In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisation

where P1, …, Pr are distinct irreducible polynomials (i.e. primes).

This paper is devoted to the study of rooted properties of phase surfaces defined by complex analytic systems. We first obtain the Rooted Theorem of Analytic Systems. Then we prove the Generalized Strong Rooted Theorem of (m ≧ 2), which implying the Strong Rooted Theorem of a Class of .

In [2], R. Arthan and S. Bullett pose the problem of representing generators of the complex bordism ring MU* by manifolds which are totally normally split; i.e. whose stable normal bundles are split into a sum of complex line bundles. This has recently been solved by Ochanine and Schwartz [8] who use a mixture of J-theory and surgery theory to establish several results, including the following.

The kernel–trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t1, k, tr,; Tt, K, Tr; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {ωt1, ωk, ωtr, εTt, εK, εTr} where ω and ε denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {εK, ωk, εT, ωt}.

1. The following note examines, more closely than the text books usually do, some results of reciprocation, and has special reference to properties of a certain infinite set of triangles and connected conies.

While Newton's Theorem on the Sums of Powers of the Roots of an equation furnishes a set of lineo-linear equations connecting the quantities s1, s2, s3, … and the quantities p1, p2, p3, … Waring gives the solution of these equations by which the s's are expressed in terms of the p's.

Giffard's injector appeared more than thirty years ago. The first serious attempt to explain its action on dynamical principles was made by the late William Froude at the Oxford Meeting of the British Association in 1860. The history of mechanical science is almost everywhere deeply marked by Rankine; and it seems, just as it ought to be, that he should be found to have contributed not a little to the literature of this particular subject in a paper presented to the Royal Society of London in 1870. As serving to show how far the problem is still interesting, even from a high standpoint, attention may be directed to the exceedingly curious procedure of Professor Greenhill, where he deals cursorily with the matter at the page numbered 448 of his article on Hydromechanics in the Encyclopædia Britannica.

1. If x, y, z are the distances of the point P from the sides BC, CA, AB of a triangle, to find P so that the product xyz may be a maximum.

A simple, but nice theorem of Banach states that the variation of a continuous function F:[a, b]→ ℝ is given by where t(y) is defined as the number of x ∈ [a, b[ for which F(x)= y (see, e.g., [1], VIII.5, Th. 3). In this paper we essentially derive a similar representation for the variation of F′ which also yields a criterion for a function to be an integral of a function of bounded variation. The proof given here is quite elementary, though long and somewhat intriciate.

Let O1, and H1 be two points, in the plane of any triangle of reference ABC, so related that if 01P, O1Q, O1R be the perpendiculars drawn to the sides of ABC, then AP, BQ, CR meet in H1. We shall find that O1, and H1 describe respectively two cubics which are related to each other in a remarkable manner. We shall show, for instance, that points in each curve may be derived from each other by two sets of three alternative rational quadric transformations, and that the join of correspondents passes through a fixed point as in plane projection. We shall then discuss the homographic relation between corresponding pencils formed by rays through pairs of related points—not direct correspondents—and investigate the relation between these latter points.

Let s, sn(n = 0, 1, …) be arbitrary complex numbers, and let

be a polynomial, with complex coefficients, which satisfies the normalizing condition

Associated with such a polynomial is a Nörlund method of summability Np: the sequence {sn} is said to be Np-convergent to s, and we write sn →s (Np), if