bluuebunny/crossref_metadata_2025 huggingface dataset:
Constructing an Automorphism From an Anti-Automorphism
Authors: Ayoub Christine
Published: December 2010

We consider the following problem: Let G be a group with distinct automorphisms β and σ and an anti-automorphism α such that

What can be said about G?

If σ = α, σ is both an automorphism and an anti-automorphism so that G is abelian. Hence we assume that σ ≠ α. In this case, we show that G is non-abelian, but has an abelian subgroup of index 2. Conversely, for such a group G there always exist distinct automorphisms β and σ and an anti-automorphism α such that (1) holds.
Some Possible and Some Impossible Tripartitions of the Plane
Authors: Meyers Leroy F.
Published: December 2010

For each positive integer n it is possible to partition the Euclidean plane into n (disjoint) congruent connected sets [1], but if n > 2, it is impossible to partition the plane into n congruent continuumwise connected sets such that some one of the sets can be translated onto another one [2]. This paper is concerned with the possibility of partitioning the plane into three congruent sets without any topological restrictions whatever.
Some Characterizations of the Ultraspherical Polynomials
Authors: Al-Salam N.A. , Al-Salam W. A.
Published: December 2010

Let be the nth ultraspherical polynomial. Also let . The following generating relation is well known (3, p.98).

It can also be written as

1.1

This suggests the consideration of the class of polynomial sets {Qn(x), n = 0, 1, 2,…}, Qn(x) is of exact degree n and

1.2
Unisolvence on Multidimensional Spaces
Authors: Dunham Charles B.
Published: December 2010

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.
More Problems Connected with Convexity
Authors: Melzak Z.A.
Published: December 2010

This is a continuation of the author's article [3], and it contains further problems connected with the theory of convex sets in En. To the list of general references in [3] may be added the recent book [2] on convex polyhedra.

1) Let A and B be two convex bodies in E2 and let a packing P = {B1, B2, …} be an infinite sequence of homothetic images of B sucn that:

a)

each Bn is a subset of A,

b)

no two of them share interior points,

c)

Area (A) =

The existence of such packings is guaranteed by Vitali's Theorem. Let D(X) be the diameter of the set X and put .
A Note on the Diameter of a Graph
Authors: Bondy J. A.
Published: December 2010

The distance d(x, y) between vertices x, y of a graph G is the length of the shortest path from x to y in G. The diameter δ(G) of G is the maximum distance between any pair of vertices in G. i.e. δ(G) = max max d(x, y). In this note we obtain an upper bound

x ε G y ε G

for δ(G) in terms of the numbers of vertices and edges in G. Using this bound it is then shown that for any complement-connected graph G with N vertices

where is the complement of G.
On Lattice Paths with Several Diagonal Steps
Authors: Mohanty S.G. , Handa B.R.
Published: December 2010

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.
A Note on Polar Topologies
Authors: Robertson J. M. , Wiser H. C.
Published: December 2010

Let (E, E') be a dual pair and G be a collection of σ(E, E') bounded sets in E with the following properties:

i)

G contains all singletons.

ii)

If A, B are in G there is a C in G with A ∪B ⊆ C.

iii)

If A is in G then λ A ε G for any scaler λ.

Then sets of the form the polar of Ai, form base of neighbourhoods in E' for the polar topology TG of G convergence.
A Generalization of a Theorem of Hilton
Authors: Hoo C. S.
Published: December 2010

Let f: A×B → X be a map. Let J(f): ∑(A ∧ B)→ ∑X be the map obtained from f by means of the Hopf construction. Let P(f) denote the space obtained from ∑X by attaching a cone on ∑(A ∧ B) by means of J(f). Let l: ∑X→P(f ) be the inclusion and T(l): X→ΩP(f) the adjoint of l. Let h1,: A1, → A, h2: B1,→ B be maps. Let c:ΩP(f)× ΩP(f) → ΩP(f) be the basic commutator. Then we prove that there exists a map ∑A1 × ∑B1 →P(f) with axes l∑(fi1 h1), l∑(fi2 h2) if and only if , where i1: A→ A × B and i2 B → A × B are the inclusions. This generalizes a result of Hilton. Also, by letting f be an H-space multiplication and h and h the identity maps, we obtain a well known criterion of Stasheff for an H-space to be homotopy-commutative. Finally, appropriate duals of these results are given.
A Three Dimensional Analogue of Pappus's Theorem
Authors: Phadke B.B.
Published: December 2010

If P, Q, R are any three points on a line a and p, q, r are any three lines that pass through a point O and lie in a plane n. which meets the line a in a point A distinct from O, then the three lines l = Qr ⋅ Rq, m = Rp ⋅ P r, n = P q ⋅ Qp lie in a plane (through O).
An Isoperimetric Inequality for Convex Polyhedra with Triangular Faces
Authors: Kömhoff Magelone
Published: December 2010

H. T. Croft [1] has conjectured that among all tetrahedra with fixed total edge length the regular one has the greatest surface area. In this note we prove the following result, which includes this conjecture as a special case
A Note on the Jacobson And Brown-McCoy Radicals
Authors: Anderson T. , Heinicke A.
Published: December 2010

Let J(R) and G(R) respectively denote the Jacobson and Brown-McCoy radicals of the ring R and recall that R = G(R) if and only if R can not be homomorphically mapped onto a simple ring with unity [1, p. 120].

In general one knows that J(R) ⊆ G(R) [1, p. 118], while there do exist rings R for which J(R) ≠ G(R) (see [1, p. 120]).
Periodic Solutions by Picard's Approximations
Authors: Burton T.A.
Published: December 2010

In [1] Demidovic considered a system of linear differential equations

with A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.
Some Remarks on a Paper of Wong
Authors: Houston R. , Kaul S.K.
Published: December 2010

The aim of this note is to point out a mistake in the proof of Theorem 2 of Wong's paper [1], We first give an example to show that the theore as stated is not true.

Example 1. Consider the cartesian plane and the graph H with equation .
Spaces A × B of Conilpotency ≤ 1
Authors: Hoo C.S.
Published: December 2010

Let A and B be spaces having the homotopy type of countable CW-complexes. Then we prove the following theorems.

Theorem 1. If conil(A × B) ≤ 1, then for each integer n ≥ 1, the inclusion j: ∑n A ∨ ∑n B → ∑nA × ∑n B is a homotopy equivalence.

This result is obtained as a corollary of Theorem 2.
λ(n)-Parameter Families
Authors: Mathsen Ronald M.
Published: December 2010

I is an interval of R, the set of real numbers, n is a positive integer and F ⊂ Cj (I) for j ≥ 0 large enough so that the following definitions are possible:

(i) Let λ(n) = (λ1, λ2,…,λk) where k, λ1, λ2,…, λk, are positive integers and λ1 + λ2 +… +λk = n. Then λ(n) is an ordered partition of n. The set of all such partitions of n is denoted by P(n).
An Application of a Theorem of J. Czipszer and G. Freud to a Problem of Simultaneous Approximation
Authors: Varma A.K.
Published: December 2010

In an earlier work [12] we considered the case of (0, 2, 3) interpolation by trigonometric polynomials at the points , i = 0, 1,…, n-1. By (0, 2, 3) interpolation we mean the problem of finding a trigonometric polynomial of suitable order whose values, second and third derivatives are prescribed at some given points. An interesting distinction between the (0, 2) interpolation studied by the Hungarian mathematician O.
Polynomials with Irreducible Factors of Specified Degree
Authors: Williams Kenneth S.
Published: December 2010

Let d be a positive integer and let p be a prime > d. Set q = pm, where m ≥ 1, and let I (q, d) denote the number of distinct primary irreducible polynomials of degree d over GF(q). It is a simple deduction from the well-known expression for I(q, d) that

(1)

where d* is the largest positive integer < d which divides d if d > 1, and d* is 0 if d = 1.
-Δ is Positive Definite on a "Spiny Urchin"
Authors: Adams Robert A.
Published: December 2010

In a recent note [3] in this department C. Clark has shown that Rellich's theorem on the compactness of the imbedding is valid if G is the "spiny urchin" domain obtained by removing from the plane the union of the sets Sk (k = 1, 2,…) defined in polar coordinates by
A Note on Quasi-Frobenius Rings and Ring Epimorphisms
Authors: Storrer H. H.
Published: December 2010

In this note, we characterize quasi-Frobenius rings by a weakened form of the usual condition, that every ideal is an annihilator ideal.

We then apply this result to pure rings in the sense of Cohn and to dominant rings, a concept arising in the study of ring epimorphisms. All rings considered have a unit element.
A Note on a Theorem of Dynkin on Necessary and Sufficient Statistics
Authors: Tan Peter
Published: December 2010

In his paper "Necessary and sufficient statistics for a family of probability distributions", Dynkin (1951) establishes the important concept of rank for such a family with this conclusion: "If the rank is infinite, then the family has no non-trivial sufficient statistic in any size of sample." His concept of rank is based on a theorem, Theorem 2 described below, which has been pointed out by Brown (1964) to be invalid under its hypotheses. This note shows that Dynkin's Theorem 2 remains valid under its original hypotheses provided that the set (in Dynkin's notation) Δ - S is countable.
Sur Deux Classes D'Anneaux Non-Commutatifs
Authors: Menard Jean
Published: December 2010

Nous nous proposons ici d'étendre certains résultats de la théorie des anneaux commutatifs à deux classes d'anneaux non- commutatifs. Il s'agit, d'une part, des anneaux inversifs à droite, définis par Thierrin [4], et des anneaux réversifs. Les premiers possèdent la propriété xyR = yxR, les seconds la propriété xRy = yRx, ce pour tout x, y ∈ R. Nous considérons aussi la réunion des parties d'un anneau qui possèdent respectivement l'une ou l'autre propriété.
Moment Sequences and the Bernstein Polynomials*
Authors: Eisenberg Sheldon M.
Published: December 2010

The Bernstein polynomials

(1.1)

and the Bernstein power series

(1.2)

have been the subject of much research (e. g. [1; 2; 3; 6; 7; 8]). It is the purpose of this paper to demonstrate the relationship between these linear operators and certain classes of moment sequences defined below.
Rings All of whose Factor Rings are Semi-Prime
Authors: Courter R.C.
Published: December 2010

We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X.
On Product of Radón Measures
Authors: Godfrey M. C. , Sion M.
Published: December 2010

Let X, Y be locally compact Hausdorff spaces and μ, ν be Radón outer measures on X and Y respectively. The classical product outer measure ϕ on X × Y generated by measurable rectangles, without direct reference to the topology, turns out to have some serious drawbacks. For example, one can only prove that closed sets (and hence Baire sets) are ϕ-measurable. It is unknown, even when X and Y are compact, whether closed sets are ϕ-measurable.
A Property of Groups with No Central Factors
Authors: Rhemtulla A. H.
Published: December 2010

Let C1 denote the class of all groups with no non-trivial central factors. We prove the following theorem

There exist non-trivial locally solvable C1 groups; but there is no non-trivial locally k-step polynilpotent C1 group for any integer k.

It is well known that a minimal normal subgroup of a locally solvable group is abelian. Thus no non-trivial locally solvable group can be pluperfect - the class of all perfect groups in which every subnormal subgroup is also perfect.
On Strong Upper Densities
Authors: Eames W.
Published: December 2010

Let ψ be the outer measure generated by a gauge g and a sequential covering class C of closed sets of a metric space X, and let D be the resulting strong upper density function. That is, for each A ⊆ X and each x ∈ X:

the infimum being taken over all countable subcollections {Iα} of C such that and, for all Iα in the subcollection, d(Iα) < ε where d(Iα) is the diameter of d(Iα).
A New Criterion for Borel Summability of Fourier Series
Authors: Sahney B.N. , Kathal P.D.
Published: December 2010

The application of Borel summability to Fourier series has been discussed by Takahashi and Wang [8] and Sahney [5]. Sahney [6] and Sinvhal [7] obtained sufficient conditions for the Borel summability of the derived Fourier series and its conjugate series, respectively. Kathal [3] obtained different conditions in the case of the conjugate series. In this paper we give a new criterion for Borel summability of Fourier series.
Disconjugacy Conditions for the Third Order Linear Differential Equation
Authors: Erbe Lynn
Published: December 2010

An nth order homogeneous linear differential equation is said to be disconjugate on the interval I of the real line in case no non-trivial solution of the equation has more than n - 1 zeros (counting multiplicity) on I. It is the purpose of this paper to establish several necessary and sufficient conditions for disconjugacy of the third order linear differential equation

(1.1)

where pi(t) is continuous on the compact interval [a, b], i = 0, 1, 2.
Inequalities for Generalized Symmetric Functions
Authors: Menon K.V.
Published: December 2010

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined by

respectively.
Finite Topological Spaces and Quasi-Uniform Structures
Authors: Fletcher P.
Published: December 2010

In [6], H. Sharp gives a matrix characterization of each topology on a finite set X = {x1, x2,…, xn}. The study of quasi-uniform spaces provides a more natural and obviously equivalent characterization of finite topological spaces. With this alternate characterization, results of quasi-uniform theory can be used to obtain simple proofs of some of the major theorems of [1], [3] and [6]. Moreover, the class of finite topological spaces has a quasi-uniform property which is of interest in its own right. All facts concerning quasi-uniform spaces which are used in this paper can be found in [4].
Lattice Paths with Diagonal Steps
Authors: Goodman E. , Narayana T.V.
Published: December 2010

The André-Poincaré "probléme du scrutin" [9] can be stated as follows: In an election between two candidates A polls m votes, B polls n, m > n. If the votes are counted one by one what is the probability that A leads B throughout the counting? Many derivations and interpretations of the solution have been given and a convenient summary of methods till 1956 can be found in Feller [1]. So numerous are the generalizations of ballot problems and their applications since this date that we do not even attempt an enumeration here.
On a Result of A. M. Macbeath on Normal Subgroups of a Fuchsian Group
Authors: Jonsson W.
Published: December 2010

A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.

Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.

This theorem will follow from a consideration of the exact sequence

*
Remarks on Quasigroups Obeying a Generalized Associative Law
Authors: Ford Gary G.
Published: December 2010

T. Evans in [4] introduced a concept generalizing the associative law for groups. Certain very restricted cases of this generalization had occurred, somewhat as side results, in the considerations of A. K. Suškevič [7], D. C. Murdoch [5], [6], and R. H. Bruck [2] associated with quasigroups which obey theorems generalizing certain results in group theory.

In this paper we consider quasigroups obeying an instance of Evans' generalization. We show two means of constructing such quasigroups, and we demonstrate these constructions with several examples. The principal result of this paper is a theorem which reveals a peculiar trait of quasigroups obeying an instance of the generalized associative law we are considering. Such a quasigroup obeys a law expressed with permutations in which the nontrivial permutations involved either are both automorphisms of the quasigroup or else both fail to distribute over any product from the quasigroup.
A Note on the Adjoint of the Product of Operators
Authors: Lin Chia-Shiang
Published: December 2010

Cordes and Labrousse ([2] p. 697), and Kaniel and Schechter ([6] p. 429) showed that if S and T are domain-dense closed linear operators on a Hilbert space H into itself, the range of S is closed in H and the codimension of the range of S is finite, then, (TS)* = S*T*. With a somewhat different approach and more restricted condition on S, the same assertion was obtained by Holland [5] recently, that S is a bounded everywhere-defined linear operator whose range is a closed subspace of finite codimension in H.
Osculatory Packings by Spheres
Authors: Boyd David W.
Published: December 2010

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.
Solutions of Homogeneous Elliptic Equations
Authors: Dubinsky E. , Husain T.
Published: December 2010

We consider an elliptic partial differential equation with constant coefficients and zero on the right hand side. It is well known [1] that every solution of such an equation can be approximated uniformly on each compact set by a sum of products of polynomials and exponential functions which satisfy the equation. Furthermore, if one assumes that the polynomial operator is homogeneous, then the approximation can be made with polynomials alone. It is our purpose to show, in the latter case, when the number of variables is two, that each solution can be written as an infinite series in certain specific polynomials. Our method is to factor the polynomial and build up the solution in terms of solutions of first degree equations.
On Measures Determined by Continuous Functions that are not of Bounded Variation
Authors: Burry J. H. W. , Ellis H. W.
Published: December 2010

In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.
Lifting Ideals in Noncommutative Integral Extensions
Authors: Hoechsmann K.
Published: December 2010

This note is intended as a commentary on a part of the paper [1] by Gulliksen, Ribenboim, and Viswanathan. It takes its inspiration from a colloquium talk by P. Ribenboim.

Our aim is a partial generalization of the theorem of Cohen-Seidenberg (cf. [2], IX. 1, Propositions 9 and 10) to noncommutative algebras.

Definition. Let A be a ring with 1 in the center of another ring B (with the same 1). B is said to be integral over A, if each element b ∊ B satisfies an integral equation
On Polynomials with Related Level Sets
Authors: Rosenfeld M.
Published: December 2010

If p is a polynomial in one real variable and p(x) = p(-x) then p has only even powers of x and is thus a polynomial in x2. If p is a polynomial in n variables and p(x1, …, xn) = p(y1, …, yn) when x12 + … + xn2 = y12+ … + yn2 then p is a polynomial in q where q(x1, …, xn) = x12 + … + xn2.
Endomorphisms of the Quasi-Injective Hull of a Module
Authors: Wong Edward T.
Published: December 2010

R is a ring and M is a right R-module for which Rl = {m ∊ M | mR = 0} is the zero submodule. Let and be the injective hull and the quasi-injective hull of M respectively. Then where [1]. The ring plays an important role, in many cases, in the studying of R especially when D is a division ring. For x ∊ M, we denote the annihilator of x in R by xγ = {r ∊ R | xr=0}, whereas xγl={m ∊ M | mxγ=0}. If N is a submodule of M and x ∊ M, x-1(N) is the right ideal in R consisting of elements r in R where xr ∊ N.
Inversion of a Class of Convolution Transforms of Generalized
Functions
Authors: Ditzian Z.
Published: December 2010

The kernels of the transforms in the class that we shall treat satisfy

1.1

where Re ak = ak Re ck = ck, 0 ≤ ak/ck and Σa-2k (see also [1], [2], [3], [6], [7] and [8]).
On an Interpolation Theorem of Zygmund and
Koizumi
Authors: Heinig H. P.
Published: December 2010

Let (X, , μ) and
(Y, ,
ν) be two σ-finite measure spaces. An operator
T, defined by h = Tf,
which maps functions on X into functions on
Y is called quasilinear if
T(f+g) is uniquely defined whenever
Tf and Tg are defined, and if

1.1

where is
independent of f and g. If the operator T is called
sublinear.
A Counterexample to a Conjecture of D. B. Fuks
Authors: Demers Luc
Published: December 2010

In [3] D. B. Fuks defined a duality of functors in the category
of weak homotopy types. In general this duality is
more difficult to work with than the duality of functors of the category
of pointed Kelley spaces [2]. It happens however
that all so-called strong functors [2] F of induce functors of
, and if we denote the duality operators of
and by
and D respectively, then there are many cases where
.
On the Modulii of Analytic Functions
Authors: Sherman Malcolm J.
Published: December 2010

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying

*

where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.
On Rings of Fractions
Authors: Davison T. M. K.
Published: December 2010

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R. Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R, and let j = js: R → S-1R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S-1R is a topological ring in ihe j(M)-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p(S), where

Hence S-1R is a topological ring for all S if and only if M ⊂ p*(R), where
One-Dimensional Representations of the Cycle Subalgebra of a Semi-Simple Lie Algebra
Authors: Lemire F. W.
Published: December 2010

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial.
On Certain Classes of Bounded Linear Operators
Authors: Lin C-S
Published: December 2010

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.

Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.
Counterexample to a Conjecture of Greenleaf
Authors: Milnes Paul
Published: December 2010

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that
On the Ordering of Multi-Point Boundary Value Functions
Authors: Peterson A. C.
Published: December 2010

We are concerned with the nth-order linear differential equation

1

where the coefficients are continuous. Aliev [1, 2] showed, in papers unavailable to the author that for n = 4

(see Definition 2). Theorems 1 and 5 give respectively nth-order generalizations of these two results.
A Characterization and a Class of Distribution Functions for the Stieltjes-Wigert Polynomials
Authors: Chihara T. S.
Published: December 2010

In his classic memoir on the moment problem that bears his name, Stieltjes [2] exhibited

1

as an example of an indeterminate (Stieltjes) moment sequence.

Stieltjes also obtained the corresponding S-fraction and thus implicitly obtained the three-term recurrence formula satisfied by the corresponding orthogonal polynomials.
On a Characterization of Collectionwise Normality
Authors: Alas O. T.
Published: December 2010

The purpose of this note is to give some characterizations of collectionwise normality by means of the normality of certain product topological spaces; they are a consequence of a theorem proved by M. Katětov, but they appear to be new.
Construction of Quasigroups Satisfying the Identity X(XY) = YX
Authors: Lindner Charles C.
Published: December 2010

In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.
A Note on Combinatorial Identities for Partial Sums
Authors: Mohanty S. G.
Published: December 2010

For a sequence σ = (x1, …, xn) of real numbers, let σi and respectively denote the cyclic permutation (xi, xi+1, …, xi-1) and the reverse cyclic permutation (xj, xj-1, …, xj+1), and let . Also denote by Mrj(σ) and mrj(σ) the rth largest and the rth smallest numbers respectively, among the first j partial sums s1, s2, …, Sj for 1≤r≤j≤n. As usual, let the superscripts + and — respectively mean maximize and minimize with zero.
On the Henstock Strong Variational Integral
Authors: Thomson B. S.
Published: December 2010

The theory of integration in division spaces introduced by Henstock ([3], [4]) serves to unite and simplify much of the classical material on nonabsolute integration as well as to provide a new approach to Lebesgue integration. In this paper we sketch a simplified approach to the division space theory and show how it can lead rapidly to the standard Lebesgue-type theory without a substantial departure from the usual methods; some applications to integration in locally compact spaces are briefly developed.
An Embedding Theorem for Separable Locally Convex Spaces
Authors: Lohman Robert H.
Published: December 2010

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.
On the Connectedness of the Sets of Limit Points of Certain Transforms of Bounded Sequences(1)
Authors: Leviatan Dany , Lorch Lee
Published: December 2010

The transforms discussed here are the quasi-Hausdorff (Theorem 1), the [J, ƒ(x)] (Theorem 2) and the Borel integral means (Theorem 3). We are concerned here with whether or not the limit-points of these transforms of bounded sequences form connected sets. Such a set is one which cannot be decomposed into the union of two disjoint nonempty open sets.
A Special Case of the Vanishing of a (G, σ)-Product in a (G, σ)-Space
Authors: Singh K.
Published: December 2010

In [1] we constructed a (G, σ)-space and determined a condition which is both necessary and sufficient for the (G, σ)-product of the vectors v1, v2, …, vn to be zero. The purpose of the present paper is to give a criterion for v1 Δ v2 Δ … Δ vn to be zero, in the particular case when V is a unitary space and the group G belongs to a special class of groups which we shall define below.
Another Note on Sperner's Lemma
Authors: Drake David A.
Published: December 2010

Let Q be a finite partially ordered (by ≤) set with universal bounds O, I. The height function h of Q is defined by the rule: h(x) is the maximum length of a chain from O to x. Let h(I)=n. Suppose that for each k≥0, there exist positive integers a(k) and b(k) such that all elements of height k

(i)

are covered by a(k) elements of height k+1;

(ii)

cover b(k) elements of height k—1.

Then we call Q a U-poset. Call a subset S of a partially ordered set an antichain if no two elements of S are comparable.
Rank of the Sum of Certain Matrices
Authors: Gibson P. M.
Published: December 2010

In this note we give a very elementary proof of a result of Meyer's [1]. Let r(A) be the rank of the matrix A.
Convolution of Lp Functions on Non-Unimodular Groups
Authors: Milnes Paul
Published: December 2010

In this note we prove the following

Theorem. If G is a nonunimodular locally compact group and 1<p<∞, then there is an open set, U, in G and there are functions, f simultaneously in every Lr(G), p≤r ≤∞, and g simultaneously in every Lq(G), 1 ≤q≤∞, such that the convolution, f * g(y), is not defined for any y in U.
A Note on a Self Injective Ring
Authors: Satyanarayana M.
Published: December 2010

The purpose of this note is to give a direct proof of Koh's main theorem in [2, p. 31] applying well-known theorems of Faith and Utumi. While doing so, we shall mention a general theorem from which Koh's theorem may be obtained as a corollary.
Approximation of Foliations
Authors: Cohen Maurice
Published: December 2010

Let be two foliations on a Cr manifold M. We say and are Ck-conjugate if there exists a Ck diffeomorphism h:M→M such that h maps the leaves of onto the leaves of .
Triples and Localizations(1)
Authors: Heinicke A. G.
Published: December 2010

Let A be a ring (associative) with unity, and let denote the category of unital left A-modules. If is a strongly complete Serre class in then (see [3], and also [6]) there is an exact functor S: , where is the quotient category , and is an abelian category.
On Functional Representations of a Ring without Nilpotent Elements
Authors: Koh Kwangil
Published: December 2010

In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0P, for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.
Sur un Théorème D'interpolation de J. Lions et J. Peetre
Authors: Nieto J. I.
Published: December 2010

Dans cette note on donne une démonstration, fondée sur des propriétés élémentaires des opérateurs linéaires dans des espaces munis de deux normes, d'un théorème d'interpolation établi par J. Lions et J. Peetre [2] dans le cadre de leur théorie des espaces de moyenne.
Some Integral Equations with Rummer's Functions in the Kernels
Authors: Prabhakar Tilak Raj
Published: December 2010

Since 1963 several authors ([13], [2], [6], [14], [10], [11], [12], [9]) have considered integral equations each one of which is contained as a special case in one of the two equations

1.1

1.2

for Re b > 0 and x ∊ [α, β].
On a Class of Projective Modules Over Central Separable Algebras
Authors: Szeto George
Published: December 2010

In [5], DeMeyer extended one consequence of Wedderburn's theorem; that is, if R is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents except 0 and 1 or if R is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective module over a central separable R-algebra A.
On Approximately Continuous and Almost Continuous Functions
Authors: Chaika M. , Perlman S.
Published: December 2010

The purpose of this note is to discuss the relationship between the concepts of approximate continuity and almost continuity of real functions of a real variable.

Contrary to the papers of T. Husain [1], [2], [3], we shall show that there is little relationship between these concepts, that approximate continuity does not imply almost continuity without additional restrictions and we shall give one such restriction.
Totally Real Subfields of p-Adic Fields having the Symmetric Group as Galois Group
Authors: Kleiman Howard
Published: December 2010

In this paper, an elementary proof is given of the following proposition:

Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.
On Maximal Sets of Mutually Orthogonal Idempotent Latin Squares
Authors: Mendelsohn N. S.
Published: December 2010

It is a well-known trivial fact that for a given integer n there exists at most n — 2 pairwise orthogonal idempotent latin squares. In the following note we prove that for n a prime power there always exists n—2 such squares.
Wreath Product of O*-Groups that is not in O*
Authors: Modak S. V.
Published: December 2010

It is well known that the wreath product of two ordered groups is an ordered group. In [2] Fuchs asks if the same is true for O*-groups. Here we construct an example to show that the wreath product of an infinite cyclic group with a free metabelian group is not an O*-group.
On Spaces and ω-Mappings
Authors: Pareek C. M.
Published: December 2010

This note is closely related, as far as methods are concerned, to [3]. In [3] Ponomarev established “In order for a regular space X to be Lindelöf, it is necessary and sufficient that for each open covering ω of the space X there exists an ω-mapping ƒ:X → Y onto some separable metric space Y.” It is the purpose of this note to show that if the word “countable (or finite)” is inserted in the proper place we can obtain an analogous characterization for normal countably paracompact (or normal) spaces.
A Simpler Method for Getting Some Exact Distributions
Authors: Chaput Luc
Published: December 2010

In [1] Anderson uses a simple random sample of p-component vectors from N(μ, Σ) to test the hypothesis H, Σ = σ2I, where I is the identity matrix and σ2 is unknown. Let W= λ2/N, where λ is the likelihood ratio criterion for testing H.
Almost all Graphs have a Spanning Cycle
Authors: Moon J. W.
Published: December 2010

A graph is a collection of nodes some pairs of which are joined by a single edge. A k-path, or a path of length k, is a sequence of nodes {p1 p2,… Pk+1} such that Pi is joined to pi+1 for 1 ≤i≤k; we assume the nodes are distinct except that p1 and pk+1 may be the same in which case we call the path a k-cycle or a cycle of length k. (Notice that two nodes joined by an edge determine a 2-cycle according to this definition; it will also be convenient to regard a single node as a 1-cycle.) A spanning path or cycle is one that involves every node of the graph. One of the unsolved problems of graph theory is to characterize those graphs that have a spanning path or cycle.
On the Solutions of the Matrix Equation f(X,X*)=g(X,X*)
Authors: Basavappa P.
Published: December 2010

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.
On Some Functional Equations of Carleman
Authors: Costley Charles G.
Published: December 2010

The celebrated Fredholm theory of linear integral equations holds if the kernel K(x, y) or one of its iterates K(n) is bounded. Hilbert utilizing his theory of quadratic form was able to extend the theory to the kernels K(x, y) satisfying

a

b

where k is independent of u(x).

These theories were extended considerably by T. Carleman who deleted condition (b) above.

Equations involving this Carleman kernel have been found useful in connection with Hermitian forms, continued fractions, Schroedinger wave equations (see [1], [2]) and more recently in scattering theory in quantum physics, etc. [3]. See also [5] for a variety of applications and extensions.
Graphs Suppressible to an Edge
Authors: Harary F. , Krarup J. , Schwenk A.
Published: December 2010

An application of graph theory to automatic traffic control [2] gave rise to the problem of deciding which connected graphs have points of degree 2 which can be successively suppressed until only a single edge remains.
A Flat Nonmetrizable Connexion
Authors: Helfenstein H. G.
Published: December 2010

A linear connexion ∇ on a smooth manifold M with vanishing torsion and curvature is called flat. It is well known that such a connexion need not be the Levi-Civita connexion of a Riemannian (nor pseudo-Riemannian) structure (cf. [1]).

The following example with M homeomorphic to a cylinder is particularly simple. Using polar coordinates the punctured plane M=R2\{0} is covered by a smooth atlas with two charts
Decomposition of the Multivariate Beta Distribution with Applications
Authors: Kabe D. G. , Gupta R. P.
Published: December 2010

Let L be a positive definite symmetric matrix having a noncentral multivariate beta density of an arbitrary rank, see, e.g. Hayakawa ([2, p. 12, Equation 38]). Then an explicit procedure is given for decomposing the density of L in terms of densities of independent beta variates.
An Elementary Result on Exponential Measure Spaces
Authors: Shen C. Y.
Published: December 2010

A simple but useful result in the measure theory for product spaces can be stated as follows:

Theorem A. A necessary and sufficient condition that a measurable subset E of X×Y has measure zero is that almost every X-section (or almost every Y-section) has measure zero (see [1, §36]).

We will show, in this short note, that a similar result also holds for the exponential of measure spaces. Before proceeding any further, we describe briefly here the exponential construction of a measure space.
On a Generalization of a Theorem of Wiener
Authors: Siddiqi Jamil A.
Published: December 2010

Let V[0, 2π] denote the class of all normalized functions F of bounded variation in [0, 2π] such that F(x) = 2-1{F(x+0)+F(x-0)} and F(x+2π)-F(x) = F(2π) — F(0) for all x and let {Cn} be the sequence of Fourier-Stieltjes coefficients of F. Wiener [9] (cf. Bari [1, p. 212], Zygmund [10, p. 108]) proved the following theorem.
The Bounds Based on the Functions of Observations for Maximum of Stable Law
Authors: Basu A. K. , Wasan M. T.
Published: December 2010

Gnedenko and Kolmogorov [3, pp. 181-182] have shown that if Xn with law F(x) belong to the domain of normal attraction of a stable law of index 0<α<2, i.e. if partial sum Sn/an1/α converges in distribution to some stable law Vα, a>0 then there exist c1 and c2 such that

and
A Note on Measures Determined by Continuous Functions
Authors: Bruckner A. M.
Published: December 2010

Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.
Carathéodory's Theorem
Authors: Cook W. D. , Webster R. J.
Published: December 2010

Despite the abundance of generalizations of Carathéodory's theorem occurring in the literature (see [1]), the following simple generalization involving infinite convex combinations seems to have passed unnoticed. Boldface letters denote points of Rn and Greek letters denote scalars.
The Singular Congruence and the Maximal Quotient Semigroup
Authors: McMorris F. R.
Published: December 2010

It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.
Some Properties of Equations in Integers
Authors: Faierman M.
Published: December 2010

In certain boundary value problems associated with two-parameter ordinary differential equations defined and having 2p, (p>1), turning points in a given interval, there arises certain equations in integers whose solutions determine the coefficients in the asymptotic expansions for the eigenvalues [1,2, 3, pp. 134-139].
Une Caracterisation de la Categorie des Groupes
Authors: Leroux Pierre
Published: December 2010

Nous présentons dans cette note une caractérisation catégorique de la catégorie Gr des groupes et homomorphismes. Pour cela nous introduisons la notion de "catégorie projective de groupes" et montrons qu'elle correspond exactement à celle de "demi-variété" de Gr; cette démonstration dépend essentiellement du théorème de caractérisation des catégories algébriques Lawvere [4]. De même la notion de "monomorphisme prénormal" est inspirée de ce théorème et permet de formuler un critère un peu plus faible que celui de Lawvere sur les précongruences et distinguant les variétés parmi les demi-variétés de Gr. Finalement, la caractérisation de Gr que nous donnons fait appel à la classification des variétés de Shreier de Gr de Neumann et Wiegold [7] et au théorème de Scott [8] permettant de plonger tout groupe dans un groupe simple.
On a Class of Multivalued Mappings in Banach Spaces
Authors: Rhee C. J.
Published: December 2010

A. Granas [4] has studied single-valued compact vector fields in Banach spaces. In [3], he extended the fixed point theorems of Roth, Boknenblust and Karlin to the case of multi-valued functions. Closely following [4], we give here some general theorems in a class of multi-valued functions in Banach spaces.
On a Class of Projective Modules Over Central Separable Algebras II
Authors: Szeto George
Published: December 2010

The purpose of this paper is to continue the work of [7]. Throughout the paper all notations shall have the same meanings as those in [7]; that is, the ring R is commutative with identity, B(R) is the set of idempotents in R, Spec B(R) is the set of prime ideals in B(R), and Ue for e in B(R) denotes the set {x/x in Spec B(R) with 1—e in x}. Then from [6] we know that {Ue/e in B(R)} forms a basic open set for a topology imposed in Spec B(R) and this topological space is totally disconnected, compact and Hausdorff.
Modular Annihilator, A*-Algebras
Authors: Tomiuk B. J.
Published: December 2010

We find several equivalent conditions for an A*-algebra with dense socle to be completely continuous. Such an A*-algebra is modular annihilator [10]. We also study modular annihilator A*-algebras with the weak (βk)-property and obtain a necessary and sufficient condition for such algebras to be dual.
Completions of Ordered Sets
Authors: Ballinger B. T.
Published: December 2010

Completions of categories were studied by Lambek in [3], using the contravariant Horn functor to embed a small category C into the functor category (C*, S), where C* is the opposite category of C, and S is the category of sets. Three completions of C were considered; the completion (C*, S), the full subcategory (C*, C)inf⊆(C*, S) whose objects consist of all inf-preserving functors, and the full sub-category B⊆(C*, S)inf consisting of all subobjects of products of representable functors of the form HomC(—, C), C an object of C.