Let G be a group of automorphisms of a commutative ring K, and let KG denote the Galois subring consisting of all elements left fixed by every g in G. An ideal M is G-stable, or G-invariant, provided that g(x) lies in M for every x in M, that is, g(M) ⊆ M, for every g in G. Then, every g in G induces an automorphism in the residue ring , and if is the group consisting of all , trivially

1

When the inclusion (1) is strict, then G is said to be cleft at M, or by M, and otherwise G is uncleft at (by) M. When G is cleft at all ideals except 0, then G is cleft, and uncleft otherwise.

The purpose of this paper is to produce explicit realizations of supercuspidal representations of Sp4(k) where k is a p-adic field with odd residual characteristic. These representations will be constructed using the Weil representation of Sp4(k) associated with a certain 4-dimensional compact orthogonal group OQ over k. The main problem addressed in this paper is the analysis of this representation; we need to find how the supercuspidal summands decompose into irreducible pieces.

The problem of decomposing Weil representations has been studied quite a bit already. The Weil representations of SL2(k) associated to 2-dimensional orthogonal groups were used by Casselman [4] and Shalika [9] to produce all supercuspidals of SL2(k). The explicit formulas for these representations were used by Sally and Shalika ([10]) to compute the characters and finally to write down a Plancherel formula for that group.

A great part of mathematical analysis relies directly on the methods of separation of variables and on the successive reduction of several variables problems to one-dimensional equations and to the theory of classical special functions; for example, the theory of elliptic or parabolic equations with regular coefficients (even with non constant coefficients) can be done because we know explicitly the fundamental solutions of the Laplace operator or of the heat equation; these fundamental solutions are functions of one variable; pseudodifferential or parametrices methods are thus basically small perturbations of an explicitly known problem in one variable.

On the other hand, there are many problems which are not of this type: they are related to the questions of operators with singular coefficients and to the global behaviour of the solutions; in that case, the local model cannot be reduced to a one variable problem but is fundamentally a several variables problem which cannot be treated in a detailed way by one variable methods or perturbation analysis of a one variable problem.

Besides the simple ones, there are several other kinds of C*-algebras which it has proved interesting to try to classify. For instance, a large body of results relates to the extensions of one given C*-algebra, possibly simple, by another. Extrapolating in this direction, we have considered the class of C*-algebras which can be decomposed in the strongest possible nontrivial sense in terms of their simple subquotients, and such that these simple subquotients in turn are as uncomplicated as possible.

We have found that the classification of these C*-algebras, namely, the lexicographic direct sums of elementary C*-algebras, is to a large degree tractable, and yet involves an interesting new invariant in the antiliminary case, which is the case of no minimal ideals. Even the postliminary case, which is the case that the ordered set of simple subquotients satisfies the decreasing chain condition, is not without interest as an extension of the case of finitely many simple subquotients, analysed in the earlier papers [1] and [2].

Let the function

1

be analytic in the unit disk Δ = {z│ │z│ ≤ 1), with

there, and let α be a real number. Then f(z) is said to be α-convex in Δ if and only if the inequality

holds in Δ. The class of α-convex functions was introduced in [8] and was studied in detail in the series [5]–[10], where in particular it is shown that α-convex functions are univalent and starlike for all α (−∞≦ α ≦ + ∞), that is, the inequality

holds in Δ.

Comma-free codes were first introduced in [1] in 1957 as a possible genetic coding scheme for protein synthesis. The general mathematical setting of such codes was presented in [3], and the biochemical and mathematical aspects of the problem were later summarized and extended in [4].

Using the notation of [3], a set D of k-tuples or k-letter words, (a1a2 … ak), where

for fixed positive integers k and n, is said to be a comma-free dictionary if and only if, whenever (a1a2 … ak) and (b1b2 … bk) are in D, the “overlaps”

are not in D. This precludes codewords having a subperiod less than k; and two codewords which are cyclic permutations of one another cannot both be in D.

Let (Q, +, ·) be a finite quasifield of dimension d over its kernel K = GF(q), where q = pk with p a prime and k ≧ 1. (See p. 18-22 and p. 74 of [7] or Section 5 of [9] for the definition of quasifield.) For the remainder of this article we will follow standard conventions and omit, whenever possible, the binary operations + and · in discussing a quasifield. For example, the notation Q will be used in place of the triple (Q, +, ·) and Q* will be used to represent the multiplicative loop (Q − {0}, ·).

Let m be a non-zero element of the quasifield Q; the right multiplicative mapping ρm:Q → Q is defined by

1

The motivation for this paper was the work of Thurston and Jørgensen on volumes of hyperbolic 3-manifolds. They prove, among other things, that the set of all volumes of complete hyperbolic 3-manifolds is well-ordered. In particular, there is a hyperbolic 3-manifold which has minimum volume among all complete hyperbolic 3-manifolds. Further, there is a minimum volume member in the collection of complete hyperbolic 3-manifolds with one cusp; and similarly for n cusps. Computer studies to date show that the manifold obtained by performing (5,1) Dehn surgery on the figure-eight knot in the 3-sphere is the leading candidate for the minimum volume hyperbolic 3-manifold. Its volume is about 0.98. The leading one-cusp minimum volume candidate is the figure-eight knot complement in the 3-sphere. Its volume is about 2.03.

The class of univalent close-to-convex functions, K, was introduced by Kaplan [4] and first studied by him. The first important extension to the class of multivalent close-to-convex functions, K(p) where p is a positive integer, was considered by Livingston [7]. Somewhat later, Styer [15] introduced the more general class, Kw(p), of weakly close-to-convex functions by simply taking the closure of Livingston's class K(p) in the topology of locally uniform convergence in B = {z: |z| ≤ 1}.

In 1936 Biernacki [2] introduced his class of linearly accessible functions. A function f is linearly accessible if f is univalent in B, f(0) = 0, and C – f(B) where C is the complex plane, is a union of closed (Euclidean) rays with disjoint interiors. In an interesting result, Lewandowski [6] showed that the classes of univalent close-to-convex functions and linearly accessible functions are equal.

Let be a nest algebra of operators on some Hilbert space H. Weakly closed -modules were first studied by J. Erdos and S. Power in [4]. It became apparent that many interesting classes of non self-adjoint operator algebras arise as just such a module. This paper undertakes a systematic investigation of the correspondence which arises between such modules and order homomorphisms from Lat into itself. This perspective provides a basis to answer some open questions arising from [4]. In particular, the questions concerning unique “determination” and characterization of maximal and minimal elements under this correspondence, are resolved. This is then used to establish when the determining homomorphism is unique.

Throughout this paper, we work in the PL and pseudosimplicial categories, for which we refer to [17] and [10] respectively. For the graph theory involved see [9].

An h-coloured graph (Γ, γ) is a multigraph Γ = (V(Γ), E(Γ)) regular of degree h, endowed with an edge-coloration γ by h colours. If is the colour set, for each we set

For each set . For n ∊ Z, n ≧ 1, set

Δn will be mostly used to denote the colour set for an (n + 1)-coloured graph.

Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a categorical context [4]. They have also shown that if the set of morphisms is saturated then the Adams completion of an object is characterized by a certain couniversai property. We want to prove a stronger version of this result by dropping the saturation assumption on the set of morphisms; we also prove that the canonical map from an object to its Adams completion is an element of the set of morphisms under very moderate assumptions. These two results are fairly general in nature and are applicable to most cases of interest. Further using these two results and introducing “modulo a Serre class C of abelian groups” [9] we have obtained the mod-C Postnikov approximation of a 1-connected based CW-complex, with the help of a suitable set of morphisms.

This note is about the interplay between two classes of radical Banach algebras, and we begin by describing the algebras in question.

A weight sequence is a positive sequence w = (wn) defined on Z+ (the non-negative integers) and satisfying w0 = 1 and wm+n ≦ wmwn for all m and n in Z+. For such a sequence w, the Banach space

is a Banach algebra with respect to the convolution product, defined by

In 1924 at the Toronto meeting of the International Congress of Mathematicians, B. N. Delone introduced his empty sphere method for lattices. We have titled our paper after this method as a tribute to his memory.

We have studied the sets of integer solutions of equations of the form

1

where f satisfies the following condition in which Z denotes the integers,

2

and have resolved this problem using the theory of L-types of lattices [3, 4, 11]. We have been able to give a complete description of all such integer solutions when n ≦ 4.

The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.

Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.

For a graph G, let V(G) denote its vertex set and E(G) its edge set. Here we shall only consider reflexive graphs, that is graphs in which every vertex is adjacent to itself. These adjacencies, i.e., the loops, will not be depicted in the figures, although we always assume them present. For graphs G and H, an edge-preserving map (or homomorphism) of G to H is a mapping of V(G) to V(H) such that f(g) is adjacent to f(g′) in H whenever g is adjacent to g′ in G. Because our graphs are reflexive, an edge-preserving map can identify adjacent vertices, i.e., possibly f(g) = f(g′) for some g adjacent to g′, cf. Figure 1(a).

Several authors have considered eigenvalue problems for differential equations where the eigenvalue parameter also appears in the boundary conditions. Such problems do not appear to arise from any spectral problem associated with a linear operator on a Hilbert space. However, it is possible to reset such problems in this context. This has been done for certain second order cases by Walter [4] using a special measure on the interval in question, and by Fulton [1, 2] using the type of space indicated in the title of this article.

It is our purpose here to consider a general class of operators on L2(I) ⊕ Cr, which are based on a differential expression τ of order n on I. We shall first investigate adjoints, boundary conditions, and self-adjointness for such operators. We shall then show that all eigenvalue problems of the form τy = λy, with boundary conditions which involve λ in a linear fashion, can be reset in the context of such operators.

A single-valued function f(z) is said to be univalent in a domain if it never takes on the same value twice, that is, if f(z1) = f(z2) for implies that z1 = z2. A set is said to be starlike with respect to the line segment joining w0 to every other point lies entirely in . If a function f(z) maps onto a domain that is starlike with respect to w0, then f(z) is said to be starlike with respect to w0. In particular, if w0 is the origin, then we say that f(z) is a starlike function. Further, a set is said to be convex if the line segment joining any two points of lies entirely in . If a function f(z) maps onto a convex domain, then we say that f(z) is a convex function in .

One of the common uses of summability theory is found in its applications to power series. A partial listing such as [l]-[5], [13], and [15]-[17] might serve to remind us of the many instances of summability theory applied to power series. In some studies, the summability transformation is applied to the sequence of partial sums of the power series, while in others it is applied to the general term akzk as a series-to-series transformation. In [2] and [5] the transformation is applied to the coefficient sequence of the Taylor series. In the present study we investigate matrix transformations that are applied to the sequence {akzk), but we are not concerned with the usual preservation of convergence or sums. At a point within its disc of convergence, a power series exhibits more than ordinary convergence; it converges very rapidly, being dominated by a convergent geometric series.

In [4] N. Steinmetz used Nevanlinna theory to establish remarkably versatile theorems on the factorization of ordinary differential equations which implied numerous previous results of various authors. (Here factorization is taken in the sense of function composition as introduced by F. Gross in [2].) The thrust of Steinmetz’ central results on factorization is that if g(z) is entire and f(z) is meromorphic in C such that the composite fog satisfies an algebraic differential equation, then so do f(z) and, degenerate cases aside, g(z). In addition, the more one knows about the equation for fog (e.g. degree, weight, autonomy), the more one can conclude about the equations for f and g.

In this note we generalize Steinmetz’ work to show the following:

• a) Steinmetz’ two basic results, Satz 1 and Korollar 1 of [4] can be seen as one-variable specializations of a single two variable result, and

• b) the function g(z) can itself be allowed to be a function of several variables.