bluuebunny/crossref_metadata_2025 huggingface dataset:
The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs
Authors: Bishwal J. P. N.
Published: April 2009

The Bernstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) as the number of Fourier coefficients in the expansion of the solution increases to infinity. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent, asymptotically normal and locally asymptotically minimax (in the Hajek-Le Cam sense), and asymptotically equivalent to the maximum likelihood estimator as the number of Fourier coefficients become large. Unlike in the classical finite dimensional SDEs, here the total observation time and the intensity of noise remain fixed.
On n+1 intersecting hyperspheres in n-space
Authors: Mandan Sahib Ram
Published: April 2009

The interesting results arising from the study of ‘Four intersecting spheres’ [9] in a solid made the author think of an analogous picture in higher spaces too and thus the present paper arose.
A special simplex
Authors: Mandan Sahib Ram
Published: April 2009

Let S = A0 hellip An be an n-simplex and Aih the foot of its altitude from its vertex Ai to its opposite prime face Si; O, G the circumcentre and centroid of S and Oi, Gi of Si. Representing the position vector of a point P, referred to O, by p, Coxeter [2] defines the Monge pointM of S Collinear with O and G by the relation so that the Monge point Mi of Si is given by If the n+1 vectors a are related by oi be given by Aih is given by Since Aih lies in Si, If Ti be a point on MiAih such that i.e. That is, MTi is parallel to ooi or normal to Si at Ti:. Or, the normals to the prime faces Si of S at their points Ti concur at M. In fact, this property of M has been used to prove by induction [3] that an S-point S of S lies at M. Thus M = 5, M = S or .
A theorem on a finite difference operator and its connection with the Poisson distribution
Authors: Yahya Q. A. M. M.
Published: April 2009

The well-known Taylor expansion of a function around a point a can be formally written as The last expression is just a symbolic form and is valid, as we know, under certain restrictive conditons. The last expression is just a symbolic form and is valid, as we know, under certain restrictive conditons. We shall study the situation when the differential operator d/da is replaced by the finite difference operator Δh/h, where the operator Δh, is defined by In general, Then we have the following theorem.
Infinite capacity storage processes
Authors: John Floyd I.
Published: April 2009

In this paper we shall discuss infinite capacity storage processes in which periods of input and output alternate. The length of a period of input and the length of the period of output immediately following may be statistically dependent and the change in storage level during an input or output period may depend on the length of the time interval in a rather general manner. However, we do not exploit either of these facts in the present paper.
On commutation semigroups of a group
Authors: Gupta N. D.
Published: April 2009

Let G be a group. With each element a in G we associate the mappings ρ(a) and λ(a) of G into itself defined as follows, The product of mappings is defined as usual. Let P(G) and ∧(G) denote respectively the semigroups generated by the set of all ρ's and λ's. These semigroups will be called the commutation semigrowps of G.
Congruence relations on orthomodular lattices
Authors: Finch P. D.
Published: April 2009

We denote lattice join and meet by ∨ and ∧ respectively and the associated partial order by ≧. A lattice L with 0 and I is said to be orthocomplemented if it admits a dual automorphism x → x′, that is a one-one mapping of L onto itself such that which is involutive, so that for each x in L and, further, is such that for each x in L.
The modular representation algebra of groups with Sylow 2-subgroup Z2 × Z2
Authors: Conlon S. B.
Published: April 2009

Let k be a field of characteristic 2 and let G be a finite group. Let A(G) be the modular representation algebra1 over the complex numbers C, formed from kG-modules2. If the Sylow 2-subgroup of G is isomorphic to Z2×Z2, we show that A(G) is semisimple. We make use of the theorems proved by Green [4] and the results of the author concerning A(4) [2], where 4 is the alternating group on 4 symbols.
A generalisation of Lorentz-equivalence in a substratum theory
Authors: Prokhovnik S. J.
Published: April 2009

It has been shown that the assumption of a specified cosmological substratum leads to the Lorentz-equivalencc of observers whose uniform motion in the substratum lies along a common straight line. The result is generalised for any pair of unaccelerated observers thus confirming that only the relative velocity of such observers is relevant in relating their observations of an event.
Lattice isomorphisms of associative algebras
Authors: Barnes D. W.
Published: April 2009

Let A be an associative algebra over the field F. We denote by ℒ(A) the lattice of all subalgebras of A. By an ℒ-isomorphism (lattice isomorphism) of the algebra A onto an algebra B over the same field, we mean an isomorphism of ℒ(A) onto ℒ(B). We investigate the extent to which the algebra B is determined by the assumption that it is ℒ-isomorphic to a given algebra A. In this paper, we are mainly concerned with the case in which A is a finite- dimensional semi-simple algebra.
Some geometric extremal problems
Authors: Hille Einar
Published: April 2009

In a recent study of generalized transfinite dimeters [4, 5] some geometric extremal problems were encountered. These form the subject matter of this note.
The field structure of the real number system
Authors: Venkataraman M. , Soundararajan T.
Published: April 2009

It is well-known that the real number system can be characterised as a topological space [1], [3], as an ordered set [2], and as an ordered field [4]. It is the aim of this note to give two characterisations of the system purely as a field (see Theorems 4 and 9) without any extra notion of order, topology, et cetera.
The iterated equation of generalized axially symmetric potential theory. II. General solutions of Weinstein's type
Authors: Burns J. C.
Published: April 2009

In the first paper of this series [1] which will be designated I, particular solutions of various kinds have been found for the iterated equation of generalized axially symmetric potential theory (GASPT) which, in the notation defined in I, is (1) where the operator is defined by
An Md-class of sets indexed by a regressive function
Authors: Barback Joseph
Published: April 2009

This paper deals with the study of a particular md-class of sets. The underlying theory was introduced and studied by J. C. E. Dekker in [4]. We shall assume that the reader is familiar with the terminology and main results of this paper; in particular with the concepts of md-class of sets, gc-class of sets, gc-set, gc-function and the RET of a gc-class of sets. We also use the following notations of [4]: ε = the set of all non-negative integers (numbers), R = Req (ε).
Semigroups acting on continua
Authors: Day J. M. , Wallace A. D.
Published: April 2009

A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication. (The latter phrase will generally be abbreviated to CAM and the multiplication in a semigroup will be denoted by juxta position unless the contrary is made explicit.)
Approximation in algebraic function fields of one variable
Authors: Coates John
Published: April 2009

In his paper (4), Mahler established several strong quantitative results on approximation in algebraic number fields using the geometry of numbers. In the present paper I derive analogous results for algebraic function fields of one variable using an analogue of the geometry of numbers.
Finitely generated -groups
Authors: Lewin Tekla
Published: April 2009

This paper is taken from the author's doctoral dissertation at New York University; the work was supported by the National Science Foundation, Grant G19674. The author would like to thank Professor Gilbert Baumslag for suggesting the questions treated in this paper, and for his very helpful guidance.
Minimal dependent sets
Authors: Asche D. S.
Published: April 2009

The subject matter of this note is the notion of a dependence structure on an abstract set. There are a number of different approaches to this topic and it is known that many of these lead to precisely the same structure. Axioms are given here to specify the minimal dependent sets for such a structure. They are closely related to conditions introduced by Hassler Whitney in [1] and to a certain “elimination axiom” for the independent sets.
On integral operators associated with Poisson transforms and the operator Hα
Authors: Okikiolu G. O.
Published: April 2009

In this paper we study certain operators allied to the Poisson operator and the transforms Hα(f) considered by the author in [2]. We define the integrals ψ(α)α(f) and θ(α)a(f) as follows:
Interval semirings on R1 with ordinary multiplication
Authors: Pearson K. R.
Published: April 2009

The study of topological semirings, initiated by Selden [5], arises naturally from the theory of topological semigroups. It is of interest to take a known multiplication and investigate the possible additions. Selden has done this in [5] for several compact semirings.
Linear operators which commute with translations. Part I: Representation theorems
Authors: Brainerd B. , Edwards R. E.
Published: April 2009

In Part I of this paper we shall be concerned with the representation as convolutions of continuous linear operators which act on various function- spaces linked with a locally compact group and which commute with left — or right — translations; cf. the results in [12]. For completeness some known results are included whenever they follow from the general procedure. We have tried to follow simple general approaches as much as possible.
On metabelian groups
Authors: Gupta N. D. , Newman M. F.
Published: April 2009

In this note we present some results on relationships between certain verbal subgroups of metabelian groups. To state these results explicitly we need some notation. As usual further [x, 0y] = x and [x, ky] = [x, (k—1)y, y] for all positive integers k. The s-th term γs(G) of the lower central series of a group G is the subgroup of G generated by [a1, … as] for all a1, … as, in G. A group G is metabelian if [[a11, a2], [a3, a4]] = e (the identity element) for all a1, a2, a3, a4, in G, and has exponent k if ak = e for all a in G.
String vibrating at finite amplitude
Authors: Stuart I. M.
Published: April 2009

Equations of motion of a vibrating string are established in terms of the transverse and longitudinal displacements. These equations contain the terms of lowest order which are neglected in the classical treatment with vanishing amplitude. These extra terms lead to the natural modes being dependent on amplitude. By a simple procedure a solution of these equations is obtained which separates, as in the classical theory. The familiar circular functions are replaced by a Mathiew Function of position and a Jacobi elliptic function of time. Agreement with a previous study is shown.
The Fitting subgroup of a linear solvable group
Authors: Dixon John D.
Published: April 2009

Let G be a group. The Fitting subgroup F(G) of G is defined to be the set union of all normal nilpotent subgroups of G. Since the product of two normal nilpotent subgroups is again a normal nilpotent subgroup (see [10] p. 238), F(G) is the unique maximal normal, locally nilpotent sungroup of G. In particular, is G is finite, then F(G) is the unique maximal normal nilpotent subgroup of G. If G is a notrivial solvable group, then clearly F(G) ≠1.
A note on the independence of the axioms for a vector space
Authors: Van Der Poorten A. J.
Published: April 2009

It is customary to define a vector space in some such manner as: A vector space over a field F is a set V of elements a, b, … called vectors, having the following properties: For arbitrary a, b, c ∈ V; λ, μ ∈ F.
A note on de Sitter type solutions of Einstein's field equations
Authors: Klotz A. H.
Published: April 2009

A method of deriving a class of solutions of Einstein's field equations with symmetry of de Sitter type is investigated, and the most general solution of this kind is derived.
Some properties of locally compact groups
Authors: Rickert Neil W.
Published: April 2009

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.
On the transient behaviour of a Poisson branching process
Authors: Seneta E.
Published: April 2009

In general, information concerning the distribution of the time to absorption,T, of a simple branching (Galton-Watson) process for which extinction in finite mean time is certain, is difficult to obtain. The process of greatest biological interest is that for which the offspring distribution is Poisson, having p.g.f.F(s) =em(s-1),m< 1.
The extreme coverings of 4-space by spheres
Authors: Dickson T. J.
Published: April 2009

The object of this paper is to apply, in the case n = 4, the results of Barnes and Dickson [1] concerning extreme coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice.
On the height of trees
Authors: Rényi A. , Szekeres G.
Published: April 2009

In this note we shall deal with the enumeration of labelled trees of given order and given height over a selected point.
Sequences of integers satisfying congruence relations and Pisot-Vijayaraghavan numbers
Authors: Gerig Stephen
Published: April 2009

We consider infinite sequences of positive integers having exponential growth: and becoming ultimately periodic modulo each member of a rather sparse infinite set of integers. If sufficient, natural conditions are placed on the growth and periodicities of , we find that a is an algebraic integer having all its algebraic conjugates within or on the unit circle, and fn has a special representation involving an. The result is a kind of dual to the theorem of Pisot (cf. Salem [2], p. 4, Theorem A).
Normal p-subgroups of solvable linear groups
Authors: Dixon John D.
Published: April 2009

In his paper [8], N. Itô gives an elegant proof that the Sylow p-group of a finite solvable linear group of degree n over the field of complex numbers is necessarily normal if p > n+1. Moreover he shows that this bound on p is the best possible when p is a Fermat prime (i.e. a prime of the form 2sk + 1) but that the bound may be improved to p > n when p is not a Fermat prime.
Finite groups with a self-centralizing subgroup of order 4
Authors: Wong Warren J.
Published: April 2009

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.
The square root of a positive self-adjoint operator
Authors: Bernau S. J.
Published: April 2009

One elementary proof of the spectral theorem for bounded self-adjoint operators depends on an elementary construction for the square root of a bounded positive self-adjoint operator. The purpose of this paper is to give an elementary construction for the unbounded case and to deduce the spectral theorem for unbounded self-adjoint operators. In so far as all our results are more or less immediate consequences of the spectral theorem there is little is entirely new. On the other hand the elementary approach seems to the author to provide a deeper insight into the structure of the problem and also leads directly to the spectral theorem without appealing first to the bounded case. Besides this, our methods for proving uniqueness of the square root and of the spectral family seem to be new even in the bounded case. In particular there is no need to invoke representation theorems for linear functionals on spaces of continuous functions.
Transitivity in integral symplectic forms
Authors: James D. G.
Published: April 2009

A Symplectic lattice L is a free Z-module of finite rank endowed with a non-degenerate alternating bilinear form. Thus we have a bilinear mapping Φ of L × L into the domain of integers Z; we donote Φ (α, β) by α · β (where α, β ∈ L). Then α2 = 0 and α·β = −β·α.
Two problems on finite groups with k conjugate classes
Authors: Poland John
Published: April 2009

Let G be a finite group of order g having exactly k conjugate classes. Let π(G) denote the set of prime divisors of g. K. A. Hirsch [4] has shown that By the same methods we prove g ≡ k modulo G.C.D. {(p–1)2 p ∈ π(G)} and that if G is a p-group, g = h modulo (p−1)(p2−1). It follows that k has the form (n+r(p−1)) (p2−1)+pe where r and n are integers ≧ 0, p is a prime, e is 0 or 1, and g = p2n+e. This has been established using representation theory by Philip Hall [3] (see also [5]). If then simple examples show (for 6 ∤ g obviously) that g ≡ k modulo σ or even σ/2 is not generally true.
A non-embeddable composite of embeddable functions
Authors: Ran A.
Published: April 2009

Let Ω be the group of the functions ƒ(z) of the complex variable z, analytic in some neighborhood of z = 0, with ƒ(0) = 0, ƒ′(0) = 1, where the group operation is the composition g[f(z)](g(z), f(z) ∈ Ω). For every function f(z) ∈ Ω there exists [4] a unique formal power series where the coefficients ƒq(s) are polynomials of the complex parameter s, with ƒ1(s) = 1, such that and, for any two complex numbers s and t, the formal law of composition is valid.
An existence theorem for optimal stochastic programming
Authors: Stoddart A. W. J.
Published: April 2009

In [4], Hanson has obtained necessary conditions and sufficient conditions for optimality of a program in stochastic systems. However, in many cases, especially in a general treatment, a program satisfying these conditions cannot be determined explicitly, so that the question of existence of an optimal program in such systems is significant. In this paper, we obtain conditions sufficient for existence of an optimal program by applying the direct methods of the calculus of variations [9], [6] and the theory of optimal control [7], [5].
An unbounded spectral mapping theorem
Authors: Bernau S. J.
Published: April 2009

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.
General relativistic fluid spheres with variable polytropic index
Authors: van der Borght R.
Published: April 2009

In this paper we derive solutions of the field equations of general relativity for a compressible fluid sphere which obeys density-temperature and pressure-temperature relations which allow for a variation of the polytropic index throughout the sphere.
Radially symmetric distribution of matter
Authors: Mehra A. L.
Published: April 2009

In this paper a solution of the Einstein field equations for a spherically symmetric distribution of a perfect fluid of variable density has been obtained.
Effect of Cosserats' couple-stresses on the stress distribution in a semi-infinite medium with varying modulus of elasticity
Authors: Paria Gunadhar
Published: April 2009

The theory of Cosserats' couple-stresses is briefly described in a cartesian system of coordinates, and is applied to the problem of stress distribution in a semi-infinite medium which possesses a non-homogeneous elastic property of an exponential type. Effects of couple-stresses on the stress concentration factors are determined both in homogeneous and non-homogeneous materials.
On a characterisation of conformally-flat Riemannian spaces of class one
Authors: Sen R. N.
Published: April 2009

In a previous paper [1] we considered those conformally-flat Riemannian spaces which satisfy the tensorial characterisation where, as usual, gij, Rhijk, Rij are the fundamental tensor, the curvature tensor, the Ricci tensor and E ≠ 0, F are certain scalars. The tensor g is always supposed to be real and analytic. A special form of the metrics of these spaces was seen to be where f is any real analytic function, subject to a restriction, of the argument θ. Writing f, f′, f″,… for f(θ), df|dθ, d2f|dθ2, … the quantities E, F and the scalar curvature R of the type of spaces (1.2) were seen to be
Self-polar double-N's defined by certain pairs of normal rational curves
Authors: Kirkpatrick P.
Published: April 2009

This paper is a sequel to T. G. Room's “Self-polar double configurations in projective geometry, I and II” ([2]). I would like to thank Professor Room for supervising and inspiring my work, and to acknowledge the financial assistance of the C.S.I.R.O.
Particles with 2 states and Lorentz transformations
Authors: Blatt John M. , Hurst C. A.
Published: April 2009

In this note, we draw attention to a natural connection between a group closely related to the homogeneous Lorentz group, and the most general set of measurements possible on particles with only two discrete states. We may think of these two states as “spin up” and “spin down’, represented by the vectors α = (1, 0) and β = (0, 1), respectively.
A note on a result of Mahler's
Authors: Luthar Indar S.
Published: April 2009

In a recent paper, Maher [2] proved that for any algebraic number field K of degree n and discriminant d there exists a constant C depending only on n and d such that for any ceiling λP of K there exists a basis α1 …, αn of the corresponding ideal αλ such that .
On Rotor calculus, I
Authors: Buchdahl H. A.
Published: April 2009

It is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called “rotors”. Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities τAkl which transform as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space.
On Rotor calculus, II
Authors: Buchdahl H. A.
Published: April 2009

In the development of the rotor calculus presented in a previous paper space-time was taken to be flat. This work is now extended to the case of curved space-times, which, in the first instance, is taken to be Riemannian. (The calculus bears at times a strong formal resemblance to the spinor analysis of Infeld and van der Waerden, but it is in fact developed quite independently of this.) Owing to the fact that all general relations had earlier already been written in a generally covariant form they may be taken over unchanged into the present context. In particular,

now serves as a defining relation for the connecting quantities τAkl. A linear rotor connection is introduced, and the covariant derivative of a rotor defined. The covariant constancy of the τAkl establishes the relation between the linear connections in w-space and in r-space. A rotor curvature tensor is considered alongside a number of other curvature objects. Next, conformal transformations are dealt with, of which duality rotations may be considered as a special case. This leads naturally to a gauge-covariant generalization of the whole calculus. A so-called rotor-derivative is defined, and some general relations involving such derivatives investigated. The relation of the rotor curvature tensor to the spin-curvature tensor is touched upon, after which the introduction of “geodesic frames” is considered. After this general theory some points concerning the Maxwell field are dealt with, which is followed by some work concerning basic quadratic and cubic invariants. Finally, a certain basic symmetric rotor is re-considered in the context of the classification of Weyl tensors, and the idea of a canonical representation suggested.
A note on doubly transitive groups
Authors: Lorimer Peter
Published: April 2009

W. Feit [1], N. Itô [2] and M. Suzuki [3] have determined all doubly transitive groups with the property that only the identity fixes three symbols. It is of interest to the theory of projective planes to determine whether any of these groups contain a sharply doubly transitive subset (see Definition 1). It is found that if such a group G contains such a subset R then R is a normal subgroup of G, i.e. R is a doubly transitive normal subgroup of G in which only the identity fixes two symbols.
Finite simple groups with nilpotent third maximal subgroups
Authors: Gagen T. M. , Janko Z.
Published: April 2009

We say that a subgroup H is an n-th maximal subgroup of G if there exists a chain of subgroups G = G0 > G1 > … > Gn = H such that each Gi is a maximal subgroup of Gi-1, i = 1, 2, …, n. The purpose of this note is to classify all finite simple groups with the property that every third maximal subgroup is nilpotent.
On the lattice equivalence of topological spaces
Authors: Finch P. D.
Published: April 2009

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.
Another Law for the 3-Metabelian Groups
Authors: Macdonald I. D.
Published: April 2009

Journal of the Australian Mathematical Society 4 (1964), 452–453

The second paragraph should be deleted. The alleged commutator identity (3) is false and is certainly not due to Philip Hall. The correct form is

as Dr. N. D. Gupta of Canberra has pointed out to me. According to Professor B. H. Neumann, this identity appeared in his (Professor Neumann's) thesis.

Nevertheless the theorem is valid and the proof given is correct.
A stability result for the linear differential equation x”+f(t)x = 0
Authors: Chang K. W.
Published: April 2009

Suppose that the real-valued function ƒ(t) is positive, continuous and monotonic increasing for t ≧ t0. If x = x (t) is a solution of the equation for for t ≧ t0, it is known that the solution x(t) oscillates infinitely often as t → ∞ and that the successive maxima of |x(t)| decrease, with increasing t. In particular x(t) is bounded as t → ∞.
A note on vector lattices
Authors: Yamamuro Sadayuki
Published: April 2009

Let E be a vector lattice in the sense of Birkhoff [1]. We use the following notations:
Identical relations and decision procedures for groups
Authors: Heineken Hermann , Neumann Peter M.
Published: April 2009

Although varieties of groups can in theory be determined as well by the identical relations which the groups all satisfy as by some structural property inherited by subgroups, factor groups and cartesian products which the groups have in common, it seems in practice just as hard to answer questions about properties of a group from knowledge of identical relations as it is from, say, a presentation. Many of the important questions connected with Burnside's problems exemplify this difficulty: we still do not know if there is a bound on the derived length of finite groups of exponent 4, nor whether there is a bound on the nilpotency class of finite groups of exponentp(p≧ 5, a fixed prime).
Projective-symmetric spaces
Authors: Reynolds R. F. , Thompson A. H.
Published: April 2009

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].
On metabelian p-groups
Authors: Weichsel Paul M.
Published: April 2009

In this paper we investigate finite metabelian p-groups from the point of view of varieties and identical relations. In other words we shall be interested in those properties of such groups which are preserved under the operations of fonning direct products, taking subgroups and taking factor groups. Ideally one would like to characterize such properties in terms of numerical invariants and with the sole restriction that the p-group in question have class less than p we accomplish this.
On varieties of metabelian p-groups, and their laws
Authors: Brisley Warren
Published: April 2009

The purpose of this article is to present some results on varieties of metabelian p-groups, nilpotent of class c, with the prime p greater than c. After some preliminary lemmas in § 3, it is established in § 4, Theorem 3, that there is a simple basis for the laws of such a variety, and this basis is explicitly stated. This allows the description of the lattice of such varieties, and in § 5, Theorem 4, it is shown that each such variety has a two-generator member which generates it; this is established by the help of Theorem 5, which states that each critical group is a two-generator group, and Theorem 6, which gives explicitly the varieties generated by the proper subgroups, by the proper quotient groups, and by the proper factor groups of such a critical group.
Semigroup structures for families of functions, I. Some Homomorphism Theorems
Authors: Magill Kenneth D.
Published: April 2009

This is the first of several papers which grew out of an attempt to provideC (X, Y), the family of all continuous functions mapping a topological spaceXinto a topological spaceY, with an algebraic structure. In the eventYhas an algebraic structure with which the topological structure is compatible, pointwise operations can be defined onC (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ringC (X, R)of all continuous, real-valued functions defined on X [3]. Now, one can provideC(X, Y)with an algebraic structure even in the absence of an algebraic structure onY. In fact, each continuous function fromYintoXdetermines, in a natural way, a semigroup structure forC(X, Y). To see this, let ƒ be any continuous function fromYintoXand for ƒ andginC(X, Y), define ƒgbyeachxinX.
Semigroup structures for families of functions, II. Continuous functions
Authors: Magill Kenneth D.
Published: April 2009

This is a continuation of [5] and we begin by recalling two definitions and a result of that paper which are needed here. Let be a family of functions with domains contained in a set X and ranges contained in a set Y and let be a function with domain D()= Y and range with the property for each pair of elements ƒ and g of . Since the composition operation is associative, is a semigroup if for ƒ and g in , we define the product ƒg by .
Extreme coverings ofn-space by spheres
Authors: Barnes E. S. , Dickson T. J.
Published: April 2009

It is well known that the problem of determining the most economical covering ofn-dimensional Eucidean space, by equal spheres whose centres form a lattice, may be formulated in terms of positive definite quadratic forms, as follows: Let

be positive definite, and

For real α, set

(the minimum being taken over integralx),
M. L. Urquhart
Authors: Elliott David
Published: April 2009

To assess the mathematical work of the late M. L. Urquhart is, paradoxically, both easy and extremely difficult. It is easy, in that he never published a mathematical paper in any of the journals. Thus one does not have to read a large volume of published work. On the other hand, Urquhart was far from being mathematically inactive, but he communicated his ideas verbally to his associates. At this point of time it is well nigh impossible to recall all the ideas that he discussed so freely during his lifetime. Conversations have long since been forgotten and ideas are now only vaguely remembered. Consequently any objective assessment of Urquhart's mathematical work is very difficult.
On dams of finite capacity
Authors: Takács Lajos
Published: April 2009

We shall consider the following mathematical model of dams of finite capacity. In the time interval (0, ∞) water is flowing into a dam (reservoir). Denote by χ(u) the total quantity of water flowing into the dam in the time interval (0, u). The capacity of the dam is a finite positive number h. If the dam becomes full, the excess water overflows. Denote by δ(u) the total quantity of water demanded in the time interval (0, u). If there is enough water in the reservoir the demand is satisfied, if there is not enough water the difference is supplied from elsewhere Denote by η(t) the content of the dam at time t. η(0) is the initial content.
Embedding semirings in semirings with multiplicative unit
Authors: Pearson K. R.
Published: April 2009

A topological semiring is a system (S, +, ·) where (S, +) and (S, ·) are topological semigroups and · distributes across + as in a ring; that is, for all x, y, z in S, The operations + and · are called addition and multiplication respectively.
On the number of topologies definable for a finite set
Authors: Shafaat A.
Published: April 2009

No general rule for determining the number N(n) of topologies definable for a finite set of cardinal n is known. In this note we relate N(n) to a function Ft(r1,…, rt+1) defined below which has a simple combinatorial interpretation. This relationship seems useful for the study of N (n). In particular this can be used to calculate N(n) for small values. For n 3, 4, 5, 6 we find N(3) = 29, N(4) = 355, N(5) = 7,181, N(6) = 145,807.
A note on approximation by automorphic images of functions in L1 and L∞
Authors: Harasymiv S. R.
Published: April 2009

In [1], it was shown that if ƒ ∈ Lp(Rn), where 1 < p < ∞, then the closed subspace of Lp (Rn) spanned by functions of the form [where a1, …, an, b1, …, bn, are real numbers; ak, ≠ 0; k = 1, …, n] coincides with the whole of Lp(Rn). In the present note, analogous results are derived for the spaces of integrable functions, essentially bounded measurable functions, bounded continuous functions, and continuous functions vanishing at infinity.
Infinite product expansions for matrix n-th roots
Authors: Smith R. A.
Published: April 2009

In this paper a denotes a square matrix with real or complex elements (though the theorems and their proofs are valid in any Banach algebra). Its spectral radius p(a) is given by with any matrix norm (see [4], p. 183). If p(a) < 1 and n is a positive integer then the binomial series converges and its sum satisfies S(a)n = (1−a)−1. Let where q is any integer exceeding 1. Then u(a) is the sum of the first q terms of the series (2). Write and let a0, a1, a2,…be the sequence of matrices obtained by the iterative procedure Defining polynomials φ0(x), φ1(x), φ2(x),…inductively by we have aν = φν (a) and therefore aμaν = aνaμ for all 4 μ, ν. The following is proved in section 2: Theorem 1. If ρ(a) < 1 thenconverges and P(a) = S(a). Furthermore, if p(a) < r < 1, thenfor all ν, where M depends on r and a but is independent of ν and q.
Gravity effects in the initial value problems of water wave theory
Authors: Low A. H.
Published: April 2009

The theory described in this paper is directed towards obtaining a general expression for the development of the free surface of a fluid, subsequent to a given initial state and prescribed boundary conditions, as a power series in g, the gravitational acceleration. In an earlier paper [4], a result, applicable to the particular case of the entry of a thin wedge into an incompressible fluid, was obtained and gave the shape of the free surface as such a power series. This series was valid for values of the ratio ut/x < 1, where u, was the (constant) velocity of entyr of the wedge; x the horizontal distance from the vertex of the wedge; and t the times elapsed after entry. This particular problem was first investigated by Mackie [6] who derived an asymptotic solution.
Positive values of inhomogeneous quaternary ouadratic forms, II
Authors: Dumir Vishwa Chander
Published: April 2009

In a previous paper [4] we showed that Γ3,1 = 16/. For the definition of Γr, s for an indefinite quadratic form in n = r + s variables of the type (r, s) see the above paper. Here we shall show that Γ2,2 = 16. More precisely we prove:
Theorem. Let Q (x, y, z, t) be an indefinite quaternary quadratic form with determinant D > 0 and signature (2, 2). Then given any real numbers x0, y0, z0, t0 we can find integers x, y, z, t such thatEquality is necessary if and only if either where ρ ≠ 0. For Q1 equality occurs if and only if
A note on group invariant continua
Authors: Gray William J.
Published: April 2009

Let (X, T, π) be a topological transformation group, where X is a Hausedorff continuum. We will say that X is irreducibly T-invariant if no proper subcontinuum of X is T-invariant. Wallace, [6], has shown that if T is abelian and X is irreducibly T-invariant, then X has no cut point; he then asked if this statement remains true if “abelian” is replaced by “compact”. In this paper we answer this question in the affirmative, and prove a related result when T satisfies a recursive property.
An unsolved problem on the powers of 3/2
Authors: Mahler K.
Published: April 2009

Let α be an arbitrary positive number. For every integer n ≦ 0 we can write where is the largest integer not greater than, i.e the integral part of, and rn is its fractional part and so satisfies the inequality
Damping loss in a cascade of fluttering phased aerofoils
Authors: Thornton B. S.
Published: April 2009

A study is made of the aerodynamic damping in a cascade of oscillating aerofoils in subsonic compressible flow with the system mode described by a constant interblade phasing predicted by Lane (9). The integral downwash equation is obtained as an extension of Possio's equation for an isolated aerofoil. Procedures for a practical solution of the equation have allowed the aerodynamic reactions and damping derivatives to be evaluated with a digital computer after minimisation of the critical flutter velocity with respect to interbiade phase angle. The effect of aerodynamic lag in separation conditions as a function of reduced frequency and chordwise location is compared with results without separation.
The order topology in a bicompactly generated lattice
Authors: Kent D. C. , Atherton C. R.
Published: April 2009

The concept of a compactly generated lattice has been studied extensively in connection with decomposition theory (see [1]). This paper investigates the order topology in a lattice which is, along with its dual, compactly generated (hence, bicompactly generated). We show that order convergence is topological and that the order topology is Hausdorff, totally disconnected, and has an open subbase of ideals and dual ideals in any bicompactly generated lattice; furthermore, with an additional restriction, the lattice operations are continuous in the order topology. Next we consider the order topology in certain special types of compactly generated lattices, namely atomic Boolean algebras and sub-complete lattices of atomic Boolean algebras in the former structures the order topology is uniformizable, in the latter, compact.
A further generalization of the arc-sine law
Authors: Heyde C. C.
Published: April 2009

Let Xi, i = 1,2,3,… be a sequence of independent and identically distributed random variables and write S0 = 0, Sn = ∑ni=1Xi, n ≧ 1. Let In(0), In(1), …, In (n) be that unique permutaion of 1, 2, …, n such that SIn(0) ≦ SIn(1) ≦ … ≦ SIn(n) and such that if Si = Sk with i < k then In(k) < In(j). Thus, In(j) is an index of the j-th largest partial sum.
On the Equation ax—xb = c
Authors: Berghout R. F.
Published: April 2009

Beniamino Segre, in his memorial lecture of 1958 [5], [6], inaugurated the study of non-linear geometry in three dimensions over a division ring. In his treatment of sections of quadrics by planes, he is naturally led to consider conics and the problem of tangency. Now in the commutative case the locus of intersection of a quadric and a plane containing a generator is the line-pair consisting of this generator and one from the other family. Such a plane is then the tangent plane of the point of intersection of the two generators. Segre extends this notion to the non-commutative case, where the locus of intersection is not always a line-pair. He joins up the remaining points of intersection in pairs, and calls the points where the lines so formed cut the base generator, the ‘points of contact’ of the plane (π) and the quadric (Q). A line in π is called a ‘tangent’ if it passes through a point of contact, but does not contain any of the points of intersection of Q and π.
A note on d-ideals in some near-algebras
Authors: Yamamuro Sadayuki
Published: April 2009

Let E be a real Banach space. The set of all continuous linear mappings of E into E is a Banach algebra under the usual algebraic operations and the operator bound as norm. We denote this Banach algebra by ℒ, if E is a separate Hilbert space.
Commutation near-rings of a group
Authors: Gupta N. D.
Published: April 2009

Let G be a group and let Ω(G) denote the semigroup of all mappings of G into G with the usual composition of mappings as multiplication, namely g(θ1θ22) = (gθ1)θ2.
The summability of formal solutions of functional equations
Authors: Gray J. D.
Published: April 2009

In 1923 Nörlund [1] considered the difference equation

and showed that the formal solution of (1)

obtained by iteration, although in general divergent, is in fact Abel summable to a solution of (1). He writes the arbitary constant c as

and thus the “principal solution” of the difference equation is
On the spaces of mappings on Banach spaces
Authors: Yamamuro Sadayuki
Published: April 2009

Let E be a real Banach space. A mapping f of E into E is said to be bounded if f maps every bounded set into a bounded set.
Finite solvable c-groups
Authors: Polimeni Albert D.
Published: April 2009

A group G is called a c-group if each of its subnormal subgroups is characteristic in G. It is the object of this note to give a characterization of finite solvable c-groups.
Minimum determinant of asymmetric quadratic forms
Authors: Worley R. T.
Published: April 2009

Let f = f (x1, x2,… xn) be an indefinite n-ary quadratic form of signature s and let m+(f), m−(f) denote the infimum of the non-negative values taken by f and —f respectively for integral (x1, x2,…, xn) ≠ (0, 0,…, 0). Furthermore let f satisfy the condition m+ (f) ≠ 0 and let for some integer k. Then Segré [3] has shown that, for n = 2, f must have determinant det (f) satisfying with equality if and only if f is equivalent under an integral unimodular transformation (denoted ˜) to a multiple of the form f1(x, y) = x2−kxy−ky2, while Oppenheim [2] has shown that, for n ≧ 3, is of the order of k2n−2
Asymmetric minima of indefinite ternary quadratic forms
Authors: Worley R. T.
Published: April 2009

Let f = f(x) = f(x1, x2,…, xn) be an indefinite n-ary quadratic form of determinant det (f); that is, f(x) = x' Ax where A is a real symmetric matrix with determinant det (f). Such a form is said to take the value v if there exists integral x ≠ 0 such that f(x) = v.
Criteria for Fourier transforms
Authors: Edwards R. E.
Published: April 2009

Salem [1] gave the following criterion for Fourier-Lebesgue sequences .

Denote by E the set of continuously differentiate periodic functions u such that u' has an absolutely convergent Fourier series, and let E1 denote the set of uσE satisfying ∥u∥∞≦1.
Thomas MacFarland cherry
Authors: Love E. R.
Published: April 2009

Professor Sir Thomas Cherry, F.A.A., F.R.S., died at his home in Melbourne on 21st November 1966 at the age of 68. He was widely known and highly respected as Australia's most distinguished mathematician and a leader in university affairs. He was associated with the University of Melbourne for most of his life, and latterly with La Trobe University as first chairman of its Academic Planning Board. He was a foundation member and a president of both the Australian Academy of Science and the Australian Mathematical Society. His greatest contributions to knowledge were probably made in the mathematics of air flow in trans-sonic flight, simultaneously with Lighthill in Britain; but he also made major contributions to global differential equation theory and general dynamics, and solved some difficult special problems in various branches of applied mathematics. He was a most distinguished teacher, amongst whose students are numbered two Fellows of the Royal Society and several professors in Australian and overseas universities. He was a man of wide interests and great ability, of keen insight and broad vision. He knew much more than he ever wrote, and his influence will live on in the minds of innumerable people with whom he worked.
On Bare Points
Authors: Bernau S. J.
Published: April 2009

This note shows that the set of bare points of a compact convex subset of a normed linear space is, in general, a proper subset of its set of exposed points.
Toeplitz Determinants and Szegö's Formula
Authors: Hartwig R. E.
Published: April 2009

In this paper the Toeplitz determinant of order s ≧ 1 generated by the rational function , with , and , is evaluated exactly for all values of s ≧ m, as , where in with and , thus proving Szegö's formula for the function fm, n(z).

By forming the rational approximation of the generating function the formula is then extended to enabling the evaluation of the limit of Toeplitz determinants generated by certain classes of complex valued functions.
On The Maximum of the Geometric Moving Average
Authors: Hasofer A. M.
Published: April 2009

By the geometric moving average of the independent, identically distributed random variables {Xn}, we mean the stochastic process , where a is a real number such that 0≦a≦1.