Krall and Frink [4] aroused interest in what they term Bessel polynomials. They studied in some detail what may, in hypergeometric form, be written as

Let L be a σ-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a “dilatator operator on L.” More specifically, he shows that the set of all dilatator operators , when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Hausdorff space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of . Moreover, when L is Dedekind complete, it is an ideal in , and the topological space is extremally disconnected.

1. Introduction. The purpose of this note is to establish some limit theorems for the non-linear recurrence relations

1.1, i= 1,2, …, N; n ≥ 0

under certain assumptions concerning the initial values ci = xi(0), and the coefficient matrices A (q) = (aij(q)).

In an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-number λ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the case λ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations.

In differential analysis, it is very useful to have the local behavior of a differentiable map be determined by the derivatives of the map at a point. Hence we have the theories of finite and infinitely determined germs. Let mnp be the space of germs of C∞ maps f: (Rn, 0) → (Rp, 0) and G a group operating on mnp. A germ f is called finitely G-determined if there exists an integer k such that every germ having the same k-jet as f is G-equivalent to (i.e., in the same G-orbit as) f. A germ f is called ∞-G-determined if every germ having the same formal power series as f is G-equivalent to f.

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that

1.1

Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

We prove a number of results about the stable and particularly the real ranks of tensor products of ${{C}^{*}}$ -algebras under the assumption that one of the factors is commutative. In particular, we prove the following:

1. (1) If $X$ is any locally compact $\sigma $-compact Hausdorff space and $A$ is any ${{C}^{*}}$-algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le \text{dim(}X\text{)+RR(}A\text{)}$ .

2. (2) If $X$ is any locally compact Hausdorff space and $A$ is any purely infinite simple ${{C}^{*}}$ -algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le 1$ .

3. (3) $\text{RR(}C([0,\,1]\,)\otimes \,A)\,\ge \,1$ for any nonzero ${{C}^{*}}$-algebra $A$, and $\text{sr(}C({{[0,\,1]}^{2}})\,\otimes \,A\text{)}\,\ge \,2$ for any unital ${{C}^{*}}$-algebra $A$.

4. (4) If $A$ is a unital ${{C}^{*}}$-algebra such that $\text{RR(}A\text{)}\,\text{=}\,\text{0,}\,\text{s}r\text{(}A\text{)}\,\text{=}\,\text{1}$, and ${{K}_{1}}(A)=0$ , then $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

5. (5) There is a simple separable unital nuclear ${{C}^{*}}$-algebra $A$ such that $\text{RR(}A\text{)}\,\text{=}\,\text{1}$ and $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the $n$-centre of that unit group. The $n$-centre of the unit group is either the centre or the second centre (for $n\,\ge \,2$).

In this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

We let T2 be the torus that is the cartesian product of 2 unit circles in C. The usual Lebesgue spaces, with respect to the Haar measure m of T2, are denoted by Lp = Lp(T2), and Hp = Hp(T2) is the space of all f in LP whose Fourier coefficients

are 0 as soon as at least one component of (j, ℓ) is negative.

A closed subspace M of L2 is said to be invariant if

Whenever this is the case, it follows that fM ⊂ M for every f in H∞. One can ask for a classification or an explicit description (in some sense) of all invariant subspaces of L2, but this seems out of reach.

This paper is a continuation of the Waterloo Research Report CORR 81-12, (see [1]) referred to in what follows as I. That Report is entitled “Chromatic Solutions”. It is largely concerned with a power series h in a variable z2, in which the coefficients are polynomials in a “colour number” λ. By definition the coefficient of z2r, where r > 0, is the sum of the chromatic polynomials of the rooted planar triangulations of 2r faces. (Multiple joins are allowed in these triangulations.) Thus for a positive integral λ the coefficient is the number of λ-coloured rooted planar triangulations of 2r faces. The use of the symbol z2 instead of a simple letter t is for the sake of continuity with earlier papers.

In I we consider the case

(1)

where n is an integer exceeding 4.

If ${{\Phi }_{\lambda }}$ is a master function corresponding to a hyperplane arrangement $\mathcal{A}$ and a collection of weights $\lambda $, we investigate the relationship between the critical set of ${{\Phi }_{\lambda }}$, the variety defined by the vanishing of the one-form ${{\omega }_{\lambda }}=\text{d}\log {{\Phi }_{\lambda }}$, and the resonance of $\lambda $. For arrangements satisfying certain conditions, we show that if $\lambda $ is resonant in dimension $p$, then the critical set of ${{\Phi }_{\lambda }}$ has codimension at most $p$. These include all free arrangements and all rank 3 arrangements.

The genus parameter for graphs has been studied extensively in recent years, with impetus given primarily by the Ringel-Youngs solution to the Heawood Map-coloring Problem [15]. This solution involved the determination of 𝛄(Kn), the genus of the complete graph Kn.

Let $\Gamma$ be a torsion free lattice in $G=\text{PGL}\left( 3,\mathbb{F} \right)$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $B$ of $G$ and there is an induced action on the boundary $\Omega$ of $B$. The crossed product ${{C}^{*}}$ -algebra $\mathcal{A}\left( \Gamma \right)=C\left( \Omega \right)\rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}\left( \Gamma \right)$ and of the larger class of rank two Cuntz-Krieger algebras.

If n is a non-negative integer, define pr(n) by otherwise define pr(n) as 0. (Here and in what follows all sums will be extended from 0 to ∞ and all products from 1 to ∞ unless otherwise stated.) pr(n) is thus generated by the powers of ,

We construct a generalization of Courant algebroids that are classified by the third cohomology group ${{H}^{3}}(A,\,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and ${{\text{ }\!\!\varepsilon\!\!\text{ }}^{1}}(M)$ structures are examples of them. Finally we introduce generalized $\text{CR}$ structures on a manifold, which are a generalization of generalized complex structures, and show that every $\text{CR}$ structure and contact structure is an example of a generalized $\text{CR}$ structure.

If a convex polytope P is centrally symmetric, and has the property that all its faces (of every dimension) are centrally symmetric, then P is called a zonotope. Zonotopes have many interesting properties which have been investigated by Coxeter and other authors (see (4, §2.8 and §13.8) and (5) which contains a useful bibliography). In particular, it is known (5, §3) that a zonotope is completely characterized by the fact that all its two-dimensional faces are centrally symmetric.

Some oscillation criteria are given for the second order neutral delay differential equation

where τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $\left( X,\,\sum ,\,\mu \right)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group ${{L}^{\infty }}\left( X,G \right)$, which is a Fréchet-Lie group if $G$ is so. We also show that the weak direct product $\prod{_{i\in I}^{*}{{G}_{i}}}$ of an arbitrary family ${{\left( {{G}_{i}} \right)}_{i\in I}}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum ${{\oplus }_{i\in I}}L\left( {{G}_{i}} \right)$ .

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.