conics in the Euclidean Plane and an Associated Representational Geometry
Authors: Blum R. , Guinand A. P.
Published: December 2010

The study of systems of conies and other algebraic curves was initiated in the middle of the nineteenth century by Cayley, Hesse, Cremona, and others. Most of the investigations from that time to the present have been concerned with extensions to algebraic varieties and systems of higher orders or dimensions, or with associated algebraic curves such as Jacobians and Hessians.
On the Solution of Moser's Problem in Four Dimensions
Authors: Chandra Ashok K.
Published: December 2010

The problem of finding the largest set of nodes in a d-cube of side 3 such that no three nodes are collinear was proposed by Moser. Small values of d (viz., d ≤,3) resulted in elegant symmetric solutions. It is shown that this does not remain the case in 4 dimensions where at most 43 nodes can be chosen, and these must not include the center node.
On Factorization of Polynomials Modulo n
Authors: Gilmer Robert
Published: December 2010

Let A be an ideal of the commutative ring R with identity. There is a canonical homomorphism ϕA from the polynomial ring R[X] onto (R/A)[X], obtained by reducing all coefficients modulo A. If fR[X], then we say that f is reducible (irreducible) modulo A if ϕA(f) is reducible (irreducible) in (R/A)[X].
Sur Le Calcul Des Zeros D'un Operateur Discontinu Par Iteration
Authors: Măruster S.
Published: December 2010

On considère un opérateur U(x) qui transforme l'espace euclidien En en soimême, en général discontinu, et on étudie la convergence d'un processus itératif de la forme xp+1 = xp-μU(x) (μ est une constante numérique positive). Processus de ce type, avec U(x) discontinu, se rencontrent par exemple à l'algorithme de relaxation pour la résolution des systémes d'inéquations [1], [2], de même qu'au calcul des polynômes de la meilleure approximation sur un ensemble fini de points [3].
Pointwise Compact Spaces
Authors: Morales Pedro
Published: December 2010

In 1962, J. M. G. Fell [5] indicated the important role played by certain topological spaces which, though locally compact in a specialized sense, do not, in general, satisfy even the weakest separation axiom. He called them "locally compact". These were called "punktal kompakt" by Flachsmeyer [6] and to avoid confusion, we shall call them pointwise compact spaces.
The Eigenvalues and Singular Values of Matrix Sums and Products. VII (1)
Authors: Thompson R. C. , Therianos S.
Published: December 2010

This paper is a continuation of the two series, Eigenvalues of Sums I, II, III [2], [3], [4] and Singular Values of Products I, II, III [5], [6], [7].
Absolute Continuity for Group-Valued Measures
Authors: Traynor Tim
Published: December 2010

In this note we generalize the following classical theorem: If μ and ν are finite real-valued measures such that ν(A) = 0 implies μ(A) = 0, then for every ε > 0, there exists δ > 0 such that μ(A)<ε whenever ν(A)< δ.
m-Bounded Extensions of Topological Spaces
Authors: Weston J. H.
Published: December 2010

An m-bounded extension of a topological space is an m-bounded space which contains the original as a dense subspace. m-bounded spaces have been studied by Gulden, Fleischman, and Weston [4], Saks and Stephenson [6], and Woods [8].
On the Absolute Nörlund Summability of a Fourier Series
Authors: Goel D. S. , Sahney B. N.
Published: December 2010

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write

(1.1)

If

(1.2)

as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,

(1.3)
Complementarity Problem and Duality Over Convex Cones
Authors: Berman Abraham
Published: December 2010

The complementarity problem is defined and studied for cases where the constraints involve convex cones, thus extending the real and complex complementarity problems. Special cases of the problem are equivalent to dual, linear or quadratic, programs over polyhedral cones.
On the Problem of R. De Vore
Authors: Freud Geza
Published: December 2010

Let f∈Lp (1≤p<∞) 2π-periodic,

(1)

and let us consider the rth order moduli of continuity of f

(2)

R. de Vore stated the following conjectures:

(a) Let ηv→0 decreasingly,

(3)
Multiplicative Properties of Jensen Measures
Authors: Siddiqi Jamil A.
Published: December 2010

Let A be a uniform algebra on the compact set X and let ψ be a non-trivial linear functional on A. A finite non-negative measure μ on X is called a Jensen measure for ψ if

(1)
A Problem of Herstein on Group Rings
Authors: Formanek Edward
Published: December 2010

Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.

This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let

1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].
On Free Semigroups and Ramsey Numbers
Authors: Lallement Gerard
Published: December 2010

If the length of a word w in a free semigroup F(X) satisfies , then for every partition of F(X) into k classes, w has n consecutive factors of length ≥p in the same class. As a consequence, the diagonal Ramsey numbers R(pn+1, p+1, k) have 1+pnk as lower bound.
Endomorphism Semigroups of Sums of Rings
Authors: Maxson Carlton J.
Published: December 2010

Let R= 〈R, +, •〉 be the cartesian sum of the rings Ri, i= 1, 2,…, n denoted by , and recall that R is a ring under the componentwise operations. It is well-known (e.g. [1], p. 212) that the endomorphisms of the group 〈R, + 〉 form a ring HomZR (under function addition and composition) and moreover HomZR is isomorphic to the matrix ring under the usual matrix operations of addition and multiplication.
The Generalized Rayleigh Quotient
Authors: Pattabhiraman M. V.
Published: December 2010

In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient

(1)

considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that

(2)

where qi is a characteristic vector corresponding to the characteristic value λi
On Completeness With Respect to the Carathéodory Metric*
Authors: Selby M. A.
Published: December 2010

Let X be a bounded region in the plane. Define

where f is an analytic function on X bounded in modulus by 1. We call d the Carathéodory metric.

In this note we give necessary and sufficient conditions to ensure that this metric be complete.
On the Complete Ring of Quotients
Authors: Koh Kwangil
Published: December 2010

In [2: p. 415], P. Gabriel proves that if R is a ring with 1 and S is a non-empty multiplicative set such that 0∉S, then S-1R exists if and only if for every pair (a, s)∈R×S, there is a pair (b, t)∈R×S such that at=sb and if s1a=0 for some s1 in S then as2=0 for some s2 in S. The purpose of this note is to give a self contained elementary proof of Gabriel’s result.
Connected Order 3 Standard Representations of Simple Lie Algebras
Authors: Lemire F. W.
Published: December 2010

The concept of standard representations of simple Lie algebras was introduced by I. Z. Bouwer [1], One of the difficulties was that of existence. The order zero standard representations are simply those having a dominant weight vector and these have been completely characterized, for example in [2].
On Singular Points of Normal Arcs of Cyclic Order Four
Authors: Spoar G. , Lane N. D.
Published: December 2010

In [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.
The Stability of Solutions of Generalized Emden-Fowler Equations
Authors: Teufel Hugo
Published: December 2010

This paper gives several monotonicity properties of all oscillatory solutions of equations with separable and nonseparable nonlinearities which are more general than the Emden- Fowler equations

*

Principally, if x(t) is an oscillatory solution, conditions are given such that; if a(t)↑ ∞ as t → ∞, then x(t) → 0; and, if a(t) ↓ 0 as t → ∞, then lim sup | x(t) | = ∞.
A Note on Contractive Selfmaps of ℓ∞
Authors: McCabe T. F.
Published: December 2010

In this note it is shown that a contractive matrix selfmap T of l∞ has T2 a contraction; hence the techniques used in showing ITNC for lp(1 ≤ p < ∞), c0, and c do not apply to l∞.
Vector Bundles Over Suspensions
Authors: Chan W. M. , Hoffman P.
Published: December 2010

We consider finite dimensional complex vector bundles over a compact connected Hausdorff space X, as defined, for example, in [1]. It is well known that if ξ is such a bundle, then there is a bundle η such that ξ⊕η is trivial.
A Cosine Functional Equation with Restricted Argument
Authors: Etigson L. B.
Published: December 2010

We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.

The purpose of this paper is to present a functional equation satisfied only by cosine functions.
On Best Simultaneous Approximation in Normed Linear Spaces
Authors: Goel D. S. , Holland A. S. B. , Nasim C. , Sahney B. N.
Published: December 2010

Let S be a non-empty family of real valued continuous functions on [a, b]. Diaz and McLaughlin [1], [2], and Dunham [3] have considered the problem of simultaneously approximating two continuous functions f1 and f2 by elements of S. If || • || denotes the supremum norm, then the problem is to find an element * ∈ S if it exists, for which
A Remark on a Theorem of M. Hall
Authors: Minc Henryk
Published: December 2010

Let A = (aij) be an m×n (0, 1)-matrix, m≤n. The permanent of A is defined by

where the summation is over all one-one functions σ:{1, 2, . . . , m}→{1, 2, . . . , n}. If A is regarded as the incidence matrix of a configuration of m subsets of an n-set, then Per (A) is the number of systems of distinct representatives in the configuration.
On The 4-Dimensional Poincarè Conjecture for Manifolds with 2-Dimensional Spines
Authors: Osborne R. P.
Published: December 2010

We shall work in the piecewise-linear category, so that all manifolds and subsets thereof, as well as all maps are assumed to be piecewise-linear. If M is a manifold, denote by #kM the k-fold connected sum of copies of M and by 2M the double of M, that is the manifold obtained by sewing two copies of M together by the identity map on their boundaries.
A Non-Reflexive Smooth Space with a Smooth Dual
Authors: Whitfield J. H. M.
Published: December 2010

Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.
Unicyclic Graphs Satisfy Harary′s Conjecture
Authors: Arjomandi E. , Corneil D. G.
Published: December 2010

Ulam in [7] has conjectured that any graph G with p≥3 nodes is uniquely reconstructable from its collection of subgraphs Gi=G-vi, i=1,2, … p. This conjecture has been proved for various finite graphs including regular, Eulerian, unicyclic, separable, trees and cacti. Since Ulam′s conjecture seems difficult to prove or disprove, some authors have tried to strengthen the conjecture [3]. One of these stronger conjectures is Harary′s conjecture [2].
Sequences Realizable by Graphs with Hamiltonian Squares
Authors: Chungphaisan V.
Published: December 2010

Let d=(d1,…,dn) be a sequence of positive integers. In this note we show that d is realizable by a graph whose square is hamiltonian if and only if (i) d is realizable by some graph, (ii) n≥3, and (iii) d1+…+dn≥2(n-1). In fact, we prove that if d is realizable by a connected graph, then d is realizable by a graph with a spanning caterpillar. From this it follows that if d is realizable by a connected graph, it is realizable by a graph whose square is pancyclic. We also prove that d is realizable by a graph with a spanning wreath if and only if d is realizable by some graph and d1+…+dn≥2n. (A wreath is a connected graph that has exactly one cycle and all vertices not in the cycle monovalent.)
Asymptotic Formulae for the Eigenvalues of a Two Parameter System of Ordinary Differential Equations of the Second Order
Authors: Faierman M.
Published: December 2010

The origin and importance of multiparameter Sturm-Liouville problems in mathematical physics has recently been discussed by Atkinson [1, sects. 3 and 4], [2, introduction]. In spite of the importance of such problems, and in spite of the work done in this field by such early investigators as Klein and Hilbert, Atkinson points out that in recent years this field has been relatively neglected in contrast to the single parameter case. As an example, he states that as opposed to the one parameter case, the detailed behaviour of the eigenvalues and eigenfunctions in the multiparameter Sturm-Liouville case is still far from clear.
An Existence Theorem for Nonlinear Boundary Value Problems
Authors: McCandless W. L.
Published: December 2010

Boundary value problems for ordinary differential equations have long been the subject of extensive research activity. In particular, questions concerning the existence and uniqueness of solutions for these problems have received much attention, and algebraic fixed-point theorems have served as important tools in such investigations. For example Picard [8] based his pioneering work in this area on the use of successive approximation techniques, and recently his classical methods have been refined and extended to more general nonlinear problems (see [1] and [4]). The standard procedure for applying these techniques requires that the boundary value problem under consideration first be converted into an equivalent integral equation through the choice of a suitable Green’s function. The resulting theory is consequently limited to problems for which such a formulation is possible.
On a Sheaf of Division Rings*
Authors: Szeto George
Published: December 2010

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.
LP Spaces from Matrix Measures
Authors: Binding P. , Browne P. J.
Published: December 2010

It is known that a Hilbert space, L2(μij), can be constructed from an n × n positive matrix measure (μij), [5, pp. 1337–1346]. The aim of this note is to show that Banach spaces, corresponding to the usual LP spaces, can also be constructed and to investigate their properties.
Convergence of Averaged Occupation Times
Authors: Lamb✝ Charles W.
Published: December 2010

Let X = {Xt, t ≥ 0} be a stationary Markov process with values in a measurable space (S, ℬ), transition function p, and initial distribution concentrated at a point x ∊ S. The occupation times of a set A ∊ ℬ are defined for t ≥ 0 by

where 1A is the indicator function of A. The expected occupation times are given by
On Weakly Almost-Periodic Families of Linear Operators
Authors: Rao Aribindi Satyanarayan
Published: December 2010

In this note first the weak almost-periodicity of the action of a weakly almost-periodic family of linear operators on an almost-periodic function is established. Then an application of this result is given.
A Counter-Example to Coherence in Cartesian Closed Categories
Authors: Szabo M. E.
Published: December 2010

It follows from [3] that all morphisms of free closed categories on finite discrete categories are components of natural or “generalized” natural transformations, and from [8] that all hom-sets of such categories are finite. The purpose of this paper is to show that neither statement remains true if the categories are also assumed to be cartesian.
On Property of Families of Sets
Authors: Abbott H. L.
Published: December 2010

A family of sets is said to have property if there exists a set B such that B ∩ F ≠ Ø and B ⊅ F for every F ∊ . Such a B will be called suitable with respect to F. It is known (see [3]) that for each positive integer n there exists a family of sets satisfying the following conditions:

(a)

|F| = n for each F ∊

(b)

|F ∩ G| ≤ for F, G ∊ F ≠ G

(c)

does not have property
On the Closure of the Convex Hull of a Set
Authors: Lin C.-S.
Published: December 2010

Let Y be a linear space over the complex plane C, and let F be a mapping on the complex linear space Y ⊕ C into subsets of C with the following properties: for y ∊ Y, λ and μ ∊ C, F(y + μ) is a nonempty and bounded subset of C, F(λy + μ) = λF(y) + μ and F(μ) = {μ}. We shall write f(y + μ) = sup{|λ + μ|: λ ∊ F(y)}, the radius of F(y + μ), y ∊ Y and μ ∊ C. The convex hull (resp. the closure) of a subset M of C is denoted by conv M (resp. ).
A Note on a Matrix Result of Ryser
Authors: Murphy I. S.
Published: December 2010

The purpose of this note is to give a short proof of a generalisation of a theorem of Ryser, Theorem 10.2.3 of [1], concerning matrices that occur in the theory of symmetric block designs.

The two main results of matrix theory required in the proof given below are:

(1)

If B, C are square matrices such that BC = zI where z is a non-zero complex number, then CB = zI

(2)

A matrix S which is both symmetric (i.e. S' = S) and skew-symmetric (i.e. S' = —S)is zero.
On the Spheres Carrying an Almost Contingent Structure
Authors: Duggal K. L.
Published: December 2010

It is well-known that odd dimensional spheres carry a normal contact structure [6], Blair, Ludden and Yano [2] have recently studied a more general structure whose non-trivial example is an even dimensional sphere. Recently, the present author introduced the notion of almost contingent structures [8] with a view to develop a unified theory of various existing structures on a differentiable manifold. It is the purpose of this paper to show that even as well as odd dimensional spheres carry an almost contingent structure. In the sequel, each manifold introduced is C∞, arcwise connected and satisfies the second axiom of countability.
A Canonical Form For Fully Indecomposable (0,1)-Matrices
Authors: Hartfiel D. J.
Published: December 2010

This paper develops another canonical form for (0, 1)-matrices which may be used in the same spirit as the nearly decomposable matrix [5] or the k-nearly decomposable matrix [1], This form is intrinsic in each fully indecomposable matrix and does not require the replacement of any of its non-zero entries by 0's.
Separation Principles and Bounded Quantification
Authors: Dawes A. M.
Published: December 2010

This note is concerned with the implication SepII(Q)→SepI(Q) where Q is a class of subsets of some set S.

where cZ denotes S—Z.

It is well-known that in general the above implication is false (e.g. let Q be the closed subsets of the reals).
Permanents of (0, 1)-Matrices Having at Most Two Zeros Per Line
Authors: Henderson J. R.
Published: December 2010

LetUndenote thenth ménage number. Within the class of ordernmatrices of zeros and ones with at most two zeros in every row and column the minimum permanent isUnwhennis even and–1+Unwhennis odd.
Injectivity and Equational Compactness in the Class of ℵ0-Semilattices
Authors: Nelson Evelyn
Published: December 2010

This note presents characterizations of the injective and of the equationally compact ℵ0-semilattices, which are analogous, respectively, to the characterizations of the injective semilattices given by Bruns and Lakser [2] and of the equationally compact semilattices given by Grätzer and Lakser [3].
A Property of Maximum Likelihood Estimators for Invariant Statistical Models
Authors: Tan Peter , Drossos Constantin
Published: December 2010

This paper generalizes some results on pivotal functions of maximum likelihood estimators of location and scale parameters and the related ancillary statistics obtained by Antle and Bain, and Fisher. It shows that the maximum likelihood estimator of the parameter in an invariant statistical model is an essentially equivariant estimator or a transformation variable in a structural model. In the latter case, ancillary statistics in the sense of Fisher used in conjunction with the maximum likelihood estimators can be easily recognized. It is also remarked that the values of maximum likelihood estimators from samples having the same “complexion” are simply related to those of other, perhaps simpler, transformation variables. In the development it also points out the importance of using the correct definition of the likelihood function originally proposed by Fisher.
A Uniqueness Theorem on the Degenerate Cauchy Problem(1)
Authors: Wang Chung-Lie
Published: December 2010

In [4] Carroll and the author have treated the following problem

(1)

where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).
An Equational Spectrum Giving Cardinalities of Endomorphism Monoids1
Authors: Gould Matthew
Published: December 2010

By determining the spectrum of a particular set of equations of type 〈2, 2, 0, 0,0〉 it is shown that a positive integer n is the cardinality of the endomorphism monoid of a universal algebra of the form if and only if n is square.
Caracterisation des Espaces Parfaits
Authors: Leblanc Raymond
Published: December 2010

Dans cette note, nous employons la terminologie et les définitions de [1] et [3].
A un sous-espace vectoriel W d'un espace vectoriel V pré-ordonné par un cône convexe pointé V+ avec élément unité e, nous associons la semi-norme suivante: pour x ϵ V, pw(x)=inf{ε>0|∃ωε ϵ W tel que -ωε-εe≤x≤ωε+εe}. Posons J(W)={x|pw(x)=0}={x ϵ V|∀ε > 0, ∃ωε ϵ W, -ωε-εe≤x≤ωε+εe}. J(W) est un idéal.
The Pn-Integral
Authors: Cross George
Published: December 2010

In the definition of the Pn-integral [2] there is a difficulty with the condition Bn-2 ([2], p. 150) since it is not linear on the set of major and minor functions. As a result, the proof of Lemma 5.1 [2] fails since the difference Q(x)—q(x) need not satisfy the conditions of Theorem 4.2, [2].
Weak Convergence and One-Sample Rank Statistics Under ϕ-mixing*
Authors: Mehra K. L.
Published: December 2010

Let {Xi:i=1, 2,…} be a real strictly stationary process (defined on a probability space (Ω, A, P)) which has absolutely continuous finite dimensional distributions (with respect to Lebesgue measure) and satisfies the ϕ-mixing condition: Let and denote the sub-cr-fields generated, respectively, by {Xi:i≤k} and {Xi:i≥k+n}; then, for each k≥1 and n≥l, and together imply.
On Some Non-Hyperfinite Factors of Type III
Authors: Ching Wai-Mee
Published: December 2010

In 1967, Powers [7] proved that there exists a one-parameter family of pairwise non-isomorphic hyperfinite factors of type III. Powers′ result on hyperfinite factors has been extended by Araki and Woods [1]. Connes [4], and Williams [11], with different proofs, showed that there exists a continuous family of mutually non-isomorphic non-hyperfinite factors of type III.
A Short Proof of Vladimirskii′s Theorem on Precompact Perturbations in Locally Convex Spaces
Authors: Chu Le Quang
Published: December 2010

Let T, P denote two continuous operators from E into F, where E and F are locally convex spaces. It is proved by L. Schwartz [8] and G. KÖthe [6] that if E and F are Fréchet spaces, T is a ϕ_-operator and P a compact operator, then T+P is a ϕ_-operator.
The Open Mapping and Closed Graph Theorem for Embeddable Topological Semigroups
Authors: Grant Douglass L.
Published: December 2010

Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.
Extension of a Tight Set Function with Values in a Locally Convex Space
Authors: Morales Pedro
Published: December 2010

The purpose of the paper is to extend a tight set function on a lattice with values in a locally convex space of special type to a measure on the cr-ring generated by . This result generalizes the extension theorem of Thomas [12, p. 151], which in turn contains the extension theorems of Pauc [9, p. 710], Fox [4, p. 525] and J. J. Uhl, Jr. [14, Corollary 2].
On Two Problems Concerning Neutral Polyverbal Operations on Groups
Authors: Macedonska-Nosalska O. N.
Published: December 2010

In his paper [2] O. N. Golovin introduced the notion of a neutral polyverbal operation on groups, of which Moran's verbal operations [4] and Gruenberg's and Šmelkin's operations [3,5] are special cases. The free and direct multiplication we shall call here trivial operations. It is known [5,4] that Mal'cev's postulate does not hold for nontrivial verbal operations but the postulate of associativity does.
A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi
Authors: Bogin Joseph
Published: December 2010

In [7], Goebel, Kirk and Shimi proved the following:

Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:

(1)

where ai≥0 and Then F has a fixed point in K.

In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).
The Topological Extension of a Product
Authors: Broverman S.
Published: December 2010

IfEis a topological space, then according to Mrowka in [7], a spaceXisE-incompletely regular ifXcan be embedded as a subspace of a topological product of copies ofE, andXisE-compact ifXcan be embedded as a closed subspace of a product of copies ofE. The following is [7, Theorem 4.14].
Rank k Vectors in Symmetry Classes of Tensors
Authors: Lim Ming-Huat
Published: December 2010

Let F be a field, G a subgroup of Sm, the symmetric group of degree m, and χ a linear character on G, i.e., a homomorphism of G into the multiplicative group of F. Let V1,...,Vm be vector spaces over F such that Vi = Vσ(i) for i=1,…,m and for all σ∈G. If W is a vector space over F, then a m-multilinear function is said to be symmetric with respect to G and χ if

for any σ ∊ G and for arbitrary xi ∊ Vi.
Some Results on Matrices with Prescribed Diagonal Elements and Singular Values
Authors: Sing Fuk-Yum
Published: December 2010

Let A be an n × n complex matrix. The singular values of A are the non-negative square-roots of the eigenvalues of A*A. G. N. De Oliviera [4] gave a necessary condition for the existence of a matrix A with a1..., an as diagonal elements and α1,..., αn as singular values. We shall give another necessary condition which implies the above author’s condition and we show that this is also a sufficient condition for the case n =2.
A Note on H-Closed Extensions of a Product
Authors: D’Aristotle Anthony J.
Published: December 2010

Any Hausdorff space X is a dense subspace of an H-closed space κX, called the Katětov extension of X, with the property that any H-closed extension Y of X is a continuous image of κX under a mapping which leaves X pointwise fixed [8], [10]. In [8], Liu has shown that the extensions κ(X×Y) and κX×κY of X × Y are equal iff (1) X or Y is finite, or (2) X and Y are H-closed. In this note, we examine whether homeomorphism of these two extensions implies equality. We give a condition under which homeomorphism implies equality and an example to show that this relation does not hold in general.
Quadratic Functionals of nth Order
Authors: Easwaran S.
Published: December 2010

It is well known that disconjugacy theorems for self-adjoint differential equations are very closely related to the study of the positivity of certain associated quadratic functionals. In this paper, this relationship is closely examined for quadratic functionals of nth order associated with selfadjoint differential equations of order 2n. The motivation of this work comes from a recent paper of W. Leighton [8] on quadratic functionals of second order. It is shown that the method used by W. Leighton extends in a straightforward manner to the study of quadratic functionals of nth order, provided one makes use of an identity due to G. Cimmino [1]. Several authors have considered related problems. In particular we wish to mention the recent papers of G. Ladas [6], W. Simons [13], D. B. Hinton [5]. The book by C. A. Swanson [14] also contains many other related results. As a consequence of our results we can obtain several known and some new criteria for oscillation of 2nth order self-adjoint differential equations.
The Gibbs Phenomenon for [S, αn] Means and [T, αn] Means
Authors: Swaminathan V.
Published: December 2010

The Gibbs phenomenon of the Fourier series for different summability methods has been investigated by various authors. In this note, we study the same for the [S, αn] method of summability introduced by Meir and Sharma [3]. The corresponding result for the [T, αn] method of summability due to Powell [5] can be worked out in exactly the same way.

The elements cnk of the [S, αn] matrix are defined by the relations:

1

where 0<αn<1. The [S, αn] matrix is regular if and only if
Cover-Avoidance Properties in Finite Soluble Groups
Authors: Tomkinson M. J.
Published: December 2010

We give a general method for constructing subgroups which either cover or avoid each chief factor of the finite soluble group G. A strongly pronorrnal subgroup V, a prefrattini subgroup W, an -normalizer D and intersections and products of V, W, and D axe all constructable. The constructable subgroups can be characterized by their cover-avoidance property and a permutability condition as in the results of J. D. Gillam [4] for prefrattini subgroups and -normalizers.
On Criticalr, λ-Systems
Authors: Vranch J.
Published: December 2010

Criticalr, λ-systems are introduced such that any nontrivialr, λ-system must be an extension of some critical system. It is shown that parametric values for which critical systems can exist are restricted to λ(v-1) ≤r(r-1) and, further, to λ(v-1)r(r-1) if the critical system is extendible.
A Convexity Result for Weak Differential Inequalities
Authors: Zaidman S.
Published: December 2010

In this note we present a natural “weak” form of a certain convexity estimate for evolution inequalities as given in Agmon-Nirenberg’s paper [1], p. 139 (see also A. Friedman [2], Theorem 4.2 and 4.3). Our proof will follow that given in [1] and [2] with the natural modifications due to the enlargement of the class of solutions which are taken into account.
Tangent Cones and Convexity
Authors: Borwein J. , O’Brien R.
Published: December 2010

The study of general multiplier theorems (Kuhn-Tucker Conditions) for constrained optimization problems has led to extensions of the notion of a differentiable arc. Abadie [1], Varaiya [10], Guignard [5], Zlobec [11] and Massam [12] investigated the so called cone of tangent vectors to a point in a set for optimization purposes.
On Commutator Equalities and Stabilizers
in Free Groups
Authors: Burns R. G. , Edmunds C. C. , Farouqi I. H.
Published: December 2010

A simple proof is given of a result of Hmelevskiï on the solutions of the equation [x, y] = [u, υ] over a free group for any specified u, υ. To illustrate, the equation is solved explicitly for (u, υ) = (a, b), (a2, b), ([a, b], c) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commutators in the automorphism group of this free group are determined.
Stalking the Souslin Tree—A Topological Guide
Authors: Tall Franklin D.
Published: December 2010

It has long been known that the existence of a Souslin line entails (and is entailed by) the existence of a Souslin tree; indeed such a tree can be built from the open subsets of the line in a natural way. It will be shown that less onerous restrictions on a topological space than orderability allow the construction to proceed. For example, to the expected requirements-that the space satisfy the countable chain condition and not be separable, one can add the hypothesis of local connectivity, and that either first category sets be nowhere dense or that nowhere dense sets be separable.
Jeffery-Williams Lecture, 1976 Non-Static Radially Symmetric Distributions of Matter
Authors: Wyman Max
Published: December 2010

When I agreed to give the 1976 Jeffery-Williams Lecture to the Canadian Mathematical Congress, I did not envisage that it would be my lot to give what might now be called the first memorial lecture in this series of lectures. During the past year, Ralph Jeffery and Lloyd Williams both died, and their deaths brought forward many deserved tributes to these two wonderful men. My lecture today will forge a link with my mathematical past, a past in which both of these men played important roles.
Lp Spaces Generated by Certain Operator Valued Measures
Authors: Binding Paul , Browne Patrick J.
Published: December 2010

In this paper we investigate the structure of certain spaces of operator valued measures and the Lp spaces they generate. The work is motivated by our earlier paper [1] in which we studied the Lp spaces generated by matrix valued measures. The present results can thus be regarded as a generalization of this “finite dimensional” situation.
On the Zeros of the Power Series with an Application to Discontinuous Riesz-Summability
Authors: Borwein D. , Kratz W.
Published: December 2010

On the zeros of . If not stated otherwise, we assume throughout that κ > 0, c > 1, and that k < κ ≤ k + 1 where k = 0, 1, 2, … We reserve the symbol x to denote real numbers, and define C* = C-{x:x <-1}, C being the complex plane. Let
There are 2ℵα Friendship Graphs of Cardinal ℵα
Authors: Chvátal Václav , Kotzig Anton , Rosenberg Ivo G. , Davies Roy O.
Published: December 2010

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbour. Finite friendship graphs were characterized by Erdös, Rényi, and Sós [1] as those for which the vertices can be enumerated as u, υ1,…υk, w1,…wk in such a way that the only edges are uυiuwi and υiwi (i = 1,…,k). Thus finite friendship graphs are rather rare. In contrast, we shall show that there are as many nonisomorphic friendship graphs of given infinite cardinal as there are nonisomorphic graphs of that cardinal altogether. In fact, we do a little more.
Inductive Limits of Banach Spaces
Authors: Bellenot Steven F.
Published: December 2010

If X and Y are infinite-dimensional Banach spaces, then Y is the inductive limit of Banach spaces each isomorphic to X.
Inequalities with Weights for Discrete Hilbert Transforms✝
Authors: Andersen Kenneth F.
Published: December 2010

Let Z(Z+) denote the set of all (positive) integers and let T denote the discrete Hilbert transform defined for suitable sequences a = {ak}kϵz by

where as usual the prime denotes omission of the term k = n.
A Commutativity Theorem for Near-Rings
Authors: Bell Howard E.
Published: December 2010

A ring or near-ring R is called periodic if for each xϵR, there exist distinct positive integers n, m for which xn = xm. A well-known theorem of Herstein states that a periodic ring is commutative if its nilpotent elements are central [5], and Ligh [6] has asked whether a similar result holds for distributively-generated (d-g) near-rings. It is the purpose of this note to provide an affirmative answer.
A Three-Fold Non-Lattice Covering
Authors: Blundon W. J.
Published: December 2010

Let be the density of thinnest k-fold covering of the plane by equal circles (of radius 1, say). Let Dk be the corresponding density when the centres of the circles are at the points of a lattice Λ. It is clear that
Remarque Sur un Article de T. M. Apostol
Authors: de Koninck Jean-Marie , Mercier Armel
Published: December 2010

Dans un récent article [1], T. M. Apostol obtient une identité liant et , où f est une fonction complètement multiplicative, |f(n)| ≤ l, et p est un nombre premier. Il utilise cette identité pour prouver que

pour tout nombre premier p. Par la suite, T. M. Apostol obtenait d'autres identités [2].
A Dominated Ergodic Theorem for Contractions with Fixed Points
Authors: De La Torre A.
Published: December 2010

Letbe a finite measure space, and letTbe a contraction in realLp(X). (i.e.Tis linear and ||T||≤1). It is said that the Dominated Ergodic Theorem holds forT, if there exists a constantcpsuch that, ifM(T)f(x) =supn1/nthen||M(T)f||p≤ cp||f||pfor everyfinLp.
Conjugacy of Elements in a Normal Ring
Authors: Lim Taw-Pin
Published: December 2010

Let (R, *) be a ring R with an involution *, i.e., * is a map RϵR such that for all a, bϵR

The trace and norm of an element a in (R, *) are respectively

(R, *) is said to be a normal ring if for all aϵR

or equivalently,
Singular Isometries in Orthogonal Groups
Authors: Gunther Georg
Published: December 2010

In this paper, we study the behaviour of singular isometries in orthogonal groups. These are isometries whose path is a singular subspace. We shall prove that the path of such a singular isometry is always even-dimensional. We shall use this result to show that the subgroup of the orthogonal group On(K, Q) which is generated by the singular isometries is the commutator subgroup Ωn(K, Q).
On the Dimension of a Complete Metrizable Topological Vector Space
Authors: Popoola J. O. , Tweddle I.
Published: December 2010

The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.
On the Duals of Flat Banach Spaces
Authors: Bick Abraham
Published: December 2010

We give a simpler proof to a theorem of L. A. Karlovitz that the dual of a flat Banach space is flat, and also study some geometric properties of the dual space.
Nonexpansive Mappings in Locally Convex Spaces
Authors: Hicks Troy L. , Kubicek John D.
Published: December 2010

Recently Bruck initiated the study of the structure of the fixed-point set of a nonexpansive selfmap T of a Banach space, where T satisfies a conditional fixed point property. We generalize many of his results to a Hausdorff locally convex space X. Also, we generalize a result of Holmes and Narayanaswami and use it, along with a procedure of Kiang, to obtain a fixed point theorem for families of asymptotically nonexpansive mappings in X.
Density of Non Quasi-Analytic Classes of Functions
Authors: Pham-Gia Thu
Published: December 2010

In the study of quasi-analytic classes (see [3], pp. 372-379), a class C{Mn} is shown to have the following properties:

(a)

If M0= 1 and (i.e. {Mn} is log convex), C{Mn} forms an algebra.

(b)

C{Mn} is invariant under affine transformations.

(c)

C{Mn} is quasi-analytic iff it contains non non-trivial function with compact support.
On Semicommuting Automorphisms of Rings
Authors: Chung L. O. , Luh Jiang
Published: December 2010

Let R be a prime ring possessing a nontrivial automorphism T such that xTx = ±xxT. If R is not of characteristic 3 or R has nonzero center, then R is a commutative integral domain.
Reducible Rational Fractions of the type of Gaussian Polynomials with only Non-Negative Coefficients
Authors: Grosswald Emil
Published: December 2010

The following problem arose in connection with the study of Poincaré polynomials for homogeneous spaces. Let

gi hi positive integers and set d = (h1, h1,…, hk), the greatest common divisors of the exponents in the denominator. Let hi = dri and assume that the ri′s are comprime in pairs, i.e.
On Intertwining and Factorization by Self-Adjoint Operators
Authors: Lin C.-S. , Radjabalipour M.
Published: December 2010

In this paper we first study the equality of two operators whose values at each point satisfy certain inequalities, and then, somehow related, we examine the possibility of writing certain operators as products of two self-ad joint operators.
Maximum Modulus Theorems and Schwarz Lemmata for Sequence Spaces, II
Authors: Shawyer B. L. R.
Published: December 2010

In this note, we continue the investigations of [3], proving another analogue of the maximum modulus theorem, this time for the sequence space bv, and we investigate maximal functions for such theorems. As in [3], we use the notation f∈MM if f is analytic in the disk |z| <1, continuous for |z| ≤ 1 and satisfies |f(z)| ≤ 1 on |z| = 1. We also write f∈SL if f∈MM and f(0) = 0. Whenever x={xk} is a sequence of complex numbers, we write f(x) = {f(xk)}.

In [3], we proved analogues of the maximum modulus theorem for the sequence spaces 5, m and c, and analogues of the Schwarz Lemma for the sequence spaces c0, lp and bv0. We begin this note with the sequence space bv.
Covering Finite Sets by Ergodic Images
Authors: Steele J. Michael
Published: December 2010

For any ergodic transformation T a set A of measure less than ∈ is constructed with the property that for every finite set F there is a j = j(F) such that F ⊂ T-iA. The basic tool used to prove this is a purely combinatorial result which says there is a small subset of { l, 2, …, n } which can be shifted a small amount to cover any k set in {j: δn ≤j≤n}. Applications are given to the theory of combinatorial entropy.
A Sum of Reciprocals of Least Common Multiples
Authors: Borwein D.
Published: December 2010

The purpose of this note is to prove the following theorem conjectured by P. Erdös.

Theorem. Let a0, a1, …, ak be integers satisfying 1 ≤ a0 a1 … k, and let [ai-1 ai] denote the least common multiple of ai-1 and ai Then

1

with equality occurring if and only if ai = 2i for 1 ≤ i ≤ k.