bluuebunny/crossref_metadata_2025 huggingface dataset:
The Exponent of a Primitive Matrix*
Authors: Dulmage A. L. , Mendelsohn N. S.
Published: December 2010

Let A be an n by n matrix with non-negative entries. If a permutation matrix P exists such that P-1 is of the form

where A and C are square matrices and O is a zero-matrix then A is said to be reducible. Otherwise, A is irreducible. If A is irreducible, then A is said to be primitive if there is an integer k such that Ak0, i.e., Ak has no zero entries. If A is primitive, the least integer m for which Am0 is called the exponent of A. In (4), Wielandt has shown that for n by n primitive matrices the exponent is at most (n-1)2+1. In this paper, the theory of directed graphs is used to determine conditions under which the exponent is less than (n-1)2+1. The following results are obtained. If A contains r non-zero entries along its main diagonal then its exponent is at most 2n-r-1, and this result is the best possible. If all the diagonal entries of A are zero but its graph KA (see next section for definition) contains a cycle of length d, then the exponent of A is at most d(2n-d-1). For the case where this improves Wielandt's result.
A Characterization of Block-Graphs
Authors: Harary Frank
Published: December 2010

The purpose of this note is to characterize "block-graphs", a collection of
graphs defined by a construction involving certain subgraphs called
"blocks". A related operation on a graph leads to the study of
"cut-point-graphs". The precise relationship between these two operations is
made explicit. In order that this characterization be self-contained, we
include the necessary definitions.
Another Remark on a Result of K. Goldberg
Authors: Marcus Marvin
Published: December 2010

In [3] K. Goldberg showed that if A is a 0-1 matrix that satisfies

(1.1)

then for some permutation matrix P, PAP* is a direct sum of matrices each of
which is either zero or consists only of ones. More recently J. L. Brenner
[1] proved that if A ∦ 0 (i.e. A has non-negative entries) and satisfies (1)
then there exists a permutation matrix P such that PAP* = A1⊕ …
⊕An in in which each Ai is either 0 or all
positive, Ai > 0, and satisfies (1) as well.
A Generalization of an Inversion Formula for the Gauss
Transformation
Authors: Rooney P. G.
Published: December 2010

In an earlier paper [3] we considered an inversion formula for the Gauss
transformation G defined by

1.1

We noted there that formally G is inverted by,

1.2

and we showed that if e-D2 is interpreted via the power series for the exponential function,
that is if

1.3

then under certain conditions on φ,

1.4
The Lebesgue Constants for (γ, r) Summation of Fourier Series
Authors: Lorch Lee , Newman Donald J.
Published: December 2010

The (γ, r) summation method, 0 < r < 1, is the "circle method" employed by G. H. Hardy and J. E. Littlewood. It is also known as the Taylor method. Its Lebesgue constants, say L(Tr, n), n = 1, 2, …, were studied by K. Ishiguro [1] in the notation L*(n;1-r). He noted that

1

where Im{z} denotes the imaginary part of the complex number z, and proved that

2

Here

3
Remarks on a Problem of Obreanu
Authors: Erdös P. , Rényi A.
Published: December 2010

Let a1 < a2 < … be any sequence of integers. Assume that the infinite sequence of numbers un satisfies the following condition: To every ɛ > 0 there is an no = no (ɛ) such that for all n > no and all k

1

Obreanu asked (Problem P. 35 Can. Math. Bull.) under what conditions on the sequence a1 < a2 < … does (1) imply that the sequence u is convergent. N. G. de Bruijn and P. Erdos proved that a necessary and sufficient condition for (1) to imply the convergence of un is that the sequence {an} be infinite and that the greatest common divisor of the a1 should be 1.
On a Paper by M. Iosifescu and S. Marcus
Authors: Neugebauer C. J.
Published: December 2010

In this paper we will construct an example showing that the problem posed in [1] has a negative answer. Two more theorems on the subject treated in [1] will be included.

Let Io = [0, 1], R the reals, and let, for A ⊂ R, Ao be the interior of A. Let {xn} be a sequence in [0, 1> such that 0 = x1 < x2 < … and lim xn = 1. For each n, let In be closed interval having x as its midpoint (except for n = 1 in which case x1 is the left endpoint of I1) such that In ∩ Im= ϕ, and the metric density relative to Io of at 1 is zero. Let Jn be a closed interval in In concentric with In (except for n = 1, where J1 has x1 as its left endpoint) whose length is half that of In
On the Reduction of Permutation Representations
Authors: Puttaswamaiah B. M.
Published: December 2010

In this paper an attempt is made to generalize the rather special role played by certain "Young subgroups" of Sn. In Theorem 1, the substitutional equation involving Young's raising operators for Sn is generalized to apply to an arbitrary subgroup H of a finite group G. Applying this to Sn, it is shown how the multiplicities mλH can be calculated in terms of the raising operators in Theorem 2, which calculations are illustrated by a simple example.
On a Factorisation of Positive Definite Matrices
Authors: Majindar Kulendra N.
Published: December 2010

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).

A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.
Remarks on a Paper by S. Trott
Authors: Moser W. O. J.
Published: December 2010

There are several statements in [3] which require clarification.

Theorem 1 [3, p. 246] states that U3= U2U4 U2-1U4 U2 U4. In fact this is (essentially) the relation O = PUPU-1PU given in [1, p. 91, (7.35)]. To see this we note that 0 = U3, P = U4, U = U4U2U4 (as explained in [1, p. 88]); since U42, = E,
On the Number of Triangles Contained in Certain Graphs
Authors: Erdös P.
Published: December 2010

Let G(n;m) denote a graph of n vertices and m edges. Vertices of G will be denoted by x1, …, y1 …; edges will be denoted by (x, y) and triangles by (x, y, z). (G - x1-x2 - … - xk) will denote the graph G from which the vertices x1, …, xk and all edges incident to them have been omitted G - (xi, xj) denotes the graph G from which the edge (xi, xj) has been omitted.
An Informal Arithmetical Approach to Computability and
Computation, II
Authors: Melzak Z.A.
Published: December 2010

In the first part of this paper [l] there was introduced a hypothetical
computing device, the Q-machine. It was derived by abstracting from the
process of calculating carried out by a man on his fingers, assuming an
adequate supply of hands and the ability to grow fingers at will. The
Q-machine was shown to be equal in computing power to a universal Turing
machine. That is, the Q-machine could compute any number regarded as
computable by any theory of computability developed so far. It may be
recalled here that Turing machines were obtained by Turing [2] by
abstracting from the process of calculating carried out by a man on some
concrete 'symbol space' (tape, piece of paper, blackboard) by means of fixed
but arbitrary symbols. Hence the contrast between the Q-machine and the
Turing machines is that between arithmetical manipulation of counters and
logical manipulation of symbols. In particular, one might say, loosely, that
in a Turing machine, as in arithmetic, numbers are represented by signs
whereas in the Q-machine, as on a counting frame, numbers represent
themselves.
A Remark on the Theorems of Lusin and Egoroff
Authors: Zakon Elias
Published: December 2010

In this note we do not intend to establish new results but only to suggest a
very simple proof of Lusin's theorem, direct for σ-finite regular measures,
a proof that bypasses the usual procedure of first establishing this theorem
for sets of finite measure only. The proposed proof utilizes the notion of
subuniform convergence, a method which seems not yet to have been used,
despite its simplicity and adaptability. Simultaneously, a useful supplement
to Egoroff's theorem will be obtained.
Notes on Numerical Analysis IV. On Accelerating Iteration Procedures With Superlinear Convergence
Authors: Kotzé Wesley
Published: December 2010

In the study of algorithms for the iterative solution of an arbitrary analytic equation f(z) = 0, acceleration procedures are of importance in practice and of considerable interest in the theory of the subject. Let

.

be an iteration formula which has a zero ξ of f(z) as attractive fixed point. An algorithm of this type is said to converge towards a root ξ of f(z) = 0 for all initial approximations z = z0 in a vicinity of ξ, of order k > 0, when

1.
A Generalization of Jacobi's Theorem to Hyperadjugates
Authors: Lawrence Peter A.
Published: December 2010

Jacobi's theorem expresses the adjugate of a determinant as a power of the determinant, or in a more general form a minor of the adjugate as the product of the complementary minor by a power of the determinant.
On Some Divisibility Properties of
Authors: Erdös P.
Published: December 2010

L. Moser [3] recently gave a very simple proof that

1.

has no solutions. In the present note we shall first of all prove that for , which by the fact that there is a prime p satisfying n < p ≤ 2n immediately implies that

2.

has no solutions. It is easy to see on the other hand that

3.

has infinitely many non-trivial solutions.
Some Inequalities for Polynomials and Related Entire Functions II
Authors: Rahman Q. I.
Published: December 2010

Let be a polynomial of degree n. Then clearly

(1.1).

(1.2).

and for R > 1

(1.3).

Note that if w = p(z) maps |z|<1 on a domain D of the w-plane then the area of D is given by
On the Relation between a Cluster set Introduced by Constantinescu and Cornea, and the Fine Cluster set of Cartan, Brelot, and Naïm
Authors: Jackson H. L.
Published: December 2010

The field of boundary limit theorems in analytic function theory is usually considered to have begun about 1906, with the publication of Fatou's thesis [8]. In this remarkable memoir a theorem is proved, that now bears the author's name, which implies that any bounded holomorphic function defined on the unit disk possesses an angular limit almost everywhere (Lebesgue measure) on the frontier. Outstanding classical contributions to this field can be attributed to F. and M. Riesz, R. Nevanlinna, Lusin, Privaloff, Frostman, Plessner, and others.
On Differentiable Functions having an Everywhere Dense set of Intervals of Constancy
Authors: Bruckner A. M. , Leonard John L.
Published: December 2010

The Cantor function C [2; p. 213], which appears in analysis as a simple example of a continuous increasing function which is not absolutely continuous, has the following properties:

(i)

C is defined on [0,1], with C(0) = 0, C (l) = l;

(ii)

C is continuous and non-decreasing on [0,1];

(iii)

C is constant on each interval contiguous to the perfect Cantor set P;

(iv)

C fails to be constant on any open interval containing points of P;

(v)

The set of points at which C is non-differentiable is non-denumerable.
The Local Diffusions of a Harmonic Sheaf
Authors: Dawson Donald A.
Published: December 2010

It is well known that the Laplacian operator is the infinitesimal generator of Brownian motion in Rn. Moreover, the classical harmonic measures are the hitting measures of Brownian motion. In other words, there is a natural correspondence between the Brownian motion and the classical harmonic functions. In this paper we will show that any family of abstract harmonic functions satisfying the axioms of M. Brelot [4] are annihilated by the infinitesimal generator of a diffusion, and that the corresponding harmonic measures are the hitting measures of the diffusion. This answers a question raised by P. A. Meyer [11] who has proved the existence of a Markov semi-group associated with a harmonic sheaf.
Restricted Choices
Authors: Abramson M.
Published: December 2010

A subset consisting of k elements chosen, from n distinct ordered elements, with given restrictions is called are stricted choice. For example, one restriction on the k elements may be that no two consecutive elements appear, while another may be that no two alternate elements appear. Certain restricted choices may be used to obtain solutions to permutation problems ([1, p. 349]; [4]). Each restricted choice corresponds to a "restricted sequence of Bernoulli trials" as described in [1], In this paper an elementary method of obtaining more general types of restricted choices is given. Some special cases of the restricted choices and restricted Bernoulli trials are presented in the form of examples.
A Family of Distributions with the same Ratio Property as Normal Distribution
Authors: Fox Charles
Published: December 2010

If U and V are independent random variables, both drawn from the Normal distribution, then it is known that the distribution of U/V follows the Cauchy law, i. e. has frequency function 1/π(1+x2). Conversely if U and V are independently drawn from the same distribution and U/V is known to follow the Cauchy law of distribution must U and V be necessarily drawn from a Normal distribution? This question has been considered by several authors ([3], [4], [5], [7]) who have obtained several examples to show that the ratio property above is not confined to the Normal distribution.
On Eulerian and Hamiltonian Graphs and Line Graphs
Authors: Harary Frank , Nash-Williams C. St. J. A.
Published: December 2010

A graph G has a finite set V of points and a set X of lines each of which joins two distinct points (called its end-points), and no two lines join the same pair of points. A graph with one point and no line is trivial. A line is incident with each of its end-points. Two points are adjacent if they are joined by a line. The degree of a point is the number of lines incident with it. The line-graph L(G) of G has X as its set of points and two elements x, y of X are adjacent in L(G) whenever the lines x and y of G have a common end-point. A walk in G is an alternating sequence v1, x1, v2, x2, …, vn of points and lines, the first and last terms being points, such that xi is the line joining vi to vi+1 for i=1, …, n-1.
On the General Theory of Differentiable Manifolds with Almost Tangent Structure
Authors: Eliopoulos Hermes A.
Published: December 2010

Some of the most important G-Structures of the first kind [1] are those defined by linear operators satisfying algebraic relations. If the linear operator J acting on the complexified space of a differentiable manifold V satisfies a relation of the form

where I is the identity operator, the manifold has an almost complex structure ([2] [3]). The structures defined by

are the almost product structures ([3] [4]).
A Generalization of the Bang-Bang Principle of Linear Control Theory*
Authors: Datko Richard
Published: December 2010

In a paper by LaSalle [l] on linear time optimal control the following lemma is proved:

Let Ω be the set of all r-dimensional vector functions U(τ) measurable on [ 0, t] with |ui(τ)≦1. Let Ωo be the subset of functions uo(τ) with |uoi(τ) = 1. Let Y(τ) be any (n × r ) matrix function in L1([ 0, t]).
Lebesgue Constants for Regular Taylor Summabllity
Authors: Forbes R. L.
Published: December 2010

The nth Taylor mean of order r of a sequence {sn} is given by

1.1

where

1.2

Cowling [l] has shown that this method is regular if and only if 0 ≦ r < 1. Since r = 0 corresponds to ordinary convergence, it will be assumed here that 0 < r < 1.
On Lattices in a Module Over a Matrix Algebra
Authors: Nobusawa Nobuo
Published: December 2010

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).
Some Remarks on a Combinatorial Theorem of Erdös and Rado
Authors: Abbott H. L.
Published: December 2010

P. Erdös and R. Rado [1] proved that to each pair of positive integers n and k, with k ≥ 3, there corresponds a least positive integer φ(n, k) such that if is a family of more than φ(n, k) sets, each set with n elements, then some k of the sets have pair-wise the same intersection.
A Universal Factorization Theorem in Topology
Authors: Sharpe R. , Beattie M. , Marsden J.
Published: December 2010

The purpose of this paper is to prove and generalize the following theorem: Given any topological space X, of all the T2 spaces Z which are continuous images of X, there is a maximal one Y; that is, one over which all others factor, as in Figure 1.
On the Reducibility of Appelles Function F4*
Authors: Saxena R.K.
Published: December 2010

It is a well-known fact in the theory of Appell's hypergeometric function of two variables F4, defined by

1

where |x|1/2 + |y|1/2 < 1, that it can be expressed interms of products of ordinary hypergeometric functions when γ + γ' = α + β + 1.
A Remarkable Class of Mannheim-Curves
Authors: Blum Richard
Published: December 2010

It is well known that the determination of a (non-isotropic) curve in the euclidean 3-space with given curvature κ(S) and torsion τ(s), where s represents the arc-length, depends upon the integration of a Riccati equation; and that this can be performed only if a particular integral of the equation is known.
On Subtournaments of a Tournament
Authors: Moon J. W.
Published: December 2010

Beineke and Harary [l] recently showed that the maximum number of strong tournaments with k nodes that can be contained in a tournament with n nodes is

if 3 ≤ k ≤ n. The object of this note is to obtain some additional results of this type. We will use essentially the same terminology as was used in [ l ], so we will not repeat the standard definitions here.
A Note on Dirichlet Convolutions
Authors: Segal S. L.
Published: December 2010

In [3] Rubel proved that if h(n) is an arithmetic function such that , L finite, then where μ(n) is the Mobius function. This result was extended to functions other than μ(n) in [4]; however, (as first pointed out to the author by Benjamin Volk), the order condition imposed there is unnecessary; in fact, utilizing the result of [3], the following slightly more general theorem has an almost trivial proof.
A Global Existence Theorem
Authors: Eisen M.
Published: December 2010

The Cauchy-Peano existence theorem (1) does not allow us to decide from the form of a given system of equations whether or not its solution can be continued for the infinite interval -∞ < t < ∞. Several sufficient conditions for such a continuation were given by A. Wintner in (2). The main result of his paper is the following theorem which is not proven.
R1-Topological Spaces1
Authors: Murdeshwar M.G. , Naimpally S.A.
Published: December 2010

In his paper ″Indexed systems of neighborhoods for general topological spaces″ (Amer. Math. Monthly 68, (1961), 886–893), A.S. Davis defined a hierarchy of what he called regularity axioms. The R1 -axiom is independent of both T0 and T1, but is strictly weaker than T2. In this note, we propose to study the properties of the spaces satisfying the R1 -axiom. In particular, we will show that in many well-known results, the hypothesis can be weakened from T2 to R1, which is part of our motivation in studying R1 -spaces.
On the Strong Summability by Triangular Means of the Derived Fourier Series and its Conjugate Series
Authors: Govil Narendra K.
Published: December 2010

The triangular matrix (A) = (X ), where n = 0, 1, 2,…; k = 0, 1, 2, …; and λn, k = 0 for k > n is regular (in the sense of defining a regular sequence to sequence transform) if

for every fixed k ;

independently of n;
Numerical Linear Algebra
Authors: Kahan W.
Published: December 2010

The primordial problems of linear algebra are the solution of a system of linear equations

and the solution of the eigenvalue problem

for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.
Some Characterizations of co and ℓ1
Authors: Retherford J. R.
Published: December 2010

The space co consists of the sequences tending to zero with addition and scalar multiplication defined coordinate-wise and with the sup norm. The space ℓ1 consists of the sequences b = (bi) under coordinate»wise arithmetic for which
On Periodic Solutions of x‴ + ax″ + b′ + g(x) = 0
Authors: Kemp R.R.D.
Published: December 2010

In [1] J. O. C. Ezeilo asks whether the equation

1

has periodic solutions for a ≠ 0. Since (1) has a two-dimensional space of solutions of period 2π if sin x is approximated by x, it is plausible to conclude, by analogy with x″ + sin x = 0, that (1) does have periodic solutions. However, when one applies the standard theory of perturbation of periodic solutions (treating a as small, see [2]), one finds that the only real periodic solutions obtainable in this manner are the trivial ones x(t, a) = nπ for some integer n.
A Discrete Analogue of Opial's Inequality
Authors: Wong James S. W.
Published: December 2010

In a number of papers [1] - [7], successively simpler proofs were given for the following inequality of Opial [1], in case p=1.

Theorem 1. If x(t) is absolutely continuous with x(0)=0, then for any p ≧ 0,

(1)

Equality holds only if x(t) = Kt for some constant K.
On Finite Plane Sets Containing for Every Pair of Points an Equidistant Point
Authors: Brown W.G.
Published: December 2010

In [l] Melzak has posed the following problem: “A plane finite set Xn consists of n ≥ 3 points and contains together with any two points a third one, equidistant from them. Does Xn exist for every n ? Must it consist of points lying on some two concentric circles (one of which may reduce to a point)? How many distinct (that is, not similar) Xn are there for a given n ? …” We shall here provide a construction for uncountably many Xn for every n > 4, and a counterexample to the second question above.
On Measures Determined by Functions with Finite Right and Left Limits Everywhere
Authors: Ellis H.W. , Jeffery R. L.
Published: December 2010

In another paper [l] measures determined from base functions, that have finite right and left limits everywhere and are of generalized bounded variation in the restricted sense, are studied and used to define non absolutely convergent integrals of Denjoy type. In this paper base functions of bounded variation and the corresponding measures are studied as a background for that paper. The results supplement parts of [2].
On the Structural Properties of the Conditional Distributions
Authors: Mathai A.M.
Published: December 2010

If x, x1,…, xnare independent stochastic variables and if the conditional distribution of x given x1+…+ xnis known, what can be said about the marginal distributions of x, x1,…, xn? In this paper we will show that if the conditional distribution of x given a subset of x1, x2,…, xn-1, x+ x1+…+ xnhas a certain structural form then x, x1,…, xnare distributed as members of the linear exponential family of distributions and further x1,…, xnare identically distributed.
Some Relationships between Filters*
Authors: Baggs Ivan
Published: December 2010

A filter is a set theoretical concept and as such, its structure is independent of any topology which can be put on the given space. However, an O-filter, whose counterpart in the theory of nets is the O-nets of Robertson and Franklin [2], is defined with respect to the topology on the given space. The purpose of this paper is to give necessary and sufficient conditions for every O-filter to be an ultrafilter and for every Cauchy filter to be an O-filter.
Five Dimensional Non-Lattice Sphere Packings
Authors: Leech John
Published: December 2010

The densest lattice packings of spheres in Euclidean spaces En of n dimensions are known for n ≤ 8 (for full n — references see [6]). However, it i s not known for any n ≥ 3 whether there can be any non-lattice sphere packing with density exceeding that of the corresponding densest lattice packing.
Integral Group Rings of Finite Groups
Authors: Banaschewski B.
Published: December 2010

The main object of this paper is to show that the existence of a particular kind of isomorphism between the integral group rings of two finite groups implies that the groups themselves are isomorphic. The proof employs certain types of linear forms which are first discussed in general. These linear forms are in some way related to the bilinear forms used by Weidmann [3] in showing that groups with isomorphic character rings have the same character table, and a shorter and, in a sense, more natural proof of this result is included here as another application of these linear forms.
On Some Limit Theorems Involving the Empirical Distribution Function
Authors: Csörgo Miklós
Published: December 2010

Let X1 …, Xn be mutually independent random variables with a common continuous distribution function F (t). Let Fn(t) be the corresponding empirical distribution function, that is

Fn(t) = (number of Xi ≤ t, 1 ≤ i ≤ n)/n.

Using a theorem of Manija [4], we proved among others the following statement in [1].
On Numbers Generated by es(ex-1)
Authors: Chinthayamma  , Gandhi J.M.
Published: December 2010

The numbers generated by es(ex-1) where s is an integer are discussed by many for s = 1. The general case also is discussed by Touchard, Riordan and others.
The Problem of Apollonius1
Authors: Coxeter H. S. M.
Published: December 2010

On behalf of the Canadian Mathematical Congress, I wish to thank the University of Toronto for its hospitality, the members of the Local Arrangements Committee (especially Chandler Davis) for the many comforts and pleasures they have provided, and our nine distinguished visitors for the courses of lectures they gave at our Seminar.
On the Conversion of the Determinant into the Permanent
Authors: Botta Peter
Published: December 2010

Let Mm(F) be the vector space of m - square matrices over a field F. If X belongs to Mm(F), then xij will denote the element occuring in row i and column j of X.

Let Sm be the symmetric group of degree m and ε: Sm → F the alternating character on Sm (i.e. ε(σ) = 1 or - 1 according as σ is an even or odd permutation).
Near-Ring Homomorphisms
Authors: Malone Joseph J.
Published: December 2010

Blackett [4] introduced the concepts of near-ring homomorphism and near-ring ideal. Beidleman [1] established the fundamental homomorphism theorem and the isomorphism theorems for (left) near - rings obeying the condition that 0.a = 0 for every a in the near-ring. Several others, for example [3], [5], and [7], have taken up the study of ideals. This paper takes up the study of homomorphisms of (left) near-rings not subject to the condition 0.a = 0. It is shown that such homomorphisms can be decomposed into homomorphisms of two special sub-near-rings. Conversely, conditions are sought under which homomorphisms of the two sub-near-rings may be mated to produce a homomorphism of the sub-near-ring.
Matrix Characterizations of Topological Properties
Authors: Bonnett D.A. , Porter J.R.
Published: December 2010

In [S], H. Sharp characterizes each topology on a finite set S = {s1, s2,…sn} with a n×n zero-one matrix T = (tij) where tij=1 if and only if . In this paper we seek matrix characterizations of certain topological properties of finite spaces. Such characterizations will provide purely mechanical ways of determining if a space has a certain topological property.
On the Evaluation of Some Expressions Concerning the PBIB Designs
Authors: Cléroux Robert
Published: December 2010

Consider a two associate class partially balanced incomplete block (PBIB) design [2] with parameters of the first kind t, b, r, λ,1 λ,2 n1, n2, and parameters of the second kind . Let the letters m, p, l, s represent treatments and define associates of treatment m, λmps = number of times the treatments m, p and s all occur together in the same block, λmpls = number of times the treatments m, p, l and s all occur together in the same block.
On the Minimal Lipschitz Constant
Authors: Goebel K.
Published: December 2010

In this paper we give necessary and sufficient conditions that a continuous transformation f: A→A of a metric space A with the metric r should be a contraction with respect to an equivalent metric s. This is the solution of a problem stated by J. S. W. Wong [2].
A Note on Henselian Valuation Rings
Authors: Endler Otto
Published: December 2010

Let K be a field and Ka its algebraic closure. A valuation a ring A of K is called henselian, if there is only one valuation ring C of Ka which lies over A (i.e. such that C ∩ K = A) or, equivalently, if Hensel's Lemma is valid for K, A (see [5], F). In the following, we shall consider only rank one valuation rings.
On Banach Limit of Fourier Series and Conjugate Series I
Authors: Dayal S.
Published: December 2010

Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:

(1)

Lim (axn + byn) = a Lim xn + b Lim yn

(2)

If xn ≥ 0 for every n, then Lim xn ≥ 0

(3)

Lim xn+1 = Lim xn

(4)

If xn = 1 for every n, then Lim xn = 1

The existence of such limits is proved by Banach [1].
Note on Weight Spaces of Irreducible Linear Representations
Authors: Lemire F. W.
Published: December 2010

Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.
Extension of a Result of L. Lorch and P. Szego On Higher Monotonicity
Authors: Muldoon M. E.
Published: December 2010

We suppose that {x1, x2, …}, are increasing sequences of consecutive zeros of non-trivial solutions y(x), , of the differential equation

(1)

on an open interval I, and define, for any fixed λ > - 1,
Almost Continuity of Mappings
Authors: Lin Shwu-Yeng T.
Published: December 2010

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.

Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].
Note on Epi in
Authors: Baron S.
Published: December 2010

Burgess [l] has pointed out that in the categories and (where . is the category of Ti. spaces), epi means onto. In this paper, Burgess1 technique will be used to show that epi has a different meaning in and that this meaning reduces to onto when the range is a T1 space.
The No-Three-In-Line Problem
Authors: Guy Richard K. , Kelly Patrick A.
Published: December 2010

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.

For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that
Local Near-Rings Of Cardinality P2
Authors: Maxson Carlton J.
Published: December 2010

The main result of this paper is the determination of all nonisomorphic local near-rings  with  = which are not near-fields. Together with the fundamental paper [6] by Zassenhaus on near-fields and the corollary to Theorem 1 of [2], this 2 paper gives a complete description of all local near-rings of order p2.

We recall that a unitary near-ring N is called local if the subset L of elements in N without left inverses is an (N, N)-subgroup and N ≠ J(N). (J(N) denotes the radical of N given in [1].) In [3] it was proved that N ≠ J(N) whenever L is an ideal of N. (For previous result s concerning local near-rings we refer the reader to [3].)
A Note on Exact Colimits
Authors: Isbell John R.
Published: December 2010

This note proves another special case of a conjecture of U. Oberst. Oberst considered [1], for any small category , the X abelian category of abelian group-valued functors on , and the X functor Colim: which takes each diagram to its colimit. The question is, when is Colim exact? For its relationships, see [1], It is a sufficient condition that each component of is upward filtered. Oberst conjectured that it is also necessary, and proved this under some conditions. He mentioned particularly the case that is a monoid, i.e. a category with one object. We shall verify the conjecture in that case.
The Coreflective Subcategory of Sequential Spaces
Authors: Baron S.
Published: December 2010

Certain theorems of recent interest [1, 2] concerning sequential spaces may be deduced from the fact that the category of sequential spaces, is a coreflective subcategory of the category of topological spaces, J. A space is said to be sequential if it has the finest topology that permits the convergence of its convergent sequences.
A Remark on the Group Rings of Order Preserving Permutation Groups
Authors: LaGrange R.H. , Rhemtulla A.H.
Published: December 2010

If G and H are two groups such that their integral group rings Z(G) and Z(H) are isomorphic, does it follow that G and H are isomorphic? This is the isomorphism problem and an affirmative answer is obtained in case G is a sub group of the group of order preserving permutations of a totally ordered set.
Tensor Products of Banach Algebras*
Authors: Natzitz Boaz
Published: December 2010

In [3] Gelbaum defined the tensor product A ⊗CB of three commutative Banach algebras, A, B and C and established some of its properties. Various examples are given and the particular case where A, B and C are group algebras of L.C.A. groups G, H and K respectively, is discussed there. It is shown there that if K is compact L1(G) ⊗ L1(K) L1(H) is isomorphic to where is L.C.A. 1 L (K) 1 1 if and only if L1(G) ⊗ L1(K) L1(H) is semisimple.
On C∞ Functions Analytic on Sets of Small Measure
Authors: May L.E.
Published: December 2010

The original motivation for this work was the problem of determining whether the signum function of a real valued continuous function defined on the real line is Riemann integrable. This problem is considered in § 2 where an example of an infinitely differentiable function is presented which possesses a non-Riemann integrable signum function. Moreover, it is shown that, for any ∈ > 0, it is possible to construct such an example for which the set of points of analyticity has Lebesgue measure which is less than ∈. This appears to be a more interesting property than the one originally sought.
On the Forced Lienard Equation
Authors: Stevens R.R.
Published: December 2010

We consider the second order differential equation

(1)

with the assumptions that

(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.

Also, throughout this paper, F(x) denotes an antiderivative of f(x).
Six and Seven Dimensional Non-Lattice Sphere Packings
Authors: Leech John
Published: December 2010

The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.
A Characterization of Modularity for
Congruence Lattices of Algebras*
Authors: Day Alan
Published: December 2010

Let us call an equational class (variety) K of algebras permutable if and only if every pair of congruences on each K-algebra is permutable. Similarly, we will call K modular (distributive) if the congruence lattice of each K-algebra is modular (distributive).
A Note on an Equivalence Relation on a
Purely Inseparable Field Extension
Authors: Rygg P. , Lehman B.
Published: December 2010

We assume F is a purely inseparable field extension of the field K. The characteristic of K is p ≠ 0, and we assume F and K are not perfect. For x ∈ F, the exponent of x over K is the smallest

non-negative integer e such that and will be denoted by

e (x); ⨱ will denote . For any subset S of F, e(x; S) will denote the exponent of x over K(S); in case S = {y} we will write e(x; y) for e(x; S).
On A Conjecture of Conway
Authors: Marica John
Published: December 2010

Problem 7 of Section VI of H. T. Croft's "Research Problems" (August, 1967 edition) is by J. H. Conway:

A is a finite set of integers {ai}. A + A denotes {ai + aj}, A - A denotes {ai - aj}. Prove that A - A always has more members than A + A, unless A is symmetrical about 0.

First note that 'about 0' is redundant and erroneous.
On Sets of Arcs Containing No Cycles in a Tournament*
Authors: Reid K.B.
Published: December 2010

A tournament Tn with n nodes is a complete asymmetric digraph [2]. A set S of arcs of a tournament is called consistent if the tournament contains no oriented cycles composed entirely of arcs of S [1]. The object of this note is to provide a new lower bound for f(n), the greatest integer k such that every tournament with n nodes contains a set of k consistent arcs. Erdös and Moon [1] showed that where [x] denotes the largest integer not exceeding x, and the second inequality holds for any fixed ∈ > 0 and all sufficiently large n.
Monotone Semigroups of Operators on Cones*
Authors: Boyd David W.
Published: December 2010

In this paper we consider a special class of linear operators defined on a cone K in a Banach space X. This class of operators is the natural generalization of a class of operators which has applications in the theory of interpolation spaces. In particular, using the criteria developed in Theorem 1, it is possible to characterize those sequence spaces X such that every linear operator A of weak types (p, p) and (q, q) is a continuous mapping of X into itself. For details of this we refer the reader to [3].
A Note on Embedding
Authors: Pittas P. A.
Published: December 2010

Let Qt = [0, 1] be equipped with the topology consisting of Qt, the empty set and all subsets of Qt of the form [0, x), 0 < x ≦ 1. R. Nielsen and C. Sloyer [1, p. 514] proved that every T0-space can be embedded in for a suitable F. The purpose of this note is to generalize this result.
A Remark on Separable Orders
Authors: Roggenkamp Klaus W.
Published: December 2010

K = algebraic number field,

R = algebraic integers in K,

A = finite dimensional semi-simple K-algebra,

A. = simple K-algebra,

i = 1,…, n,

Ki = center of Ai, = 1,…, n,

G = R-order in A,

Ri = G ∩ ki.

All modules under consideration are finitely generated left modules. A G-lattice is a G-module which is R-torsion-free.
Topology of Some Kähler Manifolds II
Authors: Srinivasacharyulu K.
Published: December 2010

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1(C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.
On Subsets with Intersections of Even Cardinality
Authors: Berlekamp E.R.
Published: December 2010

This paper solves a question posed by P. Erdös:

If A1, A2, …, A are distinct subsets of n elements and if |Ai ∩ Ai| ≡ 0 mod 2 (i ≠ j), then

and for each n, there exists a collection of subsets which achieves this bound with equality.
Sums of Complexes in Torsion-Free Abelian Groups
Authors: Tarwater J. D. , Entringer R. C.
Published: December 2010

The number of elements in the sum A + B of two complexes A and B of a group G which have multiple representations a + b = a '+ b' has been investigated by Scherk and Kemperman [1]. Kemperman [2] appealed to transfinite techniques (to order G) to prove:

If G is a torsion-free abelian group with finite subsets A and B with | B | ≥ 2, then at least two elements c of A + B admit exactly one representation c = a + b.
Multiplication on Spaces with Comultiplication*
Authors: Hoo C.S.
Published: December 2010

Let A be an H-space and K a space. It is well known that [K, A] is a loop. Suppose A has a comultiplication as well, that is, cat A < 2. Then we shall prove that [K, A] is a Moufang loop. This generalises a result of C. W. Norman who proved this for the case where A is the circle, the 3-sphere or the 7-sphere. It also improves the known result that [K, A] is a diassociative loop if A has a comultiplication as well, since Moufang loops are diassociative.
On the Rank of the Sum of two Rectangular Matrices
Authors: Meyer C.D.
Published: December 2010

The purpose of this note is to present a short proof for the following theorem.

Let A and B be two complex m x n matrices. If B*A = 0 and AB* = 0 then rank(A + B) = rank(A) + rank(B).

Let A† and B† be the generalized inverses of A and B, respectively, in the sense of Penrose [ 1]. Now,
A Convergence Theorem for Certain Riemann Sums
Authors: Chui Charles K.
Published: December 2010

For a Riemann integrable function f on the interval [0,1], let

and consider the Riemann sums
A Shorter Proof of Goldie's Theorem
Authors: Zelmanowitz Julius
Published: December 2010

In this note we present an extremely short proof of Goldie's theorem on the structure of semiprime Noetherian rings [1]. The outline of the proof was given by Procesi and Small in [4]. By utilizing the concept of the singular ideal of a ring we have been able to weaken the hypotheses of many of the steps in [4]. Most significantly, we are able to avoid a reduction to the case of prime rings, and in Lemma 5 we give an informative list of the relationship between regular elements and essential ideals of semiprime rings.
An Analogue of Birkhoff's Problem III for Infinite Markov Matrices1
Authors: Kim Choo-Whan
Published: December 2010

A celebrated theorem of Birkhoff ([1], [6]) states that the set of n × n doubly stochastic matrices is identical with the convex hull of the set of n × n permutation matrices. Birkhoff [2, p. 266] proposed the problem of extending his theorem to the set of infinite doubly stochastic matrices. This problem, which is often known as Birkhoffs Problem III, was solved by Isbell ([3], [4]), Rattray and Peck [7], Kendall [5] and Révész [8].
On Continuous Functions which are Partially Differentiable Almost Everywhere
Authors: Toralballa L.V.
Published: December 2010

In the theory of surface area one meets situations where a function z = f(x, y) which is defined and continuous on a closed rectangle E, is partially differentiable on E except on a subset of E of Lebesgue measure zero.
A Conformal Proof of a Jordan Curve Problem
Authors: Spoar G. , Lane N.D.
Published: December 2010

The following theorem appears in [1].

Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.
A Generalization of a Theorem of Zassenhaus
Authors: Ligh Steve
Published: December 2010

A near-ring is a triple (R, +,.) such that (R, +) is a group, (R,.) is a semigroup and. is left distributive over +; i.e. w(x + z) = wx + wz for each w, x, z in R. A near-field is a nearring such that the nonzero elements form a group under multiplication. Zassenhaus [3] showed that if R is a finite near-field, then (R, + ) is abelian. B.H. Neumann [1] extended this result to all near-fields. Recently another proof of this important result was given by Zemmer [4]. The purpose of this note is to give another generalization of the Zassenhaus theorem. In fact, we shall prove the following.
Singular Riemannian Structures Compatible with π-Structures
Authors: Duggal K. L.
Published: December 2010

G. Legrand [1] studied a generalization of the almost complex structures [2] by considering a linear operator J acting on the complexified space of a differentiable manifold Vm satisfying a relation of the form J2 = λ2 (identity) where λ is a nonzero complex constant. Such structures are called π-structures. A π-structure is defined on Vm by the knowledge of two fields, of proper supplementary subspaces T1 and T2 of the complexified tangent space at x ∈ Vm, such that dim(T1) = n1; dim(T2) = n2; n1 + n2 = m. In the remaining case, λ= 0, H. A. Eliopoulos [3] introduced almost tangent structures and discussed euclidean structures compatible with almost tangent structures [4]. In a similar way the purpose of this paper is to study singular riemannian structures compatible with π structures, briefly Rπ-structures.
Maximal Subsets of a given Set having No Triple in Common with a Steiner Triple System on the set
Authors: Sauer N. , Schönheim J.
Published: December 2010

Let E be a finite set containing n elements, n ≡ 1, 3 (mod 6), S = S(E) a Steiner triple system on E, i.e. each unordered pair of elements of E is a subset of exactly one triple in S. Let T be a subset of E such that none of the triples of elements of T is a member of S. Erdös has asked (in a recent letter to the authors) for the maximal size of such a set T. Denote max |T| for fixed n and S by f(n, S). We prove in this note the following result:

(i)

(ii)

for every n ≡ 1, 3 (mod 6) there exists a Steiner triple system S0 such that equality holds in i.
A Characterization of Prüfer Domains
Authors: Storrer H. H.
Published: December 2010

The purpose of this note is to give a new characterization of Prüfer domains using the concept of ring epimorphism, and to indicate some connections with well-known properties of Prüfer domains. All rings are commutative and have a unit element.
On the Annihilators of the Injective Hull of a Module
Authors: Koh Kwangil
Published: December 2010

In [2, page 151], J. Lambek proposes the following exercise: With any maximal right ideal M of a ring R with 1 associate the ideal . Show that OM, is the right annihilator of the injective hull of the right R-module R/M. The purpose of this note is to show that the above statement is true for a much larger class of right ideals than that of maximal regular right ideals of a ring.
On Spectral Synthesis for One Point
Authors: Natzitz B.
Published: December 2010

In [2, page 3831], Varopoulos proves that for any ∈ > 0 there exists a function on a neighbourhood of 0. Indeed, if 0 < ∈ < π/2, then f(x) defined to be equal to 1 - e when -eix, linear on [∈, 2π - ∈] and of period 2π, is an example of such a function.

The above result can be used to give a direct proof of the following result without reference to the L2 theory [1, Theorem 2.6.4].