bluuebunny/crossref_metadata_2025 huggingface dataset:
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 written

and 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 the Projective Cover of the Stone-Čech Compactification of a Completely Regular Hausdorff Space
Authors: Park Young Lim
Published: December 2010

The main object of this paper is to give an explicit object in the study of projective covers in the category of compact Hausdorff spaces and continuous maps studied in [2] and [5]. be a projective cover of the Stone-Čech compactification βX of a completely regular Hausdorff space X. Here, it will be shown that the maximal ideal space endowed with the Stone topology of the maximal ring of quotients of the ring C(X) of all real valued continuous functions on X is homeomorphic to K.
Distribution of the Sum of Truncated Binomial Variates
Authors: Malik Henrick John
Published: December 2010

Let X be a random variable defined by a Bernoullian probability function

(1)

The probability function of the restricted random variable which is truncated away from zero is then

(2)

The divisor 1 - qN arises from the condition excluding zero.
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,…);
Compact Subsets in Function Spaces
Authors: Kaul S.K.
Published: December 2010

We wish to study the problem of finding conditions under which a family of maps from one space into another, with a suitable topology, is compact. Some of the results obtained in this direction are in [1; 2; 3]. We propose to give conditions, to be called uniformly regular and regular (the terminology is motivated by [4]), under which "Ascoli" theorems can be proved. These notions turn out to be equivalent to even continuity of Kelley [1, page 235] under such conditions that all the theorems in the section on even continuity in it still hold when in their statements even continuity is replaced by either uniform regularity or regularity (see Theorem A below).
Some Fixed Point Theorems in Metric and Banach Spaces
Authors: Belluce L. P. , Kirk W.A.
Published: December 2010

The purpose of this paper is two-fold. Sections 2 and 3 are motivated by an observation that certain theorems concerning "diminishing orbital diameters" (introduced in [1]) are true under weaker assumptions. Specifically, we investigate the relationship between that concept and alternate conditions such as "asymptotic regularity", and in the process we sharpen some metric space results established in [1;5]. Mention is made in these sections of examples which show that certain additional weakenings of our hypotheses cannot be made, but we include in detail only the one which seemed to us most intricate.
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 that

for any two points x, y ∈ E. Every contraction mapping is continuous.
La Dérivée De La Primitive De L'Intégrale De Perron
Authors: Dubuc Serge
Published: December 2010

Nous voulons donner une démonstration directe du fait que la dérivée de la primitive d' une fonction f(x) intégrable au sens de Perron existe et est égale à f(x) presque partout. Nous utiliserons essentiellement le "Iemme du soleil levant" ou plutôt un de ses corollaires immédiats.
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.
Note on Factorable Polynomials
Authors: Williams Kenneth S.
Published: December 2010

Let X1, X2,…, Xk, denote k ≥ 2 indeterminates and let f(X1,…, Xk) be a homogeneous polynomial, in GF(pn) [X1,…, Xk], which is irreducible but not absolutely irreducible over GF(pn). Thus f is irreducible in GF(pn)[X1,…, Xk] but reducible in some GF(pnm) [X1,…, Xk], m > 1. For any polynomial h(X1,…, Xk) in GF(pnℓ)[X1,…, Xk, ℓ ≥ 1, let Npn(h) denote the number of (x1,…, xk) ∈ GF(pn)×…× GF(pn) such that (hx1…, xk) = 0.
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).
Remarks on Invariant Subspace Lattices
Authors: Rosenthal Peter
Published: December 2010

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.
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.
Continuous Functions on the Sphere and Isometries
Authors: Hadwiger H. , Mani P.
Published: December 2010

The theorem of Borsuk-Ulam states that n odd functions on the n-dimensional sphere always have a common zero. We have tried to obtain a similar theorem by "slightly" changing the conditions for the functions, but it turned out that only a very weak analogue can be expected in our case. Here we want to prove a few results and mention some of the questions which have remained unanswered.
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 by

1.1
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 Künneth

qui 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.
A Remark on Contractive Mappings
Authors: Ng Kai-Wang
Published: December 2010

Much current research is concerned with the fixed points of contractive mappings (mappings which shrink distance in some manner) from a metric space into itself. In this remark we shall point out that most mappings treated in the literature are very special in the sense that all these mappings satisfy a condition which is rather severe: every periodic point must necessarily be a fixed point.
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 relation

defines 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 equation

1

subject to boundary conditions

2

The 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.
Finite Projective Distributive Lattices
Authors: Grätzer G. , Wolk B.
Published: December 2010

The theorem stated below is due to R. Balbes. The present proof is direct; it uses only the following two well-known facts: (i) Let K be a category of algebras, and let free algebras exist in K; then an algebra is projective if and only if it is a retract of a free algebra, (ii) Let F be a free distributive lattice with basis {xi | i ∊ I}; then ∧(xi | i ∊ J0) ≤ ∨(xi | i ∊ J1) implies J0∩J1≠ϕ. Note that (ii) implies (iii): If for J0 ⊆ I, a, b ∊ F, ∧(xi | i ∊ J0)≤a ∨ b, then ∧ (xi | i ∊ J0)≤ a or b.
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 function

1

where 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 Asymptotically Nonexpansive Semigroups of
Mappings
Authors: Holmes R. D. , Narayanaswami P. P.
Published: December 2010

A selfmapping f of a metric space (X, d)
is nonexpansive (ε-nonexpansive) if
d(f(x),
f(y)) ≤ d(x,
y) for all x, y ∊ X
(respectively if d(x, y) < ε). In [1],
M. Edelstein proved that a nonexpansive mapping f of
En admits a fixed point provided the f-closure of
En (i.e. the set of all points which are cluster points of
{fn(x)} for some x) is nonempty. R.
D. Holmes [2] considered commutative semigroups of selfmappings of a metric
space and obtained fixed point theorems for such semigroups under certain
contractivity conditions.
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.
The Coefficients of
Authors: Gandhi J. M.
Published: December 2010

The purpose of the present paper is to investigate some of the properties of the coefficient K2n defined by

1.1

We prove

1.2

1.3

1.4
On Point-Symmetric Tournaments
Authors: Alspach Brian
Published: December 2010

A tournament is a directed graph in which there is exactly one arc between any two distinct vertices. Let denote the automorphism group of T. A tournament T is said to be point-symmetric if acts transitively on the vertices of T. Let g(n) be the maximum value of taken over all tournaments of order n. In [3] Goldberg and Moon conjectured that with equality holding if and only if n is a power of 3. The case of point-symmetric tournaments is what prevented them from proving their conjecture. In [2] the conjecture was proved through the use of a lengthy combinatorial argument involving the structure of point-symmetric tournaments. The results in this paper are an outgrowth of an attempt to characterize point-symmetric tournaments so as to simplify the proof employed in [2].
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]

1.1

where 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.
Are One-Sided Inverses Two-Sided Inverses in a Matrix Ring Over a Group Ring?
Authors: Losey Gerald
Published: December 2010

A ring R with identity element is n-finite if for any pair A, B of n × n matrices over R, AB = In implies BA = In. In module theoretic terms, R is n-finite if and only if in a free R-module of rank n any generating set of n elements is free. If R is n-finite for all positive integers n then R is said to be strongly finite. It is known that all commutative rings, all Artinian rings and all Noetherian rings are strongly finite. These and many other interesting results appear in a paper of P. M. Cohn [1].
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.
On an Interesting Property of a Combinatorial Function
Authors: Consul P. C. , Kemp M. J.
Published: December 2010

For any two integers n and k we take, as usual,

Using the above symbol we define a new function F(q, 2n) by a sum of two finite sums given by

1.1

for i ≠ q, and whenever q is an odd integer.
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
A Note on Permanents
Authors: Abramson Morton
Published: December 2010

Let A = (aij) be an m × n matrix and let K = {s1, …, sk} be a k-subset from {1, 2, …, n}. For 0≤t≤k≤n define the (t, K)-permanent of A to be

(1)

the summation taken over all m-tuples (i1, i2, …, im) (repetitions allowed) of 1, 2, …, n each containing exactly t distinct entries from K and any number of distinct entries from the remaining n-k integers. For example, (4, 4, 7, 1, 1, 2), (4, 4, 6, 6, 6, 5) are 6-tuples, each containing exactly two distinct entries from K={2, 4, 5} for n ≥ 7.
On a Perturbation in a Two-Parameter Ordinary Differential Equation of the Second Order
Authors: Faierman M.
Published: December 2010

Let us consider the linear system in the two parameters λ and μ; i. e.,

1.1

1.2

and where for the moment we shall assume both b(x) and q(x) are real-valued, continuous functions in [0, 1].
On a Problem of Klee
Authors: Stavrakas N. M. , Jamison R. E.
Published: December 2010

Let E be a Hausdorff topological vector space. A subset A of E is a polytope iff A is the convex hull of a finite number of points. In this note a necessary condition for every maximal convex subset of a subset B of E to be a polytope is given. This is related to a problem first posed by Klee [1] for compact three-cells in Euclidean 3 space.
Some Sufficient Conditions for Maximal-Resolvability(1)
Authors: Pearson T. L.
Published: December 2010

A topological space X is called maximally resolvable if it admits a largest possible family of pairwise disjoint, “maximally dense” subsets. More precisely, if Δ(X) denotes the least among the cardinal numbers of the nonvoid open subsets of X, then X is maximally resolvable if it has isolated points or there exists a family {Rα}α < Δ(X) of subsets of X, called a maximal resolution for X, such that ∪{Rα | α < Δ(X)} = X, Rγ ∩ Rδ==ϕ if γ ≠ δ, and, for each a and each nonvoid open subset V of X, the cardinality of Rα ∩ V is not less than Δ(X).
On Finite Invariant Measure for Semigroups of Operators
Authors: Sachdevao Usha
Published: December 2010

Let Σ be a left amenable semigroup, and let {Tσ: σ ∊ Σ} be a representation of Σ as a semigroup of positive linear contraction operators on L1(X, 𝓐, p). This paper is devoted to the study of existence of a finite equivalent invariant measure for such semigroups of operators.
On Integral Inequalities Related to Hardy's
Authors: Shum D. T.
Published: December 2010

The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state as

Theorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by

1

then

2

unless f≡0. The constant is the best possible.
Matrices with Prescribed Principal Elements and Singular Values
Authors: De Oliveira Graciano N.
Published: December 2010

We shall be concerned with the following problem. Let a11, …, ann be complex numbers and λ1, …, λn nonnegative real numbers. Under what conditions does there exist an n × n complex matrix A with a11, …, ann as principal elements and λ1, …, λn as singular values? This problem has been suggested in [3] but, to our knowledge, has not yet been solved.
Estimateurs De La Moyenne Pour Les Distributions Symetriques a Domaine Borne
Authors: Dion J. P.
Published: December 2010

L'emploi de comme estimateur de la moyenne est très fréquent en statistiques pour plusieurs bonnes raisons. Ainsi pour une densité régulière de la classe exponentielle, il est l'estimateur sans biais pour la moyenne, à variance minimale. Mais il perd cette importante propriété pour la classe des densités tronquées étudiées ici: cet article démontre que lorsque la taille de l'échantillon est suffisamment grande, il faut préférer Δ = [(mini=1, …, nXi + maxi=1, …, nXi/2] à .
On Prime One-Sided Ideals
Authors: Koh Kwangil
Published: December 2010

Let R be a ring and let Lγ(R) be the lattice of right ideals. We define that I ∊ Lγ(R) is a prime right ideal provided that if AB⊆I for some A, B in Lγ(R) such that AI⊆I then either A⊆I or B⊆I. Any prime ideal of a ring R is a prime right ideal and if R is commutative then an ideal is prime if and only if it is a prime right ideal. If R is a ring and a∊R, let aR={x ∊ R | x=ar for some r∊R} and aR1={x ∊ R | x = na+ar for some integer n and r ∊ R}.
A Compact Imbedding Theorem for Functions without Compact Support
Authors: Adams R. A. , Fournier John
Published: December 2010

The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort

1

to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings

2

are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.
Topological Spaces with a Unique Compatible Quasi-Uniformity
Authors: Lindgren W. F.
Published: December 2010

In [ 2 ] P. Fletcher proved that a finite topological space has a unique compatible quasi-uniformity; C. Barnhill and P. Fletcher showed in [1] that a topological space (X, ), with finite, has a unique compatible quasiuniformity. In this note we give some necessary conditions for unique quasiuniformizability.
On the Block Structure of Quartic Designs
Authors: O'Shaughnessy C. D.
Published: December 2010

Raghavarao and Chandrasekhararao [3] introduced a family of PBIB designs having three associate classes known as cubic designs. In this paper a detailed analysis of the case of PBIB designs having four associate classes, which are called quartic designs, is given. Results are obtained pertaining to construction and existence of quartic designs. Moreover, using methods similar to those used by Shah [5], [6], [7], the block structure of certain quartic designs is studied.
R-orders in a Split Algebra have Finitely Many Non-Isomorphic Irreducible Lattices as soon as R has Finite Class Number
Authors: Roggenkamp Klaus W.
Published: December 2010

Let R be a Dedekind domain with quotient field K and ∧ an R-order in the finite-dimensional separable K-algebra A. If K is an algebraic number field with ring of integers R, then the Jordan-Zassenhaus theorem states that for every left A-module L, the set SL(M)={M: M=∧-lattice, KM≅L} splits into a finite number of nonisomorphic ∧-lattices (cf. Zassenhaus [5]).
On Robillard′s Bounds for Ramsey Numbers
Authors: Kalbfleisch J. G.
Published: December 2010

A recent issue of the Bulletin contained a paper by Robillard [13] in which results from the theory of confounded factorial designs were used to obtain some lower bounds for Ramsey numbers. We shall derive, by more elementary methods, bounds which are much better than Robillard's in every example which he considered.
On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals
Authors: Koh Kwangil , Mewborn A. C.
Published: December 2010

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that

I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.
On Conway's Conjecture for Integer Sets
Authors: Spohn William G.
Published: December 2010

Let A be a finite set of integers {ai} and A + A denote {ai + aj} with p different members and A — A denote {ai—aj) with m different members, the interesting conjecture of Conway [1] states that p < m, unless A is symmetric.
On a Problem of Random Walk in Space(1)
Authors: Elazar R. , Gutterman M.
Published: December 2010

The following theorem is well known [1, p. 66]: Suppose that, in a ballot, candidate P scores p votes and candidate Q scores q votes, where p > q. The probability that throughout the counting there are always more votes for P than for Q, equals (p-q)/(p+q).
A Generalization of the Concept of a Ring of Quotients
Authors: Luedeman John K.
Published: December 2010

Sanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.
An Existence Result for Stepanoff Almost-Periodic Differential Equations
Authors: Zaidman S.
Published: December 2010

In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.
Periodic Solutions For ẋ = Ax + G(x, t) + ∊p(t)
Authors: Ponzo Peter J.
Published: December 2010

We wish to establish the existence of a periodic solution to

1

where x, g and p are n-vectors, A is an n × n constant matrix, and ∊ is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when ∊ = 0.
Some Theorems on a Volterra Equation of the Second Kind
Authors: Thompson D. E.
Published: December 2010

In this paper we state and prove three theorems on positive solutions of a Volterra equation of the second kind. The equation considered is

1

where K(x, t) is a Volterra type kernel, that is, K(x, t) = 0 for t > x.
On the Compactification of Products
Authors: Todd Christopher
Published: December 2010

Let {Xa, a ∊ A} be a family of completely regular Hausdorff spaces, {βXa} the corresponding family of their Stone-Čech compactifications and ΠaXathe usual topological product. The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1].
Remarks on a Problem of Moser
Authors: Chvátal V.
Published: December 2010

Let M(n) be the set of all the points (x1, x2,…, xn)∈En such that xi∈{0,1,2} for each i= 1, 2,…, n and let f(n) be the cardinality of a largest subset of M(n) containing no three distinct collinear points. L. moser [4] asked for a proof of the inequality

Let us consider the set Sn of those points (x1x2,…, xn)∈M(n) which satisfy |{i:Xi= 1}| = [(n +1)/3]. As Sn is a subset of the sphere with center at (1, 1,…, 1) and radius (n-[(n+1)/3])1/2, no three distinct points of Sn are collinear. Thus we have

1
On Some Applications of Graph Theory III
Authors: Erdös P. , Meir A. , Sós V. T. , Turán P.
Published: December 2010

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.

Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].
R1, Pairwise Compact, and Pairwise Complete Spaces
Authors: Richardson G. D.
Published: December 2010

The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.

The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.
On GL2(R) Where R is a Boolean Ring
Authors: Rosenstein Joseph G.
Published: December 2010

In this paper we characterize the 2 × 2 invertible matrices over a Boolean ring, and, using this characterization, show that every invertible matrix has order dividing 6. This suggests that GL2 of a Boolean ring is built up out of copies of the symmetric group S3. Indeed, if B is a finite Boolean ring, then GL2(B) turns out to be a direct sum of copies of S3. If B is infinite, then GL2(B) is more difficult to calculate; we present here descriptions of GL2(B) for the "extreme" cases of countable Boolean rings—namely, the Boolean ring which is generated by its atoms and the atomless Boolean ring. The former provides a negative answer to the question of whether the functor GL2(⋅) preserves inverse limits; the latter is a corollary of a theorem which states that, under certain circumstances, GL2(⋅) preserves direct limits. It turns out, in addition, that every invertible matrix is a product of elementary ones, as is the case for matrices over a Euclidean domain.
Identities in Categories
Authors: Herrlich H. , Ringel C. M.
Published: December 2010

In [4] Hatcher introduced the notion of an identity in an arbitrary category and proved a characterization of quasivarietal subcategories which is similar to Birkhoff's characterization of varietal subcategories in universal algebra. The aim of this note is to show that the theorem of Hatcher as well as the categorical generalization of Birkhoff's theorem are special cases of a "relative" theorem, formulated with respect to a projective structure.
On a Problem of P. Erdös
Authors: Ruzsa I.
Published: December 2010

P. Erdös asked the following problem: Does there exist an infinite sequence of integers a1<…satisfying for every x≥1

1

so that every integer is of the form 2k+ai [1]. The analogous questions can easily be answered affirmatively if the powers of 2 are replaced by the rth power.

In this note we give a simple affirmative answer to the problem of Erdôs. Let c2 be a sufficiently small absolute constant. Our sequence A consists of all the integers of the form

2
Another Weak Stone-Weierstrass Theorem for C*-Algebras
Authors: Elliott George A.
Published: December 2010

The purpose of this article is to present a new generalization of the classical Stone-Weierstrass theorem for commutative C*-algebras.

Under the assumption that B is a sub-C*-algebra of A separating the pure states of A and zero, Kaplansky has conjectured that B=A [4, p. 246]. He gave a proof for the case that A is postliminary ([4, Theorem 7.2]; see also [2, 11.1.8]). Glimm, Akemann, and Sakai have established the conjecture in the presence of various other additional hypotheses, most of which hold in the commutative case ([3], [1], [7]).
A Note on Unconditional Bases
Authors: Holub J. R. , Retherford J. R.
Published: December 2010

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that

1.1

convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by

1.2

where then fi(xj) = δij and fi∊X* for each positive integer i.

A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).
Free Finitary Algebras in a Cocomplete Cartesian Closed Category
Authors: Howlett C. , Schumacher D.
Published: December 2010

In [2] Volger proved that the underlying functor of a category of set-valued models of an r-ary theory has a left adjoint. We want to show that his proof remains valid if instead of set valued models of an r-ary theory models of a finitary theory with values in an arbitrary cocomplete cartesian closed category are considered. As Volger for sets we show for any cocomplete cartesian closed category T C that for every finitary theory (S being a skeleton of the full subcategory of finite sets) the restriction of the left adjoint of on C(s) is a functor in ; here brackets around the exponent indicate as usual a restriction to functors which preserve finite products. We are very much indebted to the referee for pointing out that our proof of the last statement is only based on the properties of C mentioned above and the fact that S has and T preserves finite products. With this in mind and retaining only that part of the cartesian closedness which is relevant for the following considerations we can state the following.
A Note on an Iterative Test of Edelstein(1)
Authors: Nadler Sam B.
Published: December 2010

Let (X1, d1) and (X2, d2) be metric spaces. A mapping/: X1→X2 is said to be a Lipschitz mapping (with respect to d1 and d2) if and only if (*)d2(f(x), f(y))≤λ⋅ d1(x, y) for all x, y∈X1, where λ is a fixed real number. The constant λ is called a Lipschitz constant for f. If (*) is satisfied for λ=l, then f is called nonexpansive (see, for example, [21]) and if (*), again with λ=1, is replaced by a strict inequality for all x≠y, then f is called contractive [1]. If x∊X1 and X1=X2, then the sequence where f1(x)=f(x) and fn(x)=f(fn-1(x)) for each n>1, is called the sequence of iterates of f at x.
Summability Tests for Singular Points
Authors: Hartmann F. W.
Published: December 2010

King [5] devised two tests for determining when z = 1 is a singular point of the function f(z) defined by

1

having radius of convergence equal to one. The point z = 1 and radius of convergence one may be chosen without loss of generality.
The Primality of N=2A3n-1
Authors: Williams H. C.
Published: December 2010

Lehmer [3] and Reisel [7] have devised tests for determining the primality of integers of the form A2n—1. Tables of primes of these forms may be found in [7] and Williams and Zarnke [10]. Little work, however, seems to have been done on integers of the form N=2A3n—1. Lucas [6] gave conditions that were only sufficient for the primality of N. Recently Lehmer [4] has given a method for determining the primality of an integer N if the factorization of N+1 is known.
When are Quasi-Injectives Injective?
Authors: Byrd K. A.
Published: December 2010

We call a ring R (associative and with identity) for which every quasi-injective right R-module is injective a QII-ring. Similarly R is called an SSI-ring when every semisimple right R-module is injective. Clearly every semisimple, artinian ring is a QII-ring and every QII-ring is an SSI-ring. One then asks whether these inclusions among classes of rings are proper. The purpose of this note is to point out an instance when SSI implies QII It is then easy to see that an example of Cozzens shows that the class of QII-rings properly contains the class of semisimple, artinian rings.
A Singular Inverse of a Matrix by Rank Annihilation
Authors: Ahsanullah M. , Rahman M.
Published: December 2010

Edelblute [1] has given a method of finding an inverse of a nonsingular matrix by rank annihilation. The purpose of this paper is to show that the method can be extended in the case of a singular matrix. This method will produce a singular inverse satisfying condition (3) of Penrose [3].
A Fixed Point Theorem for Weakly Uniformly Strict Contractions(1)
Authors: Assad Nadim A.
Published: December 2010

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:

Given ∊ > 0, there exists δ > 0 such that

(A)
Linear Inequalities Over Complex Cones
Authors: Berman Abraham
Published: December 2010

The basic solvability theorems of Farkas [2] and Levinson [4] were recently extended in different directions by Ben-Israel [1] and Kaul [3].

The theorem stated in this note generalizes both results of Ben-Israel and Kaul and is applicable to nonlinear programming over complex cones.
Une Remarque sur les Monoïdes de Quotients Généralisés
Authors: Berthiaume Pierre
Published: December 2010

Dans [2] nous avons introduit, en nous inspirant des notions équivalentes pour les anneaux [3], les notions de monoïde maximal Q(S) et de monoïde complet B(S) des quotients d'un monoïde donné S, la premiére construction n'étant valide que lorsque S est commutatif, et nous avons montré que dans ce cas, on a toujours les inclusions (en général strictes) de monoïdes suivantes: S ⊂ Q(S) ⊂ B(S).
On the Central Series of a Ring
Authors: Biggs R. G.
Published: December 2010

The study of group types was completed by Meldrum [1]. The concept of ring type described here is based on analogous definitions.

The series R = R0 ⊃ R1 ⊃ … ⊃ Rα = Rα+1is the lower central series for the ring R if Rγ+1=RRγ + RγR for ordinal number γ and Rγ∩δ<γ Rδ if γ is a limit ordinal. The upper central series for R is the series 0=J0 ⊂ J1 ⊂ … ⊂jβ = Jβ+1 where Jγ+1={x ∊ R:xR + Rx ⊆ Jγ} for every ordinal number γ and Jγ = ∪ ∩δ<γ Jδ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which Jβ = Jβ+1.The length of the lower central series is defined similarily. We shall say the ring has type (β,α) if the length of the upper central series is β and the length of the lower central series is α.
Densities and Measures of Linear Sets
Authors: Kelly B. G. A.
Published: December 2010

If In, I∞ denote the intervals [0, n), [0, ∞) respectively, we propose to examine the properties of an upper and lower density

(1)

(1)

of a set S belonging to the power set of I∞ where m*,m* denote the outer and inner (linear) Lebesgue measures.
Bernstein's Inequality in the Bivariate Case
Authors: Mullen Kenneth
Published: December 2010

If (Xl, X2,…, Xn), is a set of n independent random variables, such that EXi=0, Var and if t is a real positive number and , then Bernstein [2] has given an upper bound for Pr when the X's are bounded. The best English language discussion of Bernstein's work is probably by Bennett [1].
On the Simple Group of J. Tits
Authors: Parrott David
Published: December 2010

In the series of simple groups 2F4(q),q =2 2m+1, discovered by Ree, Tits [4] showed that the group 2F4(2) was not simple but contained a simple subgroup of index 2. In this note we extend the characterization of obtained by the author in [3].
Totally Multiplicative Functions in Regular Convolution Rings
Authors: Yocom K. L.
Published: December 2010

McCarthy [4] generalized a necessary and sufficient condition for an arithmetic function to be totally multiplicative to the incidence algebra on a partially ordered set. Several equivalent conditions for an arithmetic function to be totally multiplicative are known [1], [2]. In this paper we generalize several of these (and some apparently new ones) to the regular convolution rings of Narkiewicz [5]. We also investigate the prime factorization of arithmetic functions in a certain subring of some of these regular convolution rings.
On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type
Authors: Zischka K. A.
Published: December 2010

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.

The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].