bluuebunny/crossref_metadata_2025 huggingface dataset:
On some two-dimensional cracks in linear isotropic elasticity
Authors: Clements D. L.
Published: April 2009

In this paper the method of conformal transformation is used to consider a class of crack problem in linear isotropic elasticity. The conformal transformation used is the Joukowski transformation so that in general the crack profile is an aerofoil. This is a useful profile to consider since as well as providing information about the general aerofoil crack it includes as special cases the circular arc crack, the straight line crack and, as will be shown, can also be used to obtain similar results to those obtained by Bowie (1956) for the crack originating at the boundary of a circular hole. Thus by varying the parameters in the one elastic solution it is possible to obtain the elastic solution for a number of crack profiles. An examination of the stresses at the crack tip is made since this gives useful information about the direction of crack extension and also gives information necessary for the application of the fracture criterion which is derived in section 6. Some particular crack profiles are considered in detail in section 7.
On a generalization of the transformation semigroup
Authors: Symons J. S. V.
Published: April 2009

In a series of papers ((1967), (1967a) and (1967b)) Magill has considered the semigroups J(X, Y; θ) (definition below), a natural, but extensive, generalization of the usual transformation semigroup J(X). They have also been studiedin Sullivan (to appear). Under the assumption that θ be onto Magill described their automorphisms and determined when one J(X, Y; θ) is isomorphic to another.
Differentiation of composities with respect to a parameter
Authors: Gray Alistair
Published: April 2009

A formula is established for differentiating the composite F(ξ)○G(ξ) with respect to the parameter ξ where F and G are maps which assumetheir values in function spaces. This composite is denoted by . Use of this notation, together with the tangent functor T, enables the formula to be written in the variable-free form where π1 is a projection map and TF(ξ) = T(F(ξ)). Formal verificationof this formula is straightforward. It has application to many small divisor problems such as those which occur in celestial mechanics.
Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic
Authors: Holt D. F. , Spaltenstein N.
Published: April 2009

The classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.
On products of modules in a topos
Authors: Tavakoli Javad
Published: April 2009

In an elementary topos if R is a ring and X is a decidable object then there exists a canonical homomorphism from the coproduct of an X-family of R-modules to the product of the same family. In this paper it is shown that this homomorphisms is a monomorphism.
Primitive spaces of matrices of bounded rank. II
Authors: Atkinson M. D.
Published: April 2009

The classification of spaces of matrices of bounded rank is known to depend upon ‘primitive’ spaces, whose structure is considerably restricted. A characterisation of an infinite class of primitive spaces is given. The result is then applied to obtain a complete description of spaces whose matrices have rank at most 3.
An approach to nonlinear elliptic boundary value problems
Authors: Tarafdar E.
Published: April 2009

Let D ⊂ Rnbe a bounded domain andL: domL⊂L2(D) →L2(D) be a self-adjoint operator of finite dimensional kernel. Letf:D×R→Rbe a function satisfying the Carathéodory condition. Assume that there are constants λ > 0 and δ ∈ [0, 1] such thatand that.

Then with the aid of a generalized Krasnosel'skii's theorem it has been proved that under conditions exactly analogous to those of Landesman and Lazer there existsu∈L2(D) such thatL(u)(x) = f(x, u(x))for∀x∈D. This result is then used to prove the existence of weak solutions of nonlinesr elliptic boundary value problems.

Other abstract results applicable to ordinary and partial differential equations have also been proved.
The theory of functional least squares
Authors: Csörgő Sándor
Published: April 2009

The functional least squares procedure of Chambers and Heathcote for estimating the slope parameter in a linear regression model is analysed. Strong uniform consistency for the family of these estimators is proved together with a necessary and sufficient condition for weak convergence in the space of continuous vector valued functions. These results are then used to develop the asymptotic normality of an adaptive version of the functional least squares estimator with minimum limiting variance.
Simultaneous Iteration by Entire or Rational Functions and Their Inverses
Authors: Baker I. N. , Rubinstein Zalman
Published: April 2009

For a non-constant entire or rational function f normalized by f(0) = 0, f′(0) = 1, f″(0) ≠ 0, which is not a Möbius tranformation, the existence of a sequence is established which has the properties . The result certainly implies f(0)= |f(0)|= 1, so these conditions cannot be omitted. The condition f″ (0)≠ 0 can be replaced by f(k)(0) ≠ 0 for some k ≥ 2.
Compactness of subsets of Tychonoff sets via exponential laws
Authors: Morales Pedro
Published: April 2009

Using the exponential map in multifunction context, the paper deduces a system of non-Hausdorff theorems which generalize all known Ascoli theorems for the space of continuous functions and the space of point-compact continuous multifunctions.
Locally defined Fitting classes
Authors: Arcy Patrick D'
Published: April 2009

Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system − B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.
Dispersive and explosive mappings
Authors: Sheng T. K.
Published: April 2009

Let Q, R be rational numbers and real numbers respectively. We use V(F) and W(F) to denote finite dimensional inner product spaces over F. Given V(Q), we use V(R) for the smallest inner space over R containing V(Q). It is known that an R-homomorphism of V(R) to W(R) is continous. We prove that if a Q-homomorphism f: V(R) → W(R), then f is dispersive, i.e., given any v0 ∈ V(Q) and ε > 0, the image set f[D(v0, ε)], where D(v0, ε) = [v: v ∈ V(Q), ¦v – v0¦ < ε], is not bounded. It is also shown that some Q-homomorphism f: V(Q) → W(Q) can be explosive in the sense that for any v0 ∈ V(Q) and ε > 0, the set f[D[v0, ε)] is dense in W(Q). As a particular case of dispersive and explosive Q-homomorphisms, we show that the algebraic number field isomorphism f: Q(a) → Q(β), where f(a) = β and α ≠ β or βmacr; (βmacr; being complex conjugates of β) is explosive.
A note on matroids and block designs
Authors: Main R. A. , Welsh D. J. A.
Published: April 2009

The close connection between certain types of matroids or combinatorial geometries and block designs is well known. The relationships previously discussed have centred on the loose analogy between the blocks of a design and the hyperplanesor flats ot the matroid or geometry. The matroids which arise in this way have had in the main a very tight regular structure. Here we show that theclass of matroids whose bases are the blocks of a design ismuch wider — indeed from Theorem 6 below we obatain a metroid in a canonical way from any balanced incomplete block design in which no pair of blocks differ by exactly one element.
On the set of limits of Riemann sums
Authors: Daniel David , Ellis H. W.
Published: April 2009

Let F map [0, 1] into a Banach space B and let R(F) denote the set of all limits of Riemann sums of F. The set R(F) need not be convex in general (Nakamura and Amemiya (1966)) but is always convex when B is finite dimensional as first shown by Hartman (1947). A proof of Hartman's result, based on a description of R(F) when the range of F is finite, was given in Ellis (1959). In this note this description is refined, the extreme points of R(F) are determined and the following complete characterization of R(F) is obtained (where Nn = {1,2, …, n}).
Rigid subsets in Euclidean and Hilbert spaces
Authors: Janos Ludvik
Published: April 2009

A subset Y of a metric space (X, p) is called rigid if all the distances p(y1, y2) between points y1, y2 ∈ Y in Y are mutually different. The main purpose of this paper is to prove the existence of dense rigid subsets of cardinality c in Euclidean spaces En and in the separable Hilbert space l2. Some applications to abstract point set geometries are given and the connection with the theory of dimension is discussed.
Automorphisms of transformation semigroups
Authors: Sullivan R. P.
Published: April 2009

We letXbe an arbitrary non-empty set throughout. Many papers have been written describing the automorphisms of various transformation semigroups defined onX: total (Lyapin (1955), Magill (1967), Malcev (1952), Schreier (1936)), partial (Gluskin (1959), Magill (1967)), partial and 1–1 (Liber (1953)), partial and shifting at most a finite number of elements (Subov (1961a)). In all these cases the automoprhisms are shown to be “inner”, and using the fact this authors deduce that the automorphism group of the given transformation semigroup is isomorphic to the groupgxof all permutations defined onX.
Congruence-free inverse semigroups with zero
Authors: Baird G. R.
Published: April 2009

A semigroup is said to be congruence-free if it has only two congruences, the identity congruence and the universal congruence. It is almost immediate that a congruence-free semigroup of order greater than two must either be simple or 0-simple. In this paper we describe the semilattices of congruence-free inverse semi-groups with zero. Further, congruence-free inverse semigroups with zero are characterized in terms of partial isomorphisms of their semilattices. A general discussion of congruence-free inverse semigroups, with and without zero, is given by Munn (to appear).
Distribution of rational points on the real line
Authors: Erdös P. , Sheng T. K.
Published: April 2009

Denote by Nn(α, β) the number of distinct fractions p/q, where 1 ≦ q ≦ n and α < p/q < β. Let .
Comments on the preceding paper of Michael's
Authors: Mostert Paul S.
Published: April 2009

It is possible, by a different approach leading to a structure theorem for the left ideal κ, to prove the main result of the preceding paper more simply and at the same time relaxing the conditions considerably. In particular we may drop the stipulation that σ be separable and metric and one of the conditions (A) or (B) and replace the other by one of the weaker conditions (A′) or (B′) below: .
A note on Green's Theorem
Authors: Craven B. D.
Published: April 2009

Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of Qx—Py is assumed, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx, and py this latter assumption being made by other authors. However, P and Q are assumed differentiable, at points interior to the curve.
Averaging operators on the ring of continuous functions on a compact space
Authors: Brainerd Barron
Published: April 2009

In this note we answer the following question: Given C(X) the latticeordered ring of real continuous functions on the compact Hausdorff space X and T an averaging operator on C(X), under what circumstances can X be decomposed into a topological product such that supports a measure m and Tf = h where By an averaging operator we mean a linear transformation T on C(X) such that: 1. T is positive, that is, if f>0 (f(x) ≧ 0 for all x ∈ and f(x) > 0 for some a ∈ X), then Tf>0. 2. T(fTg) = (Tf)(Tg). 3. T l = 1 where l(x) = 1 for all x ∈ X.
Projectivities in finite groups
Authors: Barnes D. W.
Published: April 2009

Chains of projectivities within the lattice (G) of subnormal subgroups of group G have been considered by various authors, see for example Barnes [1] and Tamaschke [2].
A generalization of z!
Authors: White-Smith W. B. , Buchwald V. T.
Published: April 2009

A generalised factorial function (z: k)! is defined as an infinite product similar to the Euler product for z!, but with the sequences of integers replaced by the roots of F(z) = sin πz+kπz. It is proved that, apart from poles in (z) < 0, (z: k)! is analytic in both variables, and that F(z) may be expressed in the form F(z) = πz/(z: k)!(—z: k)!
Doppler broadened contour functions in the complex domain
Authors: Keane A. , Clancy B. E.
Published: April 2009

The definitions of the functions used to described Doppler broadened Breit Wigner contours are extended to the complex domain. The properties of the analytic functions are then used to evaluate a number of integrals by the theory of residues.
Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time-series
Authors: Walker A. M.
Published: April 2009

Let {xt} (t = 0, ±1, ±2 …) be a stationary non-deterministic time series with E(x2t) < ∞, E(xt) = 0, and let its spectrum be continuous (strictly absolutely continuous) so that the spectral distribution functionis the spectral density function. It is well known that {xt} then has a unique one-sided movingaverage representationwhere.
Cocompactness in algebra and topology
Authors: Logan Graham J.
Published: April 2009

This paper discusses two notions, developed independently and both termed “cocompactness”. The first arises in the area of topology, where J. de Groot and others have studied spaces which are, in a certain sense, complementary to a given space. If the given space is compact then the complementary spaces are said to be cocompact. The second concept arises in the area of logic and general algebra. Loosely speaking a logic is compact if every inconsistent set of formulas has a finite inconsistent subset. This notion of compactness may be generalized to any closure algebra and the use of the term “cocompactness” to describe the generalization was suggested to the author by Dr. R. A. Bull.

It is shown here that topological and algebraic cocompactness are related in the following ways. Firstly, if a closure algebra is algebraically cocompact then its dual space is topologjcally cocompact, and conditions may be given for the implication to be reversible.3 Furthermore any cocompact topological space may be represented as the continuous 1-1 image of the dual space of a cocompact closure algebra. A final result relates another class of closure algebras with those topological spaces which are compact.
Lattice properties of the symmetric weakly inverse semigroup on a totally ordered set
Authors: Edwards C. C. , Anderson Marlow
Published: April 2009

Let T be a totally ordered set, PT the semigroup of partial transformations on T, and A(T) the l-group of order-preserving permutations of T. We show that PT is a regular left l-semigroup. Let be the set of α ∈ PT such that α is order-preserving and the domain of α is a final segment of T. Then is an l-semigroup, and we prove that it is the largest transitive l-subsemigroup of PT which contains A(T). When T is Dedekind complete, we characterize the largest regular l-semigroup of . When A(T) is also 0 − 2 transitive we show that there can be no l-subsemigroup of properly containing A(T) which is either inverse or a union of groups.
On the behaviour near the boundary of solutions of a semi-linear partial differential equation of elliptic type
Authors: Chabrowski J. , Thompson B.
Published: April 2009

Absrtact

The purpose of this note is to investigate traces of the functions In(1 +|u|), where u is a solution of a semi-linear partial differential equation of elliptic type, belonging to an appropriate Sobolev space. This article complements the results of Chabrowski and Thompson (1980), and Mihailov (1972), (1976).
Division theorems for inverse and pseudo-inverse semigroups
Authors: Pastijn F.
Published: April 2009

We show that every inverse semigroup is an idempotent separating homomorphic image of a convex inverse subsemigroup of a P-semigroup P(G, L, L), where G acts transitively on L. This division theorem for inverse semigroups can be applied to obtain a division theorem for pseudo-inverse semigroups.
Almost locally connected spaces
Authors: Mancuso Vincent J.
Published: April 2009

This paper introduces the concept of an almost locally connected space. Every locally connected space is almost locally connected, and the concepts are equivalent in the class of semi-regular spaces. Almost local connectedness is hereditary for regular open subspaces, is preserved by continuous open maps, but not generally by quotient maps. It is productive in the presence of almost-regularity.
The asymmetric product of three inhomogeneous linear forms
Authors: Woods A. C.
Published: April 2009

It is shown, given any positive real number λ and any point (x1, x2, x3) of R3 and any lattice λ R3; that there exists a point (z1, z2, z3) of λ for which

which generalizes a theorem due to Remak.
Intersections of tangent convex curves
Authors: Zamfirescu Tudor
Published: April 2009

We prove in this note that if two convex curves are internally tangent (lie on the same side of the common tangent), then in most cases they also cut each other infinitely many times. It follows that, in a certain sense, if two convex curves have an odd number of common points, then in general they are externally tangent (lie on different sides of the common tangent). The sense of the expressions most cases and in general remains to be precised.
Extensions of a problem of Paul Erdös on groups
Authors: Lennox John C. , Wiegold James
Published: April 2009

The main results are as follows. A finitely generated soluble group G is polycyclic if and only if every infinite set of elements of G contains a pair generating a polycyclic subgroup; and the same result with “polycyclic” replaced by “coherent”.
Finite varieties and groups with Sylow p-subgroups of low class
Authors: Brandl Rolf
Published: April 2009

A finite variety is a class of finite groups closed under taking subgroups, factor groups and finite direct products. To each such class there exists a sequence w1, w2,… of words such that the finite group G belongs to the class if and only if wk(G) = 1 for almost all k. As an illustration of the theory we shall present sequences of words for the finite variety of groups whose Sylow p-subgroups have class c for c = 1 and c = 2.
A generalization of Sperner's theorem
Authors: Daykin D. E. , Frankl P. , Greene C. , Hilton A. J. W.
Published: April 2009

Some generalizations of Sperner's theorem and of the LYM inequality are given to the case when A1,… At are t families of subsets of {1,…,m} such that a set in one family does not properly contain a set in another.
Discrete groups whose multiplier representations are type I
Authors: Holzherr A. K.
Published: April 2009

We determine necessary and sufficient conditions for the multiplier representations of a discrete group to be type I. This result extends the corresponding result for ordinary representation given by Kaniuth in [4].
Degree-free bounds for dependence relations
Authors: Bijlsma A. , Cijsouw P. L.
Published: April 2009

Let α1,…, αn be non-zero albegraic numbers and let l1(α1),…ln(αn) denote arbitrary fixed values of the logarithms of α1,…n, respectively Given that l1(α1),…ln(αn) are linearly dependent over Q, the existence of non-trival dependence relation between these numbers with integer coefficients of low absolute values can be proved. Existing results of this kind give bounds for the absolute values of the coefficients which are expressions in the degree D = [Q(α1…αn): Q], the heights of α1,…αn and the magnitudes of the logarithms involved.
Quasi Clifford algebras and systems of orthogonal designs
Authors: Gastineau-Hills H. M.
Published: April 2009

The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.
The structure of the group of permutations induced by Chebyshev polynomial vectors over the ring of integers mod m
Authors: Matthews Rex
Published: April 2009

In an earlier paper the author investigated the properties of a class of multivanable polynomial vectors which generalise the multivariable Chebyshev polynomial vectors. In this paper the behaviour of these polynomials over rings of the typeZ/(m) is investigated, and conditions are determined for such an n-variable polynomial vector to induce a permutation of (Z/(m))n. More detailed results on the Chebyshev polynomial vectors follow. The composition properties of these vectors imply that the permutations induced by certain subsets of them form groups under composition of mappings, and the structure of these groups is investigated.
On separable cyclic extensions of rings
Authors: Szeto George , Wong Yuen-Fat
Published: April 2009

The quaternion algebra of degree 2 over a commutative ring as defined by S. Parimala and R. Sridharan is generalized to a separable cyclic extension B[j] of degree n over a noncommutative ring B. A characterization of such an extension is given, and a relation between Azumaya algebras and Galois extensions for B[j] is also obtained.
Gelfand dualities over topological fields
Authors: Day Brian J.
Published: April 2009

The spectral duality theory of H.-E. Porst and M. B. Wischnewsky is examined in more generality, and examples based on topological fields are described.
Some applications of group Obstructions
Authors: Wu Yel-Chiang
Published: April 2009

We show that the classical interpretation of H3(G, A) is equivalent to Taylor's solutions of compound extensions of groups. It is also equivalent to the exactness to an eight term sequence. Only halves of the equivalences are fully shown in the paper but the other halves are clear.
On the surjectivity of linear maps on locally convex spaces
Authors: Yamamuro Sadayuki
Published: April 2009

The aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).
Abelian unipotent subgroups of finite orthogonal groups
Authors: Wong W. J.
Published: April 2009

If G is the special orthogonal group O+(V) of a quadratic space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G, extending some results obtained with different methods by Barry (1979).
Representation theorems for parabolic systems
Authors: Chabrowski J.
Published: April 2009

The aim of this article is to review the progress made in the last few years in the representation theory of solutions of parabolic systems in the sense of Petrowskii.
Universal varieties of semigroups
Authors: Koubek V. , Sichler J.
Published: April 2009

A category V is called universal (or binding) if every category of algebras is isomorphic to a full subcategory of V. The main result states that a semigroup variety V is universal if and only if it contains all commutative semigroups and fails the identity xnyn = (xy)n for every n ≥ 1. Further-more, the universality of a semigroup variety V is equivalent to the existence in V of a nontrivial semigroup whose endomorphism monoid is trivial, and also to the representability of every monoid as the monoid of all endomorphisms of some semigroup in V. Every universal semigroup variety contains a minimal one with this property while there is no smallest universal semigroup variety.
Saturated and epimorphically closed varieties of semigroups
Authors: Higgins P. M.
Published: April 2009

We establish a necessary condition (E) for a semigroup variety to be closed under the taking of epimorphisms and a necessary condition (S) for a variety to consist entirely of saturated semigroups. Condition (S) is shown to be sufficient for heterotypical varieties and a stronger condition (S′) is shown to be sufficient for homotypical varieties.
On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom
Authors: Hofmann Karl H. , Lawson Jimmie D.
Published: April 2009

A Hausdorff space X is said to be compactly generated (a k-space) if and only if the open subsets U of X are precisely those subsets for which K ∩ U is open in K for all compact subsets of K of X. We interpret this property as a duality property of the lattice O(X) of open sets of X. This view point allows the introduction of the concept of being quasicompactly generated for an arbitrary sober space X. The methods involve the duality theory of up-complete semilattices, and certain inverse limit constructions. In the process, we verify that the new concept agrees with the classical one on Hausdorff spaces.
Radical decompositions of idempotent algebras
Authors: Gardner B. J.
Published: April 2009

A variant of Kurosh-Amitsur radical theory is developed for algebras with a collection of (finitary) operations ω, all of which are idempotent, that is satisfy the condition ω(x, x,…,x) =x. In such algebras, all classes of any congruence are subalgebras. In place of a largest normal radical subobject, a largest congruence with radical congruence classes is considered. In congruence-permutable varieties the parallels with conventional radical theory are most striking.
Room n-cubes of lowe order
Authors: Dinitz Jeffrey H.
Published: April 2009

A Room n-cube of side t is an n dimensional array of side t which satisfies the property that each two dimensional projection is a Room square. The existence of a Room n-cube of side t is equivalent to the existence of n pairwise orthgonal symmetric Latin squares (POSLS) of side t. The existence of n pairwise orthogonal starters of order t implies the existence of n POSLS of side t. Denote by v(n) the maximum number of POSLS of side t. In this paper, we use Galois fields and computer constructions to construct sets of pairwise orthogonal starters of order t ≤ 101. The existence of these sets of starters gives improved lower bounds for v(n). In particular, we show v(17) ≥ 5, v(21) ≥ 5, v(29) ≥ 13, v(37) ≥ 15 and v(41) ≥ 9, among others.
Duality theorems for convex programming without constraint qualification
Authors: Kanniappan P.
Published: April 2009

Invoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.
A recurrence relation generalizing those of Apéry
Authors: Askey Richard , Wilson J. A.
Published: April 2009

A three term recurrence relation is found for

when a + d = b + c. This includes the recurrence relations of Apéry associated with ζ(3), ζ(2) and log 2 as special or limiting cases.
A subclass of univalent functions
Authors: Goel R. M. , Mehrok Beant Singh
Published: April 2009

Sharp results for the cofficient estimates, distortion theorems, radius of convexity, arc-length and area of the image curve are obtained for the class R(A, B) of regular functions whose derivative is subordinate to (1+AZ)/(1+Bz), -1 ≤ B ≤ A ≤ 1, in the unit disc E = {z:|z| < 1}. We also establish a convolution theorem for this class.
A note on subdirectly irreducible distributive double p-algebras
Authors: Adams M. E. , Katriňák T.
Published: April 2009

A regular double p-algebra L satisfying (i) ∩(xn(+*); n < ω) for every 1 ≠ x ∈ L and (ii) L is not subdirectly irreducible, is constructed. The construction is purely topological and the desired result is obtained via the known Priestly duality. The notion of an auxiliary regular double p-algebra is introduced and the algebras having this property are characterized.
Varieties of nilpotent groups of class four (I)
Authors: Fitzpatrick Patrick , Kovács L. G.
Published: April 2009

This is the first of three papers (the others by the first author alone) which determine all varieties of nilpotent groups of class (at most) four. The initial step is to reduce the problem to two cases: varieties whose free groups have no elements of order 2, and varieties whose free groups have no nontrivial elements of odd order. The varieties of the first kind form a distributive lattice with respect to order by inclusion (which is not a sublattice in the lattice of all group varieties). We give an embedding of this lattice in the direct product of six copies of the lattice which consist of 0 (as largest element) and the odd positive integers ordered by divisibility. The six integer parameters so associated with a variety directly match a (finite) defining set of laws for the variety. We also show that the varieties of the second kind do form a sublattice in the lattice of all varieties. That (nondistributive) sublattice will be treated, in a similarly conclusive manner, in the subsequent papers of this series.
Varieties of nilpotent groups of class four (III)
Authors: Fitzpatrick Patrick
Published: April 2009

In this paper we complete the investigation of those varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. Each such variety is labelled by a vector of sixteen parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used and directly yields a defining set of laws for the variety it labels. Moreover, one can easily recognise from the parameters whether one variety is contained in another. In view of the reduction carried out in the first paper of this series (written jointly with L. G. Kovács) this completes the determination of all varieties of nilpotent groups of class four.
Some results on the asymptotic behavior of nonoscillatory solutions of differential equations with deviating arguments
Authors: Philos Ch. G. , Sficas Y. G. , Staikos V. A.
Published: April 2009

This paper deals with some asymptotic properties of nonoscillatory solutions of a class of n-th order (n < 1) differential equations with deviationg arguments involving the so called n-th order r-derivative of the unknown function x defined by

where ri (i = 0,1…n) are positive continous functions on [t0, ∞). The fundamental purpose of this paper is to find for any integer m, 0 < m < n – 1, a necessary and sufficient condition (depending on m) in order that three exists at least one (nonoscillatory) solution x so that the exists in R – {0} The results obtained extend some recent ones due to Philos (1978a) and they prove, in a general setting, the validity of a conjecture made by Kusano and Onose (1975).
Limit theorems for samples from a finite population
Authors: Jirina Miloslav
Published: April 2009

Two theorems on limit distributions for sums of values sampled from a finite population without replacement are presented. The emphasis is on non-normal limit distributions.
On domains of partial attraction
Authors: Goldie C. M. , Seneta E.
Published: April 2009

A new necessary and sufficient condition for a distribution of unbounded support to be in a domain of partial attraction is given. This relates the recent work of [5] and [6].
Identifiability of mixtures
Authors: Tallis G. M. , Chesson P.
Published: April 2009

Let F(x, θ) be a family of distribution functions indexed by θ ∈ Ω. If G(θ) is a distribution function on Ω H(x) = ƒohm; F(x, θ) dG(θ) is a mixture with respect to G. If there is a unique G yielding H, the mixtures is said to be identifiable.This paper summarises some known results related to identifiability of special types of mixtures and then discusses the general problem of identifiability in terms of mappings. Some new results follow for mappings with special features.
A note on an ill-posed problem for the heat equation
Authors: Talenti Giorgio , Vessella Sergio
Published: April 2009

In this paper an ill-posed problem for the heat equation is investigated. Solutions u to the equation ut – uxx = 0, which are approximately known on the positive half-axis t = 0 and on some vertical lines x = x1,…, x = xn, are considered and stability estimates of these solutions are presented. We assume an a priori bound, governing the heat flow across the boundary x = 0.
Duality theorems and an optimality condition for non-differentiable convex programming
Authors: Kanniappan P. , Sastry Sundaram M. A.
Published: April 2009

Necessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.
Matroids whose ground sets are domains of functions
Authors: Oxley James , Prendergast Kevin , Row Don
Published: April 2009

From an integer-valued function f we obtain, in a natural way, a matroid Mf on the domain of f. We show that the class of matroids so obtained is closed under restriction, contraction, duality, truncation and elongation, but not under direct sum. We give an excluded-minor characterization of and show that consists precisely of those transversal matroids with a presentation in which the sets in the presentation are nested. Finally, we show that on an n-set there are exactly 2n members of .
Congruences on regular and completely regular semigroups
Authors: Trotter P. G.
Published: April 2009

Congruences on regular semigroups have been characterized in terms of normal equivalences on sets of idempotents and kernels of congruences. A revised characterization is presented here with considerably simplified expressions for the least and greatest congruences associated with normal equivalences and with a new description of kernels. The results are then applied to characterize congruences on completely regular semigroups.
Criteria for extreme forms
Authors: Barnes E. S.
Published: April 2009

A positive quadratic form, of determinantand minimum M for integral, is said to be extreme if the ratiois a (local) maximum for small variations in the coefficients.

Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.
Discontinuous solutions of variational problems
Authors: Lawden D. F.
Published: April 2009

The most elementary problem of the calculus of variations consists in finding a single-valued function y(x), defined over an interval [a, b] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y(x) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y(x) is differentiable and accordingly when y(x) is continuous. Usually y′(x) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y′ (x) to be discontinuous at a finite number of points. The arc y = y(x) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y (x) for which y′(x) is continuous except at a finite number of points, are referred to as admissible arcs. In this paper, we shall extend the range of admissible arcs to include those for which y(x) is discontinuous at a finite number of points.
Some extreme forms defined in terms of Abelian groups
Authors: Barnes E. S. , Wall G. E.
Published: April 2009

Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.
On Euler's criterion
Authors: Lehmer Emma
Published: April 2009

Euler's criterion states that if p is a prime then if and only if D is a k-th power residue of p.

However if (1) does not hold then (2)

where ακ is some κ-th root of unity modulo ρ.
On the strength of sand
Authors: Rennie B. C.
Published: April 2009

An aggregate consisting of a large number of small, equal, hard spheres might be expected to behave like ordinary sand; in this article therefore an attempt is made to find the porperties that will be exhibited by this model, and in particular to find what stresses the sand will withstand without collapsing.
Some nozzle flows found by the hodograph method
Authors: Cherry T. M.
Published: April 2009

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.
A note on the finite Fourier transform
Authors: Griffith James L.
Published: April 2009

1. One of the best known theorems on the finite Fourier transform is:—The integral function F(z) is of the exponential type C and belongs to L2 on the real axis, if and only if, there exists an f(x) belonging to L2 (—C, C) such that ( Additionally, if f(x) vanishes almost everywhere in a neighbourhood of C and also in a neighbourhood of —C, then F(z) is of an exponential type lower than C.
On the trace of the energy-momentum tensor of fields associated with particles of zero rest mass
Authors: Buchdahl H. A.
Published: April 2009

The traceTof the metrical energy-momentum tensorTklof fields associated with particles of zero rest mass may be zero, either identically or as a consequence of the field equations. This property ofTklis correlated here with the behaviour of the Lagrangian of the field under arbitrary conformal transformations. Certain classes of special fields are considered explicitly. It is shown in particular thatTvanishes for all non-zero spin fields which correspond respectively to the two-component neutrino field or the photon field.
On distributions whose moments are majorated by the moments of a known distribution
Authors: Laha R. G. , Lukacs Eugene
Published: April 2009

Let F(x) be a distribution function and denote by its characteristic function and by its moment of order k H. Milicer-Grużewska [2]has derived the following theorem: Suppose that F(x) has moments of all orders and that they satisfy the relations. where while C is a positive constant. Then f(t) is an entire function of order not exceeding two. The proof given by H. Milicer-Gruzewska is rather complicated, moreover the restriction (1.2) seems to be artificial and motivated only by the particular method of proof. We note that condition (1.1) means that the moments of F (x) are majorated by the moments of a normal distribution and we use this remark to generalize the problem.
Application of generating functions to a problem in finite dam theory
Authors: Prabhu N. U.
Published: April 2009

We consider the finite dam model due to Moran, in which the storage {Zt} is known to be a Markov chain. The method of generating functions is used to derive stationary distributions ofZtin the two particular cases where the input is of geometric and negative binomial types.
The survival of a mutant gene under selection
Authors: Moran P. A. P.
Published: April 2009

The problem of the survival of a single mutant in a haploid genetic population when there exists selection is considered for a type of population model in which the generations ar overlapping. The results are compared with the previous work of Fisherand others for other models. The need is stressed for a solution of the same problem in a diploid population with general phenotypic selection coefficients.
Groups with one defining relator
Authors: Baumslag Gilbert
Published: April 2009

The present day theory of finite groups might be regarded as the outgrowth of the algebraic theory of equations. In much the same way one might consider the modern theory of infinite groups as stemming from late nineteenth century topology. The groups that crop up in topology are of a particularly simple type in that they are both finitely generated and finitely related. This means that every element in such a group can be expressed in terms of a finite number of elements and their inverses and every relation is an algebraic consequence of a finite number of relations between these elements. In other words the legacy of topology to group theory is the estate of finitely presented groups. This talk is concerned with the seemingly simplest of the finitely presented groups, the so-called groups with a single defining relator.
Solution of a generalized Schroeder equation in two variables
Authors: Schubert C. F.
Published: April 2009

If f(z) is analytic at the origin, f(0)=0, and f′(0)=λ, where 0< λ <1, then Koenigs' [3] solution of Schroeder's equation w(f(z))=λw(z), with multiplier λ, is given by . Here fn(z) denotes the nth iterate of f(z), defined inductively as f0(z)=z, f(fn-1(z)), n=1, 2, 3, …. More generally the solution w(z) of Schroeder's equation is uniquely determined to within a multiplicative constant by the requirement that it be analytic at the origin. From the uniqueness it follows that if g(z) is analytic at the origin, vanishes there, and commutes with f(z), i.e., f(g(z)) = g(f(z)), then w(g(z)) = w(z), for some multiplier a. Since w′(0) = 1 for Koenig's' solution, it has an inverse locally, and we find that g(z) is uniquely determined by its linear part; in fact g(z) = w-1(μw(z)). In particular the integral iterates of f can be put in the form w-1(λw(z)) for integral n. Thus for any α, real or complex, we may define fα(z), consistent with the above definition when α is a positive integer, as w-1(λw(z)). In this manner any function g(z) of the above type can be considered as an iterate of /(z). Also if cc, j9, 0 are any two distinct points sufficiently close to the origin there exists an analytic function g(z) which commutes with f(z) such that g(0) = 0, g(α) = β. In fact g(z)=w-1(χw(z)), where the multiplier χ=w(β)(w(α))-1. These facts are all well known, e.g. [2] [5], and we shall establish analogous results in a more general situation.
An inequality for a pair of polynomials that are relatively prime
Authors: Mahler K.
Published: April 2009

Let f(x) and g(x) be two polynomials with arbitrary complex coefficients that are relatively prime. Hence the maximum is positive for all complex x. Since m(x) is continuous and tends to infinity with |x|, the quantity E(f, g) = min m(x) is therefore also positive.
Finite groups with a nilpotent maximal subgroup
Authors: Janko Zvonimir
Published: April 2009

Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G is only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M.
Another law for the 3-metabelian groups
Authors: Macdonald I. D.
Published: April 2009

The 3-metabelian groups are those groups in which every subgroup generated by three elements is metabelian. In [2] it was stated and it [3] it was proved that the variety of such groups may be defined by the one law (by [a, b, c, d] we mean [[a, b], [c, d]], and for other definitions and notation we refer to [1]). Recently Bachmuth and Lewin obtained in [1] the surprising and remarkable result that the same variety is defined by the law Now (2) is reminiscent of the relation which holds in all groups and which is apparently due to Philip Hall. Using the identities , etc., we find that (2) is equivalent to where u = [z, x, y] and v = [y, z, z] [z, x, y]. Note that apart from certain displeasing conjugates (4) is curiously similar to both (1) and (2).
On normaliser preserving lattice isomorphisms between nilpotent groups
Authors: Barnes D. W. , Wall G. E.
Published: April 2009

Let (G) denote the lattice of all subgroups of a group G. By an -isomorphism (lattice isomorphism) of G onto a group H, we mean an isomorphism of (G) onto (H). By an -isomorphism (normaliser preserving -isomorphism) of G onto H, we mean an -isomorphism ø such that (Aø) = (A)ø for all A ∈ (G). In this paper, we study certain properties of groups which remain invariant under -isomorphisms.
Lattic isomorphisms of lie algebras
Authors: Barnes D. W.
Published: April 2009

Let L be a finite dimensional Lie algebra over the Field F. We denote by (L) the lattice of all subalgebras of L. By a lattice isomorphiusm (whicn we abbrevite to -isomorphism) of L onto a Lie algebra M over the same field F, we mean an isomorphism of (L) onto (M). It is possible for non-isomorphic Lie algebras to -isomorphic, for example, the algebra of real vectors with product the vector product is -isomorphic to any 2-dimensional Lie algebra over the field of real numbers.