Let H be a separable complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Let π be the quotient mapping from B(H) onto the Calkin algebra B(H)/K(H), where K(H) denotes all compact operators on B(H). An operator T ∈ B(H) is said to be convexoid[14] if the closure of its numerical range W(T) coincides with the convex hull co σ(T) of its spectrum σ(T). T ∈ B(H) is said to be essentially normal, essentially G1, or essentially convexoid if π(T) is normal, G1 or convexoid in B(H)/K(H) respectively.

Let σ4 denote the group of all permutations of {a, b, c, d}. It has 24 elements, partitioned into five conjugacy classes: (1) the identity 1; (2) 6 transpositions: (ab), …, (cd); (3) 8 elements of order 3: (abc), …, (bcd); (4) 6 elements of order 4: (abcd), …, (adcb); (5) 3 elements of order 2: x = (ab)(cd), y = (ac)(bd), z = (ad)(bc).

In this paper, we study the differentiate actions of σ4 on odd-dimensional homotopy spheres modelled on the linear actions, with the fixed point set of each transposition a codimension two homotopy sphere.

A simple (2n – l)-knot is a differentiate embedding of a homotopy sphere K2n–l into a homotopy sphere Σ2n+1 such that πj(Σ – K) = πj(S1) for j < n.

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such that

The sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.

A subset A of a topological space X is an approximateretract of X if for every neighborhood U of A in X there is a retract R of X such that A ⊂ R ⊂ U. A compactum X is an absolute approximate retract (AAR-space) if whenever X is embedded as a subset of a compactum Z, then X is an approximate retract of Z. These concepts were first defined in [2] where it is shown that every AAR-space is a contractible Peano continuum. In [3] an example is given to show that there exists a contractible LC∞ compactum which is not an AAR-space.

Kan and Miller have shown in [9] that the homotopy type of a finite simplicial set K can be recovered from its R-algebra of 0-forms A0K, when R is a unique factorization domain. More precisely, if is the category of simplicial sets and is the category of R-algebras there is a contravariant functor

with

the simplicial set homomorphisms from X to the simplicial R-algebra ∇, where

and the faces and degeneracies of ∇ are induced by

and

respectively.

G. R. Allan and A. M. Sinclair proved in [1] that if a commutative radical Banach algebra possesses bounded approximate identities then for every sequence (αn) of real numbers such that limn→∞αn = 0 there exists such that

In the other direction it is shown in [6] that if is separable and if the nilpotents are dense in then for every sequence (βn) of positive reals there exists such that

(This result was given in [2] for the Volterra algebra.)

If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP/J(RP) is simple Artinian: in short it has several properties of the commutative local rings.

In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x1, x2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x1, x2, …, xj (j = 0, 1, …, n), are all distinct primes.

Let ϕ ≠ 0 be an entire function of one complex variable and of exponential type. Let B denote the set of all monomial exponentials of the form zneζ where ζ is a zero of ϕ of order greater than h. If R is a simply connected plane region and H(R) denotes the space of functions analytic in R with the topology of uniform convergence on compacta, then ϕ can be considered as an element of the topological dual H′(R) if the Borel transform of ϕ is analytic on , the complement of R. The duality is given by

where C is a simple closed curve in the common region of analyticity of ƒ and , and C winds once around the complement of a set in which is analytic.

Let E be a Banach space and be an adapted sequence on the probability space We denote by T the set of all bounded stopping times with respect to . is called a pramart if

converges to zero in probability, uniformly in τ ≧ σ. The notion of pramart was introduced in [6]. A good property is the optional sampling property (see Theorem 2.4 in [6]). Furthermore the class of pramarts intersects the class of amarts, and every amart is a pramart if and only if dim E < ∞ ([2], see also [4]). Pramarts behave indeed quite differently than amarts. Although the class of pramarts is large, they have good convergence properties as is seen in the next two results of Millet-Sucheston, [6], [7].

THEOREM 1.1. Let be a real-valued pramart of class (d), i.e.,

The first investigations on this type of problem were carried out by O. Kiš [2]. Kiš considered the problem of interpolating a function and its second derivative at the nth roots of unity (the (0, 2) problem) by 2n – 1 degree polynomials, and the convergence of such approximates. Later, Sharma [8], [9] extended the existence and uniqueness results to (0, m) interpolation, and essentially to (0, m1, m2) interpolation. In the latter case, Sharma established the convergence results for (0, 2, 3), (0, 1, 3) and (0, 1, 4) interpolation as well. Although some further special cases were considered [10] these were the essential results until very recently. Now Cavaretta, Sharma and Varga [1] have established the existence and uniqueness theorem for all possible interpolations of this type (see Theorem 2.1 below). Motivated by the work of Cavaretta, Sharma and Varga and the earlier work of Sharma [9], a different proof of this result is provided, and this proof is used to establish a convergence theorem in the general case.

Let X be any set and A be a uniformly closed algebra of bounded real valued functions on X which contains the constants and separates the points. For a lattice ℒ of subsets of X (we assume throughout that ∅ and X belong to ℒ), let MR(ℒ) denote the space of all finite, finitely additive,ℒ-regular measures defined on the field of sets generated by ℒ . Generalizing the notion of an integral representation, in [5] Kirk and Crenshaw define a standard representation of A*, the Banach dual of A, in MR(ℒ) to be a linear map I of A* into MR(ℒ) with the property that if 0 ≦ ϕ ∈ A*, then

for every W in ℒ. The space MR(ℒ) is said to represent A* if there exists a (unique) standard representation I of A* onto MR(ℒ) which is a Banach lattice isomorphism.

A (tame) knot kn in S3 is said to have period n if there exists a homeomorphism ϕ: S3 → S3, necessarily orientationpreserving, such that

• (i) the fixed point set of ϕ is a circle disjoint from kn

• (ii) ϕ(kn) = kn;

• (iii) ϕ has order n.

Several necessary conditions for a knot to have period n have already been established in the literature; see [3] [11] [14] [18]. Here we establish the following further condition, involving the signature σ(kn) of (kn).

Suppose that n competitors compete in r races and in each race they are awarded placings l, 2, 3, …, n – 1, n. After the r races each competitor has a result consisting of his r placings. Let such a result be written (αj)1≦j≦r where for convenience the positive integers αj are arranged in ascending order. For example, if n = 4 and r = 6 a typical result is (1, 2, 2, 3, 4, 4).

A final ranking of the n competitors will be determined in all circumstances if a strict ordering is assigned to all possible results. In the next section it is shown that the number or results is . Of course only some strict orderings of this set of results will be feasible in that there are criteria which a final ranking should satisfy. For example, (1, 1, …, 1) should clearly be ranked first in any feasible ranking.

An integral domain D is said to be an essential domain if D is an intersection of valuation rings that are localizations of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin has shown [2, pp. 717-718] that every v-multiplication ring is an essential domain, and that an essential domain having a defining family of valuation rings {Vα} which is of finite character (i.e., every nonzero element of D is a non-unit in at most finitely many Vα) is necessarily a v-multiplication ring. It is noted in [4, p. 860] that any localization of a v-multiplication ring is again a v-multiplication ring. In this vein, Joe Mott has asked whether a localization of an essential domain must again be an essential domain. An example of an essential domain that is not a v-multiplication ring is given in [4], however it can be seen for this example that each localization is again an essential domain [6].

A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)

It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.

In this paper we prove a structure theorem that could be said to verify this conjecture.

THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

Two aspects, and their connections, of the problem of enumerating solutions to certain systems of congruences are explored in this paper. Although slightly more general cases are mentioned, the basic object of study is a system of diagonal equations

where d1, d2, …, dt are positive integers and the coefficient matrix [aij] has entries from Fp = GF(p), and for which solutions x = (x1, x2 …, xt) ∈ Fpt are sought. Speaking loosely, such a system usually has approximately pt–n solutions in the sense that the difference between pt–n and the correct value becomes small in comparison with pt–n as p becomes large. A parameter is introduced which measures the extent to which the matrix [aij] is non-singular over Fp. The effect of this parameter on the size of the error term (the difference between the number of solutions to (1) and pt–n) is the first aspect of the enumeration problem to be treated.

In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.

The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua.

The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these mappings were invented for quite another purpose [1]. The author believes that these maps are likely to provide even more new information about homogeneous continua.

A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h: M → M such that h(p) = q.

(R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.