In this paper we study second order linear differential systems. We examine the relationship between oscillation of n-dimensional systems and certain associated n-dimensional systems, where m ≧ n. Several theorems are presented which unify and encompass in the linear case a number of results from the literature. In particular, we present a transformation which extends an oscillation theorem due to Allegretto and Erbe [1], and a comparison theorem due to Kreith [9], and explains some work of Howard [7].

Formerly Hua [1] proved that if A is a division ring with centre Z and if there exists a natural number n such that an ∈ Z for every a ∈ A, then A is commutative; this generalizes Wedderburn's theorem on finite division rings. Another generalization of Wedderburn's theorem, due to Jacobson [3], asserts that every algebraic division algebra over a finite field is commutative. On the other hand, a theorem of Noether and Jacboson [3] states that every noncommutative algebraic division algebra contains an element which is not contained in the centre Z and is separable over Z.

Computation of the velocity of a given motion depends on measurement of nearby position changes only. Computation of acceleration, on the other hand, depends on measurement of nearby changes in velocity. But since velocity vectors are attached to positions so that even nearby ones are not a priori comparable, acceleration is not computable until a rule for comparison of vectors along a curve is given. Such a rule-parallel translation or linear connexion - exists automatically in Euclidean spaces. For motions in more general manifolds, for example (semi-) Riemannian ones, parallel translation is a less obvious consequence of the metric properties.

We prove a conjecture of Kudla and Rallis about the first occurrence in the theta correspondence, for dual pairs of the form $\left( U\left( p,q \right),\,U\left( r,s \right) \right)$ and most representations.

We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types {3, 6} and {6, 3}.

Although the literature on splines has grown vastly during the last decade [11], the study of polynomial splines on the circle seems to have suffered neglect. The first to study the subject in depth seem to be Ahlberg, Nilson and Walsh [1]. Almost at the same time I. J. Schoenberg [8] studied the problem of interpolation at the roots of unity by splines and its relation to quadrature on the circle. For discrete polynomial splines on the circle we refer to [5]. M. Golomb [3] also considers interpolation by a class of “spline” functions in the complex plane but his point of view is based on minimum norm properties of spline functions. Perhaps the reason for this neglect may be attributed to the fact that one can pass from the circle to the line by means of the transformation z → exp 2-πix. This changes the problem on the circle into periodic interpolation on the line with the difference that instead of interpolation by piecewise polynomial, we now consider piecewise exponential polynomials with complex exponents.

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in ${{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296 ± .010.

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by

(1.1)

where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is

(1.2)

Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).

Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

For a locally compact group $G$ and $1\,<\,p\,<\,\infty $, let ${{A}_{p}}\left( G \right)$ be the Herz-Figà-Talamanca algebra and let $P{{M}_{p}}\left( G \right)$ be its dual Banach space. For a Banach ${{A}_{p}}\left( G \right)$-module $X$ of $P{{M}_{p}}\left( G \right)$, we prove that the multiplier space $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$ is the dual Banach space of ${{Q}_{X}}$, where ${{Q}_{X}}$ is the norm closure of the linear span ${{A}_{p}}\left( G \right)X\,\text{of}\,u\,f\,\text{for}\,u\,\in \,{{A}_{p}}\left( G \right)\,\text{and}\,f\,\in \,X$ in the dual of $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$. If $p\,=\,2$ and $P{{F}_{p}}\left( G \right)\subseteq X$, then ${{A}_{p}}\left( G \right)X$ is closed in $X$ if and only if $G$ is amenable. In particular, we prove that the multiplier algebra $M{{A}_{p}}\left( G \right)\,\text{of}\,{{A}_{p}}\left( G \right)$ is the dual of $Q$, where $Q$ is the completion of ${{L}^{1}}\left( G \right)$ in the $||\cdot |{{|}_{M}}$-norm. $Q$ is characterized by the following: $f\,\in \,Q$ if an only if there are ${{u}_{i}}\,\in \,{{A}_{p}}\left( G \right)$ and ${{f}_{i}}\in P{{F}_{p}}\left( G \right)\left( i=1,2,... \right)$ with $\sum\nolimits_{i=1}^{\infty }{||}\,{{u}_{i}}\,|{{|}_{{{A}_{p}}\left( G \right)}}||fi|{{|}_{P{{F}_{p}}\left( G \right)}}\,<\,\infty $ such that $f=\sum{_{i=1}^{\infty }\,{{u}_{i}}{{f}_{i}}}$ on $M{{A}_{p}}\left( G \right)$. It is also proved that if ${{A}_{p}}\left( G \right)$ is dense in $M{{A}_{p}}\left( G \right)$ in the associated ${{w}^{*}}$-topology, then the multiplier norm and $||\cdot |{{|}_{{{A}_{p}}\left( G \right)}}$-norm are equivalent on ${{A}_{p}}\left( G \right))$ if and only if $G$ is amenable.

We give a characterization of univalent positively oriented harmonic mappings ƒ defined on an exterior neighbourhood of the closed unit disk { z: | z| ≤1} such that .

We study skeleton C*-subalgebras of a given C*-algebra. We show that if A is a unital (non-unital but σ-unital) simple C*-algebra, ℳ is any unital (nonunital) matroid C* -algebra, then A contains a skeleton C*-subalgebra B with a quotient which is isomorphic to ℳ. Other results for skeleton C*-subalgebras are also obtained. Applications of these results to the structure of quasi-multipliers and perturbations of C*-algebras are given.

L'enveloppe de translations d'un demi-groupe présente un intérêt particulier pour les deux raisons suivantes : d'une part, elle apparaît d'une façon naturelle dans la construction des extensions idéales des demi-groupes, et d'autre part, elle est souvent utilisée pour des caractérisations abstraites des demi-groupes. Afin de l'appliquer aux problèmes particuliers, il est important de fournir des constructions qui, pour un demi-groupe donné sous une forme ou une autre, soient aussi explicites que possible. Un exemple d'une telle construction est celle de l'enveloppe de translations d'un demi-groupe de matrices de Rees (voir [4]) et ses nombreuses applications [5 ; 6 ; 7 ; 8].

It was in the year 1931 that Dieudonné [4] proved the following necessary and sufficient condition for a polynomial to be univalent in the unit disk.

THEOREM A (Dieudonné criterion). The polynomial

(1)

is univalent in |z| < 1 if and only if for every θ in [0, π/2] the associated polynomial

(2)

does not vanish in |z| < 1. For θ = 0, ϕ(z, θ) is to be interpreted as Pn'(z).

We study the symplectic geometry of the moduli spaces ${{M}_{r}}={{M}_{r}}\left( {{\mathbb{S}}^{3}} \right)$ of closed $n$-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in $\text{SU}\left( 2 \right)$ by the diagonal conjugation action of $\text{SU}\left( 2 \right)$. Here the fusion product of $n$ conjugacy classes is a Hamiltonian quasi-Poisson $\text{SU}\left( 2 \right)$-manifold in the sense of $\left[ \text{AKSM} \right]$. An integrable Hamiltonian system is constructed on ${{M}_{r}}$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on ${{M}_{r}}$ relates to the symplectic structure obtained from gauge-theoretic description of ${{M}_{r}}$. The results of this paper are analogues for the 3-sphere of results obtained for ${{M}_{r}}\left( {{\mathbb{H}}^{3}} \right)$, the moduli space of $n$-gons with fixed side-lengths in hyperbolic 3-space $\left[ \text{KMT} \right]$, and for ${{M}_{r}}\left( {{\mathbb{E}}^{3}} \right)$, the moduli space of $n$-gons with fixed side-lengths in ${{\mathbb{E}}^{3}}\left[ \text{KM}1 \right]$.

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

The main theorem of this paper characterizes the class of essentially T1 spaces having bases of countable order as those spaces of the class of essentially T1 spaces having primitive bases in which closed sets are sets of interior condensation. In addition we deduce some corollaries of this theorem, derive some other characterizations, and prove a lemma concerning primitive sequences which is a key to the proof of the main theorem and has other applications.

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property:

(*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

How large a matching must a graph have?

We consider graphs G (finite, undirected, with no loops or multiple edges), with order nG (always ≧ 1) and mG the maximum number of edges in a matching of G. The matchability μG of G is the fraction (2m/n) of nodes covered by a maximum matching.

Motived by an intrinisic necessary condition for the purifiability of subgroups of primary abelian groups due to K. Benabdallah and T. Okuyama we introduce new functors on the category of pairs (G, A),where A is a subgroup of G,to the category of Z/pZ-vector spaces. The vanishing of these functors leads to the notion of vertical subgroup which is a weakening of purity but also an essential component of the latter. In fact, a vertical subgroup is pure if and only if it is neat. We establish various facts about vertical subgroups and “maximal” vertical subgroups and apply the resulting theory to the problem of purifiability. We show that the class of quasi-complete groups is precisely the class of reduced groups in which every subgroup satisfying the intrinsic necessary condition for purifiability is in fact purifiable. This is also the class of reduced p-groups in which the maximal vertical subgroups are precisely the pure subgroups.

The /z-cobordism theorem in [8], the generalized Poincaré conjecture in higher dimensions in [20] and several other results in differential topology are proved by using the following theorems of Morse theory:

(1) the elimination of critical points;

(2) the existence of nondegenerate functions for which the descending and ascending bowls have normal intersection;

(3) the alteration of function values at critical points. (For the details see below.)

We shall give short and elementary pr∞fs of these theorems together with some stronger statements than the ones given in [8-13] or [19].

The theorems are proved for noncompact manifolds rather than for compact manifolds since, by a trivial modification of the manifold (deleting the boundary or one point) the case of compact manifolds is included.