bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical Range and Convex Sets*
Authors: Pollack Fredric M.
Published: December 2010

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined by

W(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.
On Permutation Groups with Regular Subgroup
Authors: Bercov R. D.
Published: December 2010

W. Burnside [3, p. 343] showed that a cyclic group of order pm (p prime, m > l) cannot occur as a regular subgroup of a simply transitive primitive group. (For definitions and notation see [9].) Groups which are contained regularly in a primitive group G only when G is doubly transitive are therefore called B-groups [9, p. 64].
The Derivation Algebra of M48(C)
Authors: Goodaire Edgar G. , Snell Roy C.
Published: December 2010

Let C by a Cayley-Dickson algebra over an algebraically closed field F of characteristic 0. A multiplication table for a basis of this 8-dimensional alternative algebra can be found in [3], page 137, where we take α = β = γ = — 1.
An Elementary Proof of the Frobenius Factorization Theorem for Differential Equations
Authors: Kennedy E. W.
Published: December 2010

Consider the 1st order linear system

1

when An(t) is real valued and continuous on I, ai, i+1(t) ≠ 0, aij(t) ≡ 0 for j ≥ i + 2, t ∊ I and i = 1, …, n — l. In what follows, components cj of a vector c will be identifiable by the superscript. The purpose of this note is to give a simple induction proof of the following theorem of Frobenius [1].
Quasi-Differentiable Norms
Authors: Whitfield J. H. M.
Published: December 2010

Let E be a real Banach space with norm ρ. Let S={x ∊ E: ρ(x) = 1}. A norm on E is admissible if it generates the same topology as ρ.

The normρ is Gateaux differ-entiable if for each x ∊ S and u ∊ E

exists.
A Definition of Separation Axiom
Authors: Cramer T.
Published: December 2010

Several separation axioms, defined in terms of continuous functions, were examined by van Est and Freudenthal [3], in 1951. Since that time, a number of new topological properties which were called separation axioms were defined by Aull and Thron [1], and later by Robinson and Wu [2], This paper gives a general definition of separation axiom, defined in terms of continuous functions, and shows that the standard separation axioms, and all but one of these new topological properties, fit this definition.
Lyapunov Inequalities and Bounds on Solutions of Certain Second Order Equations*
Authors: Eliason Stanley B.
Published: December 2010

In this paper we consider the equation

(1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0

under the conditions

((H0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J;

and

(H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation

(1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0

under the conditions (H1) and

(H2): the real valued functions m and n are continuous on a non-trivial interval J of reals.
A Note on Weakly Symmetric Rings
Authors: Horn P. J.
Published: December 2010

T. Nakayama showed in [2, Theorem 13] that symmetric algebras have the property that the left and right annihilators of their two-sided ideals are equal. He also gave examples [2, p. 630] to show that QF algebras with this property are not necessarily symmetric, and that weakly symmetric algebras need not have this property.
Integral Domains which have Finite Character Locally
Authors: Pacholke Kenneth
Published: December 2010

In recent papers Brewer and Mott have studied integral domains which have finite character globally. This paper concentrates on domains which have finite character locally. Examples include global finite character domains plus Prufer, almost Dedekind, and almost Krull domains. General properties are given, including a valuation-theoretic characterization. The effect of requiring essential and/or rank one valuations is also studied.
Fixed Point Theorems for Point-to-Set Mappings
Authors: Wong Chi Song
Published: December 2010

Let (X, d) be a metric space. Let Xc be the family of all non-empty closed subsets of X endowed with the Hausdorff metric D induced by d. Let f be a non-expansive mapping of X into Xc (D(f(x), f(y))≤d(x, y) for all x, y in X). Some fixed point theorems are obtained by imposing certain conditions on f and X.
A Note on Mal′cevian Varieties
Authors: Shafaat Ahmad
Published: December 2010

By a homomorphic relation over an algebra A we mean a subalgebra of A × A. A variety [1[ of algebras will be called Mal′cevian [2[ if the identities of include two identities of the form f (x, y, y)=x, f(x, x, y)=y. In [3[ many examples and interesting properties of Mal′cevian varieties have been quoted or proved. In [4[ it is noted that every reflexive homomorphic relation over an algebra of a Mal′cevian variety is a congruence. The purpose of this short note is to observe that the property of Mal′cevian varieties noted in [1] is in fact characteristic of such varieties.
The Degrees of Radical Extensions
Authors: Ursell H. D.
Published: December 2010

The results obtained here must have been known and settled centuries ago. However, they have proved impossible to locate in the available literature. H. K. Farahat has asked for proofs of the linear independence over the rationals of certain infinite sequences of real numbers such as √2, √3, √5.... He also raised the general question of determining the degree of the field extension generated over the rationals by a family of positive irrational numbers of the form x=a1/mwhere a, m are positive integers.
Synchronous and Asynchronous Reversible Markov Systems(1)
Authors: Dawson D. A.
Published: December 2010

The relationships between synchronous and asynchronous reversible Markov systems are investigated. It is shown that the invariant measure of such systems is a second order Markov random field. The conditions under which the invariant measure is a first order Markov random field are obtained.
On Involutions of Quasi-Division Algebras
Authors: Sweet Lowell
Published: December 2010

All algebras are assumed to be finite dimensional and not necessarily associative. An involution of an algebra is an algebra automorphism of order two. A quasi-division algebra is any algebra in which the non-zero elements form a quasi-group under multiplication. The purpose of this short paper is to determine the structure of all involutions of quasi-division algebras and to give an application of this result.
Group Partitions and Mixed Perfect Codes
Authors: Lindström Bernt
Published: December 2010

Let G be a finite abelian group of the order pr and type (p, …, p), where p is a prime. A necessary and sufficient condition is determined for the existence of subgroups G1, G2, ⋯, Gn, one of the order pa and the rest of the order pb, such that G = G1 ∪ G2 ∪ ⋯ ∪ Gn and Gi, ∩ Gj,= {θ} when i ≠ j.
Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type
Authors: Rao V. Venugopal
Published: December 2010

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ = x+iy, y > 0), such that f(τ + λ)= f(τ) where λ is a positive real number and f(—1/τ) = γ(—iτ)kf(τ), k being a complex number. The function (—iτ)k is defined as exp(k log(—iτ) where log(—iτ) has the real value when —iτ is positive. Every such function is said to have signature (λ, k, γ) in the sense of E. Hecke [1] and has a Fourier expansion of the type f(τ) = a0 + σ an exp(2πin/λ), (n = 1,2,…), if we further assume that f(τ) = O(|y|-c) as y tends to zero uniformly for all x, c being a positive number.
Regular, Commutative, Maximal Semigroups of Quotients
Authors: Rompke Jurgen
Published: December 2010

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.
Some Admissible Estimators in Extreme Value Densities
Authors: Singh R.
Published: December 2010

Let X be a random variable having the extreme value density of the form

(1)

where r is assumed to be a positive Lebesgue measurable function of x and the function q is defined by

for all θ in Ω = (0, ∞). It is further assumed that q(θ) approaches zero as θ → ∞.

In this note we are concerned with estimating parametric functions g(θ) of the form [1/q(θ)]α, α any real number. The loss function is assumed to be squared error and the estimators are assumed to be functions of a single observation X.
Perfectly Homogeneous Bases in Banach Spaces
Authors: Casazza P. G. , Lin Bor-Luh
Published: December 2010

A bounded basis {xn} of a Banach space X is called perfectly homogeneous if every bounded block basic sequence {yn} of {xn} is equivalent to {xn}. By a result of M. Zippin [4], a basis in a Banach space is perfectly homogeneous if and only if it is equivalent to the unit vector basis of c0 or lp, 1 ≤ p < + ∞. A basis {xn} of a Banach space X is called symmetric, if every permutation {xσ(n)} of {xn} is a basis of X, equivalent to the basis {xn}. It is clear that every perfectly homogeneous basis is symmetric.
A Note on Fibonacci Type Groups
Authors: Campbell C. M. , Robertson E. F.
Published: December 2010

Let Fn be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of Fn by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).
Note on a Subring of C*(X)(1)
Authors: Eng-Ung Choo
Published: December 2010

Throughout, topological spaces are assumed to be completely regular. C(X) (resp. C*(X)) will denote the ring of all (resp. all bounded) continuous real-valued functions. βX will denote the Stone-Cech compactification of X. In [2], Nel and Riorden defined C≠(X) to be the set of all f ∊ C(X) such that M(f) is real in the residue class ring C(X)/M for every maximal ideal M in C(X). C≠(X) is a subalgebra as well as a sublattice of C*(X).
L1-Sequential Convergence on Stone Spaces and Stone's Theorem on Unitary Semigroups
Authors: Cooper J. B.
Published: December 2010

The following theorem is a well-known tool in the study of measurable functions:

Theorem. Let (M; μ) be a finite measure space and let (xn) be a sequence of functions in L1(M; μ) so that xn→0 in the norm of L1(M; μ). Then there is a sub-sequence so that xnk →0 pointwise almost everywhere on M.
On the Structure of Quotient Rings Which are QFX Rings
Authors: Dunn Samuel L.
Published: December 2010

The object of this paper is to consider the relationships between matrix rings and rings having classical quotient rings which are quasi-Frobenius X (QFX) rings. The main result of this paper is Theorem 12, which shows that if S is a ring with a QFX right classical quotient ring T, then T is isomorphic to a direct sum of a finite number of matrix rings over local rings Ui, while S is almost a direct sum of matrix rings over rings Ci, the Ui being right classical quotient rings of the Ci.
W-Groups and Near-Ring Modules
Authors: Mason Gordon
Published: December 2010

W-groups are treated in the more general context of near-ring modules. It is shown that they share properties of d.g. modules with respect to central series. In particular, results of Magomaev are obtained and used to show that the group-theoretic and module-theoretic central series coincide for W-groups.
Interpolation by Linear Sums of Harmonic Measures
Authors: Ortel Marvin
Published: December 2010

Let α be an open arc on the unit circle

and for z=reiθ, 0 ≤ r < 1, let

(1)

The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.
Decomposition of Projections on Orthomodular Lattices(1)
Authors: Rüttimann G. T.
Published: December 2010

The set of projections in the BAER*-semigroup of hemimorphisms on an orthomodular latticeLcan be partially ordered such that the subset of closed projections becomes an orthocomplemented lattice isomorphic to the underlying latticeL. The set of closed projections is identical with the set of Sasaki-projections onL(Foulis [2]). Another interesting class of (in general nonclosed) projections, first investigated by Janowitz [4], are the symmetric closure operators. They map onto orthomodular sublattices where Sasaki-projections map onto segments of the latticeL.
Sequences by Number of w-Rises
Authors: Abramson Morton
Published: December 2010

An m-permutation of n, repetitions allowed, is an m-sequence

(1)

A w-rise is a pair (ei, ei+1) such that ei+1-ei≥w>0. In this note we find an expression for Tk, w(n, m), the number of m-sequences having precisely k w-rises. The case w = 1 is given in [1] [2]. Also, when w = 1 we give the number when each of the integers 1, 2, …, r must appear at least once.
EP Operators and Generalized Inverses
Authors: Campbell Stephen L. , Meyer Carl D.
Published: December 2010

The relationship between properties of the generalized inverse of A, A†, and of the adjoint of A, A*, are studied. The property that A†A and AA† commute, called (E4), is investigated. (E4) generalizes the property of A being EPr. A canonical form and a formula for A† are given if a matrix A is (E4). Results are in a Hilbert space setting whenever possible. Examples are given.
Contraction Property of the Operator of Integration
Authors: Janos Ludvik
Published: December 2010

It is shown that the operator of integration Fy(x) = ∫x0y(t) dt defined on the space C(—∞, ∞) of all continuous real valued functions on (—∞, ∞) is a contraction relative to a certain family of seminorms generating the topology of uniform convergence on compacta. However, as a contrast to this it is proved that F is not contractive with respect to any metric on C(—∞, ∞) inducing the above topology on C(—∞, ∞).
An Application of Pairwise Independence of Random Variables to Regression Analysis
Authors: Schaufele Ronald A.
Published: December 2010

The exact probability of making an error at the first stage of a stepwise regression is found, both when the independent variables are fixed and when they are random. The probabilities differ in the two cases because the random variables involved are pairwise independent but not jointly independent.
A Result on Sums of Squares
Authors: Davison J. L.
Published: December 2010

In this note we give an elementary proof of the following.

Let be an integer. n≥1 Then, every positive even integer less than or equal to n(n2—1)/3 can be expressed as a sum ofn squares of integers from the set {0, 1, 2, …, n - 1}.
On Countably, σ-, and Sequentially Barrelled Spaces
Authors: Husain T. , Khaleelulla S. M.
Published: December 2010

The first author [2] calls a Hausdorff locally convex (abbreviated to l.c.) space (E, u) countably barrelled if each σ(E′, E)-bounded subset of E′ which is the countable union of equicontinuous subsets of E′, is itself equicontinuous.
Comments on the Spinor Structure of Space-Time*
Authors: Lee K. K.
Published: December 2010

A simpler proof of the theorem on the spinor structure of space-time is given. Some geometrical insights are provided.
The Theory of Compositions (I): the Ordered Factorizations of n and a Conjecture of C. Long
Authors: Andrews George E.
Published: December 2010

Several years ago, C. Long wrote two papers ([3], [4]) that related to F(n) the number of ordered factorizations of n. The second of these papers [4] was devoted entirely to a discussion of conjectured formula for F(n).
Concepts in Vector Spaces with Convergence Structures
Authors: De Bruyn G. F. C.
Published: December 2010

Limit vector spaces ('limitierte Vektorräume') were defined by Fischer [1] and concepts such as continuity, compactness, etc. were introduced and studied by him and others, e.g. by Kent [3]. In this paper the concepts of precompactness and boundedness in limit vector spaces are studied. It is shown that most of their properties in topological vector spaces hold in limit vector spaces.
A Functional Equation Arising from Ivory's Theorem in Geometry
Authors: Haruki Hiroshi
Published: December 2010

In previous papers (see [1, 2, 3, 4]), we solved the following
functional equation:

1

wheref=f(z) is an entire function of a complex variable z and x, y are complex variables.
Analogy in Solving Poisson's Equation
Authors: Kreith Kurt
Published: December 2010

The passage from ordinary to partial differential equations is a difficult one. While many properties of ordinary differential equations admit analogies in the framework of partial differential equations, the techniques used to establish them may bear little resemblance to those which suffice in the simpler setting.
Some Properties of a Certain set of Interpolating Polynomials
Authors: Leeming David J.
Published: December 2010

A Lidstone series provides a (formal) two-point expansion of a given functionf(x)in terms of its derivatives of even order at the nodes 0 and 1 and takes the form.
Isomorphic Group Rings
Authors: Parmenter M. M.
Published: December 2010

Let R and S be rings with 1, G a group and RG and SG the corresponding group rings. In this paper, we study the problem of when RG≃SG implies R≃S. This problem was previously investigated in [8] for the case where G is assumed to be infinite cyclic. The corresponding question for polynomial rings, namely, when does R[x]≃S[x] imply R≃S, has been considered by several authors, particularly Coleman and Enochs [3].
A Remark on Talenti's Semigroup
Authors: Seidman Thomas I.
Published: December 2010

For α>0 the Riemann-Liouville Integral J(α) is given for suitable functions g by

1

For a variety of function spaces (e.g., C[0, 1] or Lp(0, 1) with p≥1) this defines a C0 semigroup which has been extensively studied (cf., e.g., [3]).
Some Properties of Indicatrices in a Finsler Space(1)
Authors: Srivastava T. N. , Watanabe Shôji
Published: December 2010

Let (Mn, F) be an n-dimensional Finsler space where Mn is the underlying n-dimensional manifold and is the Finsler fundamental function. F being a differentiable function of the point and element of support where T∞(Mn) is the tangent space of Mn at x and is positively homogeneous of degree one with respect to X.
Sets Homothetic to their Intersection with a Translate
Authors: Wilker J. B.
Published: December 2010

Inspired by a question of J. B. Miller, L. Fejes Tóth asked for a catalogue of those subsets of Euclidean n-space which are homothetic (similar and similarly situated) to their intersection with a suitably translated copy of themselves. For example, a triangle is homothetic to its intersection with an arbitrarily translated replica provided only that the intersection has non-void interior. In 3-space a cube is homothetic to its intersection with a replica translated part way along any body diagonal. With these two preliminary examples for motivation, let us make a definition.
Using Rolle's Theorem in Exponential Function-Derivative Approximation
Authors: Keener L.
Published: December 2010

For a continuously differentiable function g defined on an interval [α,β], define ||g|| to be the uniform norm of g, i.e. ||g||=sup ∊[α,β]|g(x)|. Define ||g||1, by ||g||1=max||g||, ||g'||}. We call the norm ||.||x the function-derivative norm.
An Application of a Generalization of Terquem's Problem
Authors: Tanny Stephen M.
Published: December 2010

Moser and Abramson [4] proved: given m≥2 and 0≤kl, k2, …, kp<m, the number ofp-combinations

satisfying

is

1

([x] denotes the greatest integer ≤x).
Some Results on Quasi-Uniform Spaces
Authors: Carter Karen S. , Hicks T. L.
Published: December 2010

Constructions are made of a T1 space which does not have a T1 completion and of a quasi-uniform space which is complete, but not strongly complete. An example relating to a completion due to Popa is given. An alternate definition for Cauchy filter, called C-filter, is examined and a construction of a C-completion is given. We discuss quasi-pseudometrics over a Tikhonov semifield RΔ. Every topological space is quasi-pseudometrizable over a suitable RΔ. It is shown that if a quasi-pseudometric space over RΔ is complete, the corresponding quasi-uniform structure is C-complete. A general method for constructing compatible quasiuniform structures is given.
On the Representation of Mappings of Tychonov Spaces as Restrictions of Linear Transformations
Authors: Chew Kim-Peu , Tan Kok-Keong
Published: December 2010

Let (X, τ) be a Tychonov space and the collection of all families of pseudometrics on X generating the topology τ on X. Let f:X→X and c>0. Then f is said to be a topological c-homothety if there exists some B in such that d(f(x), f(y))=cd(x, y) for all d ∈ B and all x, y in X (see [4]). We say that f can be linearized in L as a c-homothety if there exists a linear topological space L, and a topological embedding i:X→L such that i(f(x))=ci(x) for all x in X (see [4]).f is said to be squeezing if for some a in X.
A Metrization Theorem for 2-Manifolds
Authors: Vincent Paul A.
Published: December 2010

There are few known metrization theorems for manifolds (locally Euclidean, connected, Hausdorff space). It is well known that for manifolds metrizability, second countability, Lindelöf's condition, σ-compactness and paracompactness are equivalent. Although these conditions imply separability, the latter does not imply any of the former (see Example 2.2), as is often believed. A common source of metrization for a covering manifold is that lifted from the base manifold [8; p. 181].

For 2-manifolds, the presence of a complex analytic structure gives us a metrization theorem; it has been shown [1] that such manifolds are topologically characterized as those which are orientable and second countable.
On Rings with Many Endomorphisms
Authors: Neggers Joseph
Published: December 2010

All rings have an identity, all homomorphisms map identities to identities, all homomorphisms on algebras over fields are algebra homomorphisms. A ring R is a quotient-embeddable ring (a QE-ring) if for any proper ideal a of R there is an endomorphism of R whose kernel is the ideal a. A QE-ring U is a receptor of R if for any proper ideal a of R there is a homomorphism from R to U whose kernel is the ideal a.

Theorem. A ring R has a receptor if and only if it is a K-algebra over some field K contained in the center of R. If R is a commutative K-algebra of this type, then it has a commutative receptor.
On a Group Presentation Due to Fox
Authors: Campbell C. M. , Robertson E. F.
Published: December 2010

In 1956 R. H. Fox had occasion, while investigating fundamental groups of topological surfaces, to believe that the group  was trivial. Using the Todd-Coxeter coset enumeration algorithm a proof was obtained, see [3], and this algorithmic proof was used to produce an algebraic proof, see [2]. In [1] Benson and Mendelsohn, using a similar method to that of [2] showed that  is trivial. In this note we give a direct proof for the more general problem of describing the structure of the group .
On Closure Conditions
Authors: Kannappan Pl. , Taylor M. A.
Published: December 2010

Quasigroups and groupoids with one or other of the Reidemeister or Thomsen closure conditions, the relationship among them with emphasis on their relationship to associativity viz groups, Abelian groups, have been investigated in [2], [3], [4], [5], [6], [12], and others. In [10] R- and T-groupoids, (that is, groupoids possessing one of the first two closure conditions mentioned above) which are generalizations of groups and Abelian groups were investigated. In this paper, we show that groupoids with the given identities may be described in terms of R- and T-groupoids. These results and others are used to give another proof of theorems given in [1], [7], and [5] describing the variety of all groups and Abelian groups defined by single laws.
Averaging Interpolation of Hermite-Fejér Type
Authors: Saxena R. B.
Published: December 2010

Averaging interpolation generalizes polynomials of Lagrange interpolation and also the next-to-interpolatory polynomials. This notion has recently been introduced by Motzkin, Sharma and Straus [3]. Later in [5], Saxena and Sharma have considered the convergence problem of the averaging interpolators on Tchebycheff abscissas. It appears that averaging interpolators have convergence properties similar to those of Lagrange interpolation. It is therefore reasonable to look for an extension of these operators along the lines of Hermite-Fejér interpolation. This can be done in three ways: (i) by taking assigned averages of function-values and by taking the derivatives to be zero, (ii) by taking assigned function-values and by taking the averages of derivatives to be zero, or (iii) by taking averages of function-values and by taking the averages of derivatives to be zero. The object of this note is to take the first approach. The second approach has been the subject of study by M. Botto and A. Sharma [2].
A Note on Best Simultaneous Approximation in Normed Linear Spaces
Authors: Brøndsted Arne
Published: December 2010

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that
A Note on Absolutely Pure Modules
Authors: Enochs Edgar
Published: December 2010

Fieldhouse observed that any finitely presented left R-module P is projective with respect to pure exact sequences, i.e.

can always be completed to a commutative diagram when the sequence is pure exact. A left R-module A is absolutely pure if it is a pure submodule of every module which contains it.
Iterations for Diagonally Dominant Matrices
Authors: Shivakumar P. N. , Chew Kim Ho
Published: December 2010

Matrix iterative methods of solving systems of linear algebraic equations for a class of matrices which includes strictly and irreducibly diagonally dominant matrices are considered and a convergence theorem proved.
A Generalization of the Cyclotomic Polynomial
Authors: Rao K. Nageswara
Published: December 2010

In this paper, the cyclotomic polynomial is generalized and several of its properties based on the Môbius inversion are derived. It is deduced that a polynomial whose roots are the roots of a cyclotomic polynomial multiplied by those of another cyclotomic polynomial is the product of cyclotomic polynomials. Character sums and finite Fourier series are employed for the latter result.
A Note on Erdös-Renyi Law of Large Numbers
Authors: Deo Chandrakant M.
Published: December 2010

In this note the Erdös-Renyi law of large numbers is extended to stationary Gaussian sequences.
On a Bifurcation Theorem of Hopf and Friedrichs*
Authors: Freedman H. I.
Published: December 2010

For the autonomous system x' = F(x,∊), the case where the second Hopf-Friedrichs condition fails is analyzed in that sufficient conditions for bifurcation of non-trivial periodic solutions to occur are given. An application to mathematical ecology is also discussed.
Irrational Numbers arising from Certain Differential Equations
Authors: Murty M. Ram , Murty V. Kumar
Published: December 2010

Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven's proof to show that er is irrational for every non-zero rational r. Dixon [1] made a similar modification to show that π is not algebraic of degree 2. In this note, we prove a general theorem which gives Niven's and Koksma's results as easy corollaries. A suitable modification in our proof also gives Dixon's result.
Triangularizing Solvable Groups of Unipotent Matrices over a Skew Field
Authors: Sizer Walter S.
Published: December 2010

In this note we show that a solvable group of unipotent matrices over a skew field can be simultaneously triangularized.
Approximation by Generalised Polynomials with Integral Coefficients
Authors: Tzimbalario J.
Published: December 2010

Let C[0,1] be the space of all continuous real valued functions defined in [0,1] with the supremum norm

1

The subspace of C[0,1] consisting of all functions f(x) for which f(0) and f(l) are integers will be denoted by C0[0,1], Let be a sequence of real numbers satisfying:

2
Commutative Non-Singular Semigroups
Authors: Johnson C. S. , McMorris F. R.
Published: December 2010

It is well known (see [5]) that the maximal right quotient ring of a ringRis (von Neumann) regular if and only ifRis (right) non-singular (every large right ideal is dense). In [8] it was shown that for a semigroupS, the regularity ofQ(S), the maximal right quotient semigroup [7], is independent of the non-singularity ofS. Nevertheless, right non-singular semigroups form an important class of semigroups.
On an Arithmetic Convolution
Authors: Hanumanthachari J.
Published: December 2010

In this paper the congruence (f ∘ g)(n) = 0 (mod n) and the functional equation f ∘ f ∘ … ∘ f = g, are studied, where ∘ is an exponential regular convolution.
The Spectrum of a Finite Lattice: Breadth and Length Techniques
Authors: Nowakowski Richard , Rival Ivan
Published: December 2010

Efforts to determine the orders of the sublattices of an arbitrary finite lattice date back at least to the early 1930's, and notably, in the work of Fritz Klein-Barmen [3], [4]. Nevertheless, very little that is new has appeared in the literature since that time.
Variants of the Hölder Inequality and its Inverses
Authors: Wang Chung-Lie
Published: December 2010

This paper presents variants of the Holder inequality for integrals of functions (as well as for sums of real numbers) and its inverses. In these contexts, all possible transliterations and some extensions to more than two functions are also mentioned.
Hyperbolic Differential Equations and Wave Propagation
Authors: Duff G. F. D.
Published: December 2010

Mr. President, ladies and gentlemen, I have had great pleasure in accepting the invitation to deliver this tenth Jeffery-Williams lecture. As one of that generation who were strongly influenced by Ralph Jeffery and Lloyd Williams, I also appreciate the challenge of maintaining the high standard that these lectures named in their honour have established. May I also say that the lecture earlier this morning by Professor Dieudonnè was certainly fortunate for us.
Lie Algebra as a Unifying Concept in Modern Physics
Authors: Ihrig Edwin
Published: December 2010

Lie algebras, in the form of algebras of observables, play an essential role in the formulation of classical and quantum mechanics. We discuss whether lie groups play a similar role in general relativity through the holonomy group. We also explore what interrelations these ideas provide between classical physics, relativity and quantum physics.
Probabilistic Proofs of the Little Riesz Theorem
Authors: Al-Hussaini A. N.
Published: December 2010

The following theorem is a widely used corollary of the Thorn-Riesz convexity theorem ([1], pp. 526).

Theorem. Let (Ω, a, μ) be a positive measure space, and let

be a linear operator with ‖T‖1≤1, ‖T‖∞≤1. Then ‖T‖p≤1, 1≤p≤∞
Adjoint Interior-Point Boundary Conditions for Linear Differential Operators
Authors: Bryan Robert Neff
Published: December 2010

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.
The Metrizability of Spaces whose Diagonals have a Countable Base
Authors: Ginsburg John
Published: December 2010

It is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.
Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups
Authors: Djoković Dragomir Ž.
Published: December 2010

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.

We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.
Thin Lens Spaces
Authors: Hoffman P. , Zabrodsky A.
Published: December 2010

In Theorem 1 below we study the existence of spaces whose cohomology rings are isomorphic (as ungraded rings) to those of lens spaces. The case p = 2 is very simple and instructive, so let us consider it first.

Suppose X is a space such that where dim x = 2d (for example X = RP4 with d = 1).
Critical Graphs for Acyclic Colorings
Authors: Berman David M.
Published: December 2010

The concept of acyclic colorings of graphs, introduced by Grunbaum [2], is a generalization of point-arboricity. An acyclic coloring of a graph is a proper coloring of its points such that there is no two-colored cycle. We denote by a(G), the acyclic chromatic number of a graph G, the minimum number of colors for an acyclic coloring of G. We call G k-critical if a(G) = fc but a(G′) for any proper subgraph G′. For all notation and terminology not defined here, see Harary [3].
Forms with O-Orthogonal Lie Algebras
Authors: Servedio Frank
Published: December 2010

A form P of degree r is a homogeneous polynomial in k[Yi, …, Yn] on kn, k a field; Yi are the coordinate functions on kn. Let V(n, r) denote the k-vector space of forms of degree r. Mn(k) = Endk(kn) has canonical Lie algebra structure with [A, B] = AB-BA and it acts as a k-Lie Algebra of kderivations of degree 0 on k[Yi, …, Yn] defined by setting D(A)Y= Yo(-A) for A∈Endk(kn), Y∈V(n,l) = Homk(kn, k) and extending as a k-derivation. Define the orthogonal Lie Algebra, LO(P), of P by LO(P) =
Limit Preserving Summability of Subsequences
Authors: Keagy Thomas A.
Published: December 2010

The purpose of this paper is to characterize those matrices A with the property that if x is a sequence with limit point σ, then there exists a subsequence y of x such that lim Ay = σ.
Regular Partitions of Regular Graphs
Authors: Kleinschmidt Peter
Published: December 2010

In the study of the combinatorial structure of edge-graphs of convex polytopes one may ask whether a given graph possesses a partition consisting of certain kinds of subgraphs.

In this paper we describe some special partitions of 3-valent and 4-valent graphs. These partitions can serve as examples for a type of partially ordered structures, called polystromas, which have recently been considered by Griinbaum [3].
Fixed Points can Characterise Curves
Authors: Schirmer Helga
Published: December 2010

The concept of a firm fixed point of a selfmap of a metric space is introduced. Loosely speaking a fixed point is firm if it cannot be moved to a point nearby with the help of a map which is arbitrarily close to the given map. It is shown that a continuum always admits a selfmap with a firm fixed point if the continuum contains a triod and if the vertex of the triod has a neighbourhood which is a dendrite. This condition holds in particular for local dendrites. Hence a local dendrite is an arc or a simple closed curve if and only if it does not admit a selfmap which has a firm fixed point.
Ordered Desarguesian Affine Hjelmslev Planes
Authors: Thomas L. A.
Published: December 2010

A Desarguesian affine Hjelmslev plane (D.A.H. plane) may be coordinatized by an affine Hjelmslev ring (A.H. ring), which is a local ring whose radical is equal to the set of two-sided zero divisors and whose principal right ideals are totally ordered (cf. [3]). In his paper on ordered geometries [4], P. Scherk discussed the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. We shall define an ordered D.A.H. plane and follow Scherk's methods to extend his results to D.A.H. planes and their A.H. rings i.e., we shall show that a D.A.H. plane is ordered if and only if its A.H. ring is ordered. We shall also give an example of an ordered A.H. ring. Finally, we shall discuss some infinitesimal aspects of the radical of an ordered A.H. ring.
An Inequality for Gamma Functions
Authors: Kabe D. G.
Published: December 2010

By using Bellman-Wishart distribution, Bellman [1], an inequality for gamma functions is derived. This inequality generalizes a recent inequality given by Selliah [4].
On the Convergence Vector Space ℒ,(E, F) and its Dual Space
Authors: Beattie Ronald
Published: December 2010

Let E be a locally convex tvs, F a normed space and the space of continuous linear mappings from E into F In this paper, we investigate the continuous convergence structure (c-structure) on. denotes the resulting convergence vector space (cvs).

The c-structure is by definition the coarsest cvs structure on making evaluation a continuous mapping.
On Earle's mod n Relative Teichmüller Spaces
Authors: Zarrow Robert
Published: December 2010

In this paper we answer an open question of C. J. Earle ([2] §3.3 remarks (a) and (b)) in several cases. We first give some definitions and state some results which are given in greater detail in [2].

We let X be a smooth surface of genus g ≥ 2 and let M (X) be the space of smooth complex structures with the C∞topology. If μ∈M(X) let Xμdenote the Riemann surface determined by μ.
The Chinese Remainder Theorem and the Invariant Basis Property
Authors: Anderson David F.
Published: December 2010

Abstracrt

The Chinese Remainder Theorem states that if I and J are comaximal ideals of a ring R, then A/(I∩J)A is isomorphic to A/IA×A/JA for any left R-module A. In this paper we study the converse; when does A/(I∩J)A and A/IA×A/JA isomorphic imply that I and J are comaximal?
On the Power Map and Ring Commutativity
Authors: Bell Howard E.
Published: December 2010

Let R denote an associative ring with 1, letnbe a positive integer, and letk= 1, 2, or 3. The ringRwill be called an (n, k)-ring if it satisfies the identities

for all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that forn >1, any(n,l)-ring must have nil commutator idealC(R).Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.
O-Distributive Semilattices
Authors: Pawar Y. S. , Thakare And N. K.
Published: December 2010

Sufficient conditions for a semilattice to be a 0- distributive are obtained. Some equivalent formulations of 0- distributivity in a semilattice are given. Further, disjunctive 0- distributive semilattices are also characterized.
Nonoscillatory Solutions of x(m) = (-1)mQ(t)x
Authors: Cheng Sui-Sun
Published: December 2010

A continuous real vector function is said to be nonoscillatory on an interval if at least one of its components is of constant positive or negative sign there. In this note, various existence criteria for nonoscillatory solutions of the system x(m) = (-l)mQ(t)x are established. In some cases, additional monotonicity properties for these solutions are also given.
Self-Converse Tournaments
Authors: Eplett W. J. R.
Published: December 2010

Let Tn denote a tournament with vertices labelled 1, …, n. Any undefined terms can be found in [5]. The converse of Tn is the tournament obtained by reversing the orientation of all the arcs in Tn. A tournament is called self-converse (s.c.) if . The transitive tournaments are examples of s.c. tournaments. In this paper we provide a structural characterization of s.c. tournaments and we also characterize the score vectors of s.c. tournaments.
On a Class of Polynomials Associated with the Stars of a Graph and its Application to Node-Disjoint Decompositions of Complete Graphs and Complete Bipartite Graphs into Stars
Authors: Farrell E. J.
Published: December 2010

A star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.
A Second Note on Ingham's Summation Method
Authors: Segal S. L.
Published: December 2010

A series ∑ an is said to be summable (I) to the limit A if

(*)

Clearly the limit is the same whether x→∞ through all real values or only positive integer values, and the expression whose limit is being taken can also be expressed in the two equivalent forms

where [x] is the greatest integer ≤x.
Stochastic Measure Diffusion Processes
Authors: Dawson Donald A.
Published: December 2010

The purpose of this article is to give an introduction to the study of a class of stochastic partial differential equations and to give a brief review of some of the recent developments in this field. This study has evolved naturally out of the theory of stochastic differential equations initiated in a pioneering paper of K. Itô [13]. In order to set this review in its appropriate setting we begin by considering a simple scalar stochastic differential equation.
Semilinear Second Order Elliptic Oscillation
Authors: Swanson C. A.
Published: December 2010

These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form

(1)

in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.
Sur le Dérivé du Cuboïde Eulérien
Authors: Lagrange Jean
Published: December 2010

L'objet de cette note est de montrer que le dérivé d'un cuboïde eulérien n'est jamais parfait.

(1) Appelons cuboïde entier un cuboïde dont les arêtes et les diagonales des faces sont des entiers; appelons cuboïde parfait un cuboïde entier dont la diagonale intérieure est un entier.
Permutations Related to Secant, Tangent and Eulerian Numbers
Authors: Abramson Morton
Published: December 2010

It is well known that

1

whereAndenotes the number of "up-down" or alternating permutations

2

of 1, 2, …,n. The numbersA2nandA2n+1are known as the secant and tangent numbers respectively andA2n= (—l)"E2n, whereEnis the Euler number.
Hamiltonian Cycles in Strong Products of Graphs
Authors: Bermond J. C. , Germa A. , Heydemann M. C.
Published: December 2010

Abstract. Let denote the graph (k times) where is the strong product of the two graphs G and H. In this paper we prove the conjecture of J. Zaks [3]: For every connected graph G with at least two vertices there exists an integer k = k(G) for which the graph is hamiltonian.
Bounds on Positive Integral Solutions of Linear Diophantine Equations II
Authors: Borosh I. , Treybig L. B.
Published: December 2010

Let A be an m × n matrix of rank r and B an m × 1 matrix, both with integer entries. Let M2 be the maximum of the absolute values of the r × r minors of the augmented matrix (A | B). Suppose that the system A x = B has a non-trivial solution in non-negative integers. We prove (1) If r = n - 1 then the system A x = B has a non-negative non-trivial solution with entries bounded by M2. (2) If A has a r x n submatrix such that none of its r x r minors is 0 and x ≥ 0 is a solution of Ax=B in integers such that is minimal, then .
Extension of a Result of S. Mandelbrojt
Authors: Pham-Gia T.
Published: December 2010

We extend the following result to several variables: For any sequence {Nj}, we have C{Nj} = C{Mj}, with {Mj} logarithmically convex i.e.
Modules Behaving Like Torsion Abelian Groups
Authors: Zubair Khan M.
Published: December 2010

Recently H. Marubayashi [1,2] and S. Singh [10,11,12] generalized some results of torsion abelian groups for modules over some restricted rings, like bounded Dedekind prime rings, bounded hereditary Noetherian prime rings. Singh [12] introduced the concept of h-purity for a module MR satisfying the following conditions:

(I) Every finitely generated submodule of every homomorphic image of M is a direct sum of uniserial modules.
On the Inequality
Authors: Kardos Peter
Published: December 2010

In this paper, we are concerned with the functional inequality

1

where 0 < Pi < l, 0 < qi < l, fi(p)≠0, for 0 < P < 1, (i = 1, 2,..., n) and n is a fixed positive integer, n ≥ 2.

Inequality (1) was studied by Rényi and Fischer, (see [1], [3]) in the special case

2

and this provided a characterization of Rényi's entropy.
On Linear Matrix Equations
Authors: Scobey P. , Kabe D.G.
Published: December 2010

Some results from the theory of minimization of vector quadratic forms (subjected to linear restrictions) are used to obtain particular solutions to the usual types of linear matrix equations. An answer to a question raised by Greville [1] is supplied.
Auslander-Reiten Sequences for "Nice" Torsion Theories of Artinian Algebras
Authors: Roggenkamp K. W.
Published: December 2010

Let t be a field and a finite dimensional t-algebra. Auslander-Reiten sequences [AR] play a fundamental rôle in the representation theory of ; in particular, they can be used to construct new indecomposable modules from known ones. For the latter reason I think it worthwile to point out certain torsion theories on the category of -modules, such that the category of -torsionfree modules has Auslander-Reiten sequences; thus giving another construction of indecomposable modules.
Minimally Generated Modules
Authors: Rant W. H.
Published: December 2010

A non-zero module M having a minimal generator set contains a maximal submodule. If M is Artinian and all submodules of M have minimal generator sets then M is Noetherian; it follows that every left Artinian module of a left perfect ring is Noetherian. Every right Noetherian module of a left perfect ring is Artinian. It follows that a module over a left and right perfect ring (in particular, commutative) is Artinian if and only if it is Noetherian. We prove that a local ring is left perfect if and only if each left module has a minimal generator set.
Decomposition Theorems for q*-Rings
Authors: Hill David A.
Published: December 2010

Let R be a ring with identity. The study of rings in which every left (right) ideal is quasi-injective was begun by Jain, Mohamed, and Singh (3). They called these rings left (right) q-rings. A number of structure theorems have been proved for q-rings. See, for example, (1), (2), and (5). A ring with the dual property (rings in which every homomorphic image of R as a left (right) R-module is quasi-projective) is called left (right) q*. These rings were first studied by Koehler (4), where some results connecting q* -rings with q-rings were obtained.