Solutions for the temperature distribution in a circular pipe have been given by various authors, notably Gretz, Nusselt, Goldstein; all these are cited in (1), section 266. I have already considered in one paper the temperature distribution in a circular pipe when viscous incompressible fluid is flowing through the pipe and rate of heat addition is an oscillatory function of time. In this paper expressions for the temperature distributions are derived when viscous incompressible fluid is flowing through the channel, the dissipation due to friction is neglected and the rate of heat addition is an exponential function of time. From the engineering point of view this situation has some importance.

In this paper we consider the problem of dividing a square of side N into squares of integral side. When we demand that these be unequal, the problem is soluble only for certain N, and has been treated in some detail by various authors (1,3,4,5). If we relax this condition, every side is permissible, for a square of side N may plainly be divided into N2 squares of side 1. It is the object of this paper to reduce this number: in fact we shall show that a square of side N is divisible into some number less than 6 ∛N + 1 squares of integral side.

A set Q is quasi-ordered if a reflexive and transitive relation ≤ is defined on Q. It is well-quasi-ordered if it is quasi-ordered and, for every infinite sequence u1, u2,… of elements of Q, there exist i, j such that i < j and ui ≤ uj. A lower set of Q is a subset P of Q such that, if x ≤ y ∈ P, then x ∈ P. The class of lower sets of Q, quasi-ordered by ⊂, is denoted by LQ, and L2Q = L(LQ), etc. The set (, say) of all finite trees is quasi-ordered by writing T1 ≤ T2 if T1 is homeomorphic to a subtree of T2. It is proved that is well-quasi-ordered for all n.

It is well known that, with respect to the natural partial ordering, the set of all congruences on a group forms a modular lattice. In the present paper we develop an extension of this result to the case of a regular semigroup S (α ∈ αSα for all α in S). Let Σ(ℋ) denote the set of all congruences on S with the property that congruent elements generate the same principal left ideal and the same principal right ideal of S. We show (Theorem 1) that, under the natural partial ordering, Σ(ℋ) is a modular lattice with a greatest and a least element.

1. Introduction. Consider a set P with elements a, b, c,… and a certain distinguished element e. Assume that P satisfies the following conditions.

(I) With each ordered pair (a, b) of distinct elements a and b of P there is associated a unique element of P (to be denoted by ab and called the product of a and b). Further, with the pair (e, e) there is associated a unique element ee of P.

(II)(ab)c = a(bc) for all elements a, b, c in P for which both sides are defined.

(III) ea = a = ae for all elements a ∈ P.

(IV) To each element a ∈ P, there corresponds an element a′ ∈ P such that

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements of

corresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k2-bundles on a quadric surface Q = P1 × P1 defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k2-bundles on P1 × … × P1 (n factors) is determined by that of reducible k2-bundles on P1 × P1. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k2-bundles (but with a distinguished factor of the product variety P1 × … × P1) is as nice as it could possibly be, the classification of reducible k2-bundles alone is as nasty as it could possibly be. Since every decomposable k2-bundle is just a sum of two line bundles, we consider only indecomposable reducible k2-bundles.

1. Introduction. Let ℛn denote the set of all n × n matrices with real elements, and let denote the subset of ℛn consisting of all real, n × n, symmetric positive-definite matrices. We shall use the notation to denote that minor of the matrix A = (aij) ∈ ℛn which is the determinant of the matrix

The Schur Product (Schur (14)) of two matrices A, B ∈ ℛn is denned by

where A = (aij), B = (bij), C = (cij) and

Let ϕ be the mapping of ℛn into the real line defined by

for all A ∈ ℛn, where, as in the sequel, .

1. The following results are special cases of theorems of Irwin and Zeeman, respectively (see (4), (9), (10)).

(a) Let V be a (2n − m + 1)−connected piecewise-linear m−manifold (bounded or not), where m − n ≥ 3. Then any element of πn(V) can be represented by a piecewise-linear imbedding of Sn in V.

(b) Let M be a (2n − m + l)-connected closed piecewise-linear (m − 1)-manifold, where m − n ≥ 3. Then two piecewise linear imbeddings of Sn−1 in V are isotopic if and only if they are homotopic.

Let f: X → X be a mapping of the metric space X into itself. We call f non-expansive if

0. Introduction. When f: T1 → T2 is a mapping between two topological spaces we shall say that f is sequentially continuous if and only if in T1 implies that in T2.

In this paper we shall be mainly concerned with the following three apparently widely differing questions.

(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?

In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.

(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)

Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.

(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?

1. Let AB be a plane simple arc, which, considered as a point-set, has a positive two-dimensional measure. If x be a point of AB, a point x′ ≠ x is said to be to the left or to the right from x according as it belongs to the arc Ax or xB. The upper and the lower limits of the ratio [m2{Ax C(x, r)}]/πr2, where C(x, r) denotes the closed disc with centre x and radius r, as r → 0 define the upper and the lower left densities at the point x. When the two densities are equal their common value defines the left density. The right densities are defined similarly.

Let T be a compact linear operator acting in a complex Hilbert space H. Then there is a simple resolution of the identity {Eλ} in H which reduces T (for a proof of this statement, and for definitions of the terms used, see (l), section 5, especially Theorem 6). In the present paper we give constructions for the resolvent of T, and for a complete set of principal (that is, eigen and adjoined) vectors, in terms of T, {Eλ}, and the diagonal coefficients {αλ} of T. These constructions require no technique more involved than the expansion of a resolvent operator in a Neumann series.

1. Introduction and summary. In many problems in theoretical physics one encounters situations in which it is desirable to expand an orbital centred at one point about another point.

1. The main object of this note is to evaluate two definite integrals involving the product of Meijer's G-function with Gauss's hypergeometric function. The first result established in this paper is the extension of the results recently given by Saxena ((3), page 490) in these proceedings and includes both of his results given there as particular cases.

1. Introduction and summary. There has been considerable interest in the problem of expanding a Coulomb potential l/r12 in spherical polar coordinates defined relative to two centres A(0, 0, − d) and B(0, 0, + d) where 2d = D = rAB.

We shall be concerned in this paper with a class of temporally homogeneous Markov processes, {R(t), X(t)}, in discrete or continuous time taking values in the space

The marginal process {X(t)} in discrete time is, in the terminology of Miller (10), a sequence of random variables defined on a finite Markov chain. Probability measures associated with these processes are vectors of the form

where

We shall call a vector of the form of (0·2) a vector distribution.

1. Introduction. The stability of a system of queues in series was investigated in (1) but the results obtained were not quite complete, for the situation in which all component queues are substable and at least one is critical was scarcely touched upon. The only reference to this situation, in fact, was an example on page 513 showing that necessary and sufficient conditions are not obvious. (The ideas and terminology of (1) will be used freely throughout this paper.)

It is assumed initially that only the increase of entropy defines locally the sense of advance of time. This assumption, together with the feature of global coherence of statistical processes, which is provided by the cosmic line-element, imply that there is a unique way of associating the standards with the cosmic statistical processes. There is hence a connexion between the sense of divergence of geodesics of fundamental observers and the cosmic time-sense. This is in keeping with the unmodified conservation laws of gravitation and the analogous adiabatic conservation law quite generally without dependence on the explicit form of the line element. These remarks correspond exactly to consequences of the ‘elliptic’ model in the case of de Sitter space, obtained by entirely separate geometrical considerations. A remark is made with regard to the irreversibility associated with quantum measurement.