The expectation, variance and covariance for different states of a k-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve the k latent roots of the determinantal equation. As it is not easy to determine the latent roots when k > 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).

Recent workers [1, 3] have proved density theorems about the rational points on K3 surfaces of the form

$$ V:\;X_0^4+cX_1^4=X_2^4+cX_3^4 $$

Let O denote the origin of ℝn (where n ≥ 2), let B(X, r) denote the open ball in ℝn of centre X and radius r, and define φn: (0, + ∞) → ℝ by φn(t) = t2-n if n ≥ 3 and φ2(t) = log (1/t). A sequence (Xk) in B(O, 1) is said to converge regularly to O if Xk → O and there is a constant k > 1 such that φn(|Xk+1|) < Kφn(|Xk|) for each k ∈ ℕ. If n ≥ 3, this is clearly equivalent to saying that there is a constant k′ ε (0, 1) such that |Xk+1| > k′|Xk| for each k (cf. [18], p. 149).

All -simple corank-1 germs from to , where n ≠ 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image.

1. Introduction. In this note, we consider some estimates for the modulus of infinite products of the form

where the lv are near to an arithmetic progression of common difference unity. In a subsequent note, an application will be given of these estimates to the study of a class of Cauchy exponential series which includes, as a special case, the series of Korous(l).

Throughout this paper, G will denote a locally compact, non-discrete, Abelian group (subjected to various conditions) and X wi11 denote the character group of G. All terminology and notation are as in (7). The measure algebra M(G), as is known, is a very complicated entity. We address ourselves here to some novel peculiarities of the subspace Ms(G) of continuous measures in M(G) that are singular with respect to Haar measure λ.

We study some fine arithmetic properties of the components of solutions of a decomposable form equation. Lower growth rates for the greatest prime factor of a component are obtained for density 1 of the solutions. Also, high pure powers are shown to occur rarely. Computations illustrate the applicability of our results; for example, to the study of units in abelian group rings.

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined by

where is given by

A β-ring is supplied with operations βH where Hruns over the conjugacy classes of subgroups of the symmetric groups Sn. In an earlier paper we introduced a second set of operations λH and we show here that the two sets are related by the isomorphism

formula here

We then consider the operations βH and λH as combinatorial species, in the sense of Joyal, and express their molecular decomposition as a finite sum of products of the exponential species with molecular species of degree at most n. We give combinatorial interpretations for βSn-structures and λSn-structures and derive various species isomorphisms.

In this paper we investigate the relationship between genus and localization of finitely generated (torsion-free abelian)-by-finite groups and genus and localization in integral representation theory. In particular when such a group has finite commutator subgroup we can determine the genus exactly.

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

It is proved that every locally convex space is linearly homeomorphic to a subspace of a space which admits no non-trivial continuous linear functionals.

We determine the stable decompositions of the classifying spaces of metacyclic 2-groups into wedges of indecomposable spectra. The stable decompositions of classifying spaces of finite groups with metacyclic 2-Sylow subgroups are also determined.

Williams ((3)) has transformed certain series equations in Legendre functions to Bessel integral equations; his main result being that if

then

and vice versa. (Here, n is written for Williams's n + m. The coefficients an are related to Williams's An by a factor. There is an extra factor 2 in equation (7) in Williams's paper which appears to be an error.)

We give examples of isospectral non-isometric surfaces of genus two and three with variable curvature, as well as hyperbolic orbifolds of genus two. We apply the first result to construct isospectral potentials on a Riemann surface of genus two.

The differential equation

which is of importance in boundary-layer theory, has recently been discussed in detail for all non-negative values of the parameter λ(1). I wish to point out here that when λ = − 1 the equation can be solved completely in terms of known functions.

Let A denote the mod 2 Steenrod algebra. Let ℤ2[x, x−l] be the (graded) ring of finite Laurent series over ℤ2 in the variable x with dim (x) = 1. ℤ2[x, x−1] is a module over the Steenrod algebra A by

where are binomial coefficients modulo 2 and m > 0 is large compared with |k| and i. Let M be the A-submodule of ℤ2[x, x−1 ] generated by all powers xi with i ≠ −1. It is easy to see that ℤ2 [,x, x−1]/M ≅ σ−1ℤ2 (means ℤ2 on dimension − 1). Let ρ: ℤ2[x, x−1] → σ−1 ℤ2 be the projection map.