bluuebunny/crossref_metadata_2025 huggingface dataset:
Unbiased distinguishing of translation families and integrability with respect to a convolution of measures
Authors: Konijn H. S. , Sacksteder R.
Published: April 2009

Let M1 and M2 be two sets of probability measures defined on Rn. Ameasurable R1 valued function h (l ≧1) is said to distinguish M1from M2unbiasedly if there are numbers or vectors I1 and I2 (I1≠I2) such that ∫Rnh(x)m(dx) = Ii if m is in Mi (i = 1, 2). Here we shall be concerned with the case where M1 and M2 are translation families, in that all of the elements of Mi are translates of a single measure mi. This means that if, for any t in Rn, mtt is the measure defined by , where E−t {x−t: x ∈ E}, then Mi = , where T is a subset of Rn. If M1 and M2 are of this type, we will investigate the conditions under which there does not exist a function to distinguishing M1 from M2 unbiasedly. A case of special interest arises if m2(E) = m1(BE) = m1 with B a non-degenerate n × n matrix, and particularly a nonzero multiple (scale parameter) of the identity matrix, cf. [1], [2]. For simplicity, take l = 1.
Alternative derivation of some regular continued fractions
Authors: Walters R. F. C.
Published: April 2009

In this paper we find an expression for ex as the limit of quotients associated with a sequence of matrices, and thence, by using the matrix approach to continued fractions ([5] 12–13, [2] and [4]), we derive the regular continued fraction expansions of e2/k and tan 1/k (where k is a positive integer).
Error analysis of an algorithm for summing certain finite series
Authors: Elliott David
Published: April 2009

An algorithm for summing the series , where the coefficients an are assumed known, and the quantities pn, satisfy a linear three term recurrence relation, has been given by Clenshaw [1]. If we suppose that the pn satisfy the recurrence relation where αn and βn are, in general, functions of n, then PN may be found by constructing a sequence {bn} for n = N(−1)0, where the b satisfy the inhomogeneous recurrence relation with the conditions, The sum PN is then given by This result can be readily verified by multiplying each side of equation (1.2) by pn, summing from n = 0 to N, and making use of equations (1.1) and (1.3).
Completion in the bidual
Authors: Nielsen R.
Published: April 2009

Let E, Ê, and E′ denote a locally convex linear Hausdorff space, completion of E and the dual of E, respectively. It has been observed that Ê is a subspace of E″ under certain conditions on E. It is the primary goal of this paper to give necessary and sufficient conditions for the Ê ⊂ E″ to be valid. Such conditions are found and are given Theorem 4. With a variation of the technique used, several equivalent characterizations of semi-reflexive spaces are given in Theorem 5. The nationa throughtout will follow that in [2].
Differences of functions and measures
Authors: Edwards R. E.
Published: April 2009

Throughout this paper, G will denote a locally compact Hausdorff group with a chosen left Haar measure m. For each s ∈ G, τs denotes the left translation operator which acts on functions ƒ according to the rule; τsƒ(x) = ƒ(s−1x) and on measures or distributions in the corresponding way. The associted left difference operators ts−1 is denoted by Δs. On occasions it will be more convenient to write τ(s) and Δ(s) in place of τs and Δs.
A note on the Ritz method with an application to overtone stellar pulsation theory
Authors: Andrew A. L.
Published: April 2009

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.
Coordinate and momentum representations in quantum mechanics
Authors: Lawden D. F.
Published: April 2009

On the basis of physical principles having a very general nature, A. Landé [1] has demonstrated that the mathematical structure of quantum mechanics can be derived without having recourse to the introduction of special assumptions of an ad hoc type (such as commutation rules governing canonical observables) which are not immediately suggested by our knowledge of the physical world, but which have simply originated as rules which mathematical physicists have discovered by past experience to yield conclusions in conformity with experiment.
Products of abelian hopfian groups
Authors: Baumslag Gilbert
Published: April 2009

Let ℜk be the variety of all nilpotent groups of class at most k. The purpose of this note is to prove the following Theorem 1. Letbe a variety of groups containing ℜ2, let A and B be torsion-free abelian hopfian groups and let P be the free-product2 of A and B. If P is residually torsion-free nilpotent, then P is hopfian.
Primitive homomorphic images of semigroups
Authors: Hall T. E.
Published: April 2009

Consideration of the primitive homomorphic images of semigroups arises naturally from their matrix representation theory. (See, for example, papers [1] and [2].) This is basically because a semigroup of matrices, in which all the matrices have the same rank, is necessarily primitive. (The set of idempotents in the semigroup may be empty, however).
Cubature method for the numerical solution of the characteristic initial value problem
Authors: Jain M. K. , Sharma K. D.
Published: April 2009

The resemblance of the Goursat problem for the hyperbolic partial differential equations to the initial value problem for the ordinary differential equations has suggested the extension of many well known numerical methods existing for (1.2) to the numerical treatment of (1.1). Day [2] discusses the quadrature methods while Diaz [3] generalizes the simple Euler-method. Moore [6] gives an analogue to the fourth order Runge-Kutta-method and Tornig [7] generalizes the explicit and implicit Adams-methods.
Characteristic multipliers and stationary integrals
Authors: Jones Arthur R.
Published: April 2009

In this note some rudimentary results about the characteristic multipliers of periodic solutions of differential equations are given which supplement those given by Poincaré [2], Chapitre IV, and by Wintner [4].
A characterization of lattice-ordered groups by their convex L-subgroups
Authors: Conrad Paul
Published: April 2009

In this paper we show that whether or not a group admits a lattice-order often depends upon whether or not it possesses a set of subgroups that satisfy certain algebraic conditions. Using these techniques we are able to determine large classes of groups that can be lattice-ordered.
Dynamical instability of linear canonical systems
Authors: Coppel W. A. , Howe A.
Published: April 2009

In the present paper we obtain first approximation formulae for the regions of dynamical instability of linear canonical systems. These formulae are analogous to the formulae for Hamiltonian systems stated by Krein and Jakubovič [5] and proved by Pittel' and Juzefovič [8]. Special cases were considered by Malkin [7] and Jakubovič [3]. Related papers are Hale [2] and Jakubovič [4]. However, our method differs from the methods used by these authors and seems to us to be both simpler and more general.
Factor Ideals of Some Representation Algebras
Authors: Wallis W. D.
Published: April 2009

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).
Monomorphisms, Epimorphisms, and Pull-Backs
Authors: Kelly G. M.
Published: April 2009

As the applications of category theory increase, we find ourselves wanting to imitate in general categories much that was at first done only in abelian categories. In particular it becomes necessary to deal with epimorphisms and monomorphisms, with various canonical factorizations of arbitrary morphisms, and with the relations of these things to such limit operations as equalizers and pull-backs.
On Single Equational-Axiom Systems for Abelian Groups
Authors: Padmanabhan R
Published: April 2009

It is a fascinating problem in the axiomatics of any mathematical system to reduce the number of axioms, the number of variables used in each axiom, the length of the various identities, the number of concepts involved in the system etc. to a minimum. In other words, one is interested finding systems which are apparently ‘of different structures’ but which represent the same reality. Sheffer's stroke operation and. Byrne's brief formulations of Boolean algebras [1], Sholander's characterization of distributive lattices [7] and Sorkin's famous problem of characterizing lattices by means of two identities are all in the same spirit. In groups, when defined as usual, we demand a binary, unary and a nullary operation respectively, say, a, b →a·b; a→a−1; the existence of a unit element). However, as G. Rabinow first proved in [6], groups can be made as a subvariety of groupoids (mathematical systems with just one binary operation) with the operation * where a * b is the right division, ab−1. [8], M. Sholander proved the striking result that a mathematical system closed under a binary operation * and satisfying the identity S: x * ((x *z) * (y *z)) = y is an abelian group. Yet another identity, already known in the literature, characterizing abelian groups is HN: x * ((z * y) * (z * a;)) = y which is due to G. Higman and B. H. Neumann ([3], [4])*. As can be seen both the identities are of length six and both of them belong to the same ‘bracketting scheme’ or ‘bracket type’.
On the Absolute Nörlund Summability of a Fourier series (II)
Authors: Hsiang Fu Cheng
Published: April 2009

Let be a given series with its partial sums {Sn} and {Pn} a sequence of real or complex parameters. Write. The transformation given by defines the Nörlund means of {Sn} generated by {Pn}. The series Σann is said to be absolutely summable (N, pn) or summable ∣N, pn∣, if {tn} is of bounded variation, i.e., Σ|tn—tn−1| converges.
Various Ricci Identities in Finsler Space
Authors: Pande H. D.
Published: April 2009

Ricci identities in a Finsler space have been given by C. I. Ispas [1], H. Rund [2], R. S. Mishra and H. D. Pande [3] and others. Here we shall prove some identities using the principle of mathematical induction. Considering a second order contravariant tensor depending on the element of support , we have the following theorems.
Semigroups in Which each Ideal is a Retract
Authors: Tully Edward J.
Published: April 2009

A non-empty subset I of a semigroup S is called an ideal if ab, ba ∈ I whenever a ∈I, b∈S. A subset R of S will be called a retract if there exsists a retraction of S onto R, that is a homomorphism of S onto R which leaves each element of R fixed. The purpose of this paper is to study semigroups in which every ideal is a retract. For convenience we shall call such semigroups retractable. Such semigroups seem to arise naturally; for example, it is easy to show that if the lattice of congruence relations on S is a complemented lattice then S is retractable.
A note on centre-by-finite-exponent varieties of groups
Authors: Gupta Narain , Rhemtulla Akbar
Published: April 2009

We refer the reader to Hanna Neumann [7] for notation and other undefined terms. Let and denote the varieties of groups defined by the laws (xy)n=xnyn, [x, y]n=1 and [x, yn]respectively, where n is an integer. (n)-groups were termed “n-abelian” by R. Baer [1] and have been a subject matter of investigation by various authors (see [3], [5], [6] and the references therein). Recently KaluŽnin [5] has shown that , thus clarifying the relationship between U(n) and the familiar varieties. From the elementary inequalities(n ≠ 0,1)it is easily deduced that(see for instance [5]). If G = Cm Wr C∞, then cleary but for any Thus and . It is also easy to see that in general (see for instance [6] § 5.1) and we are led to ask
A note on a paper by P. D. Finch
Authors: Adams D. H.
Published: April 2009

It is the purpose of this note to illustrate by example the results obtained in Finch [1]. All our definitions and notation are as in that paper, and we use them without reference. In his paper, Finch considered orthoposets and, in particular, orthomodular posets that had either the A-property or the B-property.
On the rate of convergence of probabilities of moderate deviations
Authors: Rohatgi V. K.
Published: April 2009

Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.
The adjoints of differentiable mappings
Authors: Yamamuro S.
Published: April 2009

The notion of symmetric (non-linear) mappings has been introduced by Vainberg [3, p. 56]. However, symmetric mappings of this type have not played any important rôle in non-linear functional analysis. Naturally, as in the case of linear mappings, the symmetric mappings should be defined in such a way that they are easy to handle and belong to the most elementary class of non-linear mappings.
A note on right invariant integrals on locally compact semigroups
Authors: Tewari U. B.
Published: April 2009

An integral on a locally compact Hausdorff semigroup S is a nontrivial, positive linear function μ on the space K(S) of real-valued continuous functions on S with compact support. If S has the property: is compact whenever A is compact subset of S and s ∈ S, then the function fa defined by fa(x) = f(xa) is in K(S) whenever f ∈ K(S) and a ∈ S An integral on a locally compact semigroup S with the property (P) is said to be right invariant if μ(fa) = μ(f) for all f ∈ K(S) and a ∈ S.
Some conformally separable metrics in flat space and their application to accelerated coordinate frameworks in special relativity
Authors: Taylor N. W.
Published: April 2009

If the metric of an n-dimensional space is taken in the form ds2 = u2dτ2+dσ2, where dτ2 and dσ2 are cartesian metrics of r and (n−r) dimensions, respectively, the various forms of u for flat space are quite simple. The study of accelerated motions in special relativity by various authors has led to four dimensional metrics of this form. Those in which u = ±1 at the space origin for all values of time are of particular interest. They are locally cartesian at the accelerated observer, and so the coordinates in the neighbourhood of the observer correspond directly to physical measurements. Hence, such metrics provide convenient means of describing physical conditions experienced by accelerated observers. If the τ-space contains the time direction and is of one or two dimensions, arbitrary rectilinear motions are allowed.
On a sequence of prime numbers
Authors: Cox C. D. , Van Der Poorten A. J.
Published: April 2009

Euclid's scheme for proving the infinitude of the primes generates, amongst others, the following sequence defined by p1 = 2 and pn+1 is the highest prime factor of p1p2…pn+1.
On a family of simple ordered groups
Authors: Dlab Vlastimil
Published: April 2009

In the present paper we shall consider some subgroups of (increasing) autohomeomorphisms of the closed real interval <0, 1>; mainly because of brevity, we shall defer discussing possible generalizations of our results to more general ordered fields.
The iterated equation of generalized axially symmetric potential theory. V. Generalized weinstein correspondence principle
Authors: Burns J. C.
Published: April 2009

Solutions of the iterated equation of generalized axially symmetric potential theory [1]

where the operator Lk is defined by

will be denoted by except that when n = 1, fk will be written instead of . It is easily shown [2, 3] that

by which is meant that any function is a solution of (1).
Gibbs phenomenon for the Hausdorff means of double sequences
Authors: Ustina Fred
Published: April 2009

let g (u) be a regular Hausdorff weight function, and let hm (Ψ x m) denote the mthe corresponding Hausdorff transform, evaluated at x m, of the sequence of partial sums of the Fourier series of Ψ (x), where . In [3], Szász investigated the Gibbs phenomenon for Ψ(x) for these means. His main results are contained in the following two theorems: . (3) THEOREM 2. Taking the limit superior as. If this maximum is attained for τ = τ′ then.
A universal semigroup
Authors: Michael J. H.
Published: April 2009

In [4] S. Ulam asks the following question.
‘Does there exist a universal compact semigroup; i.e., a semigroup U such that every compact topological semigroup is continuously isomorphic to a subsemigroup of it?’
The author has not been able to answer this question. However, in this paper, a proof is given for the following related result.

Let Q denote the Hilbert cube of countably infinite dimension and C(Q) the Banach space of continuous real-valued functions on Q with the usual norm. Let U denote the semigroup consisting of all bounded linear operators T: C(Q)→ C(Q) with ∥T∥ ≦ 1 and let U be endowed with the strong topology. Then, for every compact metric semigroup S with the property:
(1.1) for all x, y ∈ S, with x ≠ y, there exists a z ∈ S, such that xz ≠ yz or zx ≠ zy;

there exists a 1 − 1 mapping φ of S into U such that φ is both a semigroup isomorphism and a homeomorphism.

U is metrizable, but is not compact; hence it does not provide an answer to the question of Ulam. The proof of the above statement leans heavily on a result of S. Kakutani [1].
Idempotent-equivalent congruences on orthodox semigroups
Authors: Meakin John
Published: April 2009

Any congruence on a semigroup S with a nonempty set Es of idempotents induces a partition of the set Es. Two congruences ρ and σ on the semigroup S are defined to be idempotent-equivalent congruences on S if ρ and σ induce the same partition of Es. In this paper we investigate idempotent-equivalent congruences on orthodox semigroups (regular semigroups in which the set of idempotents forms a subsemigroup).
Disk-like functions
Authors: Libera Richard J.
Published: April 2009

The class s of functions f(z) which are regular and univalent in the open unit disk △ = {z: |z| < 1} each normalized by the conditionshas been studied intensively for over fifty years. A large and very successful portion of this work has dealt with subclasses of L characterized by some geometric property of f[Δ], the image of Δ under f(z), which is expressible in analytic terms. The class of starlike functions in L is one of these [3]; f(z) is starlike with respect to the origin if the segment [0,f(z)] is in f[Δ] for every z in Δ and this condition is equivalent to requiring that have a positive real part in Δ.
The convergence of Rayleigh-Ritz approximations
Authors: O'Carroll M. J. , Lush P. E.
Published: April 2009

This paper is concerned with the convergence of Rayleigh-Ritz approximations to the solution of an elliptic boundary value problem. Although the work arose in connection with the aerofoil problem (and it is to this problem that the results obtained are immediately applied), the methods here employed are suitable for use on the wider class of problem mentioned above.
Generalized intrinsic form of the characteristic relations in the steady supersonic flow of a gas
Authors: Suryanarayan E. R.
Published: April 2009

Coburn [1] has derived the intrinsic form of the characteristic relations, for the steady, supersonic, three-dimensional motion of a polytropic gas. The purpose of this paper is to obtain a generalized form of these relations and to apply them to obtain two classes of complex-screw motions [2].
Distinguished submodules
Authors: Dlab Vlastimil
Published: April 2009

Although there is no need for a ‘distinguished’ submodule to be given a formal definition in the present paper, we like to indicate the meaning attached to this concept here. Perhaps the shortest way of doing so is to say that a distinguished submodule is a (covariant idempotent) functor from the category of (left) R-modules into itself mapping each R-module into its R-submodule specified by a family of left ideals of R. If is a family left ideals of R, then all elements of an R-module M of orders belonging to , do not, of course, in general form a submodule of M; but, there are certain families such that all the elements of orders from form a submodule in any R-module (distinguished submodules defined by ). Consequently, no particular structural properties of the R-module are involved in the definition of such submodules. In this way we can define radicals (in the sense of Kuroš [4]) of a module. In particular, we feel that an application of this method is an appropriate way in defining the (maximal) torsion submodule of a module.
On asymptotic properties of sub-critical branching processes
Authors: Seneta E.
Published: April 2009

Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.
On a many server queue with non-recurrent input and negative exponential servers
Authors: Pearce C.
Published: April 2009

We consider a queueing system withkidentical servers in parallel, the services being negative exponential with parameter μ. The input is a natural generalisation of the usual general recurrent input. If we denote the sequence of arrival points by {An,n≧ 0} then the inter-arrival intervals are given bywhere the ƒi: are (integrable) non-negative functions and {Ui} is a sequence of identically and independently distributed random variables. In the simplest case,p= 0, this is just a general recurrent input. We writeU(·) for the probability distribution function of theUn.
The limiting behavior of transient birth and death processes conditioned on survival
Authors: Good Phillip
Published: April 2009

The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition
On rings of sets
Authors: Speed T. P.
Published: April 2009

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].
0-dimensional compactifications and Boolean rings
Authors: Magill K. D. , Glasenapp J. A.
Published: April 2009

A subset of a topological space which is both closed and open is referred to as a clopen subset. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. By a compactification αX of a completely regular Hausdorff space X, we mean any compact space which contains X as a dense subspace. Two compactifications αX and γX are regarded as being equivalent if there exists a homeomorphism from αX onto γX which keeps X pointwise fixed. We will not distinguish between equivalent compactifications. With this convention, we can partially order any family of compactifications of X by defining αX ≧ γX if there exists a continuous mapping from γX onto αX which leaves X pointwise fixed. This paper is concerned with the study of the partially ordered family [X] of all 0-dimensional compactifications of a 0-dimensional space X.
Abelian p-subgroups of the general linear group
Authors: Goozeff J. T.
Published: April 2009

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.
(v, k, λ) Configurations and Hadamard matrices
Authors: Wallis Jennifer
Published: April 2009

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems:
Theorem 1. If there exist
(i) a skew-Hadamard matrix H = U+I of order h,
(ii)m-type matrices M = W+I and N = NT of order m,
(iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying
(a) XYT, YZT and ZXT all symmetric, and
(b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.
A skew Hadamard matrix of order 36
Authors: Goethals J. M. , Seidel J. J.
Published: April 2009

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.
On the laws of some varieties of groups
Authors: Vaughan-Lee M. R.
Published: April 2009

D. E. Cohen has shown in [1] that a variety consisting of metabelian groups has a finite basis for its laws. In this paper I make use of Cohen's result to prove the following theorem.
Uniqueness theorems for a general class of functional equations
Authors: Ng C. T.
Published: April 2009

In a previous paper [1] J. Aczél has shown the following
Theorem 1. If in the (closed, half-closed or open, finite or infinite) interval 〈A, B〉and there f, F are there, f F are continuous, F intern (the value F(x, y) lies strictly between x and y) and u → H(u, v, x, y) or v → H(u, v, x, y) are injective (i.e. or , then the functional equation (*) with the initial conditionshas at most one solution.
Hp multipliers and inequalities of Hardy and Littlewood
Authors: Gaudry G. I.
Published: April 2009

Consider the classical Hardy spaces Hp(T) (1 ≦ p ≦ ∞) on the unit circle T. We shall ignore completely the fact that the elements of Hp(T) can be extended via the Poisson formula to certain types of functions analytic inside the unit disc. For our purposes, Hp(T) is the closed ideal in Lp(T) consisting of those functions f ∈ Lp(T) for which (n) = 0 (n= –1, –2,…).
Two problems in singular perturbations of differential equations
Authors: Chang K. W.
Published: April 2009

Let ε > 0 be a small real parameter, let y, z be real m-dimensional and n-dimensional vectors respectively and let f, g be respectively real m-dimensional and n-dimensional vector functions of their arguments. This paper aims to discuss the following two problems in singular perturbations.
A Characterization of the finite simple group L4(3)
Authors: Phan Kok-Wee
Published: April 2009

In this paper we aim to give a characterization of the finite simple group L4(3) (i.e. PSL(4, 3)) by the structure of the centralizer of an involution contained in the centre of its Sylow 2-subgroup. More precisely, we shall prove the following result.
On two early unified field theories
Authors: Klotz A. H.
Published: April 2009

It is shown that Einstein's 1929 Unified Field Theory and that of Levi-Civita are incompatible.
Semi-groups and differential equations
Authors: Gray J. D.
Published: April 2009

In this short note we shall apply the theory of semi-groups of operators, (cf: Hille and Phillips, [2]), to the problem of representing solutions of certain differential equations with non-constant coefficients. When the coefficients are constant, this representation reduces to the usual Laplace transform solution of the relevant equation.
A class of c-groups
Authors: Camina A. R. , Gagen T. M.
Published: April 2009

In a paper by Polimeni [3] the concept of a c-group was introduced. A group is called a c-group if and only if every subnormal subgroup is characteristic. His paper claims to characterize finite soluble c-groups, which we will call fsc-groups. There are some errors in this paper; see the forthcoming review by K. W. Gruenberg in Mathematical Reviews.
Note on weak dimension of algebras
Authors: Markanda R. K.
Published: April 2009

Let ∧ be a K-algebra over a commutative ring K. Harada [5] has introduced the notion of weak dimension of algebras ∧ (denoted by w. dim ∧) analogous to the dimension of algebras in Cartan and Eilenberg [3].
Congruences on ωn-Bisimple Semigroups
Authors: Warne R. J.
Published: April 2009

LetSbe a bisimple semigroup and letEsdenote its set of idempotents. We may partially orderEsin the following manner: ife,f∈Es,e≧fif and only ifef=fe=e. We then say thatEsis under or assumes its natural order. LetI0denote the non-negative integers and letndenote a natural number. IfEs, under its natural order, isomorphic to (I0)nunder the reverse of the usual lexicographic order, we callSan ωn-bisimple semigroup. (See [9] for an explanation of notation.) We determined the structure of ωn-bisimple semigroups completely mod groups in [9]. The ωn-bisimple semigroups, theI-bisimple semigroups [8], and the ωnI-bisimple semigroups [9] are classes of simple semigroups except completely simple semigroups whose structure has been determined mod groups.
Commutative Banach Algebras with Idempotent Maximal Ideals
Authors: Loy R. J.
Published: April 2009

Letbe a commutative Banach algebra over the complex fieldC,Man ideal of. Denote byM2the set of all finite linear combinations of products of elements fromM.Mwill be termed idempotent ifM2=M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.
Some Finite Nilpotent p-Groups
Authors: Gupta C. K. , Gupta N. D. , Newman M. F.
Published: April 2009

Consider the following statement: For every positive integer n and every prime p there is a finite p-group of nilpotency class (precisely) c all of whose (n−1)-generator subgroups are nilpotent of class at most n.
Some Remarks on a Class of Distributive Lattices
Authors: Speed T. P.
Published: April 2009

Distributive pseudo-complemented lattices form an extensively studied class of distributive lattices. Examples are the lattice of all open sets of a topological space, the lattice of all ideals of a distributive lattice with zero and the lattice of all congruences of an arbitrary lattice. Lattice which are just pseudo-complemented have been studied in detail by J. Varlet [6], [7] where, however, the most interesting results require at least the assumption of modularity, sometimes distributivity.
Stone algebras form an equational class: (Remarks on Lattice Theory III)
Authors: Grätzer G.
Published: April 2009

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.
The Compression of Yielding Loops
Authors: Stuart I. M. , Mann R. K.
Published: April 2009

In studying the compression of loops of fabric between flat parallel plates, the problem arises of the compression of bends. A ‘bend’ is illustrated in Figs. 1 and 2. Two bends are joined to form a loop. The material is postulated to yield in bending such that the bending moment where B is a bending rigidity, KI the current impressed curvature, and KR the ‘remanent’ curvature. This remanent curvature is the free, or natural curvature in the material caused by its past and present deformation. We study the case when the remanent curvature that is, the remanent curvature is proportional to the greatest previously impressed curvature. We call equations (1) and (2) the ‘Remanent Curvature Hypothesis’.
On Schoenberg's Rational Polygon Problem
Authors: Ang D. D. , Daykin D. E. , Sheng T. K.
Published: April 2009

A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a rational polygon?’ Many of the known results concerning this question are contained as special cases in theorem 1 below which was proved by one of us (cf. the references).
Some Classes of Indecomposable Varieties of Groups
Authors: Cossey John
Published: April 2009

A variety of groups is an equationally defined class of groups: equivalently, it is a class of groups closed under the operations of taking cartesian products, subgroups, and quotient groups. If and are varieties, then is the class of all groups G with a normal subgroup N in such that G/N is in ; is a variety, called the product of and . We denote by the variety generated by the unit group, and by the variety of all groups. We say that a variety is indecomposable if , and cannot be written as a product , with both and One of the basic results in the theory of varieties of groups is that the set of varieties, excluding , and with multiplication of varieties as above, is a free semi-group, freely generated by the indecomposable varieties. Thus one would like to be able to decide whether a given variety is indecomposable or not. In connection with this question, Hanna Neumann raises the following problem (as part of Problem 7 in her book [7]): Problem 1. Ifandprove that [] is indecomposable unless bothandhave a common non-trivial right hand factor.
A Note on Finite-Dimensional Differentiable Mappings
Authors: Palmer K. J. , Yamamuro Sadayuki
Published: April 2009

Let E be a real infinite-dimensional Banach space. Let ℒ be the Banach algebra of all continuous linear mappings of E into itself with topology defined by the norm:
A Remark on Critical Groups
Authors: Kovács L. G.
Published: April 2009

Problem 24 of Hanna Neumann's book [3] reads: Does there exist, for a given integer n > 0, a Cross variety that is generated by its k-generator groups and contains (k+n)-generator critical groups? In such a variety, is every critical group that needs more than k generators a factor of a k-generator critical group, or at least of the free group of rank k? In a recent paper [1], R. G. Burns pointed out that the answer to the first question is an easy affirmative, and asked instead the question which presumably was intended: Given two positive integers k, l, does there exist a variety 23 generated by k-generator groups and also by a set S of critical groups such that S contains a group G minimally generated by k+l elements and S/{G} does not generate B? The purpose of this note is to record a simple example which shows that the answer to the question of Burns is affirmative at least for k = 2, l = 1, and also that the answer to the second question of Hanna Neumann's Problem 24 is negative.
Automorphisms of Finite Groups and their Fixed-Point Groups
Authors: Ward J. N.
Published: April 2009

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.
Remarks on Stability Conditions for the Differential Equation x″ + a(t)ƒ(x) = 0
Authors: Wong James S. W.
Published: April 2009

Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfieswhere a(t) > 0, t ≧ t0 and = max (−a′(t), 0), andthen every solution of (1) satisfies (2).
An Upper Bound for Volumes of Convex Bodies
Authors: Firey William J.
Published: April 2009

Consider a non-degenerate convex body K in a Euclidean (n + 1)-dimensional space of points (x, z) = (x1,…, xn, z) where n ≧2. Denote by μ the maximum length of segments in K which are parallel to the z-axis, and let Aj, signify the area (two dimensional volume) of the orthogonal projection of K onto the linear subspace spanned by the z- and xj,-axes. We shall prove that the volume V(K) of K satisfies After this, some applications of (1) are discussed.
A characterization of locally connected unicoherent continua
Authors: Bacon Philip
Published: April 2009

If ε > 0, a subset M of a metric space is said to be ε-connected if for each pair p, q ∈ M there is a finite sequence a0, …, an such that each ai ∈ M, a0 = ρ an = q and the distance from ai−1 to ai is less than ε whenever 0 < i ≦n. It is known [1, p. 117, Satz 1] that a compact metric space is connected if and only if for each ε > 0 it is ε-connected. We present here a proof of an analogous characterization of locally connected unicoherent compacta.
Lattice isomorphic solvable Lie algebras
Authors: Glaeser R. C. , Kolman B.
Published: April 2009

Let L be a Lie algebra over a field κ of any characteristic, and consider the lattice ℒ(L) of all subalgebras of L. In this paper we prove that if L and M are lattice isomorphic Lie algebras, over a field of any characteristic, and L′ and M′ are nilpotent, then the difference between the orders of solvability of L and M differs by at most one.
Open mappings on spheres
Authors: Duda Edwin
Published: April 2009

In this paper we study open mappings of the sphere, Sn, onto itself. In particular, sufficient conditions are given that such a mapping be a homeomorphism. For the cases n≦ 2 many of the results could be obtained from the work of G. T. Whyburn [7], [8], and [10]. For the cases n ≦ 3 the useful results of A. V. Cernavskii, [1], [4], proved to be sufficient. An application is made concerning a finite to one open mapping of one n cell onto itself. It is interesting to note that for n ≦ 2 that we could use similar proofs to show that certain quasi-monotone mappings of Sn onto Sn are necessarily monotone mappings.
On polynomial expansions of analytic functions
Authors: Gross Fred
Published: April 2009

A set of polynomials p0(z), p1(z), … is said to form a basic set if every polynomial can be expressed in one and only one way as a finite linear combination of them.
Complex symmetric matrices
Authors: Craven B. D.
Published: April 2009

It is well known that a real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix of complex elements. Such complex symmetric matrices arise naturally in the study of damped vibrations of linear systems. It is shown in this paper that a complex symmetric matrix can be diagonalised by a (complex) orthogonal transformation, when and only when each eigenspace of the matrix has an orthonormal basis; this implies that no eigenvectors of zero Euclidean length need be included in the basis. If the matrix cannot be diagonalised, then it has at least one invariant subspace which consists entirely of vectors of zero Euclidean length.
Meromorphic functions with one deficient value
Authors: Shah S. M.
Published: April 2009

Let f(z) be a meromorphic function and write Here N(r, a) and T(r, f) have their usual meanings (see [4], [5]) and 0 ≧ |a| ≧ ∞. If δ(a, f) > 0 then a is said to be an exceptional (or deficient) value in the sense of Nevanlinna (N.e.v.), and if Δ(a, f) > 0 then a is said to be an exceptional value in the sense of Varliron (V.e.v.). The Weierstrass p(z) function has no exceptional value N or V. Functions of zero order can have atmost one N.e.v. [4, p. 114], but may have more than one V.e.v. (see [6], [8]). In this note we consider functions satisfying some regularity conditions and having one and only one exceptional value V.
A on simplicity criterion for finite groups
Authors: Bryce Nita
Published: April 2009

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.
Modular representations of C2×C2
Authors: Conlon S. B.
Published: April 2009

The representations of V4 (= C2 × C2) over characteristic 2 are put down in matrix form in sect. 2 of [1]. As such representations are of particular interest to finite group theorists, we present the following “geometric” descriptions of them which give immediate insight into their structure. Indeed, without such pictures it is difficult to see how they can be handled. Finally the relative Grothendieck algebra (relative to a copy of C2 in V4) falls out immediately from these diagrams. These diagrams have already helped towards the more general calculation of such algebras [2].
On the summability of a sequence of Walsh functions
Authors: Siddiqi A. H.
Published: April 2009

The Rademacher functions are defined by . The Walsh functions are then given by for , where the integers ni are uniquely determined by ni+1 < ni.
Residual finiteness of permutational products
Authors: Gregorac R. J.
Published: April 2009

Conditions sufficient to guarantee that a generalized free product of two residually finite groups A and B is again residually finite have been given by Baumslag [1]. We here show the same conditions guarantee that a certain permutational product of A and B is also residually finite.
Some results using general moment functions
Authors: Smith Walter L.
Published: April 2009

Let {Xn} be a sequence fo independent and identically distributed random variables such that 0 <μ = εXn ≦ + ∞ and write Sn = X1+X2+ … +Xn. Letv ≧ 0 be an integer and let M(x) be a non-decreasing function of x ≧ 0 such that M(x)/x is non-increasing and M(0) > 0. Then if ε|X1νM(|X1|) < ∞ and μ < ∞ it follows that ε|Sn|νM(|Sn|) ~ (nμ)vM(nμ) as n → ∞. If μ = ∞ (ν = 0) then εM(|Sn|) = 0(n). A variety of results stem from this main theorem (Theorem 2), concerning a closure property of probability generating functions and a random walk result (Theorem 1) connected with queues.
Note on a theorem of Magnus
Authors: Blackburn Norman
Published: April 2009

Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.
A basis for the laws of the variety ѕU30
Authors: Christensen C.
Published: April 2009

A group is called an ѕU-group if and only if it is locally finite and all its Sylow subgroups are abelian. Kovács [1] has shown that for any integerethe class ѕUeof all ѕU-groups of exponents dividing е is a variety. Little is known about the laws of these varieties; in particular it is unknown whether they have finite bases. Whenever ѕUeis soluble it is an easy matter to establish explicitly a finite basis for its laws namely the exponent law, the appropriate solubility length law and all laws of the type [xm,ym]mwheree=pαm,pis a prime andpdoes not dividem. (The significance of thelast type of law is made clear by Proposition 2 below and the obvious fact that any group that satisfies a law of this type for given primephas abelirn Sylow p-subgroups.) For e less than thirty ѕUeis clearly soluble whikt PSL(2, 5), the non-abelian simple group of order 60, is contained in ѕU30so that the casee= 30 is, in a sense, the first non-trivial case to be considered.
Two-variable laws forPSL(2, 5)
Authors: Bryant R. M. , Powell M. B.
Published: April 2009

In [2], John Cossey and Sheila Oates Macdonald give a basis for the set of laws ofPSL(2, 5) — the simple group of order 60 — and with one extreme exception the laws of their basis involve at most two variables. They raise the problem of finding a basis in which all of the laws involve only a small number of variables, and remark that they have shown that five variables will suffice. Here we give a basis consisting entirely of two variable laws.
Another characteristic conjugacy class of subgroups of finite soluble groups
Authors: Makan A.
Published: April 2009

Let name be a class of finite soluble groups with the properties: (1) is a Fitting class (i.e. normal subgroup closed and normal product closed) and (2) if N ≦ H ≦ G ∈, N ⊲ G and H/N is a p-group for some prime p, then H ∈. Then is called a Fischer class. In any finite soluble group G, there exists a unique conjugacy class of maximal -subgroups V called the -injectors which have the property that for every N◃◃G, N ∩ V is a maximal -subgroup of N [3]. 3. By Lemma 1 (4) [7] an -injector V of G covers or avoids a chief factor of G. As in [7] we will call a chief factor -covered or -avoided according as V covers or avoids it and -complemented if it is complemented and each of its complements contains some -injector. Furthermore we will call a chief factor partially-complemented if it is complemented and at least one of its complements contains some -injector of G.
On groups with special antiautomorphisms
Authors: Heineken Hermann
Published: April 2009

This note is concerned with the following question: What is the structure of those groups which possess two antiautomorphisms different from identity such that every element of the group is fixed by (at least) one of them?

C. Ayoub [1] stated this problem after having proved a statement equivalent to the following: The group G is a non-abelian extension of an abelian group by a group of order two, if, and only if, there is an automorphism α ≠ 1 and an antiautomorphism β ≠ 1 such that every element of G is fixed by a α by β. The Theorem at the end of the paper will show that the class of groups considered by C. Ayoub coincides with the class considered here.