bluuebunny/crossref_metadata_2025 huggingface dataset:
A Note on Products of Normal Subgroups
Authors: Shores T.S.
Published: December 2010

By a group theoretic class we mean a class of groups which contains the trivial group, denoted E, and any group isomorphic to a group in the class. Let I be a group theoretic class. Following P. Hall [4, p. 533], we define EI, CI, SI, QI, and NoI to be the (group theoretic) classes consisting of extensions of I groups by I groups, cartesian products of I groups, subgroups of I groups, homorphic images of I groups and products of two normal I subgroups of a group, respectively. If T is one of the above operations on classes of groups and TI = I, we say X is T closed.
Extensions of Vandermonde Type Convolutions with
Several Summations and their Applications - II
Authors: Handa B.R.
Published: December 2010

In earlier papers [3] and [4] some Vandermonde type convolution identities with several summations were established which were the extensions of Gould's work that involved one summation. Furthermore, these identities were utilized to obtain some inverse series relations for functions of several variables. As a continuation, this paper deals with some further generalizations of Gould's work in [1] and [2].
Relations between Finite Homology and Homotopy
Authors: Brown B.
Published: December 2010

For a finite abelian group G let λ(G) be the least positive integer such that λ(G)G = 0. Let be the least integer such that λ(G) | (λ(G) divides ) and if 2 | λ(G) then 4 | . For a finitely generated abelian group G let GT be the subgroup of G made up of all elements of G of finite order, and let GF = G/GT. For a simply-connected C-W complex X, let be the smallest class of abelian groups containing the groups .
On Topological Invariants of the Product of Graphs
Authors: Behzad M. , Mahmoodian S. E.
Published: December 2010

We consider ordinary graphs, that is, finite, undirected graphs with no loops or multiple lines. The product (also called cartesian product [4]) G1 × G2 of two graphs G1 and G2 with point sets V1 and V2, respectively, has the cartesian product V1 × V2 as its set of points. Two points (u1, u2) and (v1, v2) are adjacent if u1 = v1 and u2 is adjacent with v2 or u2 = v2 and u1 is adjacent with v1.
Subaffine Schemes*
Authors: Hoechsmann Klaus
Published: December 2010

Let an open, quasi-compact subscheme of an affine scheme be called subaffine. This note will centre on an elementary characterization of such schemes in terms of their topology and global sections. Thence one can obtain simplifications and generalizations of some well-known theorems, such as Serre's Criterion [2, Thm. 1].
A Commutator Formula for a Pair of Subgroups and A Theorem of Blackburn
Authors: Lam Tsit-Yuen
Published: December 2010

Let be the lower central series of a group G, where K2(G) = [G, G] and inductively Kn+1(G) = [Kn (G), G]. A theorem of Blackburn ([1], Hilfssatz) states that

Theorem 1. The exponent of Kn+1 (G)/Kn+2 (G) divides the exponent of Kn (G)/Kn+1 (G).
A Generalization of the Regular Maps of Type
{4, 4}b, c and {3, 6}b, c
Authors: Garbe Dietmar
Published: December 2010

In [1], Coxeter gave a complete enumeration of the regular maps on a torus. The maps consist of two families of type {4, 4}b, c and {3, 6}b, c (and their duals). b and c are non-negative integers, which determine the maps uniquely. The maps are irreflexible if and only if bc(b - c) ≠ 0.

On surfaces of genus h > 1, irreflexible regular maps are rather exceptional. The simplest surface of negative characteristic which admits irreflexible regular maps is the orientable surface of genus 7. This was shown by the author [4, Theorem 3. 1 ]. The corresponding map was discovered by J. R. Edmonds [2, p. 388].
Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)
Authors: Williams Kenneth S.
Published: December 2010

In 1957, Mordell [3] proved

Theorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that

(1.1)

provided (abc, p) = 1.

Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.
Separation Axioms and Direct Limits
Authors: Herrlich Horst
Published: December 2010

A topological space X is called a direct limit of a family (Xα) of subspaces of X if and only if

(1)

(2)

If X is a direct limit of an increasing sequence (Xn) of closed subspaces then it is well known and easy to prove that X is a T1-space resp. a T4-space provided all Xn are T1-spaces resp. T4-spaces.
Some Remarks on Extreme Derivates
Authors: Bruckner A. M.
Published: December 2010

In 1957 Hájek [1] proved that the extreme bilateral derivates of an arbitrary finite real valued function of a real variable, are Borel measurable of class ≦ 2. It was later shown by Staniszewska [3] that Hájek's result is the best possible (even among the class of functions satisfying a Lipschitz condition). Staniszewska exhibited a Eipschitz function whose extreme bilateral derivates are not in Borel class 1. Staniszewska's proof makes use of a result of Zahorski's [4] concerning kernel functions.
On the Matrices A and f(A)
Authors: Thompson R. C.
Published: December 2010

In a recent note [1], M. R. Embry proved that if A is an operator on a Banach space then, under a certain condition on the spectrum of A, each operator commuting with An also commutes with A, where n is a fixed positive integer. It turns out that, when A is a finite matrix, Embry′s conditions imply that A is a polynomial in An and hence plainly each operator commuting with An also commutes with A.
On Semi-Perfect Group Rings
Authors: Burgess W.D.
Published: December 2010

In what follows the notation and terminology of [7] are used and all rings are assumed to have a unity element.

The purpose of this note is to give some partial answers to the question: under which conditions on a ring A and a group G is the group ring AG semi-perfect?

For the convenience of the reader a few definitions and results will be reviewed. A ring R is called semi-perfect if R/RadR (Jacobson radical) is completely reducible and idempotents can be lifted modulo RadR (i.e., if x is an idempotent of R/RadR there is an idempotent e of R so that e + RadR = x).
On the Invariance of a Quotient Group of the Center of F/[R, R]
Authors: MacHenry Trueman
Published: December 2010

Let F be a free group of rank ⩾ 2, let F/R ≅ π, and let F0 = F/[R, R]. Auslander and Lyndon showed that the center of Fo is a subgroup of R/[R, R] = Ro, and that it is non-trivial if and only if π is finite [1, corollary 1.3 and theorem 2]. In this paper it will be shown that there is a canonically defined (and not always trivial) quotient group of the center of F which depends only on π.
Bounds on the N-th Power Residues (Mod P)
Authors: Chowla S. , London H.
Published: December 2010

For p a prime ≡ 1 (mod n), where n is an odd positive integer, let k(p, n) denote the least integer k such that the numbers xn and (-x)n, where x = 1,2,…, k, yield all the non-zero n-th power residues (mod p) (possibly with repetitions). Clearly
On the Generalized Hankel and K Transformations
Authors: Koh E. L.
Published: December 2010

The K transformation (also called the Meijer transformation) has been extended by Zemanian [1; 2] to a class of generalized functions, For , he defined the K transform of f by

(1)

In [2, Section 6.6] the following inversion theorem for the K transform of f is proven:

(2)

in the sense of weak convergence in D'(I). Here, σ is any fixed real number greater than σf, μ is zero or a complex number with positive real part and D'f(I) is the space of Schwartz distributions on I = (0, ∞).
On a Riemannian Manifold M 2n with an Almost Tangent Structure
Authors: Houh C.S.
Published: December 2010

Professor Eliopoulous studied almost tangent structures on manifolds M2n in [1; 2]. An almost tangent structure F is a field of class C∞ of linear operations on M2n such that at each point x in M2n, Fx maps the complexified tangent space into itself and that Fx is of rank n everywhere and satisfies that F2 = 0. In this note, we consider a (1,1) tensor field . on a Riemannian M2n which satisfies everywhere and is such that the rank of F is n everywhere. Such gives an almost tangent structure F on M2n.
Convergence of Solutions of Third Order Differential Equations*
Authors: Swick K. E.
Published: December 2010

Consider a system of differential equations . Solutions of this system are said to be convergent if, given any pair of solutions x(t), y(t), x(t) - y(t) → 0 as t → ∞. In this case the system is also said to be extremely stable.

In [6] a technique was developed which yielded the convergence of solutions of the forced Lienard equation. Here a similar technique i s applied to forced third order equations. A critical step in [6] was to show that a certain matrix was negative definite. This could be done directly in [6] since the matrix was only 2 × 2. With third and higher order equations, direct use of necessary and sufficient conditions is not feasible since the computations become unwieldy.
On Quotient-Like Extensions of C(X)
Authors: Park Young L.
Published: December 2010

The maximal ring of quotients (in the sense of Johnson [4] and Utumi [5[) of the ring C(X) of real valued continuous functions on a completely regular Hausdorff space X has been studied in [1] and [2]. The aim of the present paper is to provide some additional results to those, and to study the relevant extensions on the absence of the real maximal ideals.
Spreads which are Not Dual Spreads
Authors: Bruen A. , Fisher J. C.
Published: December 2010

In this note we show the existence of a spread which is not a dual spread, thus answering a question in [1]. We also obtain some related results on spreads and partial spreads.

Let ∑ = PG(2t-l, F) be a projective space of odd dimension (2t-l, ≥2) over the field F. In accordance with [1] we make the following definitions. A partial spread S of ∑ is a collection of (t-l)-dimensional projective subspaces of ∑ which are pairwise disjoint (skew).
On A Combinatorial Problem of Erdös
Authors: Abbott H. L. , Hanson D.
Published: December 2010

A family of sets is said to possess property if there exists a set such that and for every We consider the following question raised by P. Erdös |1|: let n and N be positive integers, n ≥ 2 and N ≥ 2n - 1 and let S be a set of N elements; what is the least integer (provided such an integer exists), for which there exists a family of subsets of S satisfying

(a)

|F| = n for each

(b)

(c)

does not have property

(d)

if and then has property
On some Axioms in Foundations of Cartesian Spaces
Authors: Gupta Haragauri N.
Published: December 2010

The present paper seeks to put into a clearer focus the roles of certain axioms in axiomatic studies of finite dimensional Cartesian spaces over various classes of ordered fields. Following Tarski [3], we define a Cartesian space as follows.
An Analogue of the Wave Equation and Certain Related Functional Equations
Authors: Baker John A.
Published: December 2010

Consider the functional equation

(1)

assumed valid for all real x, y and h. Notice that (1) can be written

(2)

a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].
On the Maximum Modulus and the Mean Modulus of an Entire Function
Authors: Reddy A.R.
Published: December 2010

Let be an entire function, but not a polynomial. As usual let,

1
On the Hahn-Banach Extension Property
Authors: To Ting-On
Published: December 2010

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.
On the Non-Existence of Conjugate Points
Authors: Butler G. J. , Erbe L. H. , Mathsen R. M.
Published: December 2010

In this paper we consider the types of pairs of multiple zeros which a solution to the differential equation

can possess on an interval I of the real line. The results obtained generalize those in [2] and (for n = 3) in [3].

I. Let f satisfy the condition

1.1

for all t ∊ I, u0 ≠ 0, and all u1, … un-1.
The Maximal Number of Triangles of Maximal Perimeter Length Determined by a Finite Set
Authors: Bollobás Béla
Published: December 2010

Suppose there are N not necessarily distinct points on a plane in such a position that any triangle (degenerate or non-degenerate) determined by these points has perimeter length at most 1. Denote by m the number of triangles with maximal perimeter length, (briefly, the number of maximal triangles), and put f(N) = max m where the maximum is taken over all permissible configurations. At the Colloquim on Graph Theory in Calgary, 1969, P. Erdös proposed the problem of determining f(N). He conjectured that the following construction gives the maximal number: place approximately half of the points at a point A and the others at B where AB = 1/2. The aim of this note is to prove this conjecture.
Sequences of Contractions in a Generalized Metric Space
Authors: Norris C. W.
Published: December 2010

The main aim of this paper is to study the convergence of a sequence of contractions in a generalized metric space. More specifically, we investigate the following question:

"If a sequence of contractions {fr} with fixed points ur (r = 1,2,...) converges to a mapping f with a fixed point u, under what conditions will the sequence ur converge to u?"

A partial answer to the above question has been given in metric spaces by Bonsall [1]. This result has since been improved by Russell and Singh [6]. Further results will now be given in a generalized metric space.
An Application of Ramsey's Theorem
Authors: Cockayne E. J.
Published: December 2010

By an r-graph, we mean a finite set V of elements called vertices and a collection of some of the r-subsets of V called edges with the property that each vertex is incident with at least one edge. An A-chromatic r-graph is an r-graph all of whose edges are coloured A.

Theorem. Let G1, …, Gt denote r-graphs. There exists a nonempty class of r-graphs such that for each if the edges of G are painted arbitrarily in t colours A1, …, At, then for at least one i in {1, …, t}, G has an Ai-chromatic r-subgraph which is isomorphic to Gi.
The Associative Part of a Convergence Domain is Invariant
Authors: Sember John J.
Published: December 2010

Of special interest in summability theory are those conservative matrices possessing the "mean-value property". If cA={x: Ax ∊ c} denotes the convergence domain of a conservative matrix A, then A has the mean-value property in case, for each x in cA, there exists M = M(A, x) > 0 such that

1

This property has been considered by many writers and has been shown, among other things, to be equivalent to the requirement that the matrix be associative, i.e., for each x in cA

2
Maximal Areas of Reuleaux Polygons
Authors: Sallee G. T.
Published: December 2010

In this paper we provide new proofs of some interesting results of Firey [2]
on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon
is a plane convex set of constant width whose boundary consists of a finite
(odd) number of circular arcs. Equivalently, it is the intersection of a
finite number of suitably chosen congruent discs. For more details, see [1,
p. 128].

If a Reuleaux polygon has n sides (arcs) of positive length
(where n is odd and ≥ 3), we will refer to it as a Reuleaux
n-gon, or sometimes just as an n-gon.
If all of the sides are equal, it is termed a regular
n-gon.
On Singular Normal Linear Integral Equations
Authors: Costley Charles G.
Published: December 2010

In this work we consider the equation

1

where K(x, y) is singular in the sense
that it does not properly belong to L2 and f(x) is an arbitrary L2 function.

A Lebesgue measurable function K(x, y) of
two variables, having real values on [0.1] × [0.1] is called a singular
normal kernel of

(i)

There exists approximating kernels Km(x,
y) satisfying

(ii)

(iii)

(iv)
Boundedness in a Quasi-Uniform Space
Authors: Murdeshwar M. G. , Theckedath K. K.
Published: December 2010

Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) [1]. Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics [4].

In spite of this importance, there have been only isolated attempts at developing the concept. Alexander [1] and Hu [7] tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.
Magic Valuations of Finite Graphs
Authors: Kotzig Anton , Rosa Alexander
Published: December 2010

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.

Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.
A Remark on the Krull-Schmidt-Azumaya Theorem
Authors: Osofsky B. L.
Published: December 2010

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊IMi = ⊕j∊JNj then there is a bijection f: I → J such that Mi is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.
On the Minimal Graph with a Given Number of Spanning Trees
Authors: Sedláček J.
Published: December 2010

LetGbe a finite connected graph without loops or multiple edges. A maximal tree subgraphTofGis called a spanning tree of G. Denote byk(G) the number of all trees spanning the graphG. A. Rosa formulated the following problem (private communication): Letx(≠2) be a given positive integer and denote by α(x) the smallest positive integeryhaving the following property: There exists a graph G onyvertices withxspanning trees. Investigate the behavior of the function α(x).
Integers of Biquadratic Fields
Authors: Williams Kenneth S.
Published: December 2010

Let Q denote the field of rational numbers. If m, n are distinct squarefree integers the field formed by adjoining √m and √n to Q is denoted by Q(√m, √n). Since Q(√m, √n) = Q(√m, √n) and √m + √n has for its unique minimal polynomial x4 —2(m + n)x2 + (m - n)2, Q(√m, √n) is a biquadratic field over Q. The elements of Q(√m, √n) are of the form a0 + a1√m + a2√n + a3√mn, where a1, a2, a3 ∊ Q. Any element of Q(√m, √n) which satisfies a monic equation of degree ≥ 1 with rational integral coefficients is called an integer of Q(√m, √n). The set of all these integers is an integral domain. In this paper we determine the explicit form of the integers of Q(√m, √n) (Theorem 1), an integral basis for Q(√m, √n) (Theorem 2), and the discriminant of Q(√m, √n) (Theorem 3).
Ensembles de Wiener-Hopf II
Authors: Letac Gérard
Published: December 2010

Un ensemble de Wiener-Hopf est une partie S de la droite réelle R tel que

(i)

S+ et S- sont non-vides

(ii)

S+ + S- ⊂ S

où S- ∪ S0 ∪ S+ est la décomposition de S en parties négative, nulle, et positive.

On remarque que ces ensembles présentent une parenté avec les semi-groupes de R. Ainsi, dans [1], est-il démontré que si S n'est pas concentré sur un multiple des entiers, S est dense dans un intervalle.
Properties of Equivalent Capacities
Authors: Adams R. A.
Published: December 2010

Various definitions of capacity of a subset of a domain in Euclidean space have been used in recent times to shed light on the solvability and spectral theory of elliptic partial differential equations and to establish properties of the Sobolev spaces in which these equations are studied. In this paper we consider two definitions of the capacity of a closed set E in a domain G. One of these capacities measures, roughly speaking, the amount by which the set of function in C∞(G) which vanish near E fails to be dense in the Sobolev space Wm, p(G).
A Note on Stone Lattices
Authors: Speed T. P.
Published: December 2010

Stone lattices can be considered as forming a category of abstract algebras and thus there is a forgetful functor from this category to the category of distributive lattices with zero and unit. In this note we consider Stone lattices in this light (cf. [3], [4]) and describe an adjoint to the forgetful functor. The Stone extension of a distributive lattice with zero unit which we obtain differs markedly from the one given in [1].
On the Generating Function for Permutations with Repetitions and Inversions
Authors: Abramson Morton
Published: December 2010

Let (a1, a2, …, am), ai ∊ {l, 2, …, n}, be an m-permutation of n (repetitions allowed) with exactly kj of the a's equal to j, j = 1, 2, …, n, m = k1+ …+kn, fixed nonnegative integers. An inversion is a pair i, j such that i < j, ai > aj. Denote by N(r; k, …, kn) the number of such permutations with exactly r inversions.
On D. J. Lewis's Equation x3+117y3 = 5
Authors: Finkelstein R. , London H.
Published: December 2010

In a recent publication [2], D. J. Lewis stated that the Diophantine equation x3+117y3 = 5 has at most 18 integer solutions, but the exact number is unknown. In this paper we shall solve this problem by proving the following

Theorem. The equationx3+117y3 = 5 has no integer solutions.
Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras
Authors: Lemire F. W.
Published: December 2010

Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.
Some Remarks Concerning Contraction Mappings
Authors: Reich Simeon
Published: December 2010

The following result is proved in [1, p. 6].

Theorem 1. Let X be a complete metric space, and let T and Tn(n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and un respectively. Suppose that limn → ∞ Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.
On a Relation between the “Square” Functional Equation And The “Square” Mean-Value Property
Authors: Haruki Hiroshi
Published: December 2010

We consider the following functional equation

1

where ƒ = ƒ(x, y) is a real-valued function of two real variables x, y on the whole xy-plane and t is a real variable.

With regard to the geometric meaning of (1), the equation is called the “square” functional equation.
Some Distributions of Ordered Values from Burr and Beta Distributions
Authors: Kabe D. G.
Published: December 2010

In this paper we use some known transformations available in the Theory of Multiple Integrals to give straightforward, simpler, and elegant proofs of some distributions of ordered values from Burr and beta distributions. The exact distribution (under the null hypothesis) of Wilks' ∧ criterion is obtained by considering it as a certain minimum value distribution problem.
On an Algorithm for Ordering of Graphs
Authors: Sekanina Milan
Published: December 2010

Let (G, ρ) be a finite connected (undirected) graph without loops and multiple edges. So x, y being two elements of G (vertices of the graph (G, ρ)), 〈x, y〉 ∊ ρ means that x and y are connected by an edge. Two vertices x, y ∊ G have the distance μ(x, y) equal to n, if n is the smallest number with the following property: there exists a sequence x0, x1, …, xn of vertices such that x0 = x, xn = y and 〈xi-1, Xi〉 ∊ ρ for i = 1, …, n. If x ∊ G, we put μ(x, x) = 0.
Integer-Valued Continuous Functions II
Authors: Subramanian H.
Published: December 2010

We follow [6] and [7] for all terminologies. The purpose of this note is to prove

Theorem 1. Let X and Y be any two integer-compact spaces. The following are equivalent:

(1)

X is homeomorphic to Y.

(2)

C(X, Z) and C(Y, Z) are isomorphic as rings.

(3)

C(X, Z) and C(F, Z) are isomorphic as lattices.

(4)

C(X, Z) and C(Y, Z) are isomorphic as p.o. groups.

(5)

C(X, Z) and C(Y, Z) are isomorphic as multiplicative semigroups.
On Linear Functional Equations with Nonpolynomial C∞ Solutions
Authors: Światak Halina
Published: December 2010

It is known (cf. M. A. McKiernan [6]) that the only measurably bounded solutions ƒ of the equations

1

where x ∊ Rn, t ∊ R, αi(i= 1, …, m) span the space , and Σi ∊ 1 μi ≠0 for any I ⊂ {1, …, m}, are polynomials. The degree of these polynomials and the dimension of the solution space can be estimated by numbers depending on m and n. (For estimates and other details concerning equations (1) see see [1], [2], [3], [4], [5], [6].)
On Canonical Generators of Subgroups
Authors: Fantham Peter
Published: December 2010

Let H be a cyclic group, K⊂H a subgroup and x, y generators of H, K. We shall say that x, y are related if y=xa where a is the index of K in H, in other words, y is the smallest positive power of x in K. The main purpose of this note is to show that for any group G one may, by means of the axiom of choice, choose for each cyclic group H⊂G a generator xH such that when K⊂H then xK, xH are related.
On a Generalization of One Dimensional Random Walk with a Partially Reflecting Barrier
Authors: Handa B. R.
Published: December 2010

Consider a one-dimensional random walk model where a particle starting at the origin at any instant either takes a jump through a unit distance to the right with probability p1, or stays at the same position with probability p0, or else takes a jump through either of 1, 2, … μ, units of distance to the left with probabilities p-1, p-2, …, p-μ respectively.
Some Results in a Correlated Random Walk
Authors: Jain G. C.
Published: December 2010

In connection with a statistical problem concerning the Galtontest Cśaki and Vincze [1] gave for an equivalent Bernoullian symmetric random walk the joint distribution ofgandk, denoting respectively the number of positive steps and the number of times the particle crosses the origin, given that it returns there on the last step.
On Convergence of Iterations for Fixed Points of Repulsive Type
Authors: Koneru Rao S. , Lalli B. S.
Published: December 2010

This paper deals with the convergence of iterations of the equation

1.1

with u0 ∊ Rn, both as a root and a point of repulsion. Here u and ϕ are real n-vectors and ϕ is continuously differentiate with respect to u. By taking u0 as an initial approximation to u0 in a neighbourhood of u0 we define the sequence of iterates {uk+1} by

1.1

For the sake of convenience we shall, sometimes, call the equation (1.1) an iteration.
Admissible Estimators of θr in Some Extreme Value Densities
Authors: Singh R.
Published: December 2010

Let the random variables X, Y, Z have respectively the extreme value densities as

1

2

3

where θ>0, λ>0 and μ are real numbers.
A Note on Normal Attraction to a Stable Law
Authors: Menon M. V. , Seshadri V.
Published: December 2010

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).
Characterizations of p-Spaces
Authors: Pareek C. M.
Published: December 2010

The concept of p-space is quite recent. It was introduced by Arhangel'skii [2]. The definition of p-space given in [2] involves compactification of the space. In view of the interesting properties of p-spaces obtained in [2], Alexadroff [1] suggested a problem of finding a direct intrinsic definition (without appeal to compactification). The main aim of this note is to answer the above problem.

I am grateful to Dr. S. K. Kaul for his comments.
On a Topology Generated by Measurable Covers
Authors: Eames W.
Published: December 2010

In [2] we showed how, for a certain class of outer measures on a metric space, a measurable cover could be constructed for each subset A of the space. The function is a closure operator, and in this note some of the properties of the resulting topology are investigated. In particular, we obtain a sufficient condition for the space to be connected.
Oscillation on Finite or Infinite Intervals of Second Order Linear Differential Equations(1)
Authors: Willett D.
Published: December 2010

Recently, Ronveaux [11] has shown how to use a combination of a Riccati transformation and a homographie transformation to estimate both from below and above the distance between a zero and the succeeding or preceding extremum (zero of y' ) of solutions of

1.1

In this paper, we show how such transformations can be used to derive an equation from which the distance between successive zeros of a solution y of (1.1) can be estimated directly.
Isomorphisms in Subspaces of c0
Authors: Lohman Robert H.
Published: December 2010

A Banach space X is said to be subspace homogeneous if for every two isomorphic closed subspaces Y and Z of X, both of infinite codimension, there is an automorphism of X (i.e. a bounded linear bijection of X) which carries Y onto Z. In [1] Lindenstrauss and Rosenthal showed that c0 is subspace homogeneous, a property also shared by l2, and conjectured that c0 and l2 are the only subspace homogeneous Banach spaces. In that paper no mention was made of subspaces of c0.
Orthogonal Families of Sets
Authors: Marica John
Published: December 2010

In this note we answer in the affirmative the following question.

If S is a finite family of sets such that each subset of a member of S also belongs to S, is it possible to arrange the sets in pairs so that the first member of the pair is contained in the complement of the second member?
Multiplication of Operators by C∞ Functions
Authors: Shultz Harris S.
Published: December 2010

Let (respectively, ) denote the set of all complex-valued infinitely differentiable functions defined on the reals whose supports are bounded on the left (respectively, right). Under addition and convolution is a commutative algebra.
On a Theorem of D. W. Barnes
Authors: Stitzinger Ernest L.
Published: December 2010

Let L be a solvable Lie algebra and A be an abelian ideal of L. For a ∊ A, let da be the (right) inner derivation of L generated by a and let exp da = 1 + da. Since A is abelian, exp da is an automorphism and exp da exp db = exp da+b for all a, b ∊ A. Let I(L, A) be the subgroup of the automorphism group of L generated by exp da for all a ∊ A.
A Family of Combinatorial Identities
Authors: Andrews G. E. , Subbarao M. V. , Vidyasagar M.
Published: December 2010

In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,

1.1

and obtained the following infinite family of identities:

For A= 1,2, 3,…, let

1.2

where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then

1.3

As Eden noted, Fh(b;x) is the generating function of ph(m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct:
On a Problem of Groübaum
Authors: Erdös P.
Published: December 2010

Pn will denote a set of n points in the plane. A well known theorem of Gallai- Sylvester (see e.g. [4]) states that if the points of Pn do not all lie on a line then they always determine an ordinary line, i.e. a line which goes through precisely two of the points of Pn.

Using this theorem I proved that if the points do not all lie on a line, they determine at least n lines. I conjectured that if n>n0 and no n—1 points of Pn are on a line, they determine at least 2n-4 lines. This conjecture was proved by Kelly and Moser [3], who, in fact, proved the following more general result: Let Pn be such that at most n—k of its points are collinear.
On the Product of the Primes
Authors: Hanson Denis
Published: December 2010

In recent years several attempts have been made to obtain estimates for the product of the primes less than or equal to a given integer n. Denote by the above-mentioned product and define as usual

Analysis of binomial and multinomial coefficients has led to results such as A(n)<4n, due to Erdôs and Kalmar (see [2]). A note by Moser [3] gave an inductive proof of A(n)<(3.37)n, and Selfridge (unpublished) proved A(n)<(3.05)n More accurate results are known, in particular those in a paper of Rosser and Schoenfeld [4] in which they prove Θ(n)< 1.01624n; however their methods are considerably deeper and involve the theory of a complex variable as well as heavy computations. Using only elementary methods we will prove the following theorem, which improves the results of [2] and [3] considerably.
A Remark on Coloring Integers
Authors: Spencer Joel
Published: December 2010

Color the integers 1, 2,…, n red and blue. In this note we consider arithmetic sequences in which the discrepancy between red and blue integers is maximized.

More formally, we represent the coloring by a function λ:{1,…,n}→{+1, — 1}. Set

1
Bounds for Characteristic Values of Positive Definite Matrices
Authors: Binding P. A. , Hoskins W. D. , Ponzo P. J.
Published: December 2010

We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].
Some Theorems on Strong Nörlund Summability
Authors: Cass Frank P.
Published: December 2010

Throughout this paper H, H1 etc. will denote positive constants which will not necessarily be the same at different occurrences.
If is a series, we shall use the notation For a real, define

1

Let {pn} be a sequence with p0>0 and pn≥0 for n > 0. Define

2

The following identities are immediate:

3

4

where

5
Riemannian Structures Subordinate to Certain Almost Tangent Structures
Authors: Closs M. P.
Published: December 2010

Almost tangent structures have been studied by Eliopoulos [1] and certain Riemannian structures subordinate to almost tangent structures have been studied by Closs [2]. In this paper we investigate those subordinate Riemannian structures for which the underlying almost tangent structure is without torsion and those for which the fundamental form is closed.

Similar studies have been carried out with respect to Riemannian structures subordinate to almost complex structures by Lichnerowicz [4] and with respect to Riemannian structures subordinate to almost product structures by Legrand [5].
Zeros of Linear Combinations of Polynomials
Authors: Rahman Q. I.
Published: December 2010

The following theorem is due to J. L. Walsh (see [2, Theorem 17, 2a]):

Theorem. If all the zeros of f1(z)=zn+a1zn-1+ … + an lie in or on the circle C1 with centre c1 and radius r1 and if all the zeros of f2(z)=zn+b1zn-1+ … + bn lie in or on the circle C2 with centre c2 and radius r2, then each zero of the polynomial

lies in at least one of the circles Γk with centre γk and radius ρk, where

and where the ωk (k= 1, 2,…, n) are the nth roots of λ.
On the Multiplicative Inverse Eigenvalue Problem
Authors: De Oliveira G. N.
Published: December 2010

By "multiplicative inverse eigenvalue problem" (m.i.e.p., for short) we mean the following. Let A be an n×n matrix and let s1,…, sn be n given numbers. Under what conditions does there exist an n×n diagonal matrix V such that VA has eigenvalues s1,…,sn?

In the "additive inverse eigenvalue problem" (a.i.e.p., for short) we seek the diagonal matrix V so that A + V has eigenvalues s1,…, sn?.

In the present paper we extend to the m.i.e.p. the ideas used in [7] for the a.i.e.p.

By per X we denote the permanent of the square matrix X.
Group Partition, Factorization and the Vector Covering Problem
Authors: Herzog M. , Schönheim J.
Published: December 2010

The covering problem. Let Si(i = 1, 2,…,n) be given sets containing mi elements respectively and let

1

be their cartesian product. The elements of S(n) will be called vectors. The vector (x1 x2,…, xn) covers (y1 y2,…, yn) if xi =yi for at least n—1 values of i. A subset M of S(n) is said to be a covering (perfect covering) of S(n) if each member of S(n) is covered by at least (exactly) one member of M. A covering M is said to be linear if the sets Si are groups Gi and M is a subgroup of G(n) = S(n) Denote by σ(n; m1 m2,…, mn) the value of min |M| when M runs through all coverings of S(n) and by σ(n; m1 m2,…, mn) the value of min |M| when the sets Si are given groups Gi and M runs through all linear coverings of G(n).
On the Monotone Nature of Boundary Value Functions for nth-Order Differential Equations
Authors: Peterson A. C.
Published: December 2010

We are concerned with the nth (n≥3) order linear differential equation

1

where the coefficients are continuous on (∞, ∞). Our main result is to give conditions under which the two-point boundary value function rij(t) (see Definition 2) are strictly increasing continuously differentiable functions of t. Levin [1] states without proof a similar theorem concerning just the monotone nature of the rij(t) but assumes that the coefficients in (1) satisfy the standard differentiability conditions when one works with the formal adjoint of (1). Bogar [2] looks at the same problem for an nth-order quasi-differential equation where he makes no assumption concerning the differentiability of the coefficients in the quasi differential equation that he considers. Bogar gives conditions under which the rij(t) are strictly increasing and continuous. The different approach of the author to this problem also enables the author to establish the continuous differentiability of the rij(t) and to express the derivatives in terms of the principal solutions Uj(x, t), j=0, 1,…,n-1 (see Definition 4).
Consistency in the Reconstruction of Patterns from Sample Data
Authors: Moore Marc
Published: December 2010

Let A be a k-dimensional Euclidean region having unit volume. An m-colors pattern is a partition PA of A into m sets Ai, i—1,…, m with positive volume. PA is called a random pattern if in addition the partition of A is a realization of a random process with the following stationarity and isotropy properties:

(i) for all points s ∊ A, P(s ∊ Ai) =pi9, i = 1,..., m

(ii) for all pair of points s, s'∊ A with distance |s—s|=d between them, P(s' ∊ Ai | s ∊ Aj)Pij(d) i,j= 1,..., m.
On the ith Latent Root of a Complex Matrix(1)
Authors: Al-Ani Sabri
Published: December 2010

Goodman [1] has pointed out the applications of the distributional results of the complex multivariate normal statistical analysis. Khatri [4], has suggested the maximum latent root statistic for testing the reality of a covariance matrix. The joint distribution of the latent roots under certain null hypotheses can be written as, [2], [3],

1

where

and
On f-Prefrattini Subgroups
Authors: Chambers Graham A.
Published: December 2010

The Prefrattini subgroups of a finite soluble group were introduced by Gaschutz [3]. These are a conjugacy class of subgroups which avoid complemented chief factors and cover Frattini chief factors. Gaschutz [3, Satz 7.1] showed that if G has p-length 1 for each prime p, and if U≤G avoids all complemented chief factors and covers all Frattini factors, then U is a Prefrattini subgroup of G. We begin by proving the analogous result for the f-Prefrattini subgroups introduced by Hawkes [5], If f is a saturated formation, then the f-Prefrattini subgroups of G are a conjugacy class of subgroups which avoid f-eccentric complemented chief factors of G and cover all other chief factors of G.
Colour Classes for r-Graphs
Authors: Cockayne E. J.
Published: December 2010

By anr-graph G we mean a finite setV(G) of elements called vertices and a setE(G) of some of ther-subsets ofV(G) called edges. This paper defines certain colour classes ofr-graphs which connect the material of a variety of recent graph theoretic literature in that many existing results may be reformulated as structural properties of the classes for some special cases ofr-graphs. It is shown that the concepts of Ramsey Numbers, chromatic number and index may be defined in terms of these classes. These concepts and some of their properties are generalized. The final subsection compares two existing bounds for the chromatic number of a graph.
Equivalence of Lipschitz Structures
Authors: Fraser Robert B.
Published: December 2010

In 1962, V. Ju. Sandberg gave a definition of a Lipschitz structure more general than that of a metric space. In [1], the author gave alternate definitions of a Lipschitz structure, one in terms of a family of pseudometrics and the other in terms of sequences of entourages. These definitions were proved equivalent to each other. In 1965, Sandberg gave conditions under which his Lipschitz structure was generated by a metric.
On the Homology of the n-Special Reduced Product Space of a Euclidean Space
Authors: Wakae M. , Hamara O.
Published: December 2010

In [2] and [3] the homology of reduced product spaces of certain type of polyhedra was studied. Let Xn=X×X×…×X be the Cartesian product of n copies of a topological space X.
On Sturm's Separation Theorem
Authors: Beesack Paul R.
Published: December 2010

The purpose of this note is to obtain an extension of the classical Sturm separation theorem for the second order, linear selfadjoint differential equation

1

to the case of a noncompact interval. The classical theorem (cf. [3, p. 209], [4, p. 224]) assumes that r and s are continuous with r positive on a compact interval I, and concludes that between each pair of zeros (on I) of one (nontrivial) solution of (1) there lies precisely one zero of any other linearly independent solution of (1). If I is not compact, a function y which is a solution of (1) on /may often be extended (continuously) to an endpoint, say a, of I.
A Theorem in the Partition Calculus
Authors: Erdös P. , Milner E. C.
Published: December 2010

If S is an ordered set we write tp S to denote the order type of S and |5| for the cardinal of S. We also write [S]k for the set {X:X ⊂ S, |X|=k}. The partition symbol

(1)
On Unbiased Estimation of a Vector Parameter
Authors: Mehra K. L. , Ramachandramurty P. V.
Published: December 2010

It is shown in this paper that Rao's criterion of comparing two unbiased estimators on the basis of definiteness of the difference between their dispersion matrices is equivalent to Cramer's criterion based on their concentration ellipsoids. When the estimators have normal distributions it is shown that both the criteria have a desirable property in terms of the probabilities of the estimators lying in ellipsoids with the parameter point as the center.
Local Malcev Conditions
Authors: Pixley Alden F.
Published: December 2010

Let p and q be polynomial symbols of a type of algebras having operations ∨, ∧, and; (interpreted as the join, meet, and product of congruence relations). If is an algebra, L(), the local variety of , is the class of all algebras such that for each finite subset G of there is a finite subset F of such that every identity of F is also an identity of G.

THEOREM. There is an algorithm which, for each inequality

p≤q,

and pair of integers n, k≥2, determines a set Un, k of (Malcev) equations with the property:

For each algebra, p≤q is true in the congruence lattice offor each∊L() if and only if for each finite subset F ofand integer n≥2 there is a k=k(n, F) such that Un, kare identities of F.

This generalizes a corresponding result for varieties due to Wille (Kongruenzklassengeometrien, Lect. Notes in Math. Springer- Verlag, Berlin-Heidelberg, New York, 1970) and at the same time provides a more direct proof.
On the Frattini Subgroups of Generalized Free Products with Cyclic Amalgamations(1)
Authors: Tang C. Y.
Published: December 2010

In [1] Higman and Neumann asked the questions whether the Frattini subgroup of a generalized free product can be larger than the amalgamated subgroup and whether such groups necessarily have maximal subgroups. In [4] Whittemore gave answers to the special cases of generalized free products of finitely many free groups with cyclic amalgamation and of generalized free products of finitely many finitely generated abelian groups. In this paper we shall study the Frattini subgroups of generalized free products of any groups with cyclic amalgamation.
On Fixed Point Theorems for Mappings in a Separated Locally Convex Space
Authors: Lee Cheng-Ming
Published: December 2010

The Banach contraction principle has been generalized by Tan [6] to the mappings in separated locally convex spaces. We show that the result of Sehgal [5] and also of Holmes [3] can be generalized in the same way.

Throughout this note, we let X be a separated locally convex space, U a base for the closed absolutely convex neighborhoods of the origin O in X, K a nonempty subset of X, and Ta mapping from K to K.
A Note on Generalized Polynomial Identities
Authors: Martindale Wallace S.
Published: December 2010

Let A be an algebra with 1 over a field F and let B be a fixed F-basis of A. Let F〈x〉=F〈x1,…, xn,…,〉 be the free algebra over F in noncommutative indeterminates x1,…, xn,…, and denote by AF〈x〉 the free product of A and F〈x〉 over F. The elements of AF〈x〉 of the form varies, repetitions allowed) form an F-basis of AF〈x〉. They will be referred to as basis monomials, and the involved in a particular basis monomial will be called the coefficients of that basis monomial.
Smoothing One-Dimensional Foliations on S1 x S1
Authors: Cohen Maurice
Published: December 2010

Letf:S1→S1 be an orientation preserving C1-diffeomorphism. Denote by the flow on S1 x S1 which is the suspension of f(see Smale [5]).

We consider the problem of approximating by a smoother foliation.
Oscillation Criteria for Second Order Nonlinear Delay Equations
Authors: Erbe Lynn
Published: December 2010

It is the purpose of this paper to establish oscillation criteria for second order nonlinear differential equations with retarded argument. Specifically, we consider the equation

1.1

where f ∊ C[0, + ∞) x R2, g ∊ C[0, + ∞), and

1.2

We shall restrict attention to solutions of (1.1) which exist on some ray [T, + ∞). A solution of (1.1) is called oscillatory if it has no largest zero.