We consider the shift-invariant space, 𝕊(Φ), generated by a set Φ = {Φ1,..., Φr} of compactly supported distributions on R when the vector of distributions ϕ:= {Φ1,..., Φr} T satisfies a system of refinement equations expressed in matrix form as

where a is a finitely supported sequence of r x r matrices of complex numbers. Such multiple refinable functions occur naturally in the study of multiple wavelets.

The purpose of the present paper is to characterize the accuracy of Φ, the order of the polynomial space contained in 𝕊(Φ), strictly in terms of the refinement mask a. The accuracy determines the Lp-approximation order of 𝕊(Φ) when the functions in (Φ) belong to Lp(ℝ) (see Jia [10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Heil, Strang and Strela [7], and of Plonka [16]. In addition, a counterexample is given to the statement of Strang and Strela [20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of Φ.

1. Introduction. A linearly ordered space S in which neighborhoods are segments is called a Souslin space if

(i) S is not separable, but

(ii) every collection of disjoint segments of S is countable.

Whether a Souslin space exists is not known; this is the problem referred to in the title and was proposed by Souslin in (2).

In this paper, the graphs under consideration may have multiple edges but they do not have loops. We enumerate the number N[H: n, p] of topologically distinct graphs with n vertices and p edges whose automorphism group is the permutation group H. As in (5), this enumeration is considered in the context of the theory of permutation representations of finite groups. We begin with some definitions and notation.

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.

In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equation

where a is analytic and |a| < 1, such that f(U) ⊂ D and

If K is the set of all compact bounded operators and L(H) is the set of all bounded operators on a separable Hilbert space H, then L(H) is the multiplier algebra of K. In general we denote the multiplier algebra of a C*-algebra A by M(A). For more information about M(A), readers are referred to the articles [1], [3],[7], [9], [14], [18],[20], [23], [26], [27], among others. It is well known that in the Calkin algebra L(H)/K every nonzero projection is infinite. If we assume that A is a-unital (nonunital) and regard the corona algebra M(A)/A as a generalized case of the Calkin algebra, is every nonzero projection in M(A)/A still infinite? Another basic question can be raised: How does the (closed) ideal structure of A relate to the (closed) ideal structure of M(A)/A?

In the first part of this note (Sections 1 and 2) we shall give an affirmative answer for the first question if A is a simple a-unital (nonunital) C*-algebra with FS.

With every connected graph G there is associated a metric space M(G) whose points are the vertices of the graph with the distance between two vertices a and b defined as zero if a = b or as the length of any shortest arc joining a and b if a ≠ b. A metric space M is called a graph metric space if there exists a graph G such that M = M (G), i.e., if there exists a graph G whose vertex set can be put in one-to-one correspondence with the points of M in such a way that the distance between every two points of M is equal to the distance between the corresponding vertices of G.

Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$, a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

Two families of associated Wilson polynomials are introduced. Both families are birth and death process polynomials, satisfying the same recurrence relation but having different initial conditions. Contiguous relations for generalized hypergeometric functions of the type 7F6 are derived and used to find explicit representations for the polynomials and to compute the corresponding continued fractions. The absolutely continuous components of the orthogonality measures of both families are computed. Generating functions are also given.

We prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue of Weyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

The distance between two vertices of a connected finite graph is the smallest number of edges forming a path that joins the two vertices, and the diameter of the graph is the largest integer which is realized as the distance between two vertices of the graph. We are concerned here with the diameters of two graphs associated with a d-dimensional convex polytope P (called henceforth a d-polytope). The graph Γ(P) of P is the 1- complex formed by the vertices and edges of P , and the polar graph II (P) of P is the 1-complex whose vertices correspond to the (d — 1)-faces of P, with two vertices joined by an edge in II (P) if and only if the corresponding (d — 1)- faces intersect in a (d — 2)-face of P. The diameters of Γ(P) and II (P) will be denoted respectively by δ(P) (called the diameter of P) and ϕ(Ps) (called the face-diameter of P ).

Groups in which the commutator subgroup coincides with the set of commutators have been studied to a certain extent by several authors. It was shown in (2; 4; 6; 7) that various types of known simple groups have this property. In (3), Macdonald has considered certain soluble groups with this property, and Hall has shown that any group can be embedded as a subgroup of a simple group of this type. Here we shall be concerned with the class C of groups defined as follows.

For any positive integer n, denote by Cn the class of all groups in which every element of the commutator subgroup can be expressed as a product of at most n commutators. It is not difficult to show that Cn is a proper subclass of Cn+1 for all n. Let so that a group G ∊ C if and only if G ∊ Cn for some n.

Although the notion of the maximal quotient ring of a nonsingular ring has been around for some time, not much is known about its structure in general beyond the important theorems of Johnson and Utumi [4; 11] that it is von Neumann regular and self-injective. The purpose of this paper is to study the structure of such a regular, self-injective ring R by looking at its prime ideals. Initially, we show that the primes of R separate into two types, called ‘'essential” and ‘“closed”, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, ${{L}^{1}}\left( G \right)$, and the Fourier algebra, $A\left( G \right)$, of a locally compact group $G$.

Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:

If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).

Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:

1. (i) K is supersolubly immersed in G.

2. (ii) K/ϕK is supersolubly immersed in G/ϕK.

3. (iii) If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.

Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.

Let be a completely regular Hausdorff space and C() the algebra of continuous real-valued functions on . In attempts to characterize abstractly those algebras that are isomorphic to C() for some , one produces subalgebras of C() which: (a) contain the constant functions, (b) separate points and closed sets in , (c) are closed under uniform convergence, and (d) are closed under inversion in C() (see, for example, (2; 5)).

Put

In a recent paper (4), Lohne showed that

When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and secondorder conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem's solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epi-derivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of "amenable" functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,

(xy)n = xnyn

for all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.

Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.

We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

A Blaschke product on the unit disc,

where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the condition

C

is satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product and

for some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metric

and its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜