bluuebunny/crossref_metadata_2025_split huggingface dataset:
Congruence Relationships for Integral Recurrences
Authors: Mendelsohn N. S. 
Published: December 2010

A sequence {un}, n=0, 1, 2, 3,… is said to be an integral recurrence of order r if the terms satisfy the equationfor n=r+1, r+2,…, and a1, a2,…, ar are integers, ar≠0. In this case we will say that {un} satisfies the relation [a1, a2,…, ar]. The sequence {un} is uniquely determined when u1, u2,…, ur are given specified values. If u1, u2,…, ur are integers all the terms of {un} are integers. The generating function f(t)=u1t + u2t2+… takes on the form where Q(t) depends on the values of u1, u2,…, ur and R(t)=tr-a1tr-1-a2tr-2-…-ar. We will refer to R(t) as the characteristic polynomial of the recurrence.
Indefinite Green's Functions and Elementary
Solutions
Authors: Duff G. F. D. , Ross R. A. 
Published: December 2010

Linear differential equations both ordinary and partial are often studied by
means of Green's functions. One reason for this is that linearity permits
superposition of solutions. A Green's function describes the "effect" of a
point source, and the description of line, surface, or volume sources is
achieved by superposing, that is to say, integrating, this function over the
source distribution.For equations with constant coefficients the use of integral transforms
permits the calculation of such source functions in the form of integrals.
Only in the simplest cases is explicit evaluation by elementary functions
possible, and this has perforce led to the use of asymptotic estimates,
which so thoroughly pervade the domain of applied mathematics.
Variational Aspects of the Abel and Schroder Functional Equations
Authors: McKiernan M.A. , Rényi A. 
Published: December 2010

Given an analytic function f, the successive iterates of f are defined byf[0](z) = z, f[n+1](z) = f{f[n](z)} for every z.In particular f[1](z) = {f[0](z)} f(z). Extensive study has been given [1] to the problem of generalizing the iterates f[n], for integer n, to f[t] for arbitrary t, where the iterative character of f[t] is to be preserved by the conditions,f[0](z) = z and f[s]{f[t](z)} = f[s+t](z) for arbitrary s and t.
On Subgraphs of the Complete Bipartite Graph
Authors: Erdös P. , Moon J. W. 
Published: December 2010

G(n) denotes a graph of n vertices and Ḡ(n) denotes its complementary graph. In a complete graph every two distinct vertices are joined by an edge. Let Ck(G(n)) denote the number of complete subgraphs of k vertices contained in G(n). Recently it was proved [1] that for every k1where the minimum is over all graphs G(n).
A New Proof of an Inequality of Heinz
Authors: Bullen P. S. 
Published: December 2010

In a recent paper, [l], Dixmier has proved Heinz' inequality by deducing it from a lemma due to Thorin. In this note it is proved directly from a convexity theorem.Let(M(k), ℳ(k), μ(k)), k = 0, …, n, be measure spaces and Lq(k) (M(k), ℳ(k), μ(k)) be all the functions on M(k) such that
Remarks on Quasi-Hermite-Fejér Interpolation
Authors: Sharma A. 
Published: December 2010

Let1be n+2 distinct points on the real line and let us denote the corresponding real numbers, which are at the moment arbitrary, by2The problem of Hermite-Fejér interpolation is to construct the polynomials which take the values (2) at the abscissas (1) and have preassigned derivatives at these points. This idea has recently been exploited in a very interesting manner by P. Szasz [1] who has termed qua si-Hermite-Fejér interpolation to be that process wherein the derivatives are only prescribed at the points x1, x2, …, xn and the points -1, +1 are left out, while the values are prescribed at all the abscissas (1).
Some Remarks on the Mathieu Groups
Authors: Garbe Dietmar , Mennicke Jens L. 
Published: December 2010

In the present note we shall study some properties of the Mathieu groups.We shall give an invariant characterisation of the 2-Sylow subgroups. The 2-Sylow subgroup of M24is the holomorph of the elementary abelian group of type (1, 1, 1, 1), and for the 2-Sylow subgroups of the other Mathieu groups there are similar characterisations.
Partitions of the Natural Numbers
Authors: Angel Myer 
Published: December 2010

We obtain in this article some results concerning partitions of the natural
numbers, the most important of which is a generalization of that quoted
immediately below. Some intuitive material is included.In 1954, J. Lambek and L. Moser [l] showed that "Two non-decreasing
sequences f and g (of non-negative integers) are inverses if and only if the
corresponding sets F and G of positive integers, defined by F(m) = the mth
element of F = f(m) + m and G(n) = g(n) + n are complementary."
On Abel's Binomial Identity
Authors: Majindar Kulendra N. 
Published: December 2010

The identity of Abel [1] [2] we deal with here can be stated in the
following form: If n is a positive integer,.(In order that all terms be defined we require a ≠ 0, b ≠ n.)This identity and deductions from it have been very useful in many problems,
for instance in mathematical statistics [3]. Usually this identity is
established by means of the Lagrange - Bűrman theorem [4]. Here we will
derive it very simply.
The Series Σ∞1 f(n)/n for Periodic f
Authors: Livingston Arthur E. 
Published: December 2010

We are here concerned with the problem of deciding when Σ∞n=1 f(n)/n ≠ 0, given that f is periodic and the series convergent. In particular, we considerConjecture A. Let p be a positive integer and f a (real-or complex-valued) number-theoretic function with period p.
Inequalities Associated with the Triangle
Authors: Blundon W. J. 
Published: December 2010

Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real functions. Let q(R, r) and Q(R, r) be real quadratic forms. The latter functions are thus a special case of the former. Our main result is to derive the strongest possible inequalities of the form1with equality only for the equilateral triangle.
On the Summability of a Class of the Derived Conjugate Series of a Fourier Series
Authors: Govil Narendra K. 
Published: December 2010

Let f(t) be integrable L(-π, π) and periodic with period 2π, and let1.1be its Fourier series. The series1.2obtained by the term by term differentiation of the series (1. 1), is called the Derived Fourier series of (1. 1). The series conjugate to (1. 2),1.3is called the Derived conjugate series of the Fourier series of f.
On Simple Rings with Maximal Annihilator Right Ideals
Authors: Koh Kwangil 
Published: December 2010

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.We first prove the following:Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.
The Meaning of Mono and EPI in Some Familiar Categories
Authors: Burgess W. 
Published: December 2010

This expository note was prompted by some questions asked by Professor P. Hilton during his lectures "Catégories non-abétiennes" at the University of Montréal, July 1964.The descriptions of set functions as one to one and as onto can be characterized in terms of set function composition. A set function is one to one iff it has the left cancellation property, that is, f · g = f · h implies g = h.
On the Densest Packing of Spheres in a Cube
Authors: Schaer J. 
Published: December 2010

How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed in a cube of given size? These questions are obviously equivalent to the following problem:Let d(Pi, Pj) denote the distance between the points Pi and Pj, and Γk the set of all configurations of k points Pi (1 ≤ i < j ≤ k) in a closed unit cube C.
On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem
Authors: Haruki Hiroshi 
Published: December 2010

Considering Cauchy's functional equationf(z1+z2)=f(z1)+ f(z2),where f(z) is an entire function of z, we have the following functional equation:(1) |f(x+iy)|=|f(x)+f(iy)|,where x and y are real.
On the Residues of a Cubic Polynomial (Mod p )
Authors: McCann K. , Williams K.S. 
Published: December 2010

If f(x) is a polynomial with integral coefficients then the integer r is said to be a residue of f(x) modulo an integer m if the congruenceis soluble for x; otherwise r is termed a non-residue. When m is a prime p, Mord ell [4] has shown that the least nonnegative residue ℓ of f(x) (mod p) satisfies
On Some New Generalizations of the Functional Equation of Cauchy
Authors: Fischer P. , Muszély Gy. 
Published: December 2010

Examining certain problems in physics M. Hosszu [l] obtained the functional equation(1)where x, y, f are real.In another paper M. Hosszu [2] proved that the equation (1) is equivalent to the functional equation of Cauchy; i. e., to the equation(1)under the assumption that x is real and f is real and continuous.
On A Theorem of Niven
Authors: Williams Kenneth S. 
Published: December 2010

In 1940, I. Niven [2] proved that the gaussian integer z = x + iy is the sum
of two squares of gaussian integers if, and only if, y is even and not both
of 1/2x and 1/2y are rational odd integers. In this note we calculate the
total number g2(z) of representations of z in this form.1where a, b, c, d are rational integers, if and only if2
A Remark on the Gibbs Phenomenon and Lebesgue Constants for a Summability Method of Melikov
Authors: Ustina Fred 
Published: December 2010

Let u = ∑ukbe a given series and letMelikov [4] has defined the n-th σ- transform of u bywhere ε and θ are assumed to be non-negative. This is easily shown to be equivalent toThe method is a generalization of a method used by Kaufman [l], and of another one used by Melikov [5]. It reduces to the (n-l)th (C;l) mean when θ = 0 and ε = 0, and to the n-th (C;l) mean when θ = 1 and ε = 0.
Circumscribing an Ellipsoid about the Intersection of Two Ellipsoids
Authors: Kahan W. 
Published: December 2010

An ellipsoid G is associated uniquely with a positive definite matrix A viaNote that all ellipsoids discussed here are centred at 0. Given G1, and G2 we seek another ellipsoid circumscribed about G1 ∩ G2. It is easy to see that if and only if x'hx ≤ maxi x'aix for all vectors x.
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.
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.
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].
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 >.
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.
Enclosure Theorems for Eigenvalues of Elliptic Operators
Authors: Clements John C. 
Published: December 2010

LetLbe the linear, elliptic, self-adjoint partial differential operator given bywhereDjdenotes 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.
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 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.
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 equation1possesses 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 equation2and 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 Carleman Integral Operators
Authors: Costley Charles G. 
Published: December 2010

L2(a, b)1with the property2were originally defined by T. Carleman [4]. Here he imposed on the kernel the conditions of measurability and hermiticity,3for all x with the exception of a countable set with a finite number of limit points and4where 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

Abstract.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 thatThen 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.
On a Solution of the Hammerstein Equation with Singular Normal Kernels
Authors: Costley Charles G. 
Published: December 2010

We consider here the equation1This equation was first studied by Hammerstein [4] under the assumption that the linear operator2is 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, byThe 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 = fWe prove the following result.