Let R be a prime ring with invoution *, of characteristic 0, with skew elements K and extended centroid C. Let a ∈ K be such that (ad a)n =0 on K. It is shown that one of the following possibilities holds: (a) R is an order in a 4-dimensional central simple algebra, (b) there is a skew element λ in C such that , (c) * is of the first kind, n ≡ 0 or n ≡ 3 (mod 4), and . Examples are given illustrating (c).

We consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.

Let G be a profinite group. We define an extension (E.j) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of к satisfying: for some open normal subgroup S of G, and the map is continuous (A being discrete).

Let jν, 1, jν,2, … denote the positive zeros of the Bessel function Jν(x), and similarly, let j'v,1, j'v,2, … denote the positive zeros of J'v(x), which are the positive critical points of Jv(x). It is well-known that when v is positive, both jν ,k. it and j'ν k are increasing functions of ν; see, e.g., [12, pp. 246 and 248]. Recently, Lorch and Szego [6] have attempted to show that the same is true for the positive zeros j″v,1, j″v,2, … of j″v(x), which are the positive inflection points of Jv(x).

The symmetries of manifolds are a focal point of study in low-dimensional topology and yet, outside of some totally asymmetrical 3- and 4-manifolds, there are very few cases in which a complete classification has been attained. In this work we provide such a classification for symmetries of the orientable and nonorientable 3-dimensional handlebodies of genus one. Our classification includes a description, up to isomorphism, of all of the finite groups which can arise as symmetries on these manifolds, as well as an enumeration of the different ways in which they can arise. To be specific, we will classify the equivalence, weak equivalence and strong equivalence classes of (effective) finite group actions on the genus one handlebodies.

We continue to develop the theory of previous papers concerning transforms corresponding to Mellin multipliers which involve products and/or quotients of Γ-functions. We show that, by working with certain subspaces of Lp,μ consisting of smooth functions, we can simplify considerably the restrictions on the parameters which were necessary in the Lp,μ setting. As a result, operators in our class become homeomorphisms on these subspaces under conditions of great generality.

Let A be an artin algebra. Let ( be a component of the stable Auslander- Reiten quiver of A. If is periodic, then the structure of G is known. Here, we are going to consider the case when is non-periodic: we will show that is isomorphic to Z with a valued quiver. In particular, there is no cyclic path in .

The aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus (p−1 )/ 2, for any odd prime number p, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2 cos(2π/p)) of the cyclotomic field Q(e2π i/p).

Two proofs of the fact that the constructed curves have this property will be given. One is by providing a double cover with the pth roots of unity in its automorphism group. The other is by explicitly writing down equations of a correspondence in C x C which defines multiplication by 2cos(2π/ p) on the jacobian of C. As a byproduct we obtain polynomials which define bijective maps Fℓ → Fℓ for all prime numbers in certain congruence classes.

Let G and S be finite groups. Suppose that S acts on G with (|G|, |S| ) = 1. If S is solvable, Glauberman showed the existence of a natural bijection from lrrs(G) = ﹛ χ ∈ Irr(G) | χs = χ for a11 s ∈ S﹜ on to Irr(C), where C = CG(S). If S is not solvable, and consequently | G| is odd, Isaacs also proved the existence of a natural bijection between the above set of characters. Finally, Wolf proved that both maps agreed when both were defined ([1], [3], [10]). As in [7], let us denote by *: Irrs(G) → Irr(C) the Glauberman-Isaacs Correspondence.

A natural rank function is defined on the set DS of everywhere divergent sequences of continuous functions on the unit circle T. The rank function provides a natural measure of the complexity of the sequences in DS, and is obtained by associating a well-founded tree with each such sequence. The set DF of everywhere divergent Fourier series, and the set DT of everywhere divergent trigonometric series with coefficients that tend to zero, can be viewed as natural subsets of DS. It is shown that the rank function is a coanalytic norm which is unbounded in ω1 on DF. From this it follows that DF, DT and DS are not Borel subsets of the Polish space SC(T) of all sequences of continuous functions on T. Finally an alternative definition of the rank function is formulated by using nested sequences of closed sets.

The results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.

Pierre Barrucand asks the following question (Unsolved Problems, # ASI 88:04, Banff, May 1988, Richard K. Guy, Ed.; also [2, p. 594]). Let K be a cyclic quartic field, and let ξ be a non-square element of K. Let M be the Galois closure of , and let G be the Galois group Gal(M/Q). Find (1) all possible G, (2) conditions on ξ to have such a G, and (3) a list of all possible subfields of M.

Let KG be the group ring of a group G over a field K and let U(KG) be its group of units. If K has characteristic p > 0 and G contains p-elements, then it is proved that U(KG) is hypercentral if and only if G is nilpotent and G′ is a finite p-group.

This paper is devoted to the machinery necessary to apply the general theory of monotone dynamical systems to neutral functional differential equations. We introduce an ordering structure for the phase space, investigate its compatibility with the usual uniform convergence topology, and develop several sufficient conditions of strong monotonicity of the solution semiflows to neutral equations. By applying some general results due to Hirsch and Matano for monotone dynamical systems to neutral equations, we establish several (generic) convergence results and an equivalence theorem of the order stability and convergence of precompact orbits. These results are applied to show that each orbit of a closed biological compartmental system is convergent to a single equilibrium.

In this paper we shall develop the scattering theory introduced by Lax and Phillips [5] for the following general wave equation; where Ω is an exterior domain Rn(n ≥ 3) with the smooth boundary δΩ and B is either a Dirichlet boundary condition or of the form Bu = Vi(x)aij(x)δju+σ(x)u with the unit outer normal vector v(x) = (v1 , … , vn) at x ∈ δ Ω. The precise assumptions on α(x), aij(x),q(x), σ(x) are denoted below. If Ω is an inhomogeneous medium with the density ρ (x), the propagation of waves is described by (1.1) with a(x) = a(x)2 ρ(x), aij(x) = ρ-l(x) δij and q(x) = 0 with the velocity a(x).