In [8] we classified degree one maps denned on Sp × Sg × Sr. In this paper we shall study degree one maps defined on the n-dimensional torus T = Sl × Sl × … × S1 as well as certain general properties of degree one maps. Theorem 1.1 must be known to experts; however we could not find it in the literature. Theorem 1.5 b) says that a Poincaré complex is nilpotent if it admits a degree one map from another nilpotent Poincaré complex. Theorem 1.5 a) means that certain stable properties are preserved by degree one maps and we use it later in Section 2.

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].

This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

There has recently been some interest in a class of limit points for Fuchsian groups now known as Garnett points [5], [8]. In this paper we show that such points are intimately connected with the structure of Dirichlet regions and the same ideas serve to show that the Ford and Dirichlet regions are merely examples of one single construction which also yields fundamental regions based at limit points (and which properly lies in the subject of inversive geometry). We examine in the general case how the region varies continuously with the construction. Finally, we consider the linear measure of the set of Garnett points.

2. Hyperbolic space. Let Δ be any open disc (or half-plane) in the extended complex plane C∞: usually Δ will be the unit disc or the upper half-plane. We may regard Δ as the hyperbolic plane in the usual way and the conformai isometries of Δ are simply the Moebius transformations of Δ onto itself.

In a recent paper [7] Longuet-Higgins and Parry prove that, given a general Clifford configuration of degree 5 (abbreviated to CL5), C0 say, there exist points P and Q such that the inverses of P in the circles of C0 form the points of another CL5C1, whilst the inverses of Q in the circles of C1 are the points of C0; also the inverses of Q in the circles of C0 form the points of a CL5 C–1, whilst the inverses of P in the circles of C–1 are the points of C0. This leads to an infinite chain …, C–2, C–1, C0, C1, C2, … of CL5s, each connected to the next by means of the same two points P and Q, called the poles of the chain.

The fact that Rolle's theorem on critical points of a real differentiable function does not hold in general for analytic functions of a complex variable raises a natural question [7, p. 21] as to whether or not it can be generalized to polynomials, the simplest subclass of analytic functions. While attempting to answer this and other related problems in [4, 5, 6], Lucas proved that all the critical points of a nonconstant polynomial f lie in the convex hull of the set of zeros of f (see Theorem (6.1) in [7]). Walsh [12] has shown that Lucas’ theorem is equivalent to the following result [7, Theorem (6.2)], namely: Any convex circular region which contains all the zeros of a polynomial f also contains all the zeros of its derivative f′.

Rappelons que l'opérateur de Stokes associe au couple vitesse-pression (, p) le couple force-divergence (, g) par:

Nous nous plaçons ici en dimension 2. . Voici l'écriture matricielle de l'opérateur:

ce que nous noterons: .

Ce système est elliptique au sens de [1] sur un domaine Ω avec les conditions complémentaires: (Dirichlet); c'est donc ce probléme que nous étudions ici.

1. An n-tuple S = (S1, …, Sn) of commuting bounded linear operators on a Hilbert space H is said to have commuting normal extension if and only if there exists an n-tuple N = (N1, …, Nn) of commuting normal operators on some larger Hilbert space K ⊃ H with the restrictions Ni|H = Si, i = 1, …, n. If we take

the minimal reducing subspace of N containing H, then N is unique up to unitary equivalence and is called the c.n.e. of S. (Here J denotes the multi-index (j1, …, jn) of nonnegative integers and N*J = N1*jl … Nn*jn and we emphasize that c.n.e. denotes minimal commuting normal extension.) If n = 1, then S1 = S is called subnormal and N1 = N its minimal normal extension (m.n.e.).

Let

where x: = cos θ, denote the nth degree Chebyshev polynomials of the first and second kind, respectively. Further, let

Given non-negative integers λ and μ we define

and

Nous renvoyons le lecteur à [4] pour toutes les définitions non expliquées ci-dessous.

Un demi-groupe inverse S est un demi-groupe pour lequel pour tout élément a ∊ S il existe un seul inverse a–1, c'est-à-dire un élément a–l qui satisfait aux conditions aa–1a = a et a–1aa–1 = a–1. Une application λ : S → S, x → λx est appelée une translation à gauche si λ(xy) = (λx)y; une application ρ : S → S, x → xρ est dite translation à droite si (xy)ρ = x(yρ). Les translations à gauche (à droite) forment un demi-groupe Λ(S) [P(S)]. L'ensemble

forme un sous-demi-groupe de Λ(S) × P(S) qui est appelé l'enveloppe de translations de S. Si pour tout a ∊ S on définit λa : S → S, x → ax, et ρa : S → S, x → xa, on appelle

la partie interne de Ω(S).

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)

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.

In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

In this paper, we study the intersection pattern of families of convex sets. Since we only consider finite families, we may assume that the sets are also compact.

As an example, we consider families of 5 convex sets in R2 such that every two intersect and no three intersect. One such family that comes immediately to mind is that of 5 lines in general position. However, this is not the only family which exhibits this intersection pattern. Fig. 1 shows a family of 3 lines (sides of the large triangle) and 2 triangles (inscribed in the large triangle) that has this property.

Let R > 1 and denote by ℰR the ellipse

(1)

If Pn is a polynomial of degree at most n such that

(2)

then ([2]; also see [13, p. 337] and [9, p. 158, Prob. No. 270])

(3)

The standard proof of this well known result runs as follows. The function

(4)

is entire and in view of (2) we have

Hence by the maximum modulus principle

(5)

which is clearly equivalent to (3).

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.

Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrix

has order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

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.

A well-known result of Chebyshev is that if pn ∊ Pn, (Pn is the set of polynomials of degree at most n) and

(1)

then an(pn), the leading coefficient of pn, satisfies

(2)

with equality holding only for pn = ±Tn, where Tn is the Chebyshev polynomial of degree n. (See [6, p. 57].) This is an example of an extremal problem in which the norm of a given linear operator on Pn is sought. Another example is A. A. Markov's result that (1) implies that

(3)

There are also results for the linear functionals pn(k)(x0), x0 real, k = 1, … n – 1 ([8]).

Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized to

as suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?

0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,

(0.1)

with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by

(0.2)

Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by

(0.3)

These choices are assumed throughout.

The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].

Soit T un endomorphisme continu de , nous montrons qu'il existe une fonction entière S sur Cn × Cn telle que ζ → S(x, ζ) soit de type exponential sur Cn avec croissance controlée uniformément lorsque x parcourt un compact de Cn, de telle sorte que pour on ait

( désigne la transformée de Fourier-Borel d'une fonctionnelle analytique). Une telle fonction S sera dite un symbole holomorphe sur Cn. Réciproquement nous montrons que si S est une fonctionnelle analytique, la formule précédente permet de définir un endomorphisme continu (encore noté S) de .

In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C1 ⊆ C2 could be related by a tidy exact sequence

It was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].

The constant term of certain rational functions has attracted much attention recently. For example the Dyson conjecture; that the constant term of

is the multinomial coefficient

has spawned many generalizations (see [2], [7]). In this paper we consider some other families of rational functions which have interesting constant terms. For example, Corollary 4 states that the constant term of

(1.1)

is . Here, and throughout this paper, A and B denote fixed positive integers.

In order to prove this result, we consider the rational function in two variables