bluuebunny/crossref_metadata_2025 huggingface dataset:
On Zeros of Derivatives of Polynomials
Authors: Meir A. , Sharma A.
Published: December 2010

In an earlier paper [2], we raised the question of determining the minimum span of the kth derivative of a polynomial with real zeros having a given span. More precisely let πn, s denote the class of polynomials , with x1 ≤ x2 ≤ … ≤ xn, and the span σ(P) ≡xn - x1 = 2s (fixed).
Linear Functionals on Homogeneous Polynomials
Authors: Dunkl Charles F.
Published: December 2010

The space Hm of homogeneous polynomials in n real variables x1, x2,…, xn of degree m may be considered as an inner product space with inner product ; where ds is the rotation-invariant measure on Sn-1 = {x ε Rn: |x| = 1}, . The problem solved in this paper is the following: given n-1 a linear functional ϕ on Hm, find Pϕ ε Hm so that ϕ(p) = (p, Pϕ) for all p ε Hm.
On a Theorem of Erdös and Szekeres
Authors: Subbarao M.V.
Published: December 2010

Given a sequence of r distinct real numbers such that the number of terms of every decreasing subsequence is at most m, then there exists an increasing subsequence of more than n terms, where n is the largest integer less than r/m.

An extremely simple and elegant proof of the theorem was given by A. Seidenberg [2]. This note is intended to point out that a result analogous to the above holds under a more general setting.
A Banach Space of Analytic Functions For Constant Coefficient Equations of Evolution
Authors: Marsden J.
Published: December 2010

The spaces involved in the theory of equations of evolution (that is, the theory of semi-groups) are such that the infinitesimal generators are only densely defined. For the infinitesimal generator to be everywhere defined and smooth (that is, differentiable), one must work with a Frechet space. This is especially important in the non linear theory of Moser (see Moser [3] and Marsden [2]). If the spaces were Banach spaces, the theory would reduce to the classical Picard theory for ordinary differential equations. See Lang [1], for a discussion of the classical theory. In fact in this case the existence theory is both simpler and more comprehensive, because we obtain the important fact that the flow is a diffeomorphism (the solutions depend smoothly, and not merely continuously on the initial data). This has other advantages too, since the theory for smooth flows on Banach manifolds is well developed. (For example, see Marsden [2], theorem. 6.10 for an application we have in mind.)
A Note on Absolute Geometry
Authors: Brossard Roland
Published: December 2010

Metric axioms have been given in [3] for space euclidean geometry. If we replace the "similarity axiom" by the "congruence axiom", where congruence is defined to be a similarity of ratio one, the resulting structure is absolute geometry. In order to show this we choose a suitable definition for absolute geometry. The P a s c h system of axioms, given in an improved formulation by H. S. M. Coxeter in [4], is particularly suitable; the primitive notions are points, betweenness relation, and congruence relation. We can verify that every axiom for the absolute geometry in [4] in a theorem in [3] where the similarity axiom has been replaced by the congruence axiom. The only case for which it is not obvious is axiom 15.15 in [4] which says that if ABC and A' B' C' are two triangles with BC ≡ B'C' CA ≡ C'A1, AB ≡ A ' B ', while D and D' are two further points such that [B, C, D] and [B', C' D'] and BD ≡ B' D', then AD ≡ A' D'. In that case we first prove that if two triangles ABC and A'B C are such that AB/A'B' ≡ BC/B'C' ≡ CA/C'A' ≡ 1 then they are congruent; a proof of this, independent of the similarity axiom, can be found in [2]. The proof of 15. 15 in [4] is then obvious. As every axiom in the weakened structure of [3] is a theorem of absolute geometry we have a definition for this geometry.
The Factors of a Square-Free Integer
Authors: James R.D.
Published: December 2010

This note is concerned with the number C(n) of ordered non-trivial factorizations of an integer n in the special case where n is square free. If F (m) denotes C(p1…Pm) where pi. are distinct primes, it is shown that

and that
Generalized Lipschitz Algebras
Authors: Bishop E.R.
Published: December 2010

The purpose of this paper is to generalize the results of Sherbert on Lipschitz algebras and to study the relationship between homomorphisms of these algebras and continuous maps of the underlying metric spaces. In Sections 1, 2, and 3 we associate with each metric space a class of Lipschitz-type algebras and extend Sherbert's results in [7] to this class; in particular Sherbert's theorem 5.1 is extended to non-compact metric spaces (3.3, 3.4, 3.5).
Extensions of Vandermonde Type Convolutions with Several Summations and their Applications - I
Authors: Mohanty S.G. , Handa B.R.
Published: December 2010

In an earlier paper [8], one of the authors has established some Vandermonde type convolution identities involving multinomial coefficients with several summations which evidently are regeneralizations of identities in [1] with one summation. In this paper similar identities are derived for coefficients (defined below) of a general type, in the line of the results in [2] and [3], Furthermore, in a series of papers [4], [5], [6], Gould has obtained results on inversion of series and on classical polynomials by an extensive use of these identities with one summation.
Application of a Weierstrass Theorem to the Convergence of Moments
Authors: Chan L.K. , Mead E.R.
Published: December 2010

In this note we apply a well-known theorem due to Weierstrass to show that under certain conditions convergence in distribution of a sequence of distribution functions implies the convergence of moments.

This note may be understood by an undergraduate student who has an introductory course of complex variables and a second course of statistics.
An Approximation to {X, Y}
Authors: Brown B.
Published: December 2010

All homology and cohomology groups are reduced. is the class of finite abelian groups whose orders are prime to p. is the class of abelian groups whose orders are products of primes less than a. If G is a finitely generated abelian group, Gp is the quotient of G by the subgroup of G made up of all elements whose orders are prime to p.
Incomplete Diagonals of Latin Squares
Authors: Marica J. , Schönheim J.
Published: December 2010

The following question has been asked by J. Dénes [2]: If n - 1 elements of the diagonal of an n × n array are prescribed, is it possible to complete the array to form an n × n latin square?" It is known that if n diagonal elements are given such a completion is not always possible.

That the answer to Dénes' question is yes follows directly from a theorem of M. Hall Jr. [1].
Nonoscillation Criteria for Elliptic Equations
Authors: Swanson C.A.
Published: December 2010

Sufficient conditions will be derived for the linear elliptic partial differential equation

(1)

to be nonoscillatory in an unbounded domain R in n-dimensional Euclidean space En. The boundary ∂R of R is supposed to have a piecewise continuous unit normal vector at each point. There is no essential loss of generality in assuming that R contains the origin. Otherwise no special assumptions are needed regarding the shape of R: it is not necessary for R to be quasiconical (as in [2]), quasicylindrical, or quasibounded [1].
On Two Functional Equations for the Trigonometric Functions
Authors: Haruki Hiroshi
Published: December 2010

We consider the following cosine and sine functional equations:

(1)

(2)

where f is an entire function of a complex variable z and x, y are complex variables [1; 2; 3].
On A Combinatorial Problem III
Authors: Erdős P.
Published: December 2010

A family of sets {Aα} is said by Miller [3] to have property B if there exists a set S which meets all the sets Aα and contains none of them. Property B has been extensively studied in several recent papers (see the references in [2] and the last chapter of P. Erdös and A. Hajnal, On chromatic number of graphs and set systems, Acta. Math. Acad. Sci. Hung. 17 (1966) 61–99).
Sums of Complexes in Torsion Free Abelian Groups
Authors: Heilbronn H. , Scherk P.
Published: December 2010

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,

Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).
An Existence Theorem for Room Squares*
Authors: Mullin R. C. , Nemeth E.
Published: December 2010

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.

A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.
The Frattini Subgroup of a p-Group
Authors: Manley P. L.
Published: December 2010

We prove that the Frattini subgroups are trivial for finite groups whose orders are not divisible by squares of a prime.

If G is a group, we define its Frattini subgroup ϕ(G) as the intersection of all the maximal subgroups of G. An element x ∈ G is a nongenerator of G if whenever G = < X, x >, where X ⊆ G, then G = < X >.
Le Théorème De Bernstein Sur Les Fonctions Complétement Monotones
Authors: Dubuc Serge
Published: December 2010

Nous voulons montrer que le théorème de Bernstein est une conséquence simple de la formule de Taylor et d' un théoréme de Helly. Rappelons une définition: une fonction réelle f définie sur (0, ∞) est dite complètement monotone si f est indéfiniment derivable, si f ainsi que toutes ses dérivées paires sont positive set si toutes les dérivées impaires sont négatives.
Categories Localement Artiniennes Sans Radical
Authors: Leduc P. -Y.
Published: December 2010

Le but de cette note est de donner une caractérisation purement catégorique des catégories de modules semi-simples sur un anneau. Nous allons voir que ces catégories sont précisément celles qui sont localement artiniennes au sens de P. Gabriel et dont le radical au sens de G. M. Kelly est nul.
Joins and Direct Products of Equational Classes
Authors: Grätzer G. , Lakser H. , Płonka J.
Published: December 2010

Let K0 and K1 be equational classes of algebras of the same type. The smallest equational class K containing K0 and K1 is the join of K0 and K1; in notation, K = K0 ∨ K1. The direct product K0 × K1 is the class of all algebras α which are isomorphic to an algebra of the form a0 × a1, a0 ∈ K1. Naturally, K0 × K1 ⊆ K0 ∨ K1, Our first theorem states a very simple condition under which K0 × K1 = K0 ∨ K1, and an additional condition under which the representation α ∨ a0 × a1 unique.
On Pearl's Paper "A Decomposition Theorem for Matrices"*
Authors: Thompson R. C.
Published: December 2010

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)
On Fixed and Periodic Points Under Certain Sets of Mappings
Authors: Holmes R.D.
Published: December 2010

Let (X, d) be a metric space and f a mapping of X into itself. D.F. Bailey [l] considered a class of mappings f satisfying the condition: ∀x, y ∈ X, x ≠ y,

(1.1)

where I+ denotes the set of positive integers.
Equicardinality of Bases in B-Matroids
Authors: Higgs Denis
Published: December 2010

It is very well known that any two bases of a finitary matroid (see [2] for definitions) have the same cardinality. As Dlab has shown in [1], the same does not hold for arbitrary transitive exchange spaces; indeed, since the examples Dlab constructs in [1] are matroids, it does not even hold for arbitrary matroids. Nevertheless with the aid of the generalized continuum hypothesis (G. C.H.) we are able to prove the result for B-matroids.
Enclosure Theorems for Eigenvalues of Elliptic Operators
Authors: Clements John C.
Published: December 2010

LetLbe the linear, elliptic, self-adjoint partial differential operator given by

whereDjdenotes partial differentiation with respect toxj, 1 ≤j≤n,bis a positive, continuous real-valued function ofx= (x1,…,xn) inn-dimensional Euclidean spaceEn, theaijare real-valued functions possessing uniformly continuous first partial derivatives inEnand the matrix {aij} is everywhere positive definite. A solutionuofLu= 0 is assumed to be of class C1.
Quasi-Frobenius X-Rings
Authors: Zaks Abraham
Published: December 2010

In a recent study of a specific class of quasi-Frobenius rings, Feller has found it useful to introduce the X-rings ([3]). He suggested among others the following topics:

(A)

Determine the properties of completely indecomposable rings and matrix rings over completely indecomposable rings.

(B)

Determine the properties of modules over quasi-Frobenius X-rings.

We point out that the completely indecomposable rings are the local quasi-Frobenius rings. Problems (A) and (B) then lead naturally to semi-local quasi-Frobenius rings, and to matrix algebra over local quasi-Frobenius rings. These types of rings are discussed in sections 1 and 2.
A Note on Endomorphism Semigroups
Authors: Platt Craig
Published: December 2010

If is a universal algebra, the set of endomorphisms of forms a monoid (i.e., semigroup with identity) under composition. We denote it by End (). For definitions and notations, see [1]. It is well known (e.g., [1], Theorem 12.3) that for any monoid M there is a unary algebra with M ≅ End (). E. Mendelsohn and Z. Hedrlin [3] have proved that the monoid of a subalgebra of an algebra is independent of the monoid of . In [2], Hedrlin proves the same for the monoid of a homomorphic image of . The proofs of these depend heavily on graph-theoretical and category-theoretical considerations. In this note considerably shorter direct algebraic proofs are given of these results.
On Completing Latin Rectangles
Authors: Lindner Charles C.
Published: December 2010

By an (incomplete) r × s latin rectangle is meant an r × s array such that (in some subset of the rs cells of the array) each of the cells is occupied by an integer from the set 1, 2, …, s and such that no integer from the set 1,2, …, s occurs in any row or column more than once. This definition requires that r≦s. If r=s we will replace the word rectangle by square. It is easy to see that for any n≧2 there is an incomplete n × 2n latin rectangle with 2n cells occupied which cannot be completed to a n × 2n latin rectangle. In this paper we prove the following theorem.

Theorem 1. An incomplete n × 2n latin rectangle with 2n—1 cells occupied can be completed to a n × 2n latin rectangle.
A Note on Inverse Limits of Finite Spaces
Authors: Nadler S. B.
Published: December 2010

The following lemma, which appears as Lemma 4 in [5], was used to determine certain multicoherence properties of inverse limits of continua.

Lemma. Let X denote the inverse limit of an inverse system {Xλ, fλμ, Λ} of compact Hausdorff spaces Xλ. If Xλ has no more than k components (where k < ∞ is fixed) for each λ ∊ Λ, then X has no more than k components.

In this paper we give a set theoretic analogue of this lemma and an extension which was suggested to the author by Professor F. W. Lawvere. An application to inverse limits of finite groups is then given.
On the Nonexistence of Orthogonal Steiner Systems of Order 9
Authors: Mullin R. C. , Nemeth E.
Published: December 2010

It is shown that no pair of orthogonal Steiner triple systems of order 9 exists.
Properties of the Fixed Point Set of Contractive
Multi-Functions
Authors: Schirmer Helga
Published: December 2010

A well known theorem by S. Banach states that a contractive function
f: X → X on a complete
metric space X has a fixed point, and that this fixed point
is unique. This result has a partial extension to multi-functions: every
contractive compact-valued multi-function on a complete metric space has a
fixed point (see Definition 1 and Theorem 1 below). But simple examples show
that this fixed point is no longer unique. We investigate some questions
concerned with the properties of the fixed point set Φ of a contractive
multi-function φ. Is, e.g., Φ connected if φ is connected-valued? Is Φ
convex if φ is convex-valued? The answer is yes if X is the
real line (§2), but examples in §3 and §4 show that in general the answer is
no.
On a Characterization of Maximal Ideals
Authors: Siddiqi Jamil A.
Published: December 2010

Let A be a commutative complex Banach algebra with identity
e. Gleason [1] (cf. also Kahane and Żelazko [2]) has
given the following characterization of maximal ideals in
A.

Theorem. A subspace X ⊂ A of codimension one is a maximal ideal in A
if and only if it consists of non-invertible elements.

The proofs given by Gleason and by Kahane and Żelazko are both based on the
use of Hadamard's factorization theorem for entire functions. In this note
we show that this can be avoided by using elementary properties of analytic
functions.
Vertex and Edge Transitive, but not 1-Transitive, Graphs
Authors: Bouwer I. Z.
Published: December 2010

A (simple, undirected) graphGisvertex transitiveif for any two vertices ofGthere is an automorphism ofGthat maps one to the other. Similarly,Gisedge transitiveif for any two edges [a,b] and [c,d] ofGthere is an automorphism ofGsuch that {c,d} = {f(a),f(b)}. A 1-pathofGis an ordered pair (a,b) of (distinct) verticesaandbofG, such thataandbare joined by an edge.Gis 1-transitiveif for any two 1-paths (a,b) and (c,d) ofGthere is an automorphismfofGsuch thatc=f(a) andd=f(b). A graph isregular of valency dif each of its vertices is incident with exactlydof its edges.
On ℵα-Noetherian Modules
Authors: Simis Aron
Published: December 2010

In this note we define two concepts which can be thought of as a
generalization of noetherian concepts.

The main result is as follows (Corollary A): If R is a ring whose
countably generated (left) ideals are (left) principal, then R is a
(left) principal ideal ring.

This result if obtained, more generally, for any (left)
R-module and any regular cardinal ℵα (Corollary
1); a cardinal ℵα is regular whenever W(ℵα) =
{ordinals γ | card γ < ℵα} has no cofinal subset of
cardinality less than ℵα.
A Remark on Integration of Almost-Periodic
Functions
Authors: Zaidman S.
Published: December 2010

A result proved by Favard for scalar-valued almost-periodic functions has an
immediate extension to Banach space valued functions (see [2] and [3] for
explicit details).

The result says that integrals of almost-periodic functions whose 'spectrum'
is at positive distance from 0 are again almost-periodic. Our aim here is to
indicate a more general formulation of this result using strongly
almost-periodic oneparameter groups of operators in Banach spaces.
On Equal Products of Consecutive Integers
Authors: Macleod R. A. , Barrodale I.
Published: December 2010

Using the theory of algebraic numbers, Mordell [1] has shown that the
Diophantine equation

1

possesses only two solutions in positive integers; these are given by
n = 2, m = 1, and n =
14, m = 5. We are interested in positive integer solutions
to the generalized equation

2

and in this paper we prove for several choices of k and
l that (2) has no solutions, in other cases the only
solutions are given, and numerical evidence for all values of
k and l for which max
(k, l) ≤ 15 is also exhibited.
The Order of Algebraic Linear Transformations
Authors: Putz Randee
Published: December 2010

In this paper we extend the results of an earlier note [1].

Definition. Let E be an extension field of the rationals. A
vector v = (b1, …, bn) in En is algebraic if each coordinate bi is algebraic over the rationals. A linear transformation
T: En → En is algebraic if
T(v) is an algebraic vector for every
algebraic vector v.

Definition. The degree of an algebraic linear
transformation T, denoted by deg T, is the
minimum of [K:Q] taken over all finite
algebraic extensions K of the rationals Q
such that T: Kn → Kn.
On a Multiple Integral Useful in Order Statistics Distribution Theory
Authors: Kabe D. G.
Published: December 2010

A known multiple integral from the difference calculus is used to derive certain distributions of ordered statistics from the truncated exponential, rectangular, and ordered random intervals populations.
Subequalizers
Authors: Lambek Joachim
Published: December 2010

The purpose of this exposition is threefold. Firstly, we wish to show that many concepts and arguments carry over from pre-ordered sets to categories. Secondly, we wish to make some propaganda for the notion of "subequalizer" of two functors, which appears to be more fundamental than Lawvere's so-called "commacategory", in the same sense in which equalizers are more fundamental than pullbacks. Thirdly, we wish to give simple proofs of the adjoint functor theorem and related theorems, which appear to be more economic than those in the literature. The author wishes to take this opportunity to refine some arguments that he has published earlier. He is indebted to Michael Barr, whose presentation of similar proofs in his course has provided the stimulation for preparing this note for publication, to John Isbell for his critical reading of the manuscript and to William Schelter for suggesting a shortcut in one of the proofs.
On Carleman Integral Operators
Authors: Costley Charles G.
Published: December 2010

L2(a, b)

1

with the property

2

were originally defined by T. Carleman [4]. Here he imposed on the kernel the conditions of measurability and hermiticity,

3

for all x with the exception of a countable set with a finite number of limit points and

4

where Jδ denotes the interval [a, b] with the exception of subintervals |x - ξv| < δ; here ξv represents a finite set of points for which (3) fails to hold.
On Sets of Ternary Vectors Whose Only Linear Dependencies Involve an Odd Number of Vectors
Authors: Berlekamp E. R.
Published: December 2010

Recent efforts to generalize a classic result of Hajos [3] on the decomposition of finite abelian groups into direct sums of subsets (see Fuchs [1, Chap. XV]) led B. Gordon [2] to the following conjecture. If are r-dimensional row vectors over GF(3) such that: (i) Any weighted (±) sum of any even number of 's is nonzero, (ii) For each r-dimensional , there exists an s such that

Then there exists a subset of either 1 or 4 's which satisfies the same conditions.

This paper proves Gordon's conjecture.
Ideaux a Droite Minimaux d'un Anneau Primitif
Authors: Thierrin G.
Published: December 2010

Soit R un anneau primitif à droite et soit A un idéal à droite de R. Dans un récent article [2], Ménard a montré que A est minimal si et seulement si abA = baA ∀ a, b ∊ A. Le but de cette note est de montrer que A est minimal si et seulement si aA = a2A ∀ a ∊ A.

Lemme. Soit R un anneau semi-simple [1], soit A un idéal à droite de R et soit S = Rad(A) le radical de Vanneau A. Alors A est un idéal à droite minimal de R si et seulement si Ā = A/S est un corps.
Inadmissibility of the Maximum Likelihood Estimator in the Presence of Prior Information
Authors: Kale B. K.
Published: December 2010

Lehmann [1] in his lecture notes on estimation shows that for estimating the unknown mean of a normal distribution, N(θ, 1), the usual estimator is neither minimax nor admissible if it is known that θ belongs to a finite closed interval [a, b] and the loss function is squared error. It is shown that , the maximum likelihood estimator (MLE) of θ, has uniformly smaller mean squared error (MSE) than that of . It is natural to ask the question whether the MLE of θ in N(θ, 1) is admissible or not if it is known that θ ∊ [a, b]. The answer turns out to be negative and the purpose of this note is to present this result in a slightly generalized form.
On a Solution of the Hammerstein Equation with Singular Normal Kernels
Authors: Costley Charles G.
Published: December 2010

We consider here the equation

1

This equation was first studied by Hammerstein [4] under the assumption that the linear operator

2

is selfadjoint and completely continuous. V. Nemytsky [5] and M. Golomb [3] dropped the assumption that A be selfadjoint and positive. M. Vainberg [6] considered (among other cases) the case in which A is a bounded operator generated by a Carleman kernel. The kernels considered in this work do not necessarily generate bounded, completely continuous or selfadjoint, operators.
Note on the Singular Submodule
Authors: Fieldhouse D.
Published: December 2010

One very interesting and important problem in ring theory is the determination of the position of the singular ideal of a ring with respect to the various radicals (Jacobson, prime, Wedderburn, etc.) of the ring. A summary of the known results can be found in Faith [3, p. 47 ff.] and Lambek [5, p. 102 ff.]. Here we use a new technique to obtain extensions of these results as well as some new ones.

Throughout we adopt the Bourbaki [2] conventions for rings and modules: all rings have 1, all modules are unital, and all ring homomorphisms preserve the 1.
On a 3-Dimensional Isoperimetric Problem
Authors: Kömhoff Magelone
Published: December 2010

Let L(P) denote the total edge length and A(P) the total surface area of a three-dimensional convex polyhedron P. In [5] it was shown that if P belongs to the set of all polyhedra with triangular faces then for all

with equality if and only if is a regular tetrahedron.

It is not difficult to establish the inequality
Arithmetical Functions and Distributivity
Authors: McCarthy P. J.
Published: December 2010

In this note we shall present a result about incidence functions on a locally finite partially ordered set, a result which is related to theorems of Lambek [2] and Subbarao [6]. Our terminology and notation will be that of Smith [4, 5] and Rota [7].

Let (L, ≤) be a partially ordered set which is locally finite in the sense that for all x, y ∊ L the interval [x, y] = {z | x ≤ z ≤ y} is finite. Denote by A(L, ≤) the set of functions f from L × L into some field, which is fixed once and for all, such that f(x, y) = 0 whenever x ≰ y.
Central Idempotents in Group Rings
Authors: Burns R. G.
Published: December 2010

Let R be a ring and G a group. The group ring RG consists of all functions f: G → R with finite support. Addition is pointwise and multiplication is defined for f, h ∊ RG and g ∊ G, by

The support group of f is defined to be the subgroup of G generated by the support of f. The element f is idempotent if ff = f

We prove the following result.
Space of Solutions of Homogeneous Elliptic Equations
Authors: Dubinsky E. , Husain T.
Published: December 2010

This is the continuation of our paper [1] and includes the results promised there. As in [1], we consider a homogeneous elliptic equation in two variables. In [1] we showed that all solutions of such equations can be written in a specific form, viz. in the form of an infinite series in certain specific polynomials. Here we first establish that a common solution of any two positive powers of any two linearly independent, linear elliptic polynomials can be expressed as a polynomial (Lemma 2).
A Matrix Representation of a Pair of Projections in a Hilbert Space
Authors: Giles R. , Kummer H.
Published: December 2010

Let H be a complex Hilbert space and let K = H ⊕ H. Then K can be identified with the set of all column matrices

equipped with componentwise addition and scalar multiplication and the scalar product
A Constructive Solution to a Tournament Problem
Authors: Graham R. L. , Spencer J. H.
Published: December 2010

By a tournament Tn on n vertices, we shall mean a directed graph on n vertices for which every pair of distinct vertices form the endpoints of exactly one directed edge (e.g., see [5]). If x and y are vertices of Tn we say that x dominates y if the edge between x and y is directed from x to y. In 1962, K. Schütte [2] raised the following question: Given k > 0, is there a tournament Tn(k) such that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. (Such a tournament will be said to have property Pk.)
Maximal Sum-Free Sets in Elementary Abelian p-Groups
Authors: Rhemtulla A. H. , Street Anne Penfold
Published: December 2010

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.
Coloring a Dodecahedron with Four Colors(1)
Authors: Ballard David
Published: December 2010

The purpose of this note is to show that there is one and only one coloring of the sides of a dodecahedron with four colors, under the condition that no two adjacent sides share the same color.

In [1] Busacker and Saaty give a graphical solution to the game known as “Instant Insanity”.
Quasigroups Orthogonal to a Given Abelian Group
Authors: Lindner Charles C.
Published: December 2010

In this note we prove the following theorem, which does not seem to appear explicitly in the literature.

Theorem. Let A be a finite abelian group and p the smallest prime which divides |A|. Then there are p—1 mutually orthogonal quasigroups of order |A|, one of which is A.
Some Remarks on Talenti's Semigroup
Authors: Chrysovergis A.
Published: December 2010

Let X be a Banach space. Consider the family I(α) of linear continuous operators from X into itself, depending on the parameter α ≥ 0. Suppose that I(α1+α2)=I(α1)I(α2)∀α1, α2 ≥ 0 and I(0)=I (semigroupal property). Such a semigroup is said to be strongly continuous for α > 0, if limh→0I(α+h)x =I(α)x, ∀x ∊ X.
Arc Components of Certain Chainable Continua
Authors: Nadler Sam B.
Published: December 2010

It is shown that if a chainable continuum has exactly two arc components, then one of them is an arc and the other is a half-ray.
Sets of Generators of a Commutative and Associative Algebra(1)
Authors: Djoković D. Ž.
Published: December 2010

Let A be a finite dimensional commutative and associative algebra with identity, over a field K. We assume also that A is generated by one element and consequently, isomorphic to a quotient algebra of the polynomial algebra K[X]. If A=K[a] and bi=fi(A), fi(X) ∊ K[X], 1≤i≤r we find necessary and sufficient conditions which should be satisfied by fi(X) in order that A = K[b1, …, br].

The result can be stated as a theorem about matrices. As a special case we obtain a recent result of Thompson [4].

In fact this last result was established earlier by Mirsky and Rado [3]. I am grateful to the referee for supplying this reference.
Some Remarks on Abstract Differential Operators
Authors: Malik M. A.
Published: December 2010

Let A be a closed linear operator with domain DA dense in a Banach space B; DA is also a Banach space under the graph norm. By we represent the space of continuous linear mappings from B to DA and R(λ; A) = (λI—A)-1 denotes the resolvent of A; λ ∊ C (complex plane).
On the Permanent of a Certain Class of (0, 1)-Matrices
Authors: Hartfiel D. J. , Crosby J. W.
Published: December 2010

In [3, p. 77] Ryser notes the importance of the minimum of the permanent function on the class of (0, 1)-matrices having exactly k ones in each row and column. In [4] a lower bound was found for the minimum of the permanent on the class Λn of n × n (0, 1)-matrices with exactly three l's in each row and column. The purpose of our work is to improve this result, in particular we show that minA ∊ Λn (per A) ≥ 3(n-1).
A Note about the Definition of CW-Complexes
Authors: Déruaz Marcel , O'Keefe Mary
Published: December 2010

In [1] Whitehead defined a CW-complex as a closure finite cell complex with the weak topology (i.e. a topology coherent with the family of closed cells, in Spanier's terminology). The purpose of this note is to show that these two conditions imposed on cell complexes can be replaced by a single one.
Finite Sets of Integers with Equal Power Sums
Authors: Djoković D. Ž.
Published: December 2010

This note originated in an attempt to generalize the assertion of Problem 164 which was proposed by Moser [2].

Let us first state our generalization.
A Condition for Equality of Cardinals of Minimal Generators Under Closure Operators
Authors: Hall Japheth
Published: December 2010

LetCbe an operator on the subsets of a set X with values among the subsets of X. We assume thatCis a closure operator inX, i.e. a monotone, idempotent and extensive operator inX(cf., e.g., Birkhoff [3, p. 39], Schmidt [1], [2]). IfA ⊆ XandB ⊆ X, we say thatAandBareC-equivalentifC(A) = C(B)(Bleicher- Marczewski [4, p. 210]).
The Ring Scheme Fg
Authors: Connell Ian G.
Published: December 2010

Let A be a ring and s2(A) the set of idempotents in A. It is a familiar fact that s2(A) becomes a ring if we define 0, 1 and multiplication as in A, and a new negative and new addition by

1

R. A. Melter [3] made the surprising observation that s3(A), where in general sq(A)={x∈A:xq=x}, is a ring if 2 is a unit in A and we define 0,1 minus and multiplication as in A, and a new addition by

2

The nonobvious facts are that s3(A) is closed under ⊕ and that ⊕ is associative when applied to the elements of s3(A). The ⊕ in (1) is actually a formal group over A, and so is associative when applied to any elements of A. However in (2) (and similarly in other cases we shall define) the ⊕ is not a formal group, and the associative law depends on the fact that the elements involved are in s3(A).
On Transportable Forms
Authors: Klemola Tapio
Published: December 2010

In the theory of deformations of compact complex manifolds, the hypothesis of constancy of the dimension of diverse structural cohomology groups pertaining to a fibre plays an important role (see, for instance, [3, Propositions 2.5 and 2.7, Theorems 2.2 and 2.3, and Definition 6.1]). This paper is the first of two devoted to the investigation of conditions under which constancy of the dimension of given cohomology groups is assured, and more generally, to the study of the variation of that dimension.

In [2] Griffiths introduces extendible forms in a holomorphic deformation. We consider in this paper a differentiate family, and besides extendible, also co-extendible and transportable forms (see §5), and deduce from their existence conclusions about the variation of the dimension of the corresponding structural cohomology groups. It is left to a subsequent paper to give more explicit conditions by means of cohomology operations, and to deal with some applications.
An Inequality for Elementary Symmetric Functions
Authors: Menon K. V.
Published: December 2010

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by

1

E0 = 1 and Er=0(r>m).

We define the rth symmetric mean by

2

where denote the binomial coefficient. If α1 α2,…,αm are positive reals then

we have two well-known inequalities

3

and

4

In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.
Local Group Rings
Authors: Nicholson W. K.
Published: December 2010

The purpose of this note is to generalize a result of Gulliksen, Ribenboim and Viswanathan which characterized local group rings when both the ring and the group are commutative.

We assume throughout that all rings are associative with identity. If R is a ring we call R local if R/J(R) is a division ring where J(R) denotes the Jacobson radical of R. It is well known that R is local if and only if each element of R\J(R) is a unit. We need the following.
Higher Derivations and the Jordan Canonical Form of the Companion Matrix
Authors: Roberts Leslie G.
Published: December 2010

The purpose of this note is to give a basis with respect to which the companion matrix of an equation (over a field of any characteristic) is in Jordan canonical form.

Let k be a field. Define a k-linear map Di:k[X]→k[X] by where the integer is the binomial coefficient n!;/i!;(n —i)!;. We adopt the usual convention that if i>9 or j<0. Then D=(D0, D1D2,…) is a higher derivation (see [1, p. 192]).
Explicit Solutions of Pyramidal Diophantine Equations
Authors: Bernstein Leon
Published: December 2010

Let Pm, k denote the set of pyramidal numbers

1.1

The question has been asked whether there exist elements p, q, r in Pm, k such that p+q = r or, as the problem is usually posed,

1.2

The case k=2 has been studied by Sierpinski [6] and Khatri [3]; the case k=3 by Oppenheim [4] and Segal [5]; recently Fraenkel [2] has generalized (1.1) to the larger set

1.3
The Slimming Number and Genus of Graphs
Authors: Guy Richard K.
Published: December 2010

J. Ch. Boland suggested, and Mrs. Sheehan named, the idea of the slimming number of a graph G, i.e. the minimum number, s(G), of edges, e1e2,…, es, which must be removed from G in order that G—∪ ei be planar.

For the complete graph, Kn (n≥3), it may be seen by Euler's formula that a planar subgraph contains at most 3n—6 edges; moreover one may construct such a subgraph inductively, starting from K3, and adding points successively, joining them to the three vertices of the region in which they lie, so

1
Noetherian Tensor Products
Authors: Magarian E. A. , Motto J. L.
Published: December 2010

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].

All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.
Mean Value Theorem for the m-Integral of Dinculeanu
Authors: Morales Pedro
Published: December 2010

The classical mean value theorem asserts that if f is a real, bounded, Riemann integrable function defined on a finite real interval a≤t≤b, then , where infa≤t≤bf(t)≤y0supa≤t≤bf(t). The extensions of Choquet [3], Price [15], and of this paper generalize the fact that y0 belongs to the closure of the convex hull of f([a, b]). The version of Choquet ([3, p. 38]) applies to a continuous function on a compact interval with values in a Banach space; that of Price ([15, p. 24]) applies to a bilinear integral of a special type containing the Birkhoff integral [2]. The m-integral of Dinculeanu [6] (specialization of Bartle's *- integral [1]) leaves intact the Lebesgue dominated convergence theorem and is strong enough to support an extended development. The paper is organized as follows: the object of §2 is to express the integral of a bounded m-integrable function as a limit of Riemann sums; §3 gives Price's generalization of "convex hull" [15]; the theorem of the paper is established in §4; §5 gives applications to vector differentiation which, for continuously differentiable functions, contain results of Dieudonné [5] and McLeod [13].
Kantorovitch Polynomials Diminish Generalized Length
Authors: Price Martin E.
Published: December 2010

The Kantorovitch polynomials of a summable function s, defined on [0, 1], are

where

and

They are the analogue for summable functions of the Bernstein polynomials Bnf(x), and they possess similar properties [1].
Positive Definite Sequence of Operators and a Fixed Point Theorem
Authors: Dash A. T.
Published: December 2010

The purpose of this note is to prove the following:

Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.

Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite if

for every finitely nonzero sequence {fn} of vectors in H [1].
Disk Packings which have Non-Extreme Exponents
Authors: Boyd David W.
Published: December 2010

Let U be an open set in the Euclidean plane which has finite area. A complete (or solid) packing of U is a sequence of pairwise disjoint open disks C={Dn}, each contained in U and whose total area equals that of U. A simple osculatory packing of U is one in which the disk Dn has, for each n, the largest radius of disks contained in (S- denotes the closure of the set U.) If rn is the radius of Dn, then the exponent of the packing, e(C, U) is the infimum of real numbers t for which In the sequel we refer to a complete packing simply as a packing.
A Note on Primitive Graphs
Authors: Bouwer I. Z. , LeBlanc G. F.
Published: December 2010

Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.
On a Question Posed by D. Leviatan and L. Lorch
Authors: Tzimbalario Jean
Published: December 2010

In this note we construct a pair of regular matrices T1 and T2 such that T1 is stronger than T2, and the T2-transforms of all bounded sequences are such that the sets of limit-points are connected, while there is a bounded sequence such that the set of limit-points of its T1-transform is not connected. This provides a negative answer to a question posed by Leviatan and Lorch [3, §6, (c)].
Integral Functionals in the Duals of Lλ-Spacesc(1)
Authors: Ellis H. W. , Hiscocks J. D.
Published: December 2010

Luxemburg and Zaanen [5] call an element φ of the topological dual of a normed or seminormed vector space V an integral if

We denote the space of integrals by VI, For the Lλ function spaces introduced by Ellis and Halperin [2] another Banach subspace of the dual emerges, namely the conjugate space Lλ* which is the Lλ space determined by the conjugate length function λ*-Lλ* is contained in (Lλ)I but need not coincide with it.
Normal Variations of Invariant Hypersurfaces of Framed Manifolds
Authors: Goldberg Samuel I.
Published: December 2010

A hypersurface of a globally framed f-manifold (briefly, a framed manifold), does not in general possess a framed structure as one may see by considering the 4-sphere S4 in R5 or S5. For, a hypersurface so endowed carries an almost complex structure, or else, it admits a nonsingular differentiable vector field. Since an almost complex manifold may be considered as being globally framed, with no complementary frames, this situation is in marked contrast with the well known fact that a hypersurface (real codimension 1) of an almost complex manifold admits a framed structure, more specifically, an almost contact structure.
Third-Engel 2-Groups are Soluble
Authors: Gupta Narain
Published: December 2010

It is shown that a 3rd-Engel group is an extension of a soluble group by a group of exponent 5.
On a Question of Colin Clark Concerning Three Properties of Convex Sets(1)
Authors: Klee Victor
Published: December 2010

Let C be a convex subset of a normed linear space E. The following properties of C were studied by Clark [2]:

(1)

C is of finite width/ that is, C lies between two parallel closed hyperplanes;

(2)

C is of finite width in some direction; that is, for some line L in E there is a finite upper bound for the lengths of C's intersections with lines parallel to L;

(3)

C is partially bounded; that is, there is a finite upper bound for the radii of balls contained in C.
On the Local Connectedness of βX-X
Authors: Grant Woods R.
Published: December 2010

Let X be any completely regular Hausdorff topological space, and let βX denote its Stone-Čech compactification. This note is devoted to proving the following result:

5. THEOREM. Let X be realcompact and noncompact. Then βX—X is not connected im kleinen at any point.
Représentation Des Polynômes Positifs Sur [-1,1]
Authors: Leblanc Raymond
Published: December 2010

En étudiant certains cônes de fonctionnelles affines positives, nous avons été amenés à considérer les polynômes qui sont positifs sur l'intervalle [-1, 1]. En fait, la fonctionnelle affine
Open, Connected Functions
Authors: Friedler Louis
Published: December 2010

Recall that a function f:X→ Yis called connected if f(C) is connected for each connected subset C of X. These functions have been extensively studied. (See Sanderson [6].) A function f:X → Y is monotone if for each y ∊ Y, f-1(y) is connected. We shall use the techniques of multivalued functions to prove that if f: X→ Y is open and monotone onto Y, then f-1(C) is connected for each connected subset C of Y. This result is used to prove that the product of semilocally connected spaces is semilocally connected and that the image of a maximally connected space under an open, connected, monotone function is maximally connected.
Extensions of Contractive Mappings and Edelstein's Iterative Test
Authors: Bryant Jack , Guseman L. F.
Published: December 2010

A mapping f from a metric space (X,d) into itself is said to be contractive if x≠y implies d(f(x),f(y))<d(x,y). Theorems of Edelstein [2] state that a contractive selfmapfofa metric space X has a fixed point if, for some x0, the sequence {fn(x0)} of iterates at x0 has a convergent subsequence; moreover, the sequence {fn(x0)} converges to the unique fixed point of f. Nadler [3] observes that, from the point of view of applications, it is usually as difficult to verify the condition (for some x0 …) as it is to find the fixed point directly.
Weakly Regular Rings
Authors: Ramamurthi V. S.
Published: December 2010

This paper attempts to generalize a property of regular rings, namely,I2=Ifor every right (left) ideal. Rings with this property are called right (left) weakly regular. A ring which is both left and right weakly regular is called weakly regular. Kovacs in [6] proved that, for commutative rings, weak regularity and regularity are equivalent conditions and left open the question whether for arbitrary rings the two conditions are equivalent. We show in §1 that, in general weak regularity does not imply regularity. In fact, the class of weakly regular rings strictly contains the class of regular rings as well as the class of biregular rings.
On a Class of Nonparametric Tests for Independence—Bivariate Case(1)
Authors: Srivastava M. S.
Published: December 2010

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) benmutually independent pairs of random variables with absolutely continuous (hereafter, a.c.)pdfgiven by

(1)

where f(ρ)denotes the conditionalpdfofXgivenY, g(y)the marginalpdfofY, e(ρ)→ 1andb(ρ)→0as ρ→0 and,

(2)

We wish to test the hypothesis

(3)

against the alternative

(4)

For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.
Cartesian Nets and Groupoids
Authors: Taylor M. A.
Published: December 2010

Aczel has conjectured, [1, p. 448], the possibility of developing a net theory for structures more general than quasigroups. Steps in this direction have been taken by Havel who considers nets associated with multigroupoids [2], The work presented here introduces a generalization of 3-nets and their algebraization which is wide enough to encompass most algebraic structures based on a single binary operation.
A Converse Problem in Matrix Differential Equations
Authors: Shreveo Warren E.
Published: December 2010

Suppose X and F are nxn matrix solutions of the n X n matrix differential equation

(1)

such that

(2)

where J is some interval.