In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.

We shall consider a multiparameter eigenvalue problem of the form

(1.1)

where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

Suppose throughout that {kn} is a sequence of positive integers, that

that k0 = 1 if l0 = 1, and that {un(r)}; (r = 0, 1, …, kn – 1, n = 0, 1, …) is a sequence of real numbers. We shall be concerned with the problem of establishing necessary and sufficient conditions for there to be a function a satisfying

(1)

and certain additional conditions. The case l0 = 0, kn = 1 for n = 0, 1, … of the problem is the version of the classical moment problem considered originally by Hausdorff [5], [6], [7]; the above formulation will emerge as a natural generalization thereof.

In his work on the Dirichlet problem for the Heisenberg group Greiner [5] showed that each Lα-spherical harmonic is a unique linear combination of functions of the form

with k = 0, 1,2, … and n = 0, ±l, ±2 , …, where Hk(α, n)(θiθ) is defined by the generating function

Since Hk(0,0)(eiθ) = Pk(cos θ), where Pk(x) is the Legendre polynomial of degree k, and these functions satisfy the orthogonality relation

Greiner raised the question of whether the functions Hk(0,0)(eiθ) are orthogonal or biorthogonal with respect to some complex valued weight function.

Consider an arbitrary set X and an arbitrary disjunctive lattice of subsets of X, ℒ. The algebra of subsets of X generated by ℒ is denoted by , the set of all ℒ-regular measures on , by MR(ℒ), and the associated Wallman space, a compact T1 space, by IR(ℒ); assume X is embedded in IR(ℒ) (otherwise, consider the image of X in IR(ℒ)).

In part of an earlier paper [4] the work of Knowles [15] and Gould and Mahowald [11] was generalized from the explicit topological setting of X, a Tychonoff space, with ℒ the lattice of zero sets of X, to the above setting, with the added assumption that ℒ was also δ and normal. This was done so that the important Alexandroff Representation Theorem [1] could be utilized in order to induce two associated measures , and defined on and respectively, where W(ℒ) is the Wallman lattice in IR(ℒ).

The problem of linearizing the product of two Jacobi polynomials, Pm(α, β)(x)Pn(α, β)(x), and to establish the conditions for the non-negativity of the coefficients has been of considerable interest for many years. Explicit non-negative representations were sought and found by many authors [7, 8, 13, 14], but only in the special case α = β, although Hylleraas [14] succeeded in finding a formula in another case α = β + 1. Gasper [9, 10] found the necessary and sufficient conditions for the non-negativity of the linearization coefficients by exploiting a recurrence relation obtained by Hylleraas for the above-mentioned product. Koornwinder [16] approached the same problem from a different point of view and managed to find a non-negative integral expression to these coefficients when . However, an exact formula in a hypergeometric series form for general α, β has been very elusive so far, in spite of the fact that all computation of special cases seemed to indicate that such a formula should exist.

The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua.

The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these mappings were invented for quite another purpose [1]. The author believes that these maps are likely to provide even more new information about homogeneous continua.

A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h: M → M such that h(p) = q.

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l by

where y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfying

with inner product

where . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.

Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.

In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

(R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.

Let ψ0, ψ1, … denote the Walsh-Paley functions and let ∔ denote the group operation which Fine [5] defined on the interval [0, 1). Thus, if k ≧ 0 is an integer and if u, t are points in the interval [0, 1) then

(where αk = 0 or 1 represents the kth coefficient of the binary expansion of t), and

A real-valued function ƒ, is said to be dyadically differentiable at a point x ∈ [0, 1) if ƒ is defined at x and at x ∔ 2–n–1, n = 0, 1, …;, and if the sequence

(1)

converges as N → ∞.

Find exact or asymptotic formulae for the number of labelled graphs of order n having a certain property. The property we are interested in in this note is that of being k-coloured and having connectivity at least l. Special cases of this problem have been tackled by many authors; in particular Gilbert [6], Read [9] and Robinson [11] found exact formulae, and Read and Wright [10], and Wright [12], [13] found asymptotic expressions (for many other examples see [7]). Recently Harary and Robinson [8] counted labelled bipartite blocks, that is 2-connected bipartite graphs. (For terms not defined here and general background in graph theory see [1].) Our present investigations have been prompted by [8]; in particular, as a very special case of our results, we shall prove the conjecture published in [8].

The exact formulae appearing in the enumeration of labelled graphs in general, and in the enumeration of k-coloured labelled graphs in particular, tend to be very pleasing, especially because of the functional equations relating them.

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.

A t-cap in a geometry is a set of t points no three of which are collinear. A (t, k)-cap is a set of t points, no k + 1 of which are collinear.

It has been shown in [3] that any Desarguesian PG(2n, q2) is a disjoint union of (q2n+l – l)/(q – 1) (q2n+l – l)/(q + l)-caps. These caps were obtained as intersections of 2n Hermitian Varieties of a certain kind; the intersection of 2n + 1 such varieties was empty. Furthermore, the caps in question constituted the ‘large points” of a PG(2n, q), with the incidence relation defined in a natural way.

It seemed at the time that nothing similar could be said about odd-dimensional projective geometries, if only because |PG(2n – 1, q)| ∤ |PG(2n – l, q2)|.