Suppose that U is an open set in Euclidean N-space which has a finite volume |U|. A complete packing of U is a sequence of disjoint N-spheres C = {Sn} which are contained in U and whose total volume equals that of U. In an osculatory packing, the spheres are chosen recursively so that for all n larger than a certain value m, Sn has the largest radius of all spheres contained in U\(S1– ∪ … ∪ Sn-1–) (S– is the closure of S). An osculatory packing is simple if m = 1. If rn denotes the radius of Sn, the exponent of the packing is defined by:

This quantity is of considerable interest since it measures the effectiveness of the packing of U by C.

In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αTƒ ‖q ≦ C‖ |x|αƒ ‖p, where Tƒ denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.

In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω∼ will denote a measurable function on Rn × Rn such that for each x ∈ Rn, Ω∼(x, .) is locally integrable and homogeneous of degree 0.

Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in En for n = 2m satisfies

In this paper we make systematic use of error-correcting codes to obtain sphere packings in En, including several of the densest packings known and several new packings.

In this paper the theory of Browder [2] and of Lions [3] on periodic solutions of non-linear evolution equations in Banach spaces is put in a more general framework so as to include the Navier-Stokes equations and their variants.

An abstract existence theorem is proved in § 1. Applications are given in § 2. The existence of periodic solutions of the Navier-Stokes equations without any restriction on the dimension of the space domain is established. Application of the abstract theorem to the following problem is given:

1. Let H be a Hilbert space and (., .)H the inner product in H. Let V and W be two reflexive separable Banach spaces with W ⊂ V ⊂ H. W is dense in V and V is dense in H.

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

If p is a prime, we call an element x ≠ 1 of a group G a generalized p-element if, for every n ≧ 1, there exists r ≧ 0 such that xpr ∈ Gn, where Gn is the nth term of the lower central series of G. Bovdi [1] proved that if G is a finitely generated group having a generalized p-element, and if ∩nΔn(Z(G) = 0 where Δ(Z(G)) is the augmentation ideal, then G is residually a finite p-group.

We recall that if R is a ring, then the nth dimension subgroup of G over R, denoted by Dn(R(G)), is defined to be {g | g – 1 ∈ Δn(R(G))}. In this note, we show that if G is finitely generated, then ∩nDn(Zp∧(G)) = 1 ⇔ ∩nΔn(Zp∧ (G)) = 0 ⇔ G is residually a finite p-group.

A connected Lie group H is generated by a pair of oneparameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups. If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n; otherwise define it as infinity.

For the isometry group of the spherical geometry, or equivalently for the rotation group SO(3), the order of generation is always finite.

In [6] it is shown that two (compact) orientable 3-manifolds which are irreducible, boundary irreducible and sufficiently large are homeomorphic if and only if there exists an isomorphism between the fundamental groups which respects the peripheral structure. In this note we extend this theorem to reducible 3-manifolds.

Any compact 3-manifold M has a decomposition into prime manifolds [1; 4].

1

Here the connected sum of two bounded manifolds N1, N2 is denned by removing 3-balls B1 B2 in int N1, int N2, respectively, and glueing the resulting boundary spheres together. The M1's which occur in the decomposition (1) are either irreducible or handles (i.e., a fibre bundle over S1 with fibre S2). If (1) contains a fake 3-sphere, we assume it to be Mn.

Let G be a group and F a field of arbitrary characteristic. In [4] Kaplansky asks under what conditions is F[G] primitive, where F[G] is the group algebra of G over F. We give some necessary conditions on G that F[G] be primitive and propose a conjecture.

Definition. A ring R is primitive if it has a faithful irreducible right module.

The above should really be considered as a definition of right primitive. One can analogously define left primitive and the two properties are not equivalent. For our purposes, the two concepts are equivalent, for the group algebra possesses a nice involution.

If we assume that F[G] is primitive, there are some immediate restrictions on G. First of all G cannot be Abelian since the only primitive commutative rings are fields. (I exclude of course the case when G consists of one element.)

In this note we prove that an isomorphism of p-adic group rings of finite p-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of the p-adic group ring of a finite p-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of the p-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.

G denotes a finite p-group.

Z denotes the ring of (rational) integers.

ZP denotes the ring of p-adic integers.

Qp denotes the p-adic number field.

Our aim in this paper is to demonstrate a relationship between left exact and representable functors. More precisely, in the functor category whose objects are the additive functors from the dual of an abelian category 𝔄 to the category of abelian groups and whose morphisms are the natural transformations between them, the left exact functors can be characterized as those equivalent to a direct limit of representable functors taken over a directed class. The proof will proceed in the following manner. Lambek [3] and Ulmer [7] have shown that any functor T in can be expressed as a direct limit of representable functors taken over a comma category. When T is left exact, it is easily observed that this comma category is a filtered category. When T is left exact, it is easily observed that this comma category is a filtered category.

In [1], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug–1 ∈ U, 1 ≠ u ∈ U, implies that g ∈ U. Baumslag considered the class of amalgamated products (A * B; U) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [13] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class.

Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer EG(u, U) of u relative to U in G is defined by

if u ≠ 1, and by EG(u, U) = U if u = 1.

The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].

Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).

Throughout this paper G will denote a finite group containing a nilpotent maximal subgroup S and P will denote the Sylow 2-subgroup of S. The largest subgroup of S normal in G will be designated by core (S) and the largest solvable normal subgroup of G by rad(G). All other notation is standard.

Thompson [6] has shown that if P = 1 then G is solvable. Janko [3] then observed that G is solvable if P is abelian, a condition subsequently weakened by him [4] to the assumption that the class of P is ≦ 2 . Our purpose is to demonstrate the sufficiency of a still weaker assumption about P.

A (non-archimedean) valuation v on a field K is said to be henselian if it has a unique prolongation to a valuation on Ka, the algebraic closure of K. A henselization (Kh, vh) of a valuated field (K, v) is a smallest separable extension of K containing a henselian prolongation vh of v. (Kh, vh) is unique up to K-isomorphism, and (Kh, vh) = (K, v) if and only if v is henselian. In this paper we confine ourselves to valuations of finite rank.

If v is a non-henselian rank one valuation on K, and if [Ks:K] = ∞, Ks being the separable closure of K, then it is known that v has infinitely many prolongations to Ka [1, (27.11)]. We shall see that this is no longer true if the rank of v is greater than one.

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.

After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.

Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.

Let H be a subgroup of a group G (all groups considered throughout this article are finite); then H will be called primitive if the subgroup

is distinct from H. Such subgroups, which are also called meet-irreducible, arise naturally in connection with minimal permutation representations of groups and in other contexts; for example, every subgroup of a group G can be written as an intersection of primitive subgroups of G, and the set of all primitive subgroups of G is characterized by its minimality with respect to this property. While maximal subgroups are always primitive, most groups contain non-maximal subgroups which are primitive (see remark at end of article). Note that a subgroup H of an abelian group G is primitive if, and only if, G/H is cyclic of prime-power order.