Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].

This note is an exposition of the simple and elegant approach to Grothendieck's inequality given in [2] and lsqb;4], with one further simplification. The process of factorizing through L2 ([2], p. 21) introduces a factor of into the final constant. We show that this step can be avoided.

Let l be an odd prime and let A be a commutative ring containing 1/l. Let K*(A;Z/lv) denote the mod lv algebraic K-theory of A [3]. As explained in [4] there exists a “Bott element” βv∈K21v–1(l–1)(Z[1/l];Z/lv) and, using the K-theory product we may, following [16, Part IV], form

which is defined as the direct limit of iterated multiplication by βv. There is a canonical localisation map

The best uniform approximation of a function f on [-1,1] by real algebraic polynomials satisfies

if and only if ƒ is the restriction to [- 1,1] of an entire function (Bernstein [2], p. 113, see also [12], pp. 83–85). For such functions ƒ the rate of best approximation has been characterized by Varga [24], Reddy [24], Shah [21], and Kapoor and Nautiyal [10] in terms of order and type of ƒ , lower order and type, and in terms of more general concepts of order. On the other hand, order and type of ƒ are connected with the Taylor coefficients, i.e. with the rate of growth of the sequence (see [23], p. 41 or [3], pp. 11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequence are obtained. But in such results the rate ofbest approximation is always described by a limit relation, e.g. of the form , and this causes a considerable loss of precision, as will be discussed in more detail in Section 3 (in this respect cf. also the remark by Bernstein [2], pp. 114/115).

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.

There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

Let R be a (commutative Noetherian) local ring (with identity) having maximal ideal and dimension d≧l. It is shown in [5,3.6rsqb; that the local cohomology module may be described as a module of generalized fractions: if x1…,xd is a system of parameters for R, then , where U(x)d+1 is the triangular subset [4,2.1] of Rd+1 given by

In previous papers [3, 4] the author has discussed the symmetric generalised Erdélyi–Kober operators of fractional integration defined by

where α>0, γ≧0 and the operators ℑiγ(η,α) and defined as in equations (1) and (2) respectively but with Jα−1, the Bessel function of the first kind replaced by Iα−1, the modified Bessel function of the first kind.

Let Γ be a spread in = PG(3, q); thus Γ consists of a set of q2 +1 mutually skew lines that partition the points of . Also let Λ be the group of projectivities of that leave Γ invariant: so Λ is the “linear translation complement” of Γ, modulo the kern homologies. Recently, inspired by a theorem of Bartalone [1], a number ofresults have been obtained, in an attempt to describe (Γ, Λ) when q2 divides |Λ|. A good example of such a result is the following theorem of Biliotti and Menichetti [3], which ultimately depends on Ganley's characterization of likeable functions of even characteristic [5].

The theory of joint spectra for commuting operators in a Hilbert space has recently been studied by several authors (Vasilescu [11,12], Curto [4,5], and Cho-Takaguchi[2,3]). In this paper we willuse the definition by Taylor [10] of the joint spectrum to show that thejoint spectrum is determined by the action of certain "Laplacians"(cf. Curto [4,5]) of a chain-complex of Hilbert spaces.

In this paper we analyse the electrical behaviour within systems of long and short coupled nerve axons by using a geometric approach to obtain a priori bounds on solutions. In [4[ we developed a general model for a bundle of n-uniform unmylinated nerve fibres. If FitzHugh-Nagumo dynamics, [3[ are used to describe the ionic membrane currents, then the model takes the form

Here W=(w1,…wn)T denotes the membrane action potentials for each fibre in the bundle and Z=(Z1,…Zn)T represents the recovery variables for each fibre, which control the return to the resting equilibrium after any transmission of signals.

In this paper the “hypercore” of a semigroup S is defined to be the subsemigroup generated by the union of all the subsemigroups of S without non-universal cancellative congruences, provided that at least one such subsemigroup exists: otherwise it is taken to be the empty set. It is shown first that if the hypercore of S is nonempty (which holds, for example, when S contains an idempotent) then it is the largest subsemigroup of S with no non-universal cancellative congruence, is full and unitary in S, and is contained in the identity class of every group congruence on S (Theorem 1).

Among the many semigroups which can be derived from a given compact (jointly continuous) semigroup S is the semigroup 2s consisting of its non-empty compact subsets; the product is the usual one defined by the rule EF = {xy:xεE, yεF}. The Vietoris or finite topology on 2s (in which a base for the open sets is obtained by taking all sets of the form for l ≦i ≦n} as Vl, V2,…, Vn run over all finite collections of open subsets of S) makes 2s a compact, jointly continuous semigroup. The topology has a long history, having been introduced by Vietoris in 1923 and studied by Michael[4]. The utility of the topological semigroup was established by Hofmann and Mostert [3; see especially Section 3.7]; in fact they prefer to produce directly the uniform structure on 2s rather than the topology.

Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equations

where Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.

A semigroup is said to be congruence-free if and only if its only congruences are the universal relation and the identical relation. Congruence-free inverse semigroups were studied by Baird [2], Trotter [19], Munn [15,16] and Reilly [18]. In addition, results on congruence-free regular semigroups have been obtained by Trotter [20], Hall [4] and Howie [7].

Given a function f meromorphic in the plane, and complex numbers w and a, call w an a-point of f(k) if f(k)(w) = a. Denote by Λ(a,f) the set of z∈ℂ such that every neighborhood of z contains a-points of infinitely many of the functions f(k). Adapting the terminology of Pólya [16], who introduced the sets Λ(a,f) in [15],we call Λ(a,f) the final set of f with respect to a.