bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Multiplicity of Representations of Numbers by Sums
Authors: Moser L.
Published: December 2010

Let A: {a1, ≤ a2, ≤ ߪ ≤ at} be a set of non-negative integers not exceeding n, and let rn(A;m) = rr(m) denote the numbers of representations of m as the sum of k a's, order taken into account, i. e.

1

Since 0 ≤ ai1 + ai2 + … + aik ≤ kn, we have

2
A Remark on Completely Monotonic Sequences, with an Application to Summability
Authors: Lorch Lee , Moser Leo
Published: December 2010

A sequence of non-negative numbers, μ0 μ1 … μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, …. Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].

It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to ">". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.)
The Hamilton-Jacobi Equation for a Charged Particle in a Combined Gravitational and Electromagnetic Field
Authors: Sen D. K.
Published: December 2010

The equation of motion of a charged particle in a combined gravitational and electromagnetic field is cast in the classical Hamilton-Jacobi form and then applied to the special case of a Schwarzschild metric, leading to the well established equation of planetary motion.
Composition Theorems on Dirichlet Series
Authors: Penry M. D.
Published: December 2010

When two uniform functions , are given, each with a finite radius of absolute convergence R1 R2 respectively, and {λn}, {μν} are real positive increasing sequences tending to infinity, a theorem due to Eggleston [1], which is a generalisation of Hurwitz1 s composition theorem, gives information about the position of the singularities of a composition function h(z), which is assumed to be uniform, in terms of the position of the singularities of f(z) and g(z). This result can be extended to Dirichlet series with real exponents by use of the transformation z = es.
A Property of Plane Sets of Constant Width
Authors: Melzak Z.A.
Published: December 2010

It is well known that sets of constant width share several properties with spheres. In this note we consider a simple property of the circle and we show that it is shared by every plane set of constant width. As an application we derive a stronger form of the following theorem of D. Gale, [1]: every plane set of diameter 1 is a union of three sets of diameters not exceeding √3/2, and this constant is best possible. We shall make free use of the more elementary properties of convex sets and of sets of constant width; for these properties and for the terminology see the standard reference [2], or [3].
Extensions of Topological Spaces
Authors: Banaschewski Bernhard
Published: December 2010

The undertaking of constructing spaces which contain a given space as a subspace is by no means new: the extension of the complex number plane to the complex number sphere by the addition of the one point at infinity, the extension of the real line by adjoining the two infinities ∞ and -∞, and the construction of the space of real numbers from that of the rationals by means of Cauchy sequences or Dedekind cuts are 19th Century examples of this very thing. However, only the advent of general topology made it possible to raise the general question of space extensions. It appears that the first study of problems in this area was carried out by Alexandroff and Urysohn in the early twenties [l]. Another mile stone in the history of the subject was the 1929 paper by Tychonoff in which the product theorem for compact spaces is proved and used to identify the completely regular Hausdorff spaces as precisely those spaces which can be imbedded in a compact Hausdorff space [33]. During the same period, work on certain specific extension problems was done by Freudenthal [17] and Zippin [35]. However, the first large body of systematic theory, used for the investigation of a wide range of extension problems, was presented by Stone [31] in 1937. There, one also finds the remark that "one of the interesting and difficult problems of general topology is the study of all extensions of a given space", and it appears that Stone' s own work must have convinced many others of the truth of this observation, for since that time there has been a steady succession of papers in this field. But apart from that, the study of extension spaces clearly has a very particular attraction for some mathematicians.
A Number Theory Problem Concerning Finite Groups and Rings
Authors: Connell Ian G.
Published: December 2010

Let f1(n) denote the number of abelian groups of order n and f2(n) the number of semi-simple rings with n elements. What can be said about the magnitude of fi(n)? We shall prove that one can expect, on the average, about 2.3 groups and 2.5 rings of the kind stated for a given order. First we state without proof the two relevant structure theorems (which are readily available in standard texts).
A Note on Regular Measures
Authors: Zakon Elias
Published: December 2010

As is well known, every Borel measure in a metric space S is regular, provided that S is the union of a sequence of open sets of finite measure. It seems, however, not yet to have been noticed that this theorem can be easily extended to all spaces with Urysohn' s f "F-property", i.e., spaces in which every closed set is a countable intersection of open sets (we call such spaces "F-spaces"). Indeed, various theorems are unnecessarily restricted to metric spaces, while weaker assertions are made about F-spaces. This seems to justify the publication of the following simple proof which extends the theorem stated above to F-spaces.
On Almost Uniform Convergence of Families of Functions
Authors: Zakon Elias
Published: December 2010

In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.
On Extremal Properties of the Derivatives of Polynomials and Rational Functions
Authors: Malik M.A.
Published: December 2010

Let p(z) be a polynomial of degree n, i. e. a finite sum of the form where cν are any given numbers and z=x+iy is a complex variable. To answer a question raised by the chemist Mendelieff, A. Markoff [3] proved the following theorem.
The Categories of Boolean Lattices, Boolean Rings and Boolean
Spaces
Authors: Doctor Hoshang P.
Published: December 2010

In Theorem 4 of [5] Stone stated that the theory of Boolean rings was
"mathematically equivalent" to the theory of Boolean spaces without,
however, properly defining the phrase "mathematically equivalent". It is the
main purpose of this note to establish a precise reformulation of Theorem 4
in [5]. This is accomplished by introducing special classes of maps between
Boolean lattices, Boolean rings and Boolean spaces respectively, and showing
the categories arising in conjunction with these maps to be equivalent in
the sense of Grothendieck [2]. Thus the notion of equivalence of categories
will replace the phrase "mathematically equivalent" in [5]. In addition the
well-known axiomatic characterization of meet and complementation of Boolean
lattices with unit is discussed in analogous terms.
On Equi-Cardinal, Restrictions of a Graph
Authors: Beatty J. C. , Miller R. E.
Published: December 2010

A graph G is an ordered pair (V, E) where V is a set of objects called vertices, and E is a set of unordered pairs of vertices (v, v') in which each such pair can occur at most once in E, and if (v, v') ϵ E then v ≠ v'. The order of G is the cardinality of the set V, and the degree δ(v) of an element v ϵ V is the number of elements of E which contain v. G is said to be regular of degree d if δ(v) = d for each v ϵ V. G is a complete graph if E contains every pair of elements of V. A graph H = (V', E') is a partial graph of G=(V, E) if V' ⊆ v and E' ⊆ E. H is a restriction of G if H is a partial graph of G in which V' = V. Let S = { e1, …, el} be a subset of E such that ej={vj-1, vj} for 1 ≤j ≤l.
A Room Design of Order 10
Authors: Weisner Louis
Published: December 2010

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square of side 2n - 1, so that each of the (2n - 1)2 cells of the array is either empty or contains just two distinct objects; each of the 2n objects occurs just once in each row and in each column; and each (unordered) pair of objects occurs in just one cell.
On the Projection of the Regular Polytope {5, 3, 3} into a Regular Triacontagon
Authors: Chilton B. L.
Published: December 2010

The polytope {5, 3, 3} is one of the six convex regular four-dimensional polytope s, and in some ways is the most complicated of the six, being bounded by 120 dodecahedra. The symbol {p, q, r} to denote a regular figure in four dimensions originated with L. Schláfli: p is the number of edges bounding each face, q is the number of edges of a cell which meet at a vertex of this ceil, and r is the number of ceils that share each edge. The Schláfli symbol can be generalized to denote regular figures in any number of dimensions; in particular, a regular p-gon is symbolized by {p}, and a dodecahedron by {5, 3}.
On Ring Properties of Injective Hulls1)
Authors: Osofsky B. L.
Published: December 2010

Several authors have investigated "rings of quotients" of a given ring R. Johnson showed that if R has zero right singular ideal, then the injective hull of RR may be made into a right self injective,- regular (in the sense of von Neumann) ring (see [7] and [12]). In articles by Utumi [10], Findlay and Lambek [6], and Bourbaki [2], various structures which correspond to sub-modules of the injective hull of R are made into rings in a natural manner, in [8], Lambek points out that in each of these cases the rings constructed are subrings of Utumi' s maximal ring of right quotients, which is the maximal rational extension of R in its injective hull. Lambek also shows that Utumi's ring is canonically isomorphic to the bicommutator of the injective hull of RR if R has 1. It thus appears that a "natural" definition of the injective hull of RR as a ring extending module multiplication by R has been carried out only in the case that the injective hull is a rational extension of R. (See [12], [10], or [6] for various definitions of this concept.)
On a Conjecture of Melzak
Authors: Shephard G. C.
Published: December 2010

Melzak [2] has shown that there exists a convex pseudopolyhedron Q (the convex hull of a convergent sequence of points together with its limit point) in E3 which is s-universal for triangles, that is, all possible triangles occur (up to similarity) as plane sections of Q. He conjectured that no polyhedron P has this property. In this short note we give an elementary proof of this conjecture.
Some Remarks on Ramsay's Theorem
Authors: Erdös P.
Published: December 2010

A special case of a well known theorem of Ramsay [3] states that an infinite graph either contains an infinite complete subgraph or it contains an infinite independent set; in other words there exists an infinite subset of its vertices so that either every two of them are joined by an edge or no two of them are joined by an edge. Thus if we have a graph whose vertices are the integers, and which has no infinite complete sub-graph, it certainly has an infinite independent set. The question can now be asked if there exists an independent set whose vertices n1 < n2 < … do not tend to infinity too fast.
Spiralling in Plane Random Walk
Authors: Wright E. M.
Published: December 2010

A particle is initially at the origin in the (X, Y) plane and each successive step it takes is of unit length and parallel either to the X-axis or to the Y-axis. Its path of n steps is called a spiral if (i) the particle never occupies the same position twice, (ii) any turns the path makes are all counter-clockwise or all clockwise and (iii) for every m > n, the path can be continued to m steps without violating (i) or (ii).
On a Geometric Extremum Problem
Authors: Schaer J. , Meir A.
Published: December 2010

The following problem was brought to our attention by L. Moser: Locate eight points in the closed unit square so that the minimum of the distances between any two of the points should be as large as possible.
On Iterated Limits of Measurable Mappings
Authors: Zakon Elias
Published: December 2010

Egoroff' s theorem [1] was extended by Kvačko [3] to functions with values in a separable metric space; and, as is easily seen, this result applies also to separable pseudometric spaces. In the present note we shall use this theorem to obtain some propositions on iterated limits, which, despite their simplicity, seem not yet to be known in the proposed generality.
Exact and Limiting Probability Distributions of Some Smirnov Type Statistics
Authors: Csörgo Miklós
Published: December 2010

Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variables

and and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.
Equal Sums of Like Powers
Authors: Wright E. M.
Published: December 2010

In what follows small latin letters denote rational integers (whole numbers) and we write

Let us consider the system of k equations

1

that is

These equations are obviously satisfied if the b are a permutation of the a; such a solution we call trivial.
On Sets of Consistent Arcs in a Tournament
Authors: Erdös P. , Moon J. W.
Published: December 2010

A (round-robin) tournament Tn consists n of nodes u1, u2, …, un such that each pair of distinct nodes ui and uj is joined by one of the (oriented) arcs or The arcs in some set S are said to be consistent if it is possible to relabel the nodes of the tournament in such a way that if the arc is in S then i>j. (This is easily seen to be equivalent to requiring that the tournament contains no oriented cycles composed entirely of arcs of S.) Sets of consistent arcs are of interest, for example, when the tournament represents the outcome of a paired-comparison experiment [1]. The object in this note is to obtain bounds for f(n), the greatest integer k such that every tournament Tn contains a set of k consistent arcs.
On the Ring of Quotients of a Noetherian Ring
Authors: Lambek J.
Published: December 2010

This paper is largely an expository account of known facts, but it contains at least one result believed to be new, Proposition 6.

Our main technique is the method of lifting idempotents developed in Part I. This has been treated in the literature, but not quite in the generality required here. It turns out that much of classical artinian ring theory can be done for the semi-perfect rings introduced by Bass, as will have been noticed by many other people.
On the Injective and Projective Limit of Complexes
Authors: Tewari Krishna
Published: December 2010

Let R be a commutative ring with unit. Let (Aα, φ βα) (α ≤ β) (resp. (Bα, ψαβ) (α ≤ β)) be an infective (resp. projective) system of R-algebras indexed by a directed set I; let ((Xα, dα), fβα) (α ≤ β) (resp. ((Yα, δα), gαβ) (α≤β)) be an injective (resp. projective) system of complexes, indexed by the same set I, such that for each αϵI, (Xα, dα) (resp. (Yα, δα)) is a complex over Aα (resp. over Bα). The purpose of this paper is to show that the covariant functor from the category of all such injective systems of complexes and complex homomorphisms over the R-algebra Aα is such that it associates with an injective system ((Uα, dα), hβα) of universal complexes a universal complex over Aα whereas the same is not true of the covariant functor the category of all such projective systems of complexes and their maps.
On a Combinatorial Problem in Number Theory
Authors: Lindström Bernt
Published: December 2010

Given an integer k ≤ 2 and a finite set M of rational integers. Let vi (i = 1, 2, …, n) be m-dimensional (column-)vectors with all components from M and such that the kn sums

1.1

are all different. Then we shall say that {v1, v2, …, vn} is a detecting set of vectors.
On Vector Lattice-Valued Measures
Authors: Hrycay R.
Published: December 2010

E. Hewitt [1] used the Daniell approach to define a real-valued measure function on a σ-algebra of the real line. He began by defining an arbitrary non-negative linear functional I on L∞ ∞(R), (the space of all complex-valued continuous functions on the real line R which vanish off some compact subset of R).
Axioms for Semi-Lattices
Authors: Potts D. H.
Published: December 2010

A semi-lattice (Birkhoff, Lattice Theory, p. 18, Ex. 1) is an algebra  with a single binary operation satisfying: (1) x = xx, (2) xy = yx, and (3) (xy)z = x(yz). In this note we show that the three identities may be reduced to two but cannot be reduced to one.

It is easy to see that (2), (3) imply (4) (uv)((wx)(yz)) = ((vu)(xw))(zy). Setting w = y = u and x = z = v in (4) and using (1) we get uv = vu. Setting v = u, x = w, and z =y in (4) and using (1) we get u(wy) = (uw)y. And so (1) and (4) imply (2) and (3).
Problems Connected with Convexity
Authors: Melzak Z. A.
Published: December 2010

This article consists of a list of problems connected with the theory of convex sets in the Euclidean space En, although in some cases the connection is indirect and/or inessential. The assumptions may often be relaxed or otherwise changed, thus giving rise to a wide variety of questions.
Construction of Special Edge-Chromatic Graphs
Authors: Kalbfleisch J. G.
Published: December 2010

The configuration formed by N points in general position in space, together with thelines joining them in pairs will be called an N-clique. The N-clique is coloured by assigning to each edge exactly one colour from a set of t possible colours. A theorem due to Ramsey [4] ensures the existence of a least integer M(q1, q2, …, qt) such that if N ≥ M, any such colouring of the N-clique must contain either a q1-clique entirely of colour 1, or a q2-clique of colour 2, …, or a qt-clique of colour t. Another proof of Ramsey's theorem is given by Ryser [5].
Derivable Nets1)
Authors: Ostrom T. G.
Published: December 2010

Up to a duality, the known finite projective planes which are not translation planes all are equivalent to affine planes which contain the type of structure defined below to be a "derivable net". (Insofar as the known finite planes are concerned, this means that the intimate connection between projective geometry and linear algebra still holds for non-Desarguesian planes.)
Remarks on the Uniqueness Theorem of Solutions of the Darboux Problem
Authors: Wong James S. W.
Published: December 2010

Recently B. Palczewski and W. Pawelski [l] have given some sufficient conditions of uniqueness for the solutions of the Darboux problem for equations of the form:

1

The criteria given there for equations of the form (1) are natural generalizations of the criteria given by Krasnosielski and Krein [2] in the corresponding ordinary differential case. The purpose of the present note is to give a further generalization of the above result and two other uniqueness conditions for the solutions of the Darboux problem of the same form.
On the Dual of L1
Authors: Ellis H. W.
Published: December 2010

If (X, S,μ) is an arbitrary complemented measure space and X is σ-finite then (L1)* = L∞ or, more precisely, (L1)* is isometric and isomorphic to L∞ by the correspondence

It is well known that there exist non σ-finite spaces with (L1)* ≥ L∞.
Groups in which Raising to a Power is an Automorphism
Authors: Trotter H. F.
Published: December 2010

For any group G and integer n, let Pn:G→G be the function defined by Pn(g) = gn for all g ϵ G. If G is abelian then Pn is a homomorphism for all n. Conversely, it is well known (and easy to show) that if P2 or P-1 is a homomorphism then G is abelian. As the groups Gn described below show, for every n other than 2 and -1 there exist non-abelian groups for which Pn is a homomorphism.
A Remark on Convex Polytopes
Authors: Glass A. S.
Published: December 2010

In this note we wish to present an alternative proof for the following well-known theorem [1, Theorem 16]: every convex polytope X in Euclidean n-dimensional space Rn is the intersection of a finite family of closed half-spaces. It will be supposed that the converse of this theorem has been verified by conventional arguments, namely: every bounded intersection of a finite family of closed half-spaces in Rn is a convex polytope [cf. 1, Theorem 15].
Free and Injective Lie Modules*
Authors: Kleiner Israel
Published: December 2010

We study free and injective Lie modules by investigating the relationship between Lie modules and (associative) modules. An important role is played by the universal enveloping ring of a Lie ring [4]. If L is an arbitrary Lie ring and W(L) its universal enveloping ring, we show that the category of Lie L-modules and the category of associative W(L)-module s are isomorphic (section 2). In section 3 we study free Lie modules and show how they may be obtained from free associative modules. A Lie module is free if and only if it is a direct sum of copies of the free Lie module on one generator.
Global Stability Determined by Local Properties and the First Variation
Authors: Datko R.
Published: December 2010

In this note we consider a system of autonomous differential equations

1.1

where f: En → En is a continuously differentials Le mapping for n ≥ 2. We shall assume that f(0) = 0 and that the origin is locally asymptotically stable.
Some Characterizations of the One-Parameter Family of Probability Distributions
Authors: Mathai A. M.
Published: December 2010

We consider one-parameter families of distributions which are of the general exponential type, the general series distribution type, or which can be transformed into the exponential type by a one-to-one transformation. In this paper we establish theorems to the effect that such distributions may be characterized by a simple differential equation involving the mean function. It is illustrated that almost all the classical one-parameter families of distributions are characterized by these theorems. Multivariate generalizations are given, and it is also noticed that the functional form of the normalizing factor determines the specific distribution in each general family.
Arithmetic Functions Satisfying a Congruence Property
Authors: Subbarao M.V.
Published: December 2010

This note proves (in the theorem below) a conjecture made by the author last year through the pages of the Departmental Problem Book. This arose in connection with some other investigations of arithmetic functions.
On the Injective Hulls of Cyclic Modules Over Dedekind Domains
Authors: Banaschewski B.
Published: December 2010

As is well known, any module M over a ring possesses injective hulls, i. e., injective essential extensions [3], unique up to isomorphisms which map M identically, but the various proofs for this all require some transfinite arguments and hence provide little indication as to how these hulls may actually be constructed for a given module.
Some Simple Properties of Simple Nil Rings
Authors: McWorter W. A.
Published: December 2010

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.
On Strict Monotonicity of Continuous Solutions of Certain Types of Functional Equations
Authors: Aczel J.
Published: December 2010

It is a commonplace that F is continuous on the cartesian square of the range of f if f is continuous and satisfies

1

say, for all real x, y (cf. e.g. [2]). A.D. Wallace has kindly called my attention to the fact, that this is trivial only if f is (constant or) strictly monotonic and asked for a simple proof of the strict monotonicity of f. The following could serve as such: if on an interval f is continuous, nonconstant and satisfies (1), then f is strictly monotonic there.
The Densest Packing of Five Spheres in a Cube
Authors: Schaer J.
Published: December 2010

The purpose of this paper is to locate five points Pi ( l ≤ i ≤ 5 ) in a closed unit cube C such that is as large as possible, where d(Pi, Pj) denotes the distance i j between Pi and Pj. We prove that this minimum distance cannot exceed (=m, say), and if it is equal to m, then the corresponding configuration is congruent to the set of points shown in fig. 1, namely P1 = A1 (0,0,0), P2 = A8 (1, 1, 1), P3 = B1 (0,1/2,1), P4 = B3 (1/2,1,0) and P5 = B5 (1, 0, 1/2).
On Arithmetic Convolutions
Authors: Davison T.M. K.
Published: December 2010

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given by

where the summation condition means sum over all positive integers d which divide n.
Function Space Topologies for Connectivity and Semi-Connectivity Functions
Authors: Naimpally Somashekhar Amrith
Published: December 2010

Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.
On Groups with all Composition Factors Isomorphic
Authors: Bercov R.
Published: December 2010

By the celebrated theorem of Jordan [3] and Hölder [2], there is associated with each finite group G a family of distinct simple groups Hi. such that every composition series of G has ni factor groups isomorphic to Hi and no others. We denote the collection of pairs (Hi, ni) by CF(G). Conversely, given k pairs (Hi, ni), we may construct by an easy direct product procedure a group G with CF(G) = { (Hi, ni) | i =1,…, k}. The composition factors, of course, do not in general determine the group.
Generating Functions for a Class of Arithmetic Functions
Authors: Gioia A. A. , Subbarao M.V.
Published: December 2010

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity

1.1
A Brouwer Type Coincidence Theorem
Authors: Schirmer Helga
Published: December 2010

Brouwer's celebrated fixed point theorem states that every map of a closed n-cell into itself has a fixed point.

A similar theorem is here proved for coincidences between a pair of maps (f, g): In→In, where I denotes a closed n-cell (i.e. a homeomorph of the n-ball) and a coincidence is a point x∊In for which f(x)=g(x). That two arbitrary maps (f, g): In→In need not have a coincidence is shown by the pair f:In→y0, g:In→y1, where y0, y1∊In and y0≠y1. More generally, one can immediately construct a map g so that (f, g) is coincidence free if f is not surjective.
A New Inversion and Representation Theory for the Laplace Transform
Authors: Heinig H. P.
Published: December 2010

If

1.1

and

1.2

a ≥ 0, k=1, 2, 3, …; whereis the Laguerre polynomial of order v, defined by

then we shall show that under certain conditions

1.3

Following the inversion theory, two representation theorems are given. The proofs of these theorems follow easily along the lines of Widder [4, Ch. VII] and are therefore omitted.
The Solution of Non-Homogeneous Systems of Differential Equations by Undetermined Coefficients
Authors: Brauer Fred
Published: December 2010

The solution of a linear non-homogeneous differential equation whose non-homogeneous term is of the form tkeαt can be obtained by what is usually called the method of undetermined coefficients. The application of this method may be justified in several different ways (see for example [1, pp. 114–117], [2, pp. 94–99], [3]).

We shall consider the analogous problem for a system of differential equations. It turns out that we can solve this problem using only elementary techniques of linear algebra. The solution has essentially the same form as in the case of a single equation, but may contain terms which would not be expected and may lack terms which would be expected in a straightforward extension of the theory to systems. Our method of obtaining the solution is constructive, in the sense that while our results give only the form of the solution, the solution itself may be found by substitution of this form into the system of differential equations.
A Counterexample to a Conjecture of D.F. Sanderson
Authors: Kleiner Israel
Published: December 2010

In [2, p. 511] Sanderson has shown that if every Large left ideal of a ring R with identity contains a regular element, and if the regular elements in R satisfy Ore's condition, then the complete (Utumi's) ring of quotients coincides with the classical ring of quotients. He conjectured that the above conditions are also necessary. The following is a counter example.
Limites Projectives de Chaines de Demi-Groupes Nilpotents
Authors: Gagnon F. P.
Published: December 2010

Dans cette note nous étudions les demi-groupes avec zero qui vérifient la condition T:

1

Le théorème 1 montre que ces demi-groupes sont isomorphes à la limite projective d'une chaine de demi-groupes nilpotents. Le théorème 2 donne une façon de réaliser tout demi-groupe nilpotent.
Hamiltonian Circuits and Paths on the n-Cube
Authors: Abbott H. L.
Published: December 2010

For positive integral n let Cn denote the n-dimensional unit cube with vertices (δ1, δ2,…, δn) where δi = 0 or 1 for i=1, 2,…, n. Call two vertices of Cn adjacent if the distance between them is 1.
On the Maximum Order of the Group of a Tournament
Authors: Goldberg Myron , Moon J. W.
Published: December 2010

A (round-robin) tournament Tn consists of n nodes p1, p2, …, Pn such that each pair of distinct nodes pi and pj is joined by one of the oriented arcs or . If the arc is in Tn, then we say that pi. dominates pj. The set of all dominance-preserving permutations a of the nodes T form a group, the automorphism group G(T ) of Tn. It is known (see [1]) that there exist tournaments Tn whose group n G(Tn) is abstractly isomorphic to a given group H if and only if the order g(H) of H is odd.
A Congruence for a Class of Arithmetic Functions
Authors: Subbarao M.V.
Published: December 2010

There is considerable literature concerning the century old result that for arbitrary positive integers a and m,

1.1

where μ(m) is the usual Mobius function. For earlier work on this we refer to L.E. Dickson [4, pp. 84–86] and L. Carlitz [1,2]. Another reference not noted by the above authors is R. Vaidyanathaswamy [6], who noted that the left member of (1.1) represents the number of special fixed points of the m th power of a rational transformation of the n th degree.
On the Exactness of the Eckmann-Hilton Homotopy Sequence
Authors: Pears A. R.
Published: December 2010

The theorem that the homotopy sequence is exact splits into six statements. Scherk ([4]) obviates the use of homotopy extension in the proof of one of these statements. The purpose of this note is to show that the method can be adapted to give a direct proof of the corresponding statement in the theorem that the Eckmann-Hilton homotopy sequence ([l]) is exact. The note is based on Eckmann' s exposition ([2]). We are concerned with the proof of b2, pp. 34–35.
An Integral for Cesàro Summable Series
Authors: Cross George
Published: December 2010

The pk+2 - integral of James [2] is strong enough to integrate a trigonometric series of the form

1.1

which is summable (C, k) in [0,2π], provided an extra condition holds involving the conjugate series

1.2

Considering series with coefficients o(n), Taylor [5] constructed an integral (the AP-integral) which successfully integrates series of the form (1. 1) which are Abel summable provided an extra condition holds involving the Abel means of the conjugate series (1.2).
A Correspondence Principle for Momentum Operators
Authors: Marsden J. E.
Published: December 2010

The purpose of this note is to give a precise formulation of the correspondence principle between classical and quantum mechanics, for the case of momentum operators. (See e. g. I. Segal, Journ. Math. Phys. (1960), 475).
Cycles of Each Length in Regular Tournaments
Authors: Alspach Brian
Published: December 2010

It is known that a strong tournament of order n contains a cycle of each length k, k=3,…, n, ([l], Thm. 7). Moon [2] observed that each vertex in a strong tournament of order n is contained in a cycle of each length k, k = 3,…, n. In this paper we obtain a similar result for each arc of a regular tournament, that is, a tournament in which all vertices have the same score.
A Packing Problem
Authors: Stanton R.G. , Williams H. C. , Zarnke C. R.
Published: December 2010

Consider three externally tangent circles A, B, C, with radii a, b, c, respectively. Let D be the circle which is externally tangent to all three and enclosed by them. Pack circles so that Z1 = D and Zi is externally tangent to A, B, Zi-1.
Representation Theories for the Laplace Transform
Authors: Heinig H. P.
Published: December 2010

The Widder-Post real inversion operator [4[ is defined by

1.1

k = 1, 2,…. Utilizing this inversion operator one can obtain the following representation theorem (see e.g. [4[ Chapter VII, Theorem 15a).
A Characterization of Real almost Continuous Functions
Authors: Husain T. , Dwivedi T. D.
Published: December 2010

The main purpose of this note is to give a characterization of almost continuous functions similar to that of approximately continuous functions ([3], §235, p. 312). The characterization of approximately continuous real functions has been used to show that each approximately continuous function is almost continuous ([4], §6). Thus combining these results we see that the classes of almost continuous, approximately continuous and continuous functions form a descending chain.
Which Graphs have only Self-Converse Orientations?
Authors: Harary Frank , Palmer Edgar , Smith Cedric
Published: December 2010

An orientation of a graph G is an assignment of a unique direction to each line of G. The result is called an oriented graph. Two orientations of a graph are regarded as equivalent if the resulting oriented graphs are isomorphic as directed graphs. For example, the graph C3 consisting of a cycle of length 3 (a triangle) shown in Figure 1(a), has exactly two orientations D1 and D2; see Figure 1(b) and (c).
Measurable Cover Functions
Authors: Eames W. , May L. E.
Published: December 2010

Let μ∗ be an outer measure on (X, S) with σ- algebra S and let
μ* be the inner measure induced by μ∗. A set M is a
measurable cover of a set A ⊆ X if A ⊆ M, M is measurable, and μ∗
(M-A) = 0. We assume that every subset of X has a measurable cover; this
holds, for example, if μ∗ is the outer measure induced by a
measure which is σ- finite on X [2, theorem C, p. 50].
Extension of a Bounded Vector Measure with Values in a Reflexive
Banach Space
Authors: Fox Geoffrey
Published: December 2010

A vector measure (countable additive set function with values in a Banach
space) on a field may be extended to a vector measure on the generated σ-
field, under certain hypotheses. For example, the extension is established
for the bounded variation case [2, 5, 8], and there are more general
conditions under which the extension exists [ 1 ]. The above results have as
hypotheses fairly strong boundedness conditions on the n o rm of the measure
to be extended. In this paper we prove an extension theorem of the same type
with a restriction on the range, supposing further that the measure is
merely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5
of [3]) but it is conceivable that a vector measure on a field could be
unbounded.
On the Set of Zero Divisors of a Topological
Ring
Authors: Koh Kwangil
Published: December 2010

Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Ra
are closed subsets of R. We will prove that if the set of non - trivial
right (left) zero divisors of R is a non-empty set and the set of all right
(left) zero divisors of R is a compact subset of R, then R is a compact
ring. This theorem has an interesting corollary. Namely, if R is a discrete
ring with a finite number of non - trivial left or right zero divisors then
R is a finite ring (Refer [1]).
Quelques Caractérisations Du Radical D′un Anneau
Authors: Thierrin G.
Published: December 2010

Le radical R(A) d′un anneau A peut être caract érisé de plusieurs manières (Jacobson [1]). Rappelons seulement ici deux des caractérisations les plus importantes:

(1) R(A) est l′intersection de tous les idéaux à droite (à gauche) modulaires maximaux de A.

(2) R(A) est un idéal quasi - régulier contenant tout idéal à droite (á gauche) quasi - régulier.

L′objet de cet article est de donner quelques caractérisations du radical d′un anneau.
A Note on Coverings of En by Convex Sets
Authors: Chalk J.H.H.
Published: December 2010

Let (x1, x2, …, xn) denote the coordinates of a point of Euclidean n-space En. Let be a set of n+1 points of En with the property that

form a linearly independent set and define a lattice Λ of points

by allowing u1, …, un to take all integer values.
On a Theorem in the Generalised Fourier Transform
Authors: Mitra S. C.
Published: December 2010

The function was defined by G.N. Watson, [9, (i)]in 1931 by the integral relation∗

+ another term with μ and v interchanged;
Representation of Functions as Weierstrass- Transforms
Authors: Heinig H.P.
Published: December 2010

The Weierstrass - respectively Weierstrass - Stieltjes transform of a function F(t) or μ(t) is defined by

1.1

and

1.2

for all x for which these integrals converge. In what follows we shall always assume that F(t) is Lebesgue integrable in every finite interval and that μ(t) is a function of bounded variation.
Necessity of Alteration
Authors: Dunham C.
Published: December 2010

In this note we point out a not easily fillable gap in the proof of a major result in Chebyshev approximations on an interval [α, β]. Tornheim developed a theory for approximation by families unisolvent of degree n (n-parameter families), which was extended by Rice to families unisolvent of variable degree.
A New Construction of the Injective Hull
Authors: Fleischer Isidore
Published: December 2010

The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.
On the Non-Cutpoint Existence Theorem
Authors: Ward L. E.
Published: December 2010

The theorem of the title asserts that every non-degenerate continuum (that is, every compact connected Hausdorff space containing more than one point) contains at least two non-cutpoints. This is a fundamental result in set - theoretic topology and several standard proofs, each varying from the others to some extent, have been published. (See, for example, [1], [4] and [5]). The author has presented a less standard proof in [3] where the non-cutpoint existence theorem was obtained as a corollary to a result on partially ordered spaces. In this note a refinement of that argument is offered which seems to the author to be the simplest proof extant. To facilitate its exposition, the notion of a weak partially ordered space is introduced and the cutpoint partial order of connected spaces is reviewed.
Some Distribution Problems of Order Statistics From Exponential and Power Function Distributions1
Authors: Kabe D. G.
Published: December 2010

This paper gives alternative straightforward and simpler proofs of some of the results of Laurent's [10], and Likes' [11], [13]. The derivation of the results is simplified by using the theory of Dirichlet's multiple integral and the transformation used to derive this multiple integral. Some applications of Dirichlet's transformation to order statistic theory from gamma, and normal populations, have been already given by Kabe [7].
Representation of m as
Authors: Entringer R. C.
Published: December 2010

J.H. van Lint has recently shown [1] that if A(n, m) denotes the n number of representations of m in the form , where εk = 0 or 1 for -n ≤ k ≤ n then

(1)

Using this result, the fact that A(n, m) is a non-increasing function of |m|, and a simple recurrence relation for A(n, m) we derive the following extension of (1):

(2)

where [0 (n)] is any integral valued function m(n) = 0(n).
A Genuine Topology for the Field of Mikusiński Operators
Authors: Struble Raimond A.
Published: December 2010

Let C denote the complex algebra of continuous functions of a non-negative real variable under addition, scalar multiplication and convolution. C has no divisors of zero and its quotient field F is called the field of Mikusiński operators [1]. It is well known that Mikusiński has defined a sequential convergence in F which is not topological [2]. Using a recent result due to T.K. Boehem [3] we shall provide F with a sequential convergence which is topological.
Notes on Splitting Extensions of Groups
Authors: Tang C.Y.
Published: December 2010

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.
Duo Rings: Some Applications to Commutativity Theorems
Authors: Bell Howard E.
Published: December 2010

Proofs of commutativity theorems for general rings usually employ the Jacobson structure theory; however, alternative approaches to the "xn = x theorem" [ l, 2] suggest that the power of the Jacobson theory is not required. In this note we prove two commutativity theorems of Herstein in an elementary way. Both proofs involve establishing first that the rings under consideration are duo-rings - rings in which every one-sided ideal is two-sided.
Rings of Quotients of Rings of Derivations
Authors: Kleiner Israel
Published: December 2010

The concept of a rational extension of a Lie module is defined as in the associative case [1, pp. 81 and 79]. It then follows from [3, Theorem 2.3] that any Lie module possesses a maximal rational extension (a rational completion), unique up to isomorphism. If now L and K are Lie rings with L⊆ K, we call K a (Lie) ring of quotients of L if K, considered as a Lie module over L, is a rational extension of the Lie module LL. Although we do not know if for every Lie ring L its rational completion can be given a Lie ring structure extending that of L (as is the case for associative rings), this is so, in any case, for abelian Lie rings (Propositions 2 and 4).
A Combinatorial Interpretation of Ramanujan's Continued Fraction
Authors: Szekeres G.
Published: December 2010

The purpose of the present note is to give a combinatorial interpretation of the coefficients of expansion of the Ramanujan continued fraction ([1], p. 295)

The result is expressed by formula (12) below.

The enumeration of distinct score vectors of a tournament leads to the following problem: (Erdős and Moser, see Moon [2], p. 68). Given n ≥ 1, k ≥ 0, determine the number of distinct sequences of positive integers
Sub-Manifolds of a Locally Product Riemannian Manifold
Authors: Mishra R. S.
Published: December 2010

In this paper, we have obtained the conditions that a sub-manifold of a locally product Riemannian manifold may also be locally product. Generalizations of the equations of Gauss, Mainardi-Codazzi and Ricci have been obtained for the locally product sub-manifold of the locally product Riemannian manifold.
A Pappus Type Theorem in the Affine Group
Authors: Paré R.
Published: December 2010

In [3] H. Schwerdtfeger embedded the one-dimensional affine group over the real numbers in the projective plane. The relationship between group-theoretical properties and geometrical concepts was studied.

In this paper the methods of [3] are used to prove Pappus' theorem. In the final section we give a similar theorem for (4n+2)-gons.

This paper is a generalization of part of my master's thesis, written under the direction of Professor H. Schwerdtfeger.
Elements of Packing and Covering
Authors: Oler N.
Published: December 2010

The term 'covering' is known to any student who has seen the Heine-Borel theorem and he soon learns that it denotes a very basic and widely used concept. Quite generally, a family {Xα: α ∈ A} of a subsets of X is a covering of the subset Y of X if .

The concept of packing is perhaps no less frequently encountered although the term has only a rather specialized use. In general, a packing is any family of subsets {Xα: α ∈ A} of a set X which a re pairwise disjoint. To make this definition more similar to that of covering, we might define {Xα} to be a packing of the subset Y of X if Xα ∩ Xβ ∩ Y = ϕ for α ≠ β. This is intended to suggest only a P that there is a certain parallel between the ideas of packing and covering but not a duality in any technical sense.
On Maitland's Generalised Bessel Function
Authors: Srivastava T.N. , Singh Y.P.
Published: December 2010

Maitland's generalised Bessel function [4] is defined by the equation

where u is real and positive and v is any number real or complex. If u = 1, then (1.1) reduces to the form