The object of this paper is to give generalizations of Okuyama's Theorem [Bull. Austral. Math. Soc.12 (1975), 9–21, Tôohoku Math. J. (2) 28 (1976), 563–581] on the absolute Nörlund summability factors of a Fourier series and its conjugate series.

Our theorems imply many results proved by other authors: especially Theorem 1 includes the results of Bhatt and Kishor [Indian J. Math.9 (1967), 259–267 (1968)], Dikshit [Pacific J. Math.63 (1976), 371–379], and Lal [Publ. Inst. Math. (Beograd) 20 (34) (1976), 169–178], and we can easily deduce Lal's result [Indian J. Math.16 (1974), 1–22] from our Corollary 2.

Best possible conditions are given here, under which all solutions of several delay differential equations are oscillatory.

This is a report on a family of varieties of algebraic systems that are groups with an additional unary operation satisfying certain laws. The study was inspired by the theory of Fibonacci groups, and the free one-generator algebras of our varieties turn out to have, in fact, the abelianized Fibonnaci groups as their underlying groups.

We prove that a cyclic extension of a free product of finite p-groups is residually finite, p a fixed prime. The question of cyclic extensions of free products of arbitrary cyclic groups remains open.

Let G = (X, U) be a digraph of order n. P(G) denotes the minimal cardinal of a path-partition of the arcs of G.

Oystein Ore, Theory of graphs (Amer. Math. Soc, Providence, Rhode Island, 1962) has proved that , where . We say that G satisfies Q if the preceeding inequality is an equality.

We give some properties of the digraphs satisfying Q, and in particular the case where G is strongly connected. Then we prove that P(G) ≤ [n2/4], and that this result is the best possible. Next we exhibit the existence of digraphs with circuits such that P(G) = [n2/4].

Finally we prove that if G is a strongly connected digraph of order n which satisfies Q, then there exists a strongly connected digraph H of order n + 1 having G as a sub-digraph and satisfying Q, too.

We generalize a result of Walter Rudin about the structure of a compact abelian group G for which C(G) + H∞(G) is a closed subalgebra of L∞(G).

After defining a generalized C-fraction (a kind of Jacobi-Perron algorithm) for an n-tuple of formal power series over (n ≥ 2), the connection between interruptions in the algorithm and linear dependence over [x] of the power series is studied.

Examples will be given showing that the algorithm behaves in a way similar to the Jacobi-Perron algorithm for an n-tuple of real numbers (the gcd-algorithm): there do exist n-tuples of formal power series f(1), f(2), …, f(n) with a C-n-fraction without interruptions but for which 1, f(1), f(2), …, f(n) is nevertheless linearly dependent over [x].

Moreover an example will be given of algebraic functions f, g of degree n over [x] (formally defined) for which the C-n-fraction for f, f2, …, fn has just one interruption and that for g, g2, …, gn 1 none, while of course 1. f, f2, …, fn and 1, g, g2, …, gn admit (only) one dependence relation over [x].

It has been shown by W.F. LaMartin that the Pontryagin dual of a k-space product of Hausdorff abelian k-groups is the coproduct of their duals. Here we offer a different proof, based partly on LaMartin's, that the dual of a product is the coproduct of the duals in a more general setting.

A refinement of the Newton-Kantorovich Theorem, which has many-potential applications in existence - uniqueness theory, is used to strengthen a result of Lancaster and Rokne concerning existence and uniqueness regions for zeros of operator polynomials.

A new construction of regeneration times is exploited to prove ergodic and renewal theorems for semi-Markov processes on general state spaces. This work extends results of the authors in Ann. Probability (6 (1978), 788–797).

In the paper by Aderogba and Blake [1], the first order approximation to the shape of the interface between two immiscible liquids, at which surface tension acted, was obtained in the case of a point force directed parallel to the interface. In the case of the normal force no solution was obtained due to the logarithmic singularity associated with problems of this type. However, the problem is not physically well-posed in the case of the normal force. Surface tension is not suitable because it cannot alone balance the induced stress on the interface due to a point force. We need an additional force to balance the action of the point force on the interface. The obvious solution is to include a density difference Δρ* between the two fluids

where and are the densities of the lower and upper fluid respectively.

The last 3 references in the author's note [1] should read as follows:

[4] V. Sree Hari Rao, “On boundedness of impulsively perturbed systems”, Bull. Austral. Math. Soc. 18 (1978), 237–242.

[5] V. Sree Hari Rao, “Asymptotically self invariant sets and stability of measure differential equations“, Nonlinear Anal., Theory, Methods Appl. 2 (1978), 483–489.

[6] V. Sree Hari Rao, ”Note on conditionally asymptotically invariant sets“, J. Mathematical and Physical Sci. 12 (1978), 469–471.