bluuebunny/crossref_metadata_2025 huggingface dataset:
On potential wells and stability in nonlinear elasticity
Authors: Knops R. J. , Payne L. E.
Published: November 2008

This paper is devoted to a preliminary discussion of potential wells in non-linear three-dimensional elasticity. Our interest in the subject arises from the role that potential wells play in the justification of the energy criterion as a sufficient condition for stability of an elastic equilibrium solution. It will be recalled that the energy criterion, which is a simple extension of the original Lagrange-Dirichlet version for finite-dimensional systems, states that an equilibrium solution is stable provided the potential energy achieves its minimum on the solution. No proof of this statement as it applies to three-dimensional elasticity is yet forthcoming, although when the notion of a minimum is replaced by that of a potential well, several authors have proved that the criterion thus modified is sufficient for the Liapunov stability of the equilibrium solution with respect to appropriate measures. Indeed, the proofs are applicable to many other continuum theories, apart from elasticity. (See, for instance, Coleman(5), Gurtin(11) and Koiter(21).) It thus becomes important to determine what constitutive and other conditions, if any, ensure the existence of a potential well. While we present two such conditions, our main purpose is to describe examples in support of the conjecture that non-existence rather than existence of a potential well is likely to be the generic property. In these examples, particular forms of the potential energy are chosen which have positive-definite quadratic part and yet in any W1, P-neighbourhood (1 ≤ p ≤ ∞) of the origin, have a non-positive infimum, thus violating a condition for existence of a potential well in the, Sobolev space W1, P (1 ≤ p ≤ ∞). Related results are also reported by Ball, Knops and Marsden(3) and by Knops(17) for the space W1, ∞. Another conclusion to be drawn from these examples, is that a potential well cannot be ensured by restricting only the quadratic part of the potential energy. Indeed, in our example, we show that the equilibrium (null) solution is unstable in the sense that (non-linear) motions, starting in its neighbourhood, cease to exist after finite time. The same phenomenon has been shown by Knops, Levine and Payne(19) (see also Hills and Knops(14)) to hold for a general class of materials which includes as a special case one possessing the potential energy considered in our example.
Some results concerning linear codes and (k, 3)-caps in three-dimensional Galois space
Authors: Hill Raymond
Published: November 2008

The packing problem for (k, 3)-caps is that of finding (m, 3)r, q, the largest size of (k, 3)-cap in the Galois space Sr, q. The problem is tackled by exploiting the interplay of finite geometries with error-correcting codes. An improved general upper bound on (m, 3)3 q and the actual value of (m, 3)3, 4 are obtained. In terms of coding theory, the methods make a useful contribution to the difficult task of establishing the existence or non-existence of linear codes with certain weight distributions.
Limiting behaviour of the von Mises-Fisher distribution
Authors: Kent John
Published: November 2008

The von Mises-Fisher density of parameter K is shown to be very similar to the spherical normal distribution of parameter υ = 1/K as k → ∞. In fact the two densities are closer by factor of K than one might expect.
On the exterior problems of acoustics: II
Authors: Ursell F.
Published: November 2008

The exterior problems of acoustics can be reduced to the solution of Fredholm integral equations of the second kind in many different ways. Some of these equations break down at certain values of the wavenumber but this difficulty can be reduced or even removed altogether by choosing a different (usually more complicated) integral equation. The present paper is concerned with a class of equations proposed recently by D. S. Jones. It is shown that the proof of one of his principal results can be much shortened.
A theorem on integral closures of ideals in semi-local rings
Authors: Burch A. Lindsay
Published: November 2008

Consider an ideal A of a commutative ring R; denote by Ai the integral closure of A in R.
Quaternionic analysis
Authors: Sudbery A.
Published: November 2008

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself (1) and his principal followers and expositors, Tait(2) and Joly(3), only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.
On the nth root set of an element in a connected semisimple Lie group
Authors: McCrudden M.
Published: November 2008

For any group G and x ∈ G, and any n ∈ ℕ (the natural numbers) let

i.e. the set of all nth roots of x in G.
Splitting of some more spaces
Authors: Cohen F. R. , May J. P. , Taylor L. R.
Published: November 2008

In (4) we showed how to stably split certain spaces CX built up from a ‘coefficient system’ and a ‘Π-space’ X. Via an approximation theorem relating particular examples to loop spaces, there resulted stable splittings of Ωn σnX for all n ≥ 1 and all (path) connected based spaces X.
Subsequence principles for vector-valued random variables
Authors: Garling D. J. H.
Published: November 2008

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such that

almost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.
Random evolutions and the spectral radius of a non-negative matrix
Authors: Cohen Joel E.
Published: November 2008

1.Introduction and summary. This paper offers yet another example of what probability theory can do for analysis. Using a Feynman-Kac formula derived in the theory of random evolutions (5), we find an expression (1) for the spectral radiusr(A) of a finite square non-negative matrixA. This expression makes it very easy to study howr(A) behaves as a function of the diagonal elements ofA.
A further instability theorem for a certain fifth-order differential equation
Authors: Ezeilo J. O. C.
Published: November 2008

In our previous consideration in (1) of the constant-coefficient fifth-order differential equation:

an attempt was made to identify (though not exhaustively) different sufficient conditions on a1,…,a5 for the instability of the trivial solution x = 0 of (1·1). It was our expectation that the conditions so identified could be generalized in some form or other to equations (1·1) in which a1,…,a5 were not necessarily constants, thereby giving rise to instability theorems for some non-linear fifth-order differential equations; and this turned out in fact to be so except only for the case:

with R0 = R0(a1, a2, a3, a4) > 0 sufficiently large, about which we were unable at the time to derive any worthwhile generalization to any equation (1·1) in which a1, …,a5 are not all constants.
The computational complexity of matroid properties
Authors: Robinson G. C. , Welsh D. J. A.
Published: November 2008

Knuth (12) seems to have been the first to carry out non-trivial matroid operations on a computer. However, as he remarks, there are considerable computational difficulties. In this paper we examine in detail the computational complexity of some fundamental matroid properties. The model of computation is the Hausmann–Korte oracle machine introduced in (10), (11) to deal with properties of independence spaces and fixed point problems. It was these papers (10), (11) and (12) which motivated the present work. Basically, the problems we will be discussing are matroid analogues of low-level complexity problems for graphs. For an excellent survey of this we refer to the recent monograph of Bollobés ((3), chapter 8). We shall use the approach of (3), and Bollobaé;s and Eldridge(4) rather than the more general treatment of Hausmann and Korte (11).
The infinite loop Adams conjecture via classification theorems for ℱ-spaces
Authors: Friedlander Eric M.
Published: November 2008

We prove the following generalized version of the complex Adams conjecture (see Theorem 10·4), as announced in (5).
Strict convexity, strong ellipticity, and regularity in the calculus of variations
Authors: Ball J. M.
Published: November 2008

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.
Null hypersurfaces in Lorentzian manifolds: I
Authors: Katsuno K.
Published: November 2008

This paper is concerned with geometrical properties of null hypersurfaces in Lorentzian manifolds. Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. First, the contravariant metric cannot immediately be defined, so the connection cannot be specified uniquely in the normal way. Secondly, the normal is a null vector lying in the tangent plane, which makes it necessary to look for some other vector to rig the hypersurface, and makes it impossible to normalise the normal in the usual way. These problems are considered in this paper.
On the equations defining tangent cones
Authors: Robbiano Lorenzo , Valla Giuseppe
Published: November 2008

This paper treats the local study of singularities by means of their tangent cones, more specifically the study of graded rings associated to an ideal of a local ring. We recall some basic facts: let (R,) be a local ring,I, Jideals ofR, such thatJ⊆I; thenGR/J(I/J), the graded ring associated toI/J, is canonically isomorphic to the quotient ofGR(I) modulo a homogeneous ideal, which is calledJ*, and which is generated by the so-called ‘initial forms’ of the elements ofJ. Let us consider the following example: Letkbe a field,R=k[X, Y, Z](x, y, Z),I= (X, Y, Z)R, Jthe prime ideal generated byfl,f2wheref1=Y3−Z2,f2=YZ−X4. Then

and it is easily seen thatJ*properly contains the ideal generated by the initial formsf*1f*2off1,f2; namelyf*1= −Z2,f*2=YZand (Yf1+Zf2)* =Y4∉ (−Z2,YZ).
Radiation fields and hyperbolic scattering theory
Authors: Friedlander F. G.
Published: November 2008

Let u(x, t), where x ∈ 3, t ∈ , be a C∞ solution of the wave equation on {|x| > a} × for some a > 0, and suppose that u = 0 for |x|> a, t < 0. One then also has u = 0 for |x| > a + t, t > 0, so that u(.,t) can be thought of as a wave expanding into a previously undisturbed medium. One can now ask for a description of the asymptotic behaviour of u as |x| → ∞. It turns out that there is a v0(θ, τ) ∈ C∞ (S2 × ) such that

in the topology of C∞ (S2 × ). This limit may be called the radiation field of the expanding wave u. (See (2) and the earlier papers quoted there.)
Orthomorphisms of archimedean vector lattices
Authors: Bernau S. J.
Published: November 2008

A linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.
Kneading of Lorenz type for intervals and product spaces
Authors: Rhodes Frank
Published: November 2008

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.
Some identities for terminating q-series
Authors: Bressoud D. M.
Published: November 2008

We present the following sequence of polynomial identities:

is the Gaussian polynomial denned to be zero for m < 0 or m > N, one for m = 0 or N and
Asymptotic formula for the capacitance of two oppositely charged discs
Authors: Chew W. C , Kong J. A
Published: November 2008

Asymptotic formulae for the capacitances of two oppositely charged, identical, circular, coaxial discs and two identical, infinite parallel strips separated by dielectric slabs are derived from the dual integral equations approach. The formulation in terms of dual integral equations using transforms gives rise to relatively simple Green's functions in the transformed space and renders the derivation of the asymptotic formulae relatively easy. The solution near the edge of the plates, a two-dimensional problem previously thought not solvable by the Wiener-Hopf technique, is solved indirectly using the method. Solution of the Helmholtz wave equation is first sought, and the solution to Laplace's equation is obtained by letting the wavenumber go to zero. The solution away from the edge is obtained by solving the dual integral equations approximately. The total charge on the plate is obtained by matching the solution near the edge and away from the edge giving the capacitance.
S-rings over loops, right mapping groups and transversals in permutation groups
Authors: Johnson K. W.
Published: November 2008

The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.
Fractional parts of powers of rationals
Authors: Beukers F.
Published: November 2008

In 1975 A. Baker and J. Coates published a paper bearing the same title (2). Since the introduction to their paper can be used in this paper very appropriately, I shall quote large parts of it here.
The cohomology of finite H-spaces as U(M) algebras. II
Authors: Kane Richard
Published: November 2008

This paper is a continuation of (6). We will use the definitions and notation of (6) freely. We continue our study of the restrictions imposed on the mod p cohomology of finite H-spaces by the requirement that is a U(M) algebra. As in (6) we will restrict our attention to the case where p is an odd prime. As we observed in (6), the restriction proved there, that the pth power map on is trivial, may not really require the hypothesis of . However, the restrictions proved in this paper are not valid without the assumption that . Indeed, they actually characterize the known mod odd finite H-spaces which have U(M) algebras.
Convergence of spatial birth-and-death processes
Authors: Lotwick H. W. , Silverman B. W.
Published: November 2008

Some models for spatial point processes are difficult to simulate directly and are most easily realized as the equilibrium distribution of certain spatial-temporal Markov processes. This paper examines the convergence of such processes, concentrating mainly on the ‘hard core’ case when the points represent the centres of non-overlapping discs. Coupling methods from the theory of Markov chains are used to establish sufficient conditions for the processes to converge to the required equilibrium, and to give a lower bound on the rate of convergence. One technique used is to couple processes when they become close in a suitable metric.
Threshold functions for small subgraphs
Authors: Bollobás Béla
Published: November 2008

In this note we shall study random labelled graphs. Denote by

the set of all graphs with n given labelled vertices andM(n)edges. As usual, we turn(M(n))into a probability space by giving all graphs the same probability. The question we address ourselves to is the following. Given a graphHand a constant p, 0 < p < 1, for what functionsM(n)is it true that the probability PM(n)(H ⊂ G) that a graphG∈(M(n))containsHtends topas n∞→? This question was posed by Erdös and Rényi (3), (4), who also proved several beautiful and surprising theorems. In order to state the main general result of Erdös and Rényi in this direction, and for our use later, we introduce some definitions.
Homological properties of the enveloping algebraU(Sl2)
Authors: Stafford J. T.
Published: November 2008

In this paper we will study the homological properties of the enveloping algebraU=U(Sl2(ℂ)), with particular reference to the homological dimension of simpleU-modules and the global dimension of the primitive factor rings ofU.
Two definitions of fractional dimension
Authors: Tricot Claude
Published: November 2008

The main properties of the Hausdorff dimension, here denoted by dim, are

In ℝp, in variance under a group Н of homeomorphisms: ∀HεH, dim О H = dim. The definition of H, introduced in (15) and (16), is recalled in § 2.
Differential equation for Hermite-Bell polynomials
Authors: Kalinowski M. W. , Seweryński M.
Published: November 2008

An ordinary differential equation for Hermite-Bell polynomials is derived. Bell's error concerning orthogonality of his polynomials is corrected.
Asymptotic estimates of the eigenvalues of certain positive Fredholm operators
Authors: Little G.
Published: November 2008

1. Introduction. Suppose that K is a continuous function on the square Q = [ – 1, 1] x [– 1,1] satisfying , for – 1 ≤ s, t ≤ 1; then the Fredholm operator T on L2(-1,1)

is compact and symmetric. Suppose also that T is a positive operator, i.e.

then there is an eigenfunction expansion

where (λn) is a sequence of non-negative real numbers which decreases to 0 and (φn) is an orthonormal sequence in L2( – 1,1). In this paper we shall find asymptotic estimates for λn when K takes certain specific analytic forms. In all cases K will be real-valued on Q and analytic in a neighbourhood of Q in complex 2-space; for example
Indefinite binary forms representing the same numbers
Authors: Delang Li
Published: November 2008

It is an old problem whether two binary quadratic forms representing the same numbers are equivalent. Legendre asserted without proof that two positive definite forms representing the same numbers are equivalent. In 1895, Bauer(1) proved Legendre's assertion for primitive forms with the same discriminant. In 1938 Delone (2) proved that two positive definite binary forms with real coefficients are equivalent if they represent the same numbers, with the sole exception of the pair equivalent to

and their scalings. Watson (3) rediscovered this result in 1979 and asked the same question for indefinite binary forms.
On Springer's correspondence for simple groups of type En (n = 6, 7, 8)
Authors: Alvis Dean , Lusztig George
Published: November 2008

Let G be a connected reductive algebraic group over complex numbers. To each unipotent element u ε G (up to conjugacy) and to the unit representation of the group of components of the centralizer of u, Springer (11), (12) associates an irreducible representation of the Weyl group W of G. The tensor product of that representation with the sign representation will be denoted ρu. (This agrees with the notation of (5).) This representation may be realized as a subspace of the cohomology in dimension 2β(u) of the variety of Borel subgroups containing u, where β(u) = dim . For example, when u = 1, ρu is the sign representation of W. The map u → ρu defines an injective map from the set of unipotent conjugacy classes in G to the set of irreducible representations of W (up to isomorphism). Our purpose is to describe this map in the case where G is simple of type Eu (n = 6, 7, 8). (When G is classical or of type F4, this map is described by Shoji (9), (10); the case where G is of type G2 is contained in (11).
Note on a martingale inequality of Pisier
Authors: Cox David C.
Published: November 2008

The principal purpose of this note is to give an elementary proof of an inequality, due to Pisier (proposition 2·4 of (4)), for martingales taking values in a Banach space X. This proof yields an explicit estimate, sharp when X is a Hilbert space, for one of the constants involved. The notation follows (1, 4).
Some properties of wild orbits of infinite-dimensional transformation groups
Authors: Magnus R. J.
Published: November 2008

A wild orbit is an orbit of a transformation group which has the property that its dimension appears to increase in a paradoxical way. The existence of such orbits was noted in a previous paper (1).
On the rank of horizontal maps
Authors: Rawnsley J. H.
Published: November 2008

Let M, N be smooth manifolds and ℋ ⊂ TN a smooth sub-bundle. A smooth map φ:M → N will be called horizontal if

At points where φ(M) is a submanifold of N, the tangent spaces to φ(M) will be sub-spaces of the fibres of ℋ. In other words where φ(M) is a submanifold it is an integral submanifold of ℋ. If ℋ is not integrable it will follow that the rank of Æ must be less than dim ℋx. But we can often place much stricter bounds on the rank of φ by examining the integrability tensor of ℋ. This we shall do in this note.
The order of the approximation to a Wiener process by its Fourier series
Authors: Hall Peter
Published: November 2008

LetFbe a distribution function with support confined to ( − π, π),Wa Wiener process on (0, 1), andW0a Brownian bridge. We examine the order of the approximation toW(F)andW0(F)by their Fourier series. The results we obtain are of theoretical interest since the Wiener process possesses the unusual property of satisfying a Lipschitz condition of order ∝ for each 0 < ∝ < ½, but not one of order ∝. = ½. The classical results on the rate of convergence of Fourier series are not sufficiently specific to provide an exact rate in the case of the Wiener process. The work also has practical significance because many statistical procedures can be closely approximated by functions of a Wiener process, and several procedures rely heavily on Fourier series methods. Thus, some of our results may be applied to obtain limit theorems for quantities such as trigonometric series estimates of density and distribution functions.
Homothetic ellipsoids
Authors: Goodey P. R.
Published: November 2008

We let K1 and K2 be two convex bodies in Ed (d ≥ 3) and consider translates of K2 for which ∂K1 ∩ ∂ ≠ ø and K1 ≠ . We show that if, for all such translates,∂K1 ∩ is contained in a hyperplane then K1 and K2 are homothetic ellipsoids.

If K1 and K2 are two planar non-coincident convex sets whose interiors intersect we define α(K1, K2) to be the number of connected components of ∂K1 ∩ ∂K2. The starting point for our investigations is the result of Fujiwara(S) and Bol(2) to the effect that K is a circular disc if and only if α(K, K′) = 2 for all congruent copies K′ of K. An easy extension of this result is the observation that K1 and K2 are congruent circular discs if and only if for all congruent copies of K2. When working with such results it is natural to look for analogues in the situation where only translates are allowed rather than all possible congruences. The appropriate analogue was obtained in (7) and we state it here for completeness.
Oriented percolation in dimensions d ≥ 4: bounds and asymptotic formulas
Authors: Cox J. Theodore , Durrett Richard
Published: November 2008

Let pc(d) be the critical probability for oriented percolation in ℤd and let μ(d) be the time constant for the first passage process based on the exponential distribution. In this paper we show that as d → ∞,dpc(d) and dμ,(d) → γ where γ is a constant in [e−1, 2−1] which we conjecture to be e−1. In the case of pc(d) we have made some progress toward obtaining an asymptotic expansion in powers of d−1. Our results show

The left hand side agrees, up to O(d−3), with a (nonrigorous) series expansion of Blease (1, 2):
Perturbation of a nest algebra module
Authors: Karanasios Sotirios
Published: November 2008

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.
Families of smooth hypersurfaces on certain compact homogeneous complex manifolds
Authors: Borcea Ciprian
Published: November 2008

Let X be a compact connected homogeneous complex manifold, which is Kāhlerian and has the second Betti number equal to one: b2(X) = 1; dimcX ≥ 3.

It is known that these conditions imply the following: X is a projective-rational homogeneous manifold (see (3)); X has an ‘algebraic cell-decomposition’: the 2s-dimensional closed cells are s-dimensional irreducible algebraic sets in X and they form a basis for the 2s-homology group of X, s = 1, 2, …, dimcX (see (1)); there are no holomorphic maps of X on lower dimensional (normal) analytic spaces except constants (see (9)).
Krull dimension of factor rings of the enveloping algebra of a semi-simple Lie algebra
Authors: Smith S. P.
Published: November 2008

Let g be a semi-simple complex Lie algebra with enveloping algebraU(g). It is shown that the Krull dimension ofU(g) is bounded above by dim g −r, whereris half the minimal dimension of a non-trivialGorbit in g (Gis the adjoint group of g).
Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson
Authors: Raka Madhu
Published: November 2008

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:

Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that
On the Arf invariant of links
Authors: Murasugi K.
Published: November 2008

Let Δl(x, y) be a Alexander polynomial of a link l of two components X and Y in S3. Denote by Arf (Z) the Arf invariant of Z, a knot or a proper link [9]
On the Jacobson radical of certain commutative semigroup algebras
Authors: Munn W. D.
Published: November 2008

In two previous papers the author studied the Jacobson and nil redicals of the algebra of a commutative semigroup over a field [8] and over a commutative ring with unity [9]. This work is continued here.
Spherical diagrams and equations over groups
Authors: Howie James
Published: November 2008

Spherical diagrams were introduced by Lyndon and Schupp[14] in order to study asphericity in group presentations and in 2-complexes. They have since been studied by several authors [2, 3, 5]. In particular, some technical loopholes in the original approach were closed in [5]. For many purposes the dual notion of pictures, introduced by Rourke[17], is more useful. These arise naturally through transversality. Pictures have also been studied and applied in [2, 5, 6, 12, 19].
A short elementary proof of the Bishop–Stone–Weierstrass theorem
Authors: Ransford T. J.
Published: November 2008

Fix the following notation. Let X be a compact Hausdorff space, and denote by C(X) the vector space of continuous complex-valued functions on X, equipped with the uniform norm ∥·∥x. Let A be a unital subalgebra of C(X). A non-empty subset S of X is said to be A-antisymmetric if whenever h ∈ A and h is real-valued on S then h is constant on S.
A new proof for the residual set dimension of the apollonian packing
Authors: Tricot Claude
Published: November 2008

Let (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.
Limit theorems for sums of general functions ofm-spacings
Authors: Hall Peter
Published: November 2008

Laws of large numbers and central limit theorems are proved for sums of general functions ofm-spacings from general distributions. Explicit formulae are given for the norming constants. The results enable us to describe asymptotic properties of distributional tests under fixed alternatives. A generalization of Kimball's spacings test is considered in detail.
A classification theorem for planar distributions based on the shape statistics of independent tetrads
Authors: Small Christopher G.
Published: November 2008

The uniformity of a certain shape density of a random ΠD tetrad relative to Lebesgue measure along part of the set corresponding to alignment is shown to be sufficient under certain technical conditions for the generating distribution to be the uniform distribution on a compact convex set. A counter-example is provided to show that this result fails for random IID triangles. The main theorem is used to characterize circular, elliptical and triangular uniform generators from the shape distribution of random IID tetrads.
Minimal projections in Lp-spaces
Authors: Halton E. J. , Light W. A.
Published: November 2008

Let X be a normed linear space and let W be a proper subspace of X. A projection is a surjective linear map P: X → W such that P is idempotent. It is immediately clear that P has norm at least unity. Thus the problem of calculating the number

has some interest. The number λ(W, X) is often called the relative projectiion constant of W in X. If the infimum is attained, any attaining projection is called a minimal projection. The problems of calculating λ(W, X) for a fixed X and W or finding a minimal projection turn out to be very dificult. For example, if X = C [0, 1] with the usual supremem norm and W is the subspace of polynominals of degree at most two then λ(W, X) remains unknown as does any example of a minimal projection. One of the few places where the problem shows much tractability is the case
On two summability methods
Authors: Bor Hüseyin
Published: November 2008

Let Σan be a given infinite series with partial sums sn, and rn = nan. By and we denote the nth Cesáro means of order α (α –1) of the sequences (sn) and (rn), respectively. The series Σan is said to be absolutely summable (C, a) with index k, or simply summable |C, α|k, k ≥ 1, if
Residual finiteness of certain 1-relator groups: extensions of results of Gilbert Baumslag
Authors: Allenby R. B. J. T. , Tang C. Y.
Published: November 2008

Let be the class of groups which can be presented in the form

where u, v are positive words on the generators g, h, …, k and where each generator appears in uv−1 with zero exponent sum. Let be the class of groups which can be presented in the form

where u, v are words (not necessarily positive) on the disjoint sets of generators c1, …, cm and d1, …, dn and where [u, v] = u−1v−1uv.
Growth conditions on subharmonic functions and resolvents of operators
Authors: Partington J. R.
Published: November 2008

Let X be a complex Banach space and T a bounded operator on X. The numerical range of T is defined by
Classification of five-dimensional naturally reductive spaces
Authors: Kowalski Oldřich , Vanhecke Lieven
Published: November 2008

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).
Finding fundamental units in algebraic number fields
Authors: Lettl Günter
Published: November 2008

Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].
Fourier coefficients of Eisenstein series on non-congruence subgroups
Authors: Scholl A. J.
Published: November 2008

Let Γ be a subgroup of PSL2 (ℤ) of finite index. In this note we are concerned with the arithmetic nature of the Fourier coefficients of holomorphic Eisenstein series on Γ.
Ramified coverings, orbit projections and symmetric powers
Authors: Dold Albrecht
Published: November 2008

L. Smith, in a recent paper [11], studied a class of maps X →Y which he called ramified coverings. Roughly speaking, these are maps with a multiple-valued inverse Y → SPdX; cf. 1·1. He showed that X → X/G is a ramified covering whenever a finite group G acts on X. Using results of [4] on infinite symmetric powers SP∞X of CW-complexes X he obtained transfer homomorphisms .
A note on the normal subgroups of mapping class groups
Authors: Long D. D.
Published: November 2008

0. If Fg is a closed, orientable surface of genus g, then the mapping class group of Fg is the group whose elements are orientation preserving self homeomorphisms of Fg modulo isotopy. We shall denote this group by Mg. Recall that a group is said to be linear if it admits a faithful representation as a group of matrices (where the entries for this purpose will be in some field).
An infinite family of non-Haken hyperbolic 3-manifolds with vanishing Whitehead groups
Authors: Nicas Andrew J.
Published: November 2008

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:

Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,

and its analogue in algebraic K-theory:

Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.
Non-splitting of prime divisors
Authors: Sally Judith D.
Published: November 2008

In this study of complete, or integrally closed, ideals in a two-dimensional regular local ring (R, m), Zariski established a one-to-one correspondence between prime divisors of R, i.e. rank 1 discrete valuations v birationally dominating R with residue field of transcendence degree 1 over R/m, and m-primary simple complete ideals Iv in R; cf. [17] and [18]. In this correspondence, the blow-up of such an ideal has unique exceptional prime and the localization at this prime is the valuation ring of a prime divisor of R. In this paper, we will study such ideals in a more general setting, so we begin by recalling some definitions and background results.
Bipartite completely positive matrices
Authors: Berman Abraham , Grone Robert
Published: November 2008

A non-zero n-by-n matrix A is said to be completely positive if there exist non-negative vectors b1,…, bk, such that

The smallest such integer k is called the factorization index of (completely positive) A, and is denoted by ø(A). Completely positive matrices are important in the study of block designs [4], statistics and modelling of energy demand [3].
A simple proof of Denjoy's theorem
Authors: MacKay R. S.
Published: November 2008

A simple proof of Denjoy's theorem on topological conjugacy of circle maps with irrational rotation number to uniform rotation is presented.
Almost convergence of double sequences and strong regularity of summability matrices
Authors: Móricz F. , Rhoades B. E.
Published: November 2008

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if

that is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].
Fourier coefficients of cusp forms
Authors: Rankin R. A.
Published: November 2008

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society'sProceedings[32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, theTransactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in theProceedings, it seems appropriate to include further developments in this field of study in the present survey.
Some criteria for nuclearity
Authors: Sofi M. A.
Published: November 2008

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.
Uniform bounds for the number of solutions to Yn = f(X)
Authors: Evertse J.-H. , Silverman J. H.
Published: November 2008

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the form

has attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Let

where a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples
The measure theory of random fractals
Authors: Taylor S. James
Published: November 2008

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Lévy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener[98]definition of mathematical Brownian motion. In fact the solution of that first problem in[81]used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.
Generalized inverses of matrices: a perspective of the work of Penrose
Authors: Ben-Israel A.
Published: November 2008

As a mathematical areageneralized inversion1was inaugurated in 1955 by R. Penrose [128]. Since then there have appeared about 2000 articles and 15 books2on generalized inverses of matrices and linear operators. See the annotated bibliography by Nashed and Rall [121] for the period up to 1976.
On a generalization of Barton's integral and related integrals of complete elliptic integrals
Authors: Bushell P. J.
Published: November 2008

Let K(k) and E(k) denote respectively the complete elliptic integrals of the first and second kind with modulus k, as defined by Byrd and Friedman ([5] 110·06 and 110·07), and let k′ = √(1 − k2), the complementary modulus.
Constructions of arcbodies
Authors: Lozano María Teresa
Published: November 2008

The concept of arcbody was introduced in [4] as a generalization of prime tangles used by Lickorish[3] and Bleiler[l]. The pair (A, l), where A is a compact 3-manifold with boundary ∂A, and l is a 1-submanifold properly embedded in A, is an arcbody if:

(i) the inclusion ∂ A – l ↪ A – l is monic, i.e. A – l has incompressible boundary;

(ii) no component of (A, l) is homeomorphic to (D2, 0) × I or (D2, 0) × S1; and

(iii) every sphere of ∂A intersects l.
On l1 subspaces of Orlicz vector-valued function spaces
Authors: Bombal Fernando
Published: November 2008

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.
A many-variable Landau-Kolmogorov inequality
Authors: Boyadzhiev Khristo N.
Published: November 2008

The Landau–Kolmogorov inequality

where ‖.‖ is the ‘sup’ norm, is well known and has many interesting applications and generalizations (see [1, 4–7, 13, 16]). Its study was initiated by Landau[10] and Hadamard [8] (the case n = 2). Kolmogorov [9] succeeded in finding in explicit form the best possible constants K(n, k) = Cn, k in (1) for functions on the whole real line R. The best constants for the half line R+ are not known in explicit form except for n = 2, 3, 4, but an algorithm exists for their computation (Schoenberg and Cavaretta [15]).
On the ranks of certain finite semigroups of transformations
Authors: Gomes Gracinda M. S. , Howie John M.
Published: November 2008

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations

generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which  = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.