bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some Remarks on Limits in Categories
Authors: Maranda J. M. 
Published: December 2010

The object of this paper is to give simple criteria for the existence of
direct limits in categories and for the permuting of a functor with direct
limits.The notion of direct limit of a diagram that we shall use here is
essentially that of Kan (4), which is more general than the usual notion of
direct limit of a directed diagram.Our treatment is based on the fact (Lemma 2) that the usual process for
constructing the direct limit of a diagram of modules, which consists in
taking a direct sum of the modules in the diagram and then considering a
certain homomorphic image of this direct sum (3, p. 220), is essentially,
once certain notions have been properly generalized, the only process for
constructing the direct limit of any diagram in any category.
On Algebraic Number Fields with Unique
Factorization
Authors: Connell Ian G. 
Published: December 2010

In this note we obtain some simple criteria which show that in certain
algebraic number fields factorization of elements is not unique. All the
arguments depend only on the most elementary ideas (except in §6) so that
probably many of the results are not new. However the proofs are short and
direct and therefore should be of some interest.We now list some basic facts, whose proofs can be found in any of the books
listed in the bibliography, at the same time fixing our notation.
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.1Since 0 ≤ ai1 + ai2 + … + aik ≤ kn, we have2
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.)
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 variablesand 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.
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 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 sums1.1are all different. Then we shall say that {v1, v2, …, vn} is a detecting set of vectors.
Axioms for Semi-Lattices
Authors: Potts D. H. 
Published: December 2010

A semi-lattice (Birkhoff, Lattice Theory, p. 18, Ex. 1) is an algebra <A,.> 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).
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:1The 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 correspondenceIt is well known that there exist non σ-finite spaces with (L1)* ≥ L∞.
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 satisfies1say, 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.
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 identity1.1
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

If1.1and1.2a ≥ 0, k=1, 2, 3, …; whereis the Laguerre polynomial of order v, defined bythen we shall show that under certain conditions1.3Following 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.
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:1Le 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.
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 form1.1which is summable (C, k) in [0,2π], provided an extra condition holds involving the conjugate series1.2Considering 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).
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 thatform a linearly independent set and define a lattice Λ of pointsby allowing u1, …, un to take all integer values.
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 by1.1and1.2for 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.
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 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
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.
A Commutator Formula for a Pair of Subgroups and A Theorem of Blackburn
Authors: Lam Tsit-Yuen 
Published: December 2010

Let be the lower central series of a group G, where K2(G) = [G, G] and inductively Kn+1(G) = [Kn (G), G]. A theorem of Blackburn ([1], Hilfssatz) states thatTheorem 1. The exponent of Kn+1 (G)/Kn+2 (G) divides the exponent of Kn (G)/Kn+1 (G).
A Generalization of the Regular Maps of Type
{4, 4}b, c and {3, 6}b, c
Authors: Garbe Dietmar 
Published: December 2010

In [1], Coxeter gave a complete enumeration of the regular maps on a torus. The maps consist of two families of type {4, 4}b, c and {3, 6}b, c (and their duals). b and c are non-negative integers, which determine the maps uniquely. The maps are irreflexible if and only if bc(b - c) ≠ 0.On surfaces of genus h > 1, irreflexible regular maps are rather exceptional. The simplest surface of negative characteristic which admits irreflexible regular maps is the orientable surface of genus 7. This was shown by the author [4, Theorem 3. 1 ]. The corresponding map was discovered by J. R. Edmonds [2, p. 388].
Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)
Authors: Williams Kenneth S. 
Published: December 2010

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.
Separation Axioms and Direct Limits
Authors: Herrlich Horst 
Published: December 2010

A topological space X is called a direct limit of a family (Xα) of subspaces of X if and only if(1)(2)If X is a direct limit of an increasing sequence (Xn) of closed subspaces then it is well known and easy to prove that X is a T1-space resp. a T4-space provided all Xn are T1-spaces resp. T4-spaces.
On Semi-Perfect Group Rings
Authors: Burgess W.D. 
Published: December 2010

In what follows the notation and terminology of [7] are used and all rings are assumed to have a unity element.The purpose of this note is to give some partial answers to the question: under which conditions on a ring A and a group G is the group ring AG semi-perfect?For the convenience of the reader a few definitions and results will be reviewed. A ring R is called semi-perfect if R/RadR (Jacobson radical) is completely reducible and idempotents can be lifted modulo RadR (i.e., if x is an idempotent of R/RadR there is an idempotent e of R so that e + RadR = x).
On the Generalized Hankel and K Transformations
Authors: Koh E. L. 
Published: December 2010

The K transformation (also called the Meijer transformation) has been extended by Zemanian [1; 2] to a class of generalized functions, For , he defined the K transform of f by(1)In [2, Section 6.6] the following inversion theorem for the K transform of f is proven:(2)in the sense of weak convergence in D'(I). Here, σ is any fixed real number greater than σf, μ is zero or a complex number with positive real part and D'f(I) is the space of Schwartz distributions on I = (0, ∞).
Convergence of Solutions of Third Order Differential Equations*
Authors: Swick K. E. 
Published: December 2010

Consider a system of differential equations . Solutions of this system are said to be convergent if, given any pair of solutions x(t), y(t), x(t) - y(t) → 0 as t → ∞. In this case the system is also said to be extremely stable.In [6] a technique was developed which yielded the convergence of solutions of the forced Lienard equation. Here a similar technique i s applied to forced third order equations. A critical step in [6] was to show that a certain matrix was negative definite. This could be done directly in [6] since the matrix was only 2 × 2. With third and higher order equations, direct use of necessary and sufficient conditions is not feasible since the computations become unwieldy.
Spreads which are Not Dual Spreads
Authors: Bruen A. , Fisher J. C. 
Published: December 2010

In this note we show the existence of a spread which is not a dual spread, thus answering a question in [1]. We also obtain some related results on spreads and partial spreads.Let ∑ = PG(2t-l, F) be a projective space of odd dimension (2t-l, ≥2) over the field F. In accordance with [1] we make the following definitions. A partial spread S of ∑ is a collection of (t-l)-dimensional projective subspaces of ∑ which are pairwise disjoint (skew).
On A Combinatorial Problem of Erdös
Authors: Abbott H. L. , Hanson D. 
Published: December 2010

A family of sets is said to possess property if there exists a set such that and for every We consider the following question raised by P. Erdös |1|: let n and N be positive integers, n ≥ 2 and N ≥ 2n - 1 and let S be a set of N elements; what is the least integer (provided such an integer exists), for which there exists a family of subsets of S satisfying(a)|F| = n for each (b)(c) does not have property (d)if and then has property
An Analogue of the Wave Equation and Certain Related Functional Equations
Authors: Baker John A. 
Published: December 2010

Consider the functional equation(1)assumed valid for all real x, y and h. Notice that (1) can be written(2)a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].
On the Non-Existence of Conjugate Points
Authors: Butler G. J. , Erbe L. H. , Mathsen R. M. 
Published: December 2010

In this paper we consider the types of pairs of multiple zeros which a solution to the differential equationcan possess on an interval I of the real line. The results obtained generalize those in [2] and (for n = 3) in [3].I. Let f satisfy the condition1.1for all t ∊ I, u0 ≠ 0, and all u1, … un-1.
Sequences of Contractions in a Generalized Metric Space
Authors: Norris C. W. 
Published: December 2010

The main aim of this paper is to study the convergence of a sequence of contractions in a generalized metric space. More specifically, we investigate the following question:"If a sequence of contractions {fr} with fixed points ur (r = 1,2,...) converges to a mapping f with a fixed point u, under what conditions will the sequence ur converge to u?"A partial answer to the above question has been given in metric spaces by Bonsall [1]. This result has since been improved by Russell and Singh [6]. Further results will now be given in a generalized metric space.
An Application of Ramsey's Theorem
Authors: Cockayne E. J. 
Published: December 2010

By an r-graph, we mean a finite set V of elements called vertices and a collection of some of the r-subsets of V called edges with the property that each vertex is incident with at least one edge. An A-chromatic r-graph is an r-graph all of whose edges are coloured A.Theorem. Let G1, …, Gt denote r-graphs. There exists a nonempty class of r-graphs such that for each if the edges of G are painted arbitrarily in t colours A1, …, At, then for at least one i in {1, …, t}, G has an Ai-chromatic r-subgraph which is isomorphic to Gi.
The Associative Part of a Convergence Domain is Invariant
Authors: Sember John J. 
Published: December 2010

Of special interest in summability theory are those conservative matrices possessing the "mean-value property". If cA={x: Ax ∊ c} denotes the convergence domain of a conservative matrix A, then A has the mean-value property in case, for each x in cA, there exists M = M(A, x) > 0 such that1This property has been considered by many writers and has been shown, among other things, to be equivalent to the requirement that the matrix be associative, i.e., for each x in cA2
Maximal Areas of Reuleaux Polygons
Authors: Sallee G. T. 
Published: December 2010

In this paper we provide new proofs of some interesting results of Firey [2]
on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon
is a plane convex set of constant width whose boundary consists of a finite
(odd) number of circular arcs. Equivalently, it is the intersection of a
finite number of suitably chosen congruent discs. For more details, see [1,
p. 128].If a Reuleaux polygon has n sides (arcs) of positive length
(where n is odd and ≥ 3), we will refer to it as a Reuleaux
n-gon, or sometimes just as an n-gon.
If all of the sides are equal, it is termed a regular
n-gon.
On Singular Normal Linear Integral Equations
Authors: Costley Charles G. 
Published: December 2010

In this work we consider the equation1where K(x, y) is singular in the sense
that it does not properly belong to L2 and f(x) is an arbitrary L2 function.A Lebesgue measurable function K(x, y) of
two variables, having real values on [0.1] × [0.1] is called a singular
normal kernel of(i)There exists approximating kernels Km(x,
y) satisfying(ii)(iii)(iv)
Boundedness in a Quasi-Uniform Space
Authors: Murdeshwar M. G. , Theckedath K. K. 
Published: December 2010

Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) [1]. Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics [4].In spite of this importance, there have been only isolated attempts at developing the concept. Alexander [1] and Hu [7] tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.
Magic Valuations of Finite Graphs
Authors: Kotzig Anton , Rosa Alexander 
Published: December 2010

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.
Ensembles de Wiener-Hopf II
Authors: Letac Gérard 
Published: December 2010

Un ensemble de Wiener-Hopf est une partie S de la droite réelle R tel que(i)S+ et S- sont non-vides(ii)S+ + S- ⊂ Soù S- ∪ S0 ∪ S+ est la décomposition de S en parties négative, nulle, et positive.On remarque que ces ensembles présentent une parenté avec les semi-groupes de R. Ainsi, dans [1], est-il démontré que si S n'est pas concentré sur un multiple des entiers, S est dense dans un intervalle.