An A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

The central distribution of the second largest (smallest) root following the Fisher-Girshick-Hsu-Roy distribution under certain null-hypothesis has been derived in series form by Pillai and Al-Ani [6]. In this paper the noncentral distributions of the second largest roots in the MANOVA situation, the canonical correlation, and equality of two covariance matrices are obtained. Further, the distribution of the second largest root of the covariance matrix is obtained as a limiting case. The largest root and its noncentral distributions have been considered already by Pillai and Sugiyama [7] for the situations stated above. However, in the present paper, the joint densities of the largest and the second largest roots are derived in all the above cases from which the distributions of the largest roots can be obtained, although in more elaborate forms.

In this paper we obtain a characterization for all bipartite zero-divisor graphs of commutative rings $R$ with 1 such that $R$ is finite or $\left| \text{Nil}\left( R \right) \right|\,\ne \,2$.

A collection of examples of first order linear neutral differential delay equations having a nonoscillatory solution with lim sup = ∞ and lim inf = 0 at ∞ is given. This disproves a recent conjecture about the asymptotic behavior of solutions to such equations.

Meir and Sharma [1] have suggested a modification of Romberg quadrature using Newton-Cotes and, in particular, Simpson sums in place of trapezoidal sums. By comparing the error term with that obtained by Bulirsch [3] for trapezoidal sums, they concluded that the use of Simpson sums would lead to an improvement of the results. The procedure adopted by Meir and Sharma [1] permits them to obtain an expression for the error in the numerical quadrature. However, for the purpose of numerical computation, this procedure appears to be less suitable. In section 3, we give an alternative formulation which would enable us to carry out the computation, using Simpson sums, in the same wasy as is done in the case of Romberg quadrature with trapezoidal sums. Some numerical results are discussed in section 4.

The main result in this note is the following

Theorem: Let G be a finite solvable group. There exists a permutation σ of the set G such that {g • σ(g); g∈G} = G if and only if the Sylow 2-subgroup of G is non-cyclic or trivial

Let f(x) be a real analytic function of the real variable x and α a simple root of the equation f(x) = 0. It is well known that a function ϕ(x) can be associated with the equation in many different ways such that

1. (i) α is a root of the equation ϕ(x), i.e. α is a fixed point (invariant point) of the function ϕ(x);

2. (ii)

The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. A $t$-minimal graph is a graph of thickness $t$ that contains no proper subgraph of thickness $t$. In this paper, upper and lower bounds are obtained for the thickness, $t\left( G\,\square \,H \right)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t\left( G \right)$ and $t\left( H \right)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph ${{K}_{4k,\,4k}}$, the thickness of the Cartesian product of two complete bipartite graphs ${{K}_{n,n}}$ and ${{K}_{n,n}}$ is also given for $n\,\ne \,4k\,+\,1$.

Let G be a finite group. By a Moore G-space we mean a G-space X such that for each subgroup H of G the fixed point space XH is a simply connected Moore space of type (MH,n), where MH is an abelian group depending on H, and n is a fixed integer. By a co-Hopf G-space we mean a G-space with a G-equivariant comultiplication. In this note it is shown that, in contrast to the non-equivariant case, there exist Moore G-spaces which are not co-Hopf G-spaces.

A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener–Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump–Diaconis–Tracy–Widom identity is a differentiated version of the classical Jacobi–Trudi identity.

Le but de cette note est de donner une caractérisation purement catégorique des catégories de modules semi-simples sur un anneau. Nous allons voir que ces catégories sont précisément celles qui sont localement artiniennes au sens de P. Gabriel et dont le radical au sens de G. M. Kelly est nul.

This note originated in an attempt to generalize the assertion of Problem 164 which was proposed by Moser [2].

Let us first state our generalization.

Let T, P denote two continuous operators from E into F, where E and F are locally convex spaces. It is proved by L. Schwartz [8] and G. KÖthe [6] that if E and F are Fréchet spaces, T is a ϕ_-operator and P a compact operator, then T+P is a ϕ_-operator.

Precise, elegant evaluations are given for Gauss sums of orders six and twelve.

In this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.

We prove that every continuous function on the product of certain curves can be asymptotically approximated by entire functions.

We give an algorithm for a surgery description of a $p$-fold cyclic branched cover of ${{B}^{3}}$ branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

For k-step Markov chains, factors generated by finite length codes split off with Bernoulli complement when maximal in entropy. Those not maximal are relatively finite in another factor which generates or splits off.

These results extend to random Markov chains with finite expected step size, implying that random Markov chains with finite expected step size can have only finitely many ergodic components, each of which is isomorphic to a finite rotation, a Bernoulli shift, or a direct product of a Bernoulli shift with a finite rotation. This result limits the type of zero entropy factors which occur in random Markov chains with finite expected step size, providing a counterpoint to the work of Kalikow, Katznelson, and Weiss, who have shown that each zero entropy process can be embedded in some random Markov chain.

Extending Rudolph and Schwarz, random Markov chains with finite expected step size are limits in of their canonical Markov approximants. The -closure of the class is the Bernoulli cross Generalized Von Neuman processes.

Finitary isomorphism of aperiodic ergodic random Markov chains with finite expected step size is considered.

Applications are made to a class of generalized baker's transformations.