bluuebunny/crossref_metadata_2025 huggingface dataset:
A Note on the Generalized (Positive) Cauchy Distribution
Authors: Rao B. Raja , Garg M. L.
Published: December 2010

In this note, a certain generalization of the Cauchy distribution is obtained, using the result of Malik [2].

The generalized gamma distribution having the density

(1)

is introduced by Stacy [1], who studied some of its properties. As remarked by Stacy [1], the familiar gamma, chi, chi-squared, exponential and Weibull distributions are special cases of (1), as are the distributions of certain functions of a normal variable - viz.
The Group of the Quadratic Residue Tournament
Authors: Goldberg Myron
Published: December 2010

A tournament Tn is a set of n nodes a1a2, …, an such that every pair (ai, aj) of distinct nodes is joined by exactly one of the oriented edges or . If is in Tn, then we say that ai dominates aj and write ai → aj.

The (automorphism) group G(Tn) of a tournament Tn is the group of all permutations ϕ of the nodes of Tn such that ϕ(a)→ϕ(b) if and only if a → b. It is known [9] that there exist tournaments whose group is abstractly isomorphic to a given group H if and only if H has odd order; thus all tournament groups are solvable, by the Feit-Thompson Theorem [7].
A Note on a Theorem of H. L. Abbott
Authors: Douglas Robert J.
Published: December 2010

Let In be the graph of the unit n-dimensional cube. Its 2n vertices are all the n-tuples of zeros and ones, two vertices being adjacent (joined by an edge) if and only if they differ in exactly one coordinate. A path P in In is a sequence x1, …, xm of distinct vertices in In where xi is adjacent to xi+1 for 1 ≤ i ≤ m-1; P is a circuit if it is also true that xm and x1 are adjacent. A path is Hamiltonian if it passes through all the vertices of In. Finally, for vertices x and y in In, we define d(x, y) to be the graph theorectic distance between x and y, i.e., the number of coordinates in which x and y differ.
The Automorphisms of an Algebraically Closed Field
Authors: Charnow A.
Published: December 2010

It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k.

The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.
On Knock-Out Tournaments
Authors: Capell P. , Narayanan T. V.
Published: December 2010

We define, as in [2], a random knock-out tournament with n players as a vector (m1, m2, …, mk) of positive integers satisfying

1

On the first round of the tournament 2m1 players, chosen at random, are paired off randomly; the remaining n — 2m1 players have a "bye". The m1 losers are knocked out, leaving a tournament n—m1 players with vector (m2, m3, …, mk).
A Remark on a Theorem of Lyapunov
Authors: Wong James S. W.
Published: December 2010

Consider the linear ordinary differential equation

1

where x ∊ En, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:

Theorem (Lyapunov). The following three statements are equivalent:

(I) The spectrum σ(A) of A lies in the negative half plane.

(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies

2

where ∥ ∥ denotes the Euclidean norm.

(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there
exist q1,q2>0 such that

3

satisfying

4

where I is the identity matrix.
Averages Involving Fourier Coefficients of Non-Analytic
Automorphic Forms
Authors: Rao V. Venugopal
Published: December 2010

Let f(τ) be a complex valued function, defined and analytic
in the upper half of the complex τ plane (τ=x+iy, y
> 0), such that
f(τ+λ) =
f(τ) where λ is real
and f(-1/τ) = γ(-iτ)k f(τ), k being a complex number.
The function (—iτ)k is defined as ek log(-iτ) where log(—iτ) has the real value when — iτ is
positive and γ is a complex number with absolute value 1. Such functions
have been studied by E. Hecke [4] who calls them functions with signature
(λ, k, γ). We further assume that f(τ) =
O(|y| -c) as y tends to zero uniformly for all x, c being
a positive real number. It then follows that f(τ) has a Fourier expansion of
the type f(τ) = a0 + Σ an exp(2πinτ/λ) (n = 1,2,…), the series being
convergent absolutely in the upper half plane.
On Sprott's Example of an Ancillary Statistic
Authors: Tan Peter
Published: December 2010

For statistical inference on the unknown parameter θ of a probability
distribution with a known form of its density function f(x | θ), the late
Sir R. A. Fisher suggests that inference should be based on the observed
value of a sufficient statistic when it exists. In the absence of a
sufficient statistic, fiducial probability statements about θ can still be
made, according to Fisher, if we can find an ancillary statistic so that
"the most likely estimate could be made exhaustive by means of the ancillary
values" [1, p. 138].
Lower Bounds for the Ramsey Numbers
Authors: Robillard Pierre
Published: December 2010

A lower bound for a family of Ramsey numbers is derived using a geometrical
argument.
Power-Associative Algebras in which Every Subalgebra is a Left
Ideal
Authors: Rodabaugh D. J.
Published: December 2010

By an L-algebra we mean a power-associative nonassociative
algebra (not necessarily finite-dimensional) over a field F
in which every subalgebra generated by a single element is a left ideal. An
H-algebra is a power-associative algebra in which every
subalgebra is an ideal. The H-algebras were characterized
by D. L. Outcalt in [2]. Let Sα be the semigroup with cardinality
α such that if x, y ∊ Sα then xy = y. Consider the
algebra over a field F with basis Sα. Such an algebra is an L-algebra that is not
an H-algebra unless Sα contains only one element. In this paper we will prove that an
algebra A over a field F with char. ≠ 2 is an
L-algebra if and only if it is either an
H-algebra or has a basis Sα where α is the dimension of A.
Sur une Conjecture de R. M. Robinson
Authors: Giroux André
Published: December 2010

Dans un récent article [2, p. 379], R. M. Robinson a émis une conjecture
intéressante sur le diamètre transfini des ensembles compacts et symétriques
du plan. La conjecture est la suivante:

Soit E un ensemble compact de points du plan, symétrique par rapport
á l'axe réel R. Pour n ≥ 2, choisissons n points z1, …,
zn de E et posons xi = zi,   pour i=1, 2,…, n.

Soient:
On the Noncentral Distributions of the Second Largest Roots of Three Matrices in Multivariate Analysis
Authors: Al-Ani Sabri
Published: December 2010

The central distribution of the second largest (smallest) root following the Fisher-Girshick-Hsu-Roy distribution under certain null-hypothesis has been derived in series form by Pillai and Al-Ani [6]. In this paper the noncentral distributions of the second largest roots in the MANOVA situation, the canonical correlation, and equality of two covariance matrices are obtained. Further, the distribution of the second largest root of the covariance matrix is obtained as a limiting case. The largest root and its noncentral distributions have been considered already by Pillai and Sugiyama [7] for the situations stated above. However, in the present paper, the joint densities of the largest and the second largest roots are derived in all the above cases from which the distributions of the largest roots can be obtained, although in more elaborate forms.
On a Problem in Partial Difference Equations
Authors: Long Calvin T.
Published: December 2010

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.

Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.
A Note on Undirected Graphs Realizable as P.O. Sets
Authors: Haff C. E. , Murty U. S. R. , Wilton R. C.
Published: December 2010

Let (P, ≥) be a p.o. set. The basis graph of (P, ≥) is defined to be the directed graph whose vertex set is P and in which the ordered pair 〈a, b〉 is an edge if and only if b covers a in (P, ≥).

Let D be a directed graph. All graphs considered in this note are finite and are free of loops and multiple edges.
The Rank of the Sum of Two Rectangular Matrices
Authors: Murphy Ian S.
Published: December 2010

In what follows, the transposed complex conjugate of a complex rectangular matrix D is denoted by D* and the rank of D by r(D). Meyer [1] proved the following result using generalized inverses:

Theorem. Let A and B be complex m × n matrices such that AB*=B*A=0. Then r(A+B) = r(A)+r(B).

Below we prove this result by repeated use of the fact that for every complex m × n matrix D we have r(D) = r(D*D) = r(DD*) (e.g. See [2] Theorem 5.5.4).
A Theorem on Involutions on Cyclic Peano Spaces
Authors: Hunt J. H. V.
Published: December 2010

The purpose of this note is to prove that an involutionf on a cyclic Peano space S leaves some simple closed curve in Ssetwise invariant.

We shall first define the required terms. A Peano space is a locally compact, connected and locally connected metric space. A connected space is called cyclic if it has no cut-point. An involution on a space is a periodic mapping whose period is 2; it is necessarily a homeomorphism. A mapping f: X → X is said to leave a subset E of S setwise invariant if f(E) = E. These definitions may be found, for example, in [2].
On Y. C. Wong's Conjecture
Authors: Datta D. K.
Published: December 2010

Let M be an n-dimensional connected C∞ manifold with a linear connection Γ. M is said to be of recurrent curvature with respect to Γ if the corresponding curvature tensor R satisfies [1], [4]

where Δ denotes covariant derivative with respect to Γ and W is a nonzero covector called the recurrence co-vector. Let T be the torsion of Γ.
On Free Groups of the Variety AN2 ∧ N2A
Authors: Gupta Chander Kanta
Published: December 2010

Let R be a commutative ring with unity and let M(R) be the multiplicative group of 4 x 4 triangular matrices (aij) over R, where a11 is a unit element of R and aii = 1 for i = 2, 3, 4. If V(=AN2 ∧ N2A) denotes the variety of groups which are both abelian-by-class-2 and class-2-by-abelian, then it is routine to verify that M(R) ∊ V. Here we prove the following,

Theorem. Let F(V) denote the free group of finite or countable infinite rank of the variety V. Then for a suitable choice of R, F(V) is embedded in M(R).
The Weak Weak Category of a Space
Authors: Hoo C. S.
Published: December 2010

Let X be a topological space. We say that cat X ≤ n if there exists a map ϕ: X → T1(X, …, X) such that jϕ≃Δ: X → Xn+1, where T1(X, …, X) is the “fat wedge”, j is the inclusion and Δ is the diagonal map. This is an example of a right structure system. This right structure system leads to an associated weak structure system, namely weak category in this particular case.
Orthogonal Recurrence Polynomials and Hamburger Moments
Authors: Law A. G.
Published: December 2010

A three-term recurrence

1

where An (n ≥ 0), Bn (n ≥ 0) and Cn (n ≥ 0) are real numbers for which AnCn+1 ≠0 (n ≥ 0), generates a sequence {Pn} of real polynomials in which Pn has degree exactly n.
The Generalized Singular Direct Product for Quasigroups
Authors: Lindner Charles C.
Published: December 2010

In [2], A. Sade gives a construction for quasigroups which he calls the singular direct product. In this paper we generalize Sades' construction. As an application we obtain a recursive construction for quasigroups orthogonal to their transposes. All quasigroups considered in this paper will be finite.
Elementary Remarks on Multiply Monotonic Functions and Sequences(1)
Authors: Muldoon M. E.
Published: December 2010

A function f(x) is said to be completely monotonic on (0, ∞) if

1

Familiar examples of such functions are given by f(x) = exp(—αx) and f(x) = (x+β)-α, where α ≥ 0, β ≥ 0. A discussion of completely monotonic functions is given in [5, Ch. IV].
On Covering the Unit Ball in Normed Spaces
Authors: Connett J.
Published: December 2010

By compactness, the unit ball Bn in Rn has a finite covering by translates of rBn, for any r > 0. The main theorem of this note shows that a weaker covering property does not hold in any infinite-dimensional normed space.
Paracompactness in Small Products
Authors: Willard S.
Published: December 2010

All spaces in this note are regular, Hausdorff topological spaces.

At the topology conference in Pullman, Washington, in March 1970, E. Michael posed the following question: if X is Lindelôf Y is separable and metrizable and I × Y is paracompact, must X × Y be Lindelöf?
Some Combinatorial Theorems on Monotonicity
Authors: Chvátal V. , Komlόs J.
Published: December 2010

P. Erdös and G. Szekeres [1] proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex polygon, thus answering a question due to Miss Esther Klein (who later became Mrs. G. Szekeres).
Remarks on Rings of Quotients of Rings of Functions
Authors: Gonshor Harry
Published: December 2010

This paper was inspired by [4]. The main result there suggests a close relationship between injective hulls of C* algebras as studied in [2] and [3] and rational completions as studied in [1]. We shall prove an analogue of Theorem 1 in [3] for rational completions. The latter theorem states that the injective hull of the algebra C(X) of all complex valued functions on the compact Hausdorff space X is the algebra B(X) of all bounded Borel functions modulo sets of first category.
Rings of Formal Power Series
Authors: Sankaran N.
Published: December 2010

In this brief exposition we collect several results on rings of formal power series with coefficients from a field or a ring with some special properties. The results that are catalogued below are mostly algebraic in nature.
An Approximately Continuous Perron Integral
Authors: Kubota Y.
Published: December 2010

J. C. Burkill [1] has defined the AP-integral whose indefinite integral is approximately continuous. An (approximately continuous) function which is approximately derivable at all points of an interval is necessarily an indefinite AP-integral of its approximate derivative.
The Distinct Zeros of the Product of a Polynomial and its Successive Derivatives
Authors: Rahman Q. I.
Published: December 2010

It has been conjectured that if p(z) is a polynomial of degree n then the product P(z)=p(z)p'(z)p"(z)…p(n-1)(z) has at least n+1 distinct zeros unless p(z)=c(z—a)n. Professor P. Erdös who mentioned this problem in a lecture at the University of Montreal attributed it to Tiberiu Popoviciu.
On the Representation of Modules by Sheaves of Factor Modules
Authors: Lambek J.
Published: December 2010

Throughout this paper we consider associative rings with unity elements. In §1 various results on the representation of rings by rings of sections of special rings are compared. In particular, it is shown that results enunciated by Dauns and Hofmann, Koh, and the present author may all be deduced from one statement, the proof of which appears in §3.
Note on the Number of Solutions of f(x1) = f(x2) = … =f(xr) over a Finite Field
Authors: Williams Kenneth S.
Published: December 2010

Let GF(q) denote the finite field with q = pn elements and let

(1)

where each ai ∊ GF(q) and 1 < d <p. For r=2, 3, …, d we let nr denote the number of solutions (x1, …, xr) over GF(q) of

(2)

for which x1, …, xr are all different. Birch and Swinnerton-Dyer [1] have shown that

(3)

where each vr is a nonnegative integer depending on f and q and the constant implied by the O-symbol depends here, and throughout the paper, only on d.
Q-Divisible Modules
Authors: Armendariz Efraim P.
Published: December 2010

Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].
Subdirectly Irreducible DQC Rings
Authors: Burgess W. , Chacron M.
Published: December 2010

Twenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.
On Möbius Functions and a Problem in Combinatorial Number Theory
Authors: Lindström Bernt
Published: December 2010

After the publication of the important paper by Rota [9] on Möbius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Möbius functions is now recognized as a valuable tool in combinatorial and arithmetical research.
The Lattice of Equational Classes of Semigroups with Zero
Authors: Nelson Evelyn
Published: December 2010

In contrast to the very complicated structure of the lattice of equational classes of commutative semigroups (see [5]), the lattice of equational classes of commutative monoids (semigroups with unit) is isomorphic with N × N* with a unit adjoined, where N is the lattice of natural numbers with the usual order and N* is the lattice of natural numbers ordered by division. (See [4].) However, the lattice of equational classes of commutative semigroups-with-zero is not so simple to describe.
A Lattice Point Problem Related to Sets Containing No l-Term Arithmetic Progression
Authors: Riddell J.
Published: December 2010

In 1927 van der Waerden [6] proved that given positive integers k and l, there exists an integer W such that if 1, 2, …, W are partitioned into k or fewer classes, then at least one class contains an l-term arithmetic progression (l-progression). Let W(k, l), be the smallest such integer W. It would be of interest to find a reasonable upper estimate for W(k, l), say one that could be written down.
Variations on the Hamiltonian Theme
Authors: Bondy J. A.
Published: December 2010

As its name implies, this paper consists of observations on various topics in graph theory that stem from the concept of Hamiltonian cycle. We shall mainly adopt the notation and terminology of Harary [5]. However, we use vertices and edges for what are called "points" and "lines" in [5]. V(G), E(G) respectively will denote the sets of vertices and edges of graph G, and |X| will denote the cardinal of the set X.|V(G)| is the order of G, and |E(G)| the size of G. Throughout n is reserved for the order of G.
A Constructive Definition of the Approximately Continuous Denjoy Integral
Authors: Kubota Y.
Published: December 2010

The author has defined [2] the approximately continuous Denjoy integral (AD-integral) which includes exactly the general Denjoy integral and the AP-integral defined by Burkill [2].

The aim of this paper is to give a constructive definition of the AD-integral.
Certain Extensions of the Mehler Formula
Authors: Singhal J. P.
Published: December 2010

For the Hermite polynomials Hn(z) defined by

1

it is easy to see from the Rodrigues formula that

2

where, as usual, Dz=d/dz.

In recent papers ([1], [2]) Carlitz has proved the following formulae:

3

4

where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).
The Generalized Equations of Bisymmetry Associativity and Transitivity on Quasigroups
Authors: Taylor M. A.
Published: December 2010

The generalized equations of bisymmetry, associativity and transitivity are, respectively,

(1)

(x1y)2(z3u) = (x4z)5(y6u)

(2)

(x1y)2z = x3(y4z)

(3)

(x1z)2(y3z) = x4y.

The numbers 1, 2, 3,…, 6 represent binary operations and x, y, z and u are taken freely from certain sets.

We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.)
A Compactification due to Fell
Authors: Wulfsohn Aubrey
Published: December 2010

We give an alternative construction of a Hausdorff compactification due to Fell [2]. We say that a space is compact if it has the Heine-Borel property, locally compact if each point has a fundamental system of compact neighbourhoods. The interesting spaces from the point of view of this paper, are the non-Hausdorjf ones since for locally compact Hausdorff spaces Fell's compactification is the usual one-point compactification. The motivation for the compactification comes from the theory of continuous fields of C*-algebras: the primitive spectrum of a C*- algebra A is a locally compact T0 space X and Fell [3] realizes A as an algebra of fields of operators over the compactification of X. This note is based on a discussion of the author with Professor Fell.
Invariance Theorems for First Passage Time Random Variables
Authors: Basu A. K.
Published: December 2010

LetX1X2,… be i.i.d. r.v. withEX=μ>0, andE(X-μ)2= σ2<∞.

LetSk=X1+…+Xkandvx=max{k:Sk≤x}, x≥0 andvx=0 ifX1>x. Billingsley [1] proved if X1≥0 then

converges weakly to the Wiener measure W.

Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove that

converges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.
σ-Reflexive Semigroups and Rings
Authors: Chacrono M. , Thierrin G.
Published: December 2010

We shall call a semigroup S a σ-reflexive semigroup if any subsemigroup H in S is reflexive (i.e. for all a, b ∈ S, ab ∈ H implies ba∊H ([2], [5]). It can be verified that any group G is a σ-reflexive semigroup if and only if any subgroup of G is normal. In this paper, we characterize subdirectly irreducible o-reflexive semigroups. We derive the following commutativity result: any generalized commutative ring R ([1]) in which the integers n=n(x, y) in the equation (xy)n = (yx)m can be taken equal to 1 for all x, y ∈ R must be a commutative ring.
Remarks Concerning Uniformly Bounded Operators on Hilbert Space
Authors: Istrǎțescu I.
Published: December 2010

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that

1

is similar to a unitary operator.

The following problem is an extension of this result: If T and S are two operators such that

1.

sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)

2.

TS = ST

then there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?

Also, in [7] B. Sz.-Nagy has proved that every compact operator T such that

sup ‖Tn‖<∞ (n = 1, 2, 3,…)

is similar to a contraction.
On Minimally Thin Sets in a Stolz Domain
Authors: Jackson H. L.
Published: December 2010

Let D denote the open right half plane and

a Stolz domain in D with vertex at the origin. If h is a minimal harmonic function on D with pole at the origin then E⊂D is minimally thin at the origin iff where is the reduced function of h on E in the sense of Brelot. We now define

where s shall be fixed to be 1/e. For the set E∩In we shall let cn denote the outer ordinary capacity (see [1, pp. 320-321]), An the outer logarithmic capacity, and on the outer Green capacity with respect to D. If E⊂K, Mme. Lelong [3, p. 131] was able to prove that E is minimally thin at the origin Since one cannot easily relate the classical measure theoretic properties of a plane set with its Green capacity, it would appear desirable to find some other criteria for minimal thinness.
Existence Theorem for the Initial-Boundary Value Problem for a Singular Parabolic Partial Differential Equation
Authors: Krantzberg Julius A.
Published: December 2010

We consider the initial-boundary value problem for the parabolic partial differential equation

1.1

in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).
Diagonals of Doubly Stochastic Matrices
Authors: Maxwell George
Published: December 2010

Let A be an additive abelian group and Dn(A) the set of those n x n matrices over A all of whose row and column sums are equal. Such matrices can be regarded as a possible generalization of doubly stochastic real matrices; alternatively, if A is a commutative ring, it turns out that Dn(A) is exactly the image of the permutation representation of Sn, the symmetric group of degree n, over A.
Distribution of the Sum of Variates from Truncated Discrete Populations
Authors: Ehsanes Saleh A. K. MD. , Rahim M. A.
Published: December 2010

Distributions often arise in practice where one or more values of the variate are unobserved. Various practical problems have been referred by Finney [1], David and Johnson [4], Bliss and Fisher [3]. It is of interest to know the exact distribution of the sum of variâtes from such truncated discrete population. In this paper, utilising the property of characteristic function certain general results are shown. The distribution of the sum of independent variables from a discrete population, truncated by any set of s distinct values, follows from them immediately. Using these results the exact distributions of the sum, from binomial, poisson, negative binomial and geometric population, truncated from anywhere, are derived.
Discontinuity Conditions on Transformation Groups
Authors: Thompson D. V.
Published: December 2010

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].

The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.
Almost Automorphic Integrals of Almost Automorphic Functions
Authors: Zaki M.
Published: December 2010

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:

A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that

(i) exits for every real t but no kind of uniformity of convergence is stipulated;

(ii) exits for every t;

(iii) for every t.
A Note on Group Rings of Certain Torsion-Free Groups
Authors: Burns R. G. , Hale V. W. D.
Published: December 2010

As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.
When Does Rank(A+B)=Rank(A)+Rank(B)?
Authors: Marsaglia G. , Styan G. P. H.
Published: December 2010

In a recent note in the Bulletin, Murphy [5] gave a short proof that for complex m×n matrices A and B, r(A+B)=r(A)+r(B) if the rows of A are orthogonal to the rows of B and the columns of A are orthogonal to the columns of B. His proof was elegant and simple, an improvement on an earlier proof of the same result by Meyer [4].
The Laurent Expansion Without Cauchy's Integral Theorem
Authors: Beesack Paul R.
Published: December 2010

Since Cauchy's time the theory of analytic functions of a complex variable has depended on complex integration theory, and in particular on the fundamental integral theorem (1825) and integral formulas bearing his name. Cauchy defined an analytic function to be one which had a continuous first derivative in a region D, and showed that an analytic function had derivatives of all orders in D. It was not until 1900, with E. Goursat's famous proof of Cauchy's integral theorem, that the continuity of the first derivative could be inferred from its mere existence at all points of D.
The Group of Units of the Integral Group Ring ZS3
Authors: Hughes I. , Pearson K. R.
Published: December 2010

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is

(i) abelian and the order of each element divides 4, or

(ii) abelian and the order of each element divides 6, or

(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.
On the Finite Two-Dimensional Linear Groups II.(1)
Authors: Lorimer Peter
Published: December 2010

A group G is called a T3-group if it contains subgroups K and H, HΔK, with the property that if g and gb are members of G—K there is exactly one h∊H which satisfies the equation gh=gb. In these circumstances (G, K, H) is called a T3-triple.
Permutation Functions on a Finite Field
Authors: Bruen Aiden
Published: December 2010

Using a well-known theorem of Burnside on permutation groups of prime degree we offer new and simplified proofs of Theorems A, B, B' below for the case q=p a prime.
Note on a Subalgebra of C(X)
Authors: Nel L. D. , Riordan D.
Published: December 2010

C(X) (resp. C*(X)) will denote as usual the ring of all (resp. all bounded) continuous functions into the real line R. Define C#(X) to consist of all f∊C(X) whose image M(f) in the residue class ring C(X)jM is real for every maximal ideal M in C(X). Then C# shares with C* the property of being an intrinsically determined subalgebra of C. Then C* shares with C* the property of being an intrinsically determined subalgebra of C. The compactification corresponding to C# (as uniformity determining subalgebra of C*) is thus also an intrinsically determined one. We show that this compactification is well known and "natural" in the cases of several elementary spaces X.
On the Determination of the Maximum Order of the Group of a Tournament
Authors: Alspach B. , Berggren J. L.
Published: December 2010

Let denote the automorphism group of the tournament T. Let g(n) be the maximum of taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of Sn of maximum odd order where Sn denotes the symmetric group of degree n.
Flip-Flops in Hypohamiltonian Graphs
Authors: Chvátal V.
Published: December 2010

Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.
On Density of Fourier Coefficients
Authors: Siddiqi Rafat N.
Published: December 2010

Letfbe anLintegrable real valued function of period 2π and let

(1)

be its Fourier series. It is known that iffis of bounded variation then allnanandnbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] whereV(f) is the total variation off. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.
Strong Oscillation of Elliptic Equations in General Domains
Authors: Swanson C. A.
Published: December 2010

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation

(1)

in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.
Determinateness and the Pasch Axiom
Authors: Adler Andrew
Published: December 2010

Let E be the (nonelementary) plane Euclidean geometry without the Pasch axiom. (The Pasch axiom says that a line cutting one side of a triangle must also cut another side. A full list of axoms for E is given in [5].) E satisfies in particular the full second-order continuity axiom.

Szczerba [5] has recently shown using a Hamel basis for the reals over the rationals that there exists a model of E not satisfying the Pasch axiom. It is natural to ask whether the axiom of choice plays an essential role in the proof. It will turn out that it does.
Some Functional Stable Limit Theorems
Authors: Beuerman D. R.
Published: December 2010

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,

1

where

and

where L(n) is a function of slow variation; also take S0=0, B0=l.

In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.
On Isomorphisms of Locally Convex Spaces With Similar Biorthogonal Systems
Authors: Bozel F. , Husain T.
Published: December 2010

The relationship between bases and isomorphisms (i.e. linear homeomorphisms) between complete metrizable linear spaces has been studied with great interest by Arsove and Edwards (see [1] and [2]). We prove (Theorem 1) that in the case of B-complete barrelled spaces, similar generalized bases imply existence of an isomorphism. This result was also proved by Dyer and Johnson [4], so we do not give a proof. We show (Theorem 6) that if one assumes that the bases are Schauder and similar, then Theorem 1 holds for countably barrelled spaces. We use Theorem 1 to advantage (Theorems 2-5) to show that one can improve some results due to Davis [3].
A Generalization of a Fixed Point Theorem of Reich
Authors: Hardy G. E. , Rogers T. D.
Published: December 2010

The following theorem is the principal result of this paper.

Let (M, d) be a metric space and T a self-mapping of M satisfying the condition for x,y ∊ M

1

where a, b, c, e,f are nonnegative and we set α=a+b+c+e+f.
Some Comments on Quantiles and Order Statistics
Authors: Ramachandramurty P. V. , Rao M. Sudhakara
Published: December 2010

A new concept—that of pseudoconsistency—which seems to be particularly appropriate for the estimation of a quantile is introduced. It is shown without any conditions whatsoever on the underlying distributionthat the sample quantile is strongly pseudoconsistent for the corresponding population quantile. The asymptotic distribution of the sample quantiles and order statistics is derived when the underlying distribution is discrete.
The Cardinalities of A+A and A-A
Authors: Stein Sherman K.
Published: December 2010

H. T. Croft's Research Problems, 1967, contains the following problem due to J. H. Conway. "Let A = {a1, a2,…, aN} be a finite set of integers, and define

and

Prove that A—A always has more members than A+A, unless A is symmetric about 0."Marica in [1] showed that the conjecture is false for the set A = {1, 2, 3, 5, 8, 9, 13, 15, 16}. In this case A+A has 30 elements and A—A has 29 elements.
The Kloosterman Sum Revisited
Authors: Williams Kenneth S.
Published: December 2010

Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by

(1.1)

where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to the modulus.
On the Expected Number of Visits of a Particle before Absorption in a Correlated Random Walk
Authors: Jain G. C.
Published: December 2010

Let a particle move along a straight line a unit distance during every interval of timeτ. During the first intervalτit moves to the right with probabilityρ1and to the left with probabilityρ2= 1 -ρ1. Thereafter at the end of each intervalτ, the particle with probabilitypcontinues its motion in the same direction as in the previous step and with probabilityq= l -preverses it.
On Certain Quotients of Grothendieck Groups
Authors: Small Charles
Published: December 2010

A general categorical construction is described which has as special cases the construction of the Brauer group of a field and the construction of the Witt ring of a field of characteristic ≠2.
A Note on Semiprime Rings with Torsionless Injective Envelope
Authors: Armendariz Efraim P.
Published: December 2010

Satyanarayana establishes in [6] that a semiprime right selfinjective ring with ACC on annihilator right ideals is semisimple Artinian, thereby extending a similar result of Koh [5] for prime rings. A theorem of Faith [3, Theorem 5.2], shows that the annihilator chain condition on either side implies that a right selfinjective semiprime ring is semisimple Artinian. Noting that any selfinjective ring has torsionless injective envelope we consider the possibility of replacing selfinjectivity by torsionless together with an annihilator condition.
A Note on Semihereditary Rings
Authors: Enochs E.
Published: December 2010

It's well known (see Endo [1]) that for a commutative ring A, if A is semihereditary then w.gl. dim. A ≤ 1. It seems worth recording the noncommutative version of this.
A Note on a Square Type Functional Equation
Authors: Haruki Shigeru
Published: December 2010

The following square functional equation

(1)

was considered (for example [l]-[9]) previously.
Epimorphisms of Modules which must be Isomorphisms
Authors: Djoković D. Ž.
Published: December 2010

Let R be an associative ring (not necessarily with identity).

R is a left П-ring if it has the following property: Let M be a finitely generated left R-module, N a submodule of M and ϕ:N→M an epimorphism. Then ϕ is an isomorphism.
Property (G), Regularity, and Semi-Equicontinuity
Authors: Yang J. S.
Published: December 2010

This note, motivated by [2], [3], and [4], is devoted to an investigation of properties related to equicontinuity in function spaces of topological spaces. In §2, we study the property (G) defined in [3], and the regularity defined in [4]. A sufficient condition for the simultaneous continuity of a function of two variables, which is analogous to a well known result in equicontinuity, is given at the end of the section. In §3, we relate the regularity with the semi-equicontinuity defined in [2], by localizing the semi-equicontinuity in an obvious way which leads us to weaken some of the hypotheses used in [2]. By the way of constructing an example, we also obtained a sufficient condition for a regular semitopological group to be a topological group.
Most Infinitely Differentiable Functions are Nowhere Analytic
Authors: Darst R. B.
Published: December 2010

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.
Rings with A Finitely Generated Total Quotient Ring
Authors: Adams John Conway
Published: December 2010

Let R be a commutative ring with non-zero identity and let K be the total quotient ring of R. We call R a G-ring if K is finitely generated as a ring over R. This generalizes Kaplansky′s definition of G-domain [5].

Let Z(R) be the set of zero divisors in R. Following [7] elements of R—Z(R) and ideals of R containing at least one such element are called regular. Artin-Tate's characterization of Noetherian G-domains [1, Theorem 4] carries over with a slight adjustment to characterize a Noetherian G-ring as being semi-local in which every regular prime ideal has rank one.
Rings with No Nilpotent Elements and with the Maximum Condition on Annihilators
Authors: Cornish W. H. , Stewart P. N.
Published: December 2010

Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.
An Irreducible Representation of sl(2)
Authors: Lemire F.W.
Published: December 2010

In a recent paper [1] MM. Arnal and Pinczon have classified all complex irreducible representations (ρ, V) of sl(2) having the property (P) that there exists a non-zero element x∈sl(2) such that ρ(x) admits an eigenvalue. It is the purpose of this note to demonstrate, by example, that there exist irreducible representations of sl(2) which do not have property (P). As usual, we consider sl(2) embedded in its universal enveloping algebra U and identify the representations of sl(2) and U.
Regularizers of Closed Operators
Authors: Lin C.-S.
Published: December 2010

Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.
Quasi-Uniform Spaces and Topological Homeomorphism Groups(1)
Authors: Seyedin Massood
Published: December 2010

Let X be a topological space and G a subgroup of the homeomorphism group H(X) with the topology of point-wise convergence. It is well-known that if X is uniformizable and G is equicontinuous with respect to a compatible uniformity then G is a topological group. In this paper we show that essentially this same result applies when X is only an R0-space (and hence in particular if X is T1 or regular). A corresponding result for regular spaces has been proved [2].
An Inequality for Positive Semidefinite Hermitian Matrices(1)
Authors: Merris Russell
Published: December 2010

Let A and B be positive semidefinite Hermitian n-square matrices. If A—B is positive semidefinite, write A≥B. Haynsworth [1] has proved that if A≥B then det(A+B)≥det A+n det B.

Let G be a subgroup of the symmetric group, Sn, and let λ be a character on G. Let

where A = (aij) and Er is the rth elementary symmetric function.
On Injective Near-Ring Modules
Authors: Seth V. , Tewari K.
Published: December 2010

Let N be a left near ring and let M be a right N-module. We recall [1] that M is called injective iff every diagram can be embedded into a commutative diagram where A and B are right N-modules with exact.

The purpose of this note is to show that if N is a d.g. near-ring with identity, then M is injective iff for every right ideal u of N and every N-homomorphism f:u→N, there exists an element m in M such that f(a)=ma for all a in u.
Right-Ordered Polycyclic Groups
Authors: Mura Roberta Botto
Published: December 2010

One of the features that make right-ordered groups harder to investigate than ordered groups is that their system of convex subgroups may fail to have the following property:

(*) if C and C’ are convex subgroups of G and C’ covers C, then C is normal in C’ and C’/C is order-isomorphic to a subgroup of the naturally ordered additive group of real numbers.
The Matrix EquationsA = XYZAndB = ZYXand Related Ones
Authors: Brenner J. L. , Lim M. J. S.
Published: December 2010

In [15], O. Taussky-Todd posed the problem of title, namely to findX, Y, ZwhenA, Bare given. Clearly ifX, Y, Zexist thenA, Bare either both invertible or both noninvertible.

In section 1, the problem is reviewed in caseA, Bare both invertible. The problem is seen to be fundamentally one of group theory rather than matrix theory. Application of results of Shoda, Thompson, Ree to the general group-theoretical results allows specialization to certain matrix groups.

In Section 2, examples and counterexamples are given in caseA, Bare noninvertible. A general necessary condition for solvability (involving ranks) is obtained. This condition may or may not be sufficient. For dimA=2, 3 the problem is settled: there is always a solution in the noninvertible case.
Carathéodory’s Theorem with Linear Constraints
Authors: Cook W. D.
Published: December 2010

Carathéodory has shown that if x1, x2,…, xm, (m finite) are points of Rn and if some

then ∃μ∈Ω with at most n+1 nonzero components and for which (See [5]). The authors of [2] have extended this result to include the case where m= +∞. In theorems 1 and 2 below we establish somewhat similar results for the case in which Ω is further restricted by a finite system of linear inequalities (or equalities).
Periodic Points and Contractive Mappings
Authors: Hsieh Tsu-Teh , Tan Kok-Keong
Published: December 2010

Let X be a non-empty set and f:X→X. A point x ∈ X is (i) a fixed point off f(x)=x, and (ii) a periodic point of f iff there is a positive integer N such that fN(x)=x. Also a periodic orbit of f is the (finite) set {x, f(x), f2(x),…} where x is a periodic point of f.
Boundedness of Multiplicative Linear Functionals
Authors: Husain T. , Ng S. B.
Published: December 2010

Let A be a complex sequentially complete commutative locally m-convex topological algebra which is symmetric with continuous involution. The purpose of this note is to prove that every multiplicative linear functional on A is bounded (Theorem 3). In fact, we prove a more general result for operators on real algebras (Theorem 1) from which we derive the above result.
On the Distribution of the Direction and Collinearity Factors in Discriminant Analysis
Authors: Gupta R. P. , Gupta R. D.
Published: December 2010

If the samples of sizes n1, n2,…, nq+1 are available from (q+1) normal populations with different mean vectors μα, α=l, 2,…, q+1 and the same covariance matrix ∑ and if x’=(x1, x2, …, xp) denotes the vector of p variates on which the measurements are made, then we obtain the following multivariate analysis of variance table.
Note on a Paper of S. Uchiyama
Authors: Mortimer B. C. , Williams K. S.
Published: December 2010

Let p be a rational prime and n a positive integer ≥2. We denote by an(p) the least positive integral value of a for which the polynomial xn+x+a is irreducible (mod p), and set

(1)