In the study of direct differential geometry, families of oriented arcs and curves have been employed extensively to define the differentiability of an arc at a point in various kinds of planes; cf. [2]. In [6], P. Scherk used lines in the projective plane; in [3] and [4], N. D. Lane and P. Scherk used circles in the conformai plane; conic-sections in the projective plane were employed in [5] and [7] by N. D. Lane and K. D. Singh; in [1], M. Gupta and N. D. Lane used the graphs of polynomials of degree at most n in the affine plane.

Our motivation for this note originates with consideration of a subset A of Euclidean w-space, Rn, which contains only part of its boundary. The part contained is t h a t part of the closure of A which cannot be “bettered“ within A with respect to the preference associated with a fixed closed convex cone Γ. Here b is preferred to a if and only if a — b ∊ Γ; if, for instance, Γ is the non-negative orthant of Rn, this preference is ordinary vector inequality.

In [13] we characterized the completely regular Hausdorff spaces as the class of spaces whose topology is generated by an Alexandrofï base. A space may have more than one Alexandrofï base and each such base determines a Hausdorff compactification .

The topic of homotopy pull-backs and push-outs has recently been discussed by a number of authors; Boardman and Vogt [5], Bousfield and Kan [6], Fantham [7], Mather [11], and Vogt [16]. Mather develops the theory with an eye to applications and of particular interest is his cube theorem which appears in this paper as Theorem (1.10); the significance of this theorem to applications is shown in [11]. As often occurs in homotopy theory the dual is not true. The purpose of this paper is to examine approximations to the dual in order to obtain new information concerning classical problems of duality.

Herrlich [7] has introduced the limit-operators to obtain every coreflective subcategory of the category Top of topological spaces and continuous maps. Using limit-operators, S. S. Hong [9] has constructed new reflective subcategories from a known extensive subcategory of a hereditary category of Hausdorff spaces and continuous maps.

Let be a Lie group with connected Lie subgroup , and let M(t), N(i) be real analytic curves in , the Lie algebra of , with . The main result in this paper is a Lie algebraic condition which is necessary and sufficient for

A normal Moore space is non-metrizable only if it fails to be ƛ-collectionwise normal for some uncountable cardinal ƛ [1].

For each uncountable cardinal X we present a class of normal, locally metrizable Moore spaces and a particular space Sλ in . If there is any space of class which is not X-collectionwise normal, then Sλ is such a space. The conditions for membership in make a space in behave like a subset of a product of a Moore space with a metric space. The class is sufficiently large to allow us to prove the following. Suppose F is a locally compact, 0-dimensional Moore space (not necessarily normal) with a basis of cardinality X and M is a metric space which is O-dimensional in the covering sense. If there is a normal, not X-collectionwise normal Moore space X where X ⊂ Y × M, then Sx is a normal, not λ-collectionwise normal Moore space.

Dirichlet was the first to give an ingenious proof of the exact (finite) number of elements in the basis of the multiplicative group of units in any algebraic number field of arbitrary degree n. These elements are called fundamental units. If the field is real and its generating number is a real root of a polynomial over Q of degree n﹜ having r1 real and r2 pairs of conjugate complex roots, so that r1 + 2r2 = n, then Dirichlet's famous result states that the exact number of fundamental units in Q(w) equals r1 + r2 — 1.

Dans ce travail, nous établissons une caractérisation complète des groupes quasi-purs-projectifs torsions résolvant ainsi le problème 17 de L. Fuchs dans le cas important des groupes torsions. Les groupes quasi-purs-injectifs ont été caractérisés dans le cas torsion par K. Benabdallah et A. Laroche dans [1].

By a 3-dimensional handlebody Vn of genus n, we mean an oriented 3-manifold which is a disk-sum of n copies of D2 X S1, where D2 is the unit disk and S1 is the boundary ∂D2; and by a surface Fn of genus n we mean the oriented boundary surface ∂ Vn.

In [4], De Groot and Aarts constructed Hausdorff compactifications of topological spaces to obtain a new intrinsic characterization of complete regularity. These compactifications were called GA compactifications in [5] and [7]. A characterization of complete regularity was earlier given by Frink [3], by means of Wallman compactifications, a method which led to the intriguing problem of whether every Hausdorff compactification is a Wallman compactification.

Let Σ be the projective 3-space over the field GF(q) where q = pe, p an odd prime. A spread W in ∑ is a set of q2 + 1 lines in ∑ which are such that each point of Σ lies on exactly one line of W. Thus the lines of W are all mutually skew. The notion of a spread extends to higher dimensions and also applies for arbitrary fields [1; 3; 6, p. 29; 7, p. 5]. Our concern, however, will be within the narrower but still extensive bounds indicated.

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

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.

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.

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.

In their study of the uniform approximation of the reciprocal of e2 by reciprocals of polynomials on the positive real axis, Cody, Meinardus, and Varga [3] showed that if denotes the class of all polynomials of degree at most n and

then

Ebin [8] gives a thorough study of the space of riemannian metrics on a compact manifold M and of the action of the diffeomorphism group of M on . The purpose of this paper is to study the action of the larger group of conformorphisms, or conformai transformations, on and on . On , the L2-orthogonal decomposition induced by the action of gives a splitting of symmetric tensors into three summands introduced by York [25; 26]. We find submanifolds of tangent to the pieces of this decomposition.

For a separable Hilbert space is the algebra of bounded linear operators on is the ideal of compact operators, and Π is the natural map of onto the Calkin algebra .