The chief purpose of this paper is to find all pairs (G, θ) where G is a finite special p-group, and θ is an automorphism of G acting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class of p-groups, which we call special F-groups, where F is a finite field of characteristic p. We point out that if p is odd, then an F-group has exponent p. On the other hand, every special 2-group is also a special GF(2)-group.

Let F be a formally real field, and let A be a preordering of F; that is, a subset of F satisfying Δ + Δ = Δ, Δ Δ = Δ, F2 ⊆ Δ. Denote by X Δ the set of all orderings P of F satisfying P ⊇ Δ. Thus Δ = ⋂ p ∈xΔP. This result is well known. It was first proved by Artin [3, Satz 1] in the case Δ = ∑ F2.

W. J. Stiles showed in [10, Corollary 4.5] that Banach's weak basis theorem fails in the spaces lp, 0 < p < 1. Then, J. H. Shapiro [9] indicated certain general classes of non-locally convex F-spaces with the same property, and asked whether the weak basis theorem fails in every non-locally convex F-space with a weak basis. Our purpose is to answer this question in the affirmative. In [3] we observed that, essentially, the only case that remained open is that of an F-space with irregular basis (en), i.e. such that snen →0 for any scalar sequence (sn).

One of the nicest results in elementary number theory is the following, giving the relation between quadratic residues and sums of squares.

If a is a degree of unsolvability, a is called high if a ≦ 0’ and a’ = 0”. In [1], S. B. Cooper showed that if a is high, then (i) a is not a minimal degree, and (ii) there is a minimal degree b < a. We give new proofs of these results which avoid the intricate priority and recursive approximation arguments of [1] in favor of “oracle” constructions using the recursion theorem. Also our constructions apply to degrees a which are not below 0'. Call a degree ageneralized high if a’ = (a U 0')' . Among the degrees ≦ 0', the generalized high degrees obviously coincide with the high degrees.

Let F be a non-singular metric vector space, that is, a vector space over a field F not of characteristic two, upon which is defined a non-singular symmetric bilinear form ( , ). For any a ϵ V, we define the cone with vertex a to be the set

S(X) is the semigroup under composition of all continuous selfmaps of the topological space X. For certain spaces X and Y we classify completely the homomorphisms from S(X) into S(Y). An application of the main result to S(I) the semigroup of all continuous selfmaps of the closed unit interval I results in the solution of a problem which was suggested in the closing paragraph of [6].

Let T be a (not necessarily bounded) self-adjoint operator on a Hilbert space H with the spectral resolution The set of elements x in H for which ||Etx||2 is absolutely continuous is a subspace, H a, of H which reduces T. (See H almos [1, p. 104]; Kato [2, p. 516].) If H a ≠ 0, the restriction of T to DT ⋂ H a is called the absolutely continuous part of T; in case H = H a, T is said to be absolutely continuous.

In [1] a simple but ingenious technique was developed for calculating hitting probabilities for submartingales (or martingales or supermartingales) subject to various constraints. This technique is extended here in order to find sharp bounds on upcrossing probabilities for submartingales subject to constraints.

Let be a countable atomless Boolean algebra and let X be a countable partial ordering. We prove that there exists an embedding of X into which is recursive in X, and which destroys all suprema and infima of X which can be destroyed. We show that the above theorem is false when we try to preserve all suprema and infima of X instead of destroying them.

Every monoid M is isomorphic to the monoid End0,1(L) of all (0,1)-preserving endomorphisms of a bounded lattice L, see [3]. The lattices L with M ≌ End0,1(L) that are constructed there are of an arbitrary infinite cardinality not smaller than that of M, and they all have infinite chains. The aim of the present article is to supplement these results. It will be shown that every finite monoid M is representable as End0,1(L) of a finite lattice. In addition, an account of the difficulties involved in attempting to characterize endomorphism monoids of lattices of a fixed finite height will be given.

In a recent paper [2], Bushard established and applied a comparison theorem for positive solutions to the equation:

in an arbitrary bounded domain D of Euclidean w-space Rn. The proof of these results depended on the absence of mixed derivatives of u in the equation considered.

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

Dirichlet's Theorem says that for any real α and for N ≧ 1, there exists a natural n ≦ N with

where || || denotes the distance to the nearest integer. Heilbronn [2], improving estimates of Vinogradov [3], showed that for α, N as above and for ϵ ≧ 0, there exists an n ≦ N with

Tutte [9] has given necessary and sufficient conditions for a finite graph to have a perfect matching. Different proofs are given by Brualdi [1] and Gallai [2; 3]. The shortest proof of Tutte's theorem is due to Lovasz [5]. In another paper [10] Tutte extended his conditions for a perfect matching to locally finite graphs. In [4] Kaluza gave a condition on arbitrary graphs which is entirely different from Tutte's.

Let X be a set. A function d from X X X into the nonnegative real numbers is called a ﹛non-archimedean) quasi-metric on X if

1.1. Two complex measurable functions/ and g on complex measure spaces (X, η) and (Y, v) are equimeasurable, abbreviated ƒ ∼ g, if

for every Borel set E ⊆ C. If Φ is a continuous complex function on C, then we make the following standing hypothesis (HI) which relates Φ, f, and g:

(HI) For all α, β ∊ C, we have

Dans cette note on se propose d'étendre certains résultats de L. Mirsky concernant des matrices doublement sous-stochastiques [5; 6; 7].

On commence par rappeler des définitions et notations qu'on aura à utiliser par la suite.

A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.