In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalises a theorem of Mazhar [7] on |C, 1|k summability factors of infinite series, is proved. Also we apply the theorem to Fourier series.

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

The author and Ira J. Papick have termed an integral domain R a going-down ring if R ⊂ T satisfies going-down for each domain T containing R. The present paper investigates conditions which, for an integral extension A ⊂ B of domains, imply that A (respectively B ) is going-down whenever B (respectively A ) is going-down. This explains the “descent” (respectively “ascent”) in the title. Two typical results (the first about descent, the second about ascent) are given next.

Each semiprime ring admits a unique projectable, strongly projectable, laterally complete and orthocomplete hull. Almost all of the theory for X–hulls of lattice-ordered groups in Paul Conrad, “The hulls of representable l-groups and f-rings”, J. Austral. Math. Soc. 16 (1973), 385–415, has a counterpart for semiprime rings. The proofs of these results will appear elsewhere. They come in a large part directly from the corresponding theory for lattice-ordered groups. There is also a feedback from the rings to the groups.

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

A regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the Cauchy space in a complete function algebra with the continuous convergence structure.

The oscillatory behaviour of the solutions of a functional differential equation, of more general form than an equation arising from in industrial problem, is examined. The question of whether one can maintain the oscillatory properties of this equation by adding a forcing term is also answered. The results obtained extend already known results on the subject and complete the relevant literature.

Let S and T be commuting operators of type ω and type ϖ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S, T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ϖ < π. We show that this operator is always of type μ where μ = max{ω, ϖ}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.

K.A. Bush has asked whether there is a symmetric Hadamard matrix of order m2, m even, which can be partitioned into an m × m array of m × m blocks, such that:

(i) each diagonal block has every entry 1;

(ii) each non-diagonal block has every row-sum zero?

We give two ways of constructing such matrices.

Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovskii's theorem, for every positive integer n with m → n in the Sharkovskii ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question.

Let P(z) be a polynomial of degree n and P′(z) be its derivative. Given a zero of P′(z), we shall determine regions which contains at least one zero of P(z). In particular, it will be shown that if all the zeros of P(z) lie in |z| < 1 and W1, W2, …, Wn−1 are the zeros of P′(z), then each of the disks |(z/2)–wj| < ½ and |z–Wj| < 1, j = 1, 2, …, n−1 contains at least one zero of P(z). We shall also determine regions which contain at least one zero of the polynomials mP(z) + zP′(z) and P′(z) under some appropriate assumptions. Finally some other results of similar nature will be obtained.

We consider category theory enriched in a locally finitely presentable symmetric monoidal closed category ν. We define the ν-filtered colimits as those colimits weighted by a ν-flat presheaf and consider the corresponding notion of ν-accessible category. We prove that ν-accessible categories coincide with the categories of ν-flat presheaves and also with the categories of ν-points of the categories of ν-presheaves. Moreover, the ν-locally finitely presentable categories are exactly the ν-cocomplete finitely accessible ones. To prove this last result, we show that the Cauchy completion of a small ν-category Cis equivalent to the category of ν-finitely presentable ν-flat presheaves on C.

Let $(a, b, c)$ be a primitive Pythagorean triple satisfying ${a}^{2} + {b}^{2} = {c}^{2} . $ In 1956, Jeśmanowicz conjectured that for any given positive integer $n$ the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $x= y= z= 2. $ In this paper, for the primitive Pythagorean triple $(a, b, c)= (4{k}^{2} - 1, 4k, 4{k}^{2} + 1)$ with $k= {2}^{s} $ for some positive integer $s\geq 0$, we prove the conjecture when $n\gt 1$ and certain divisibility conditions are satisfied.

An asymptotic formula is derived for the number of partitions of a large positive integer n into r unequal positive integer parts and maximal summand k. The number of parts has a normal distribution about its maximum, the largest summand an extreme-value distribution. For unrestricted partitions the two distributions coincide and both are extreme-valued. The problem of joint distribution of unrestricted partitions with r parts and largest summand k remains unsolved.

A new coincidence point theorem is proved for a pair of multivalued mappings operating between G-convex spaces. From this theorem, a generalisation of the classical Fan-Glicksberg fixed point theorem is established.

Dedicated to George Szekeres on the occasion of his 90th birthday

MacMahon devoted a significant portion of Volume II of his famous book Combinatory Analysis to the introduction of Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. In a series of papers we have shown that MacMahon's method turns into an extremely powerful tool when implemented in computer algebra. In this note we explain how the use of the package Omega developed by the authors has led to a generalisation of a classical counting problem related to triangles with sides of integer length.

There is a skew-Hadamard matrix of order 92.

Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below. The orders < 200 which are now undecided are 100, 116, 148, 156, 172, 188, 196; see [2], [3]. The existence of any Hadamard matrix of order 92 was unknown until 1962 [1].

For any pair of integers p, q such that (p, q) = 1 and p ≥ 2q, the graph has vertices {0, 1, …, p − 1} and edges {ij: q ≤ |i − j| ≤ p − q}. These graphs play the same role in the study of circular chromatic number as that played by the complete graphs in the study of chromatic number. The graphs share many properties of the complete graphs. However, there are also striking differences between the graphs and the complete graphs. We shall prove in this paper that for many pairs of integers p, q, one may delete most of the edges of so that the resulting graph still has circular chromatic number p/q. To be precise, we shall prove that for any number r < 2, there exists a rational number p/q (where (p, q) = 1) which is less than r but arbitrarily close to r, such that contains a subgraph H with and . This is in sharp contrast to the fact that the complete graphs are edge critical, that is, the deletion of any edge will decrease its chromatic number and its circular chromatic number.

In this paper, we shall prove a generalisation of Himmelberg's fixed point theorem and as applications, the existence of equilibrium points for abstract economies given by preference correspondences and utility functions have been established.

Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r, and R respectively. We establish a number of best possible upper bounds for (w−2r)d, (w−2r)R,(w−2r)p, (w−2r)A in terms of w and r. Examples are: