bluuebunny/crossref_metadata_2025_split huggingface dataset:
Identities for Multiplicative Functions
Authors: Subbarao M. V. , Gioia A. A. 
Published: December 2010

Throughout this paper the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n > 1, with L(1) = 0 and w(1) = 1. Also letWe recall that an arithmetic function f(n) is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n.
Note on Pointwise Convergence on the Choquet Boundary
Authors: Grossman Marvin W. 
Published: December 2010

In [6] J. Rainwater obtained the following theorem.Theorem. Let N be a normed linear space, {xn} a bounded sequence of elements in N and X ∊ N. for each extreme point f of the unit ball of N✶, then {xn} converges weakly to x.Now let X be a compact Hausdorff space and H a linear subspace of C(X) (all real-valued continuous functions on X ) which separates the points of X and contains the constant functions. If x∊X, then MX(H) denotes the set of positive linear functionals μ on C(X) such that μ(h) = h(x) for all h in H.
An Extremal Problem in Number Theory
Authors: Abbott H. L. , Gardner B. 
Published: December 2010

Let n and k be integers with n ≥ k ≥ 3. Denote by f(n, k) the largest positive integer for which there exists a set S of f (n, k) integers satisfying (i) and (ii) no k members of S have pairwise the same greatest common divisor. The problem of determining f(n, k) appears to be difficult. Erdős [2[ proved that there is an absolute constant c > 1 such that for every ∈ > 0 and every fixed k1provided n > no (k, ∈). In [l[ it i s proved that for every ∈ > 0 and every fixed k
On Stability of Solutions of Certain Differential Equations of the Third Order
Authors: Lalli B.S. 
Published: December 2010

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.
On the Monotonicity of Certain Functionals in the Theory of Analytic Functions
Authors: Rahman Q.I. 
Published: December 2010

Let f(z) be a function regular for | z | < R, and denote by L(r) the length of the curve Γ onto which the circle | z | = r is mapped by f(z), i. e.If Dr is the image of | z | ≤ r on the Riemann surface of f(z) then its area S(r) is given by
Rellich′s Embedding Theorem for a “Spiny Urchin”
Authors: Clark Colin 
Published: December 2010

From the plane R2 we remove the union of the sets Sk (k = 1, 2, …) defined as follows (using the notation z = x + iy):Sk = {z: arg z = nπ2-k for some integer n; |z|≥k}.
The remaining connected open set Ω we call the spiny urchin.
Note on an Application of the δ-Function in the Representation of Solutions of Algebraic Equations
Authors: Sabat J. B. 
Published: December 2010

The “function” δ(x - xo) is known as the Dirac Delta function and may be defined as zero everywhere except at xo, where it is infinite in such a way that1having property that for every continuous function φ(x) on (a, b)2It is well known [2] δ(x-xo) can be approximated as a limit of a sequence of piecewise continuous functions, and there is an abundance of such sequences.
Note on Pairs of Consecutive Residues of Polynomials
Authors: Williams Kenneth S. 
Published: December 2010

Let f(x) be a polynomial of degree d ≥ 3 with integral coefficients, 'say,In a previous paper [6] I deduced, from a deep result of Lang and Weil [2], that there is a constant k1(d), depending only on d, such that for all primes p ≥ k1(d), p ⫮ ad, f(x) has a pair of consecutive residues (mod p), that i s, there exists an integer r(0 ≤r ≤ p-1) with the property that
The Circle Problem in an Arithmetic Progression
Authors: Smith R.A. 
Published: December 2010

In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤xn as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of1
A Functional Equation for the Cosine
Authors: Kannappan PL 
Published: December 2010

It is known [3], [5] that, the complex-valued solutions of(B)(apart from the trivial solution f(x)≡0) are of the form(C)(D)In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],(E)In this paper, the functional equation(P)where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.
A Note on an Oscillation Criterion for an
Equation with a Functional Argument
Authors: Waltman Paul 
Published: December 2010

It might be thought that, as far as the oscillation of solutions is concerned, the behaviour ofandwould be the same as long as t - α(t) → ∞ as t→∞. To motivate the theorem presented in this note, we show first that this is not the case. Consider the above equation with α(t) = 3t/4, a(t) = l/2t2 i.e.This equation has the non-oscillatory solution y(t) = t1/2 although all solutions ofare oscillatory [1, p. 121].
On a Certain Set of Linear Inequalities
Authors: Kalbfleisch J.G. , Stanton R. G. 
Published: December 2010

In this paper we shall discuss the following set of n + 1 linear inequalities:If we let , and Z = (zi) (i = 0, 1,…, n) be (n+1)-dimensional column vectors, and define the n+1 by n+1 tridiagonal matrix Dn(ϕ) bythe set of inequalities (1) may be writtenwhere An= Dn(1) and zi≥ 0 (i = 0, 1,…, n). In sections 2 and 3, we
Some Elementary Converse Problems in Ordinary Differential Equations*
Authors: Seminar D. E. 
Published: December 2010

In studying differential equations, the usual task is to determine properties of the solutions of such equations from a knowledge of the coefficient functions. The converse question, namely, of determining the coefficient functions from properties of solutions, also has significance. It has been studied especially in the case of Sturm-Liouville equations.A discussion of the inverse Sturm-Liouville problem can be found in [8, Chapter 8], where references are given to the work of W.A. Ambarzumiam, G. Borg, I.M. Gelfand, M.G. Krein, B.M. Levitan, N. Levinson and W.A. Marchenko on this problem. Work of a quite different character, but dealing also with questions of a converse type arising from Sturm-Liouville equations, has been done by O. Boruvka and his colleagues and students [2].
On Polyhedral Realizability of Certain Sequences
Authors: Jucovič E. 
Published: December 2010

A finite sequence (pk) = (p3, p4,…) of non-negative integers shall be called realizable provided there exists a 3-valent 3-polytope P which has pi. i-gonal faces for every i. P is called a realization of (pk).For realizability of a sequence (pk), from Euler's formula follows(*) as a necessary condition.
A Presentation of the Mathieu Group M12
Authors: Leech John 
Published: December 2010

In the course of work on factor groups of the modular group, A. O. L. Atkin (private communication) obtained the permutationswhich satisfy the relations(1)for T = T1, T2. It was evident that neither pair S, T generates either A12 or LF(2, 11), so he suggested that they probably generate the Mathieu group M2.
On a Theorem of Cutler
Authors: Megibben Charles K. 
Published: December 2010

In [1] Cutler proved the following theorem.Theorem. If G and K are abelian groups such that nG ≅ nK for some positive integer n, then there are abelian groups U and V such that U ⊕ G ≅ V ⊕ K and nU = 0 = nV.Cutler's proof is long and fairly involved. Walker [3] obtains the theorem rather elegantly as a corollary of his results on n-extensions. We give here a proof that is extremely simple both in conception and execution. Our proof relies on the notion of p-basic subgroups introduced by Fuchs in [2]. Therefore we shall first recall certain pertinent facts from [2].
Boolean Near-Rings
Authors: Clay James R. , Lawver Donald A. 
Published: December 2010

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].
On Finite Polarized Partition Relations
Authors: Chvátal V. 
Published: December 2010

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write(1)if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is writtenand means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.
On Rationality of Algebraic Function Fields
Authors: Nobusawa Nobuo 
Published: December 2010

Let A be an algebraic function field with a constant field k which is an algebraic number field. For each prime p of k, we consider a local completion kp and set Ap = Ak ꕕ kp. Then we have the question:Is it true that A/k is a rational function field (i.e., A is a purely transcendental extension of k) if Ap/kp is so for every p ? In this note we shall discuss the question in a slightly different and hence easier case.
Scales of Logarithmic Methods of Summability
Authors: Borwein D. , Phillips R. 
Published: December 2010

We suppose throughout that p is a non-negative integer, and use the following notations:where log0x = x for x ≥ e0 = 1, and logn+1x = log(lognx) for x ≥ en+1 = een (n = 0, 1, 2,…);
A Note on a Sequence of Contraction Mappings
Authors: Singh S.P. , Russell W. 
Published: December 2010

Let E be a metric space. A mapping T of the space E into itself is said to be a contraction if there exist s a number k, with 0 ≤ k < 1 such thatfor any two points x, y ∈ E. Every contraction mapping is continuous.
On Two Conjectures of Chowla
Authors: Williams Kenneth S. 
Published: December 2010

Let p denote a prime and n a positive integer ≥ 2. Let Nn(p) denote the number of polynomials xn + x + a, a = 1, 2,…, p-l, which are irreducible (mod p). Chowla [5] has made the following two conjectures:Conjecture 1. There is a prime p0(n), depending only on n, such that for all primes p ≥ p0(n)
On Characterizing the Multivariate Linear Exponential Distribution1
Authors: Kabe D. G. 
Published: December 2010

If x and y are independent p component column vectors, and the conditional distribution of x, given x+y = z, is known, what can be said about the distributions of x and y? This problem has been solved by Seshadri (1966) in the particular case when the conditional distribution of x, given x+y = z, is multivariate normal. In fact Seshadri′s paper implicitly contains a characterization of the multivariate linear exponential distribution(1)where A(x) is a function of x not involving the p component column vector w of constant terms.
Differences of Sets and A Problem of Graham
Authors: Marica J. , Schöheim J. 
Published: December 2010

R. L. Graham has posed the following question:Given n positive integers a1, < a2 <… < an, does there exists a pair of indices i, j such that ai/(ai, aj) ⩾ n? ((ai, aj) = g.c.d. of ai and aj).
A Note on the Quotients of Indecomposable Injective Modules
Authors: Vámos P. 
Published: December 2010

Let R be a commutative domain, I an ideal of R and write E1 for the infective envelope of R / I. In this note the following theorem will be proved:Theorem. For a prime ideal P of a commutative domain R the following are equivalent:(i)Every factor module of Ep is an indecomposable infective module;(ii)Every non-zero prime ideal P′ ⊆ P is contained in only one maximal ideal M of R, and RM is an almost maximal valuation ring.
The Implicit Function Theorem in the Scalar Case*
Authors: Freedman H. I. 
Published: December 2010

The implicit function theorem has applications at all levels of mathematics from elementary calculus (implicit differentiation) to finding periodic solutions of systems of differential equations ([1, Chapter 14] and [4], for example).In 1961 W. S. Loud [3] studied the case of two equations in three unknowns. He considered only cases where up to third order derivatives were involved and only those cases where the derivative of the solutions at the critical point existed. Coddington and Levinson [1] consider a specific singular case involving n equations in n + m unknowns. In general the number of distinct critical cases involving up to third derivatives for such a general system is not known.
A Packing Inequality for Compact Convex Subsets of the Plane
Authors: Folkman J. H. , Graham R. L. 
Published: December 2010

Let X be a compact metric space. By a packing in X we mean a subset S ⊆ X such that, for x, y ∈ S with x ≠ y, the distance d(x, y) ≥ 1. Since X is compact, any packing of X is finite. In fact, the set of numbers{card(S): S is a packing in X}is bounded. The cardinality of the largest packing in X will be called the packing number of X and will be denoted by ρ(X). If A(X) and P(X) denote the area and perimeter, respectively, of a compact convex subset X of the plane, then a special case of a result conjectured by H. Zassenhaus [6] and proved by N. Oler [l] is the following.
Symmetric Forms
Authors: Menon K. V. 
Published: December 2010

Let Rm denote a m dimensional Euclidean space. When x ∊ Rm will write x = (x1, x2,..., xm). Let R+m ={x: x ∊ Rm, xi < 0 for all i} and R-m ={x: x ∊ Rm, xi < 0 for all i}. In this paper we consider a class of functions which consists of mappings, Er(K) and Hr(K) of Rm into R which are indexed by K ∊ R+m and K ∊ R-m respectively, and defined at any point α ∊ Rm by1.1
Demonstration de la Formule Künneth par la Theorie Harmonique
Authors: van Quê Ngô 
Published: December 2010

Etant données deux variétés différentiables compactes M et N, nous avons la formule de Künnethqui établit la relation entre la cohomologie réelle de la variété produit avec celle des facteurs. Cette formule est bien connue en topologie algébrique, et elle est valable dans le contexte plus général de la catégorie des espaces topologiques. Nous voulons donner ici une démonstration de cette formule par la théorie harmonique de Hodge-Kodaira [1]. Une telle démonstration n'est pas explicite, autant que je sache, dans la littérature; elle est désirable, par exemple, dans un cours de géométrie différentielle sur les formes harmoniques.
Injective Hulls of Semilattices
Authors: Bruns G. , Lakser H. 
Published: December 2010

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.
A Note on Artin's Diophantine Conjecture
Authors: Maxwell George 
Published: December 2010

A well known theorem of Hasse [1] says that every quadratic form in at least 5 variables over the field Qp of p-adic numbers has a nontrivial zero. This fact has led Artin to make the conjecture(C): "Every form over Qp of degree d in n > d2 variables has a non-trivial zero." However, a counterexample has been provided by Terjanian [2] in the case d=4.The case d=2 is distinguished by the fact that every quadratic form may be "diagonalized", i.e., assumed to be of the type Σ aiX2i. One is therefore led to the weaker conjecture(C): "Every form f= Σ aiXdi over Qp in n > d2 variables has a nontrivial zero in Qp,"which still generalizes Hasse's theorem.
A Note on the Existence of a Solution of the Falkner-Skan Equation
Authors: Tam K. Kuen 
Published: December 2010

We are concerned with the existence proof of solution of the Falkner-Skan equation1subject to boundary conditions2The first existence and uniqueness proof based on a fixed point theorem was given by Weyl [4] in 1942, with the added assumption that f' > 0. In 1960, Coppel [1] proved the existence (and uniqueness with the assumption 0 < f' < 1) by considering trajectories in the three-dimensional phase space.
On the Distribution of Sum of Independent Positive Binomial Variables
Authors: Ahuja J. C. 
Published: December 2010

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.
On a Theorem of Mitchell
Authors: Bishop E. R. 
Published: December 2010

In this note we apply a theorem of T. Mitchell on fixed
points of contraction maps to a particular set of contraction maps of metric
spaces. Also, part of Mitchell's theorem is extended to permit application,
under suitable conditions, when the space involved is not compact.We employ the following terminology due to Mitchell [1]. Let
S be a semigroup; m(S)
the space of all bounded real functions on S with sup norm;
and X a subset of m(S).
For all s ∊ S, the left translation
ls {right translation rs} of m(s) by s
is given by (lsf)(s') =
f(ss')
{(rsf)(s')=f(s's)}
where f ∊ m(S) and
s' ∊ S.
On the Borel-Cantelli Problem
Authors: Shuster Jonathan 
Published: December 2010

Let (Ω, , P) be a
probability space, and A1, A2… be a sequence of members of . The
classical Borel-Cantelli problem is to determine the probability that
infinitely many events Ak occur. The classical results may be found in Feller [2, p. 188];
while related work may be found in Spitzer [3, p. 317], and Dawson and
Sankoff [1]. The latter works are generalizations of the Borel-Cantelli
lemmas, taken in different directions.In this paper, necessary and sufficient conditions will be given for
infinitely many events Ak to occur, with probability 1. A lower bound for the probability that
only finitely many Ak occur, is developed. In addition, necessary and sufficient conditions
that only finitely many Ak occur, with probability 1, are given.
A Unitary Relation Between a Matrix and its
Transpose
Authors: Gordon William R. 
Published: December 2010

It is well known that if A is an n ×
n complex matrix and AT is its transpose, then there is an invertible n x
n complex matrix S such that
AT = S-1AS. In this note we wish to point out another simple
relation between A and AT.If A is an n × n complex
matrix and AT is its transpose then there are unitary n ×
n complex matrices U and
V such that AT = UAV.
On Subtrees of Directed Graphs with No Path of Length Exceeding One
Authors: Graham R. L. 
Published: December 2010

The following theorem was conjectured to hold by P. Erdös [1]:Theorem 1. For each finite directed tree T with no directed path of length 2, there exists a constant c(T) such that if G is any directed graph with n vertices and at least c(T)n edges and n is sufficiently large, then T is a subgraph of G.	In this note we give a proof of this conjecture. In order to prove Theorem 1, we first need to establish the following weaker result.
A Remark about Noncommutative Integral Extensions
Authors: Heinicke A. G. 
Published: December 2010

Let B be a ring with unity, A a imitai subring of the centre Cof B. Suppose further that B is A-integral. (That is, every element of B satisfies a monic polynomial with coefficients in A.) Under these assumptions, Hoechsmann [2] showed that "contraction to A" is a mapping from:(1)The prime ideals of B onto the prime ideals of A,(2)The maximal ideals of B onto the maximal ideals of A.In this note we show that, under additional assumptions, a noncommutative version of the rest of the Cohen-Seidenberg "going up theorem" can be established.
Graphs and k-Societies
Authors: Hell Pavol , Nešetřil Jaroslav 
Published: December 2010

A graph G is a couple (X, R) where X is a set, R ⊂ X × X. If G is an undirected graph without loops (R a symmetric irreflexive relation), we can interpret G as a couple (X, R), where R is a set of two-element subsets of X, i.e. . This interpretation is generalized in the notion of society.A society is a couple (X, R), where ; a k-society is a society (X, R) with |A| = k for each A ∊ R.
On the Dimension of an Irreducible Tensor Representation of the General Linear Group GL(d)
Authors: Matthews J. A. J. , de B. Robinson G. 
Published: December 2010

As has long been known, the irreducible tensor representations of GL(d) of rank n may be labeled by means of the irreducible representations of Sn, i.e., by means of the Young diagrams [λ], where λ1 + λ2 + … λr = n. We denote such a tensor representation by 〈λ〉. Using Young's raising operator Rij we can write [1, p. 42]1.1where the dot denotes the inducing process. For example, [3] . [2] is that representation of S5 induced by the identity representation of its subgroup S3 × S2.
A Note on the Σ(S)-Injectivity of R(S)
Authors: Luedeman John K. 
Published: December 2010

Let R be a ring with 1. All modules considered are to be unital left R-modules unless otherwise noted.Definition. A σ-set for R is a nonempty set Σ of left ideals of R satisfying the following conditions:(σ1)If I ∊ Σ, J is a left ideal of I, and J ⊇ I, then J ∊ Σ.(σ2)If I ∊ Σ and r ∊ R, then Ir-1 = {s ∊ R | sr ∊ I} ∊ Σ.(σ3)If I is a left ideal of R, J ∊ Σ, and It-1 ∊ Σ for each t ∊ J, then I ∊ Σ.
Comment on a Note by J. Marica and J. Schönheim
Authors: Lindner Charles C. 
Published: December 2010

In [2] it is shown that an n × n partial latin square with n — 1 cells occupied on the main diagonal can be completed to a latin square. We can use the technique in [2] to prove the following result.An n × n partial latin square with n — 1 cells occupied with n — 1 distinct symbols can be completed to a latin square if the occupied cells are in different rows or different columns.Let P be an n × n partial latin square based on 0,1, 2, …, n — 1 satisfying the above conditions, and let (x0, y0), (x1, y1), …, (xn-2, yn-2) De the occupied cells where y0, y1, … yn-2 are distinct.
A Bound for the Moduli of the Zeros of Polynomials
Authors: Rahman Q. I. 
Published: December 2010

The following theorem is due to Walsh [2]. For another proof see [1].Theorem A. All the zeros of the polynomial p(z) = a0 + a1z + … an-1zn-1+znlie on the disk.where M = Σj = 2n|an-j|1/j