The Laguerre polynomials Ln(α)(x) have the following hypergeometric and Rodrigues representation (3, pp. 188, 189):

1,

2.

By a 3-dimensional handlebody Vn of genus n, we mean an oriented 3-manifold which is a disk-sum of n copies of D2 X S1, where D2 is the unit disk and S1 is the boundary ∂D2; and by a surface Fn of genus n we mean the oriented boundary surface ∂ Vn.

The present paper is a sequel to the previous paper bearing the same title by the same authors [3] and which will be hereafter designated by the bold-face Roman numeral I. Further results are obtained in determining whether a given finite non-abelian group G has a graphical regular representation. In particular, an affirmative answer will be given if (|G|, 6) = 1.

Inasmuch as much of the machinery of I will be required in the proofs to be presented and a perusal of I is strongly recommended to set the stage and provide motivation for this paper, an independent and redundant introduction will be omitted in the interest of economy.

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere ${{\mathbb{S}}^{4}}$ by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's ${{\tau }_{3,1}}$-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for ${{\lambda }_{1}}$ on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture.

This paper considers Hilbert C*-bimodules, a slight generalization of imprimitivity bimodules which were introduced by Rieffel [20]. Brown, Green, and Rieffel [7] showed that every imprimitivity bimodule X can be embedded into a certain C*-algebra L, called the linking algebra of X. We consider arbitrary embeddings of Hilbert C*-bimodules into C*-algebras; i.e. we describe the relative position of two arbitrary hereditary C*-algebras of a C*-algebra, in an analogy with Dixmier's description [10] of the relative position of two subspaces of a Hilbert space.

The main result of this paper (Theorem 4.3) is taken from the doctoral dissertation of the third author [22], although the proof here follows a different approach. In Section 1 we set out the definitions and basic properties (mostly folklore) of Hilbert C*-bimodules. In Section 2 we show how every quasi-multiplier gives rise to an embedding of a bimodule. In Section 3 we show that , the enveloping C*-algebra of the C*-algebraA with its product perturbed by a positive quasi-multiplier , is isomorphic to the closure (Proposition 3.1). Section 4 contains the main theorem (4.3), and in Section 5 we explain the analogy with the relative position of two subspaces of a Hilbert spaces and present some complements.

In this paper, we find equations that characterize locally projectively flat Finsler metrics in the form $F\,=\,{{(\alpha \,+\,\beta )}^{2}}/\alpha $ where $\alpha \,=\,\sqrt{{{a}_{ij}}{{y}^{i}}{{y}^{j}}}$ is a Riemannian metric and $\beta \,=\,{{b}_{i}}{{y}^{i}}$ is a 1-form. Then we completely determine the local structure of those with constant flag curvature.

A classical theorem of J. Liouville states that if z is a real algebraic number of degree n ≥ 2, then there exists a constant c > 0 such that for every pair of integers a, b with b ≠ 0.

In the development of any mathematical theory it is often advisable to test the known theoretical results in particular cases, for in this way we are able to judge the state of the subject, its strengths and its weaknesses. It is the purpose of this paper to put forward three remarkable graphs which can be used to indicate the progress being made in research in graph theory. We shall undertake three tasks - the construction of the graphs themselves, the investigation of their properties (insofar as the theory allows), and the indication of possible lines for further enquiry.

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)$. The periodic ${{\text{C}}^{*}}$-algebras are characterized, and it is shown that ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)\,\simeq \,\text{C}\left( \mathbb{T} \right)\,\otimes \,\mathfrak{U}$ where $\mathfrak{A}$ is a simple ${{\text{C}}^{*}}$-algebra.

Eigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and completed by Coxeter (6; 7, p. 141). In addition Coxeter enumerated the regular maps on the simplest non-orientable surfaces (7, pp. 116, 139), and constructed tables of some interesting families of regular maps (3; 7, p. 140).

Most of the regular maps on a surface of genus 3 have appeared in these papers, but no systematic enumeration of them seems to have been attempted. The ultimate goal of this paper is a complete list of these regular maps. However, the families of maps {j.p,q}and {j.p,j.q} which are defined in § 4 and listed in Tables I and II are of considerable interest in themselves. Also of some importance is the complete list of regular maps of type {p,3} with six or fewer faces (§ 5 and Table III).

We study the cardinal invariants of analytic $P$-ideals, concentrating on the ideal $Z$ of asymptotic density zero. Among other results we prove $\min \,\{\mathfrak{b},\,\operatorname{cov}(\mathcal{N})\,\}\,\le \,\operatorname{cov}*\,(Z)\,\le \,\max \{\mathfrak{b},\,\text{non(}\mathcal{N}\text{)}\}$.

Let X be a Banach space such that X* is a von Neumann algebra. We prove that X has the analytic Radon-Nikodym property (in short: ARNP). More precisely we show that for any function ƒ in H1(X) we have This implies the ARNP for X as well as for all the Banach spaces which are finitely representable in X. The proof uses a C*-algebraic formulation of the classical factorization theorems for matrix valued H1-functions. As a corollary we prove (for instance) that if A ⊂ B is a C*-subalgebra of a C*-algebra B, then every operator from A into H∞ extends to an operator from B into H∞ with the same norm. We include some remarks on the ARNP in connection with the complex interpolation method.

We study the singular integral operator

$${{T}_{\Omega ,\alpha }}f\left( x \right)\,=\,\text{p}\text{.v}\text{.}\,{{\int }_{{{R}^{n}}}}\,b\left( \left| y \right| \right)\Omega \left( {{y}'} \right){{\left| y \right|}^{-n-\alpha }}\,f\left( x\,-\,y \right)\,dy,$$

defined on all test functions $f$, where $b$ is a bounded function, $\alpha \ge 0,\,\Omega \left( {{y}'} \right)$ is an integrable function on the unit sphere ${{S}^{n-1}}$ satisfying certain cancellation conditions. We prove that, for $1\,<\,p\,<\infty$, ${{T}_{\Omega ,\alpha }}$ extends to a bounded operator from the Sobolev space $L_{\alpha }^{p}$ to the Lebesgue space ${{L}^{p}}$ with $\Omega$ being a distribution in the Hardy space ${{H}^{q}}\left( {{S}^{n-1}} \right)$ where $q=\frac{n-1}{n-1+\alpha }$. The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for ${{T}_{\Omega ,\alpha }}$ on the Hardy spaces, as well as the boundedness for the truncated maximal operator $T_{\Omega ,m}^{*}$.

The definition of the genus γ(G) of a graph G is very well known (König 2): it is the minimum genus among all orientable surfaces in which G can be drawn without intersections of its edges. But there are very few graphs whose genus is known. The purpose of this note is to answer this question for one family of graphs by determining the genus of the n-cube.

The graph Qn called the n-cube has 2n vertices each of which is a binary sequence a1a2. . . an of length n, where ai = 0 or 1.

We prove that pre-classifiable (see 3.1) simple nuclear tracially $\text{AF}\,\,{{C}^{*}}$-algebras $\left( \text{TAF} \right)$ are classified by their $K$-theory. As a consequence all simple, locally $\text{AH}$ and $\text{TAF}\,\,\,{{C}^{*}}$-algebras are in fact $\text{AH}$ algebras (it is known that there are locally $\text{AH}$ algebras that are not $\text{AH}$). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and ${{K}_{0}}{{\left( A \right)}_{+}}$ is locally finitely generated, then $A\,\otimes \,Q\,\cong \,B\,\otimes \,Q$, where $Q$ is the $\text{UHF}$-algebra with the rational ${{K}_{0}}$. Classification results (with restriction on ${{K}_{0}}$ - theory) for the above ${{C}^{*}}$-algebras are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with the unique normalized trace satisfying the $\text{UCT}$ and with ${{K}_{1}}\left( A \right)\,=\,{{K}_{1}}\left( B \right)$, and $A$ and $B$ have the same rational (scaled ordered) ${{K}_{0}}$, then $A\,\cong \,B$. Similar results are also obtained for some cases in which ${{K}_{0}}$ is non-divisible such as ${{K}_{0}}\left( A \right)\,=\,\mathbf{Z}\left[ 1/2 \right]$.

In [2], [8] and [10], Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi [6] noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski [4] observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear).

2. Preliminaries. Horowitz [3] first defined the weighted Bergman space Ap,α(0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy

(1)

If a < u < b and n > 0 then

(1)

is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely

(2).

Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.

This paper is primarily a brief elaboration on the axioms for a bicategory introduced in (3). From this point of view, the main aim is the development of the structure of certain systems of topological and uniform spaces, and the present paper merely points out some very general properties which follow from axioms so weak that they are satisfied by any system likely to be considered.

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,

for a connected graph G of n vertices and m edges, where

and ﹛x﹜ denotes the smallest integer ≧ x.

We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.

Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.