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LetGF(q) be the finite field containingq=plelements, wherepis a prime andla positive integer. LetP(x)be a monic polynomial inGF[q, x] of degreem. In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types:(I)P(xr), whereris a positive integer (r-polynomials).(II)xmP(x+x−1). (Reciprocal polynomials.) These have the form(III)xrmP(xr+x−r). (r-reciprocal polynomials.) These have the formQ(xr), whereq(x) satisfies (1·1).

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

When n points are distributed at random (equally likely anywhere) over a large area or volume, they may form small clusters or aggregates. In this paper a k-aggregate is defined more precisely as k points which can be covered by a small figure of given size, shape and orientation, but whose position is not otherwise specified. Exact formulae for the expected number of such k-aggregates are given in some cases, while a good approximate formula (accurate in most practical cases) and limits are stated, covering all cases.

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.

The object of this paper is to extend the range of validity of Ingham's Tauberian theorem for partitions (2). Ingham's theorem deals with many types of partitions, but it cannot be used in some cases of importance. With a slight modification of Ingham's notation we definewhere P(u) is the number of solutions ofin integers ni ≤ 0, and the λi are a given set of numbers, not necessarily integers, such that 0 < λ1 < λ2 N(u), the number of λi not exceeding u, is given bywhere B > 0, β > 0, and as u → ∞then, according to Ingham's theorem, when u → ∞where

Throughout this paper q1,q2,… are positive and x1,x2,… non-negative real numbers. When q1+…+ qn = 1, we define Δn, Sn and Σn as follows:andWe use the term ‘comparable’ in Kober's(2) sense. Two functions f(x1, …, xn) and g(x1, …, xn) are comparable if and only if αg ≤ f ≤ βg for some positive constants α and β.

1. In this paper I discuss the expression of the general form F of order n, homogeneous in r + 1 variables z0, z1,…, zl,…, zr, as the sum of the nth powers of h + H linear forms in these variables. I take h (> 0) of these linear forms to be undetermined, namely the formswhose coefficients are undetermined; and I take the remaining H(≥0) linear forms to be assigned, namely the formswhose coefficients are assigned.

Let c0, c1,…, cn,…be a sequence of real constants, and ƒ0(x), ƒ1(x),…, ƒn(x),… a sequence of functions defined, for example, in the interval (0, 1). In this paper we shall investigate some of the properties of the serieswhich may be obtained from the standard seriesby interchanging the signs of the terms in a quite arbitrary way.

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

In a recent paper (1) Nagata proved that a (linear associative) algebra, not necessarily of finite dimension, over a field of characteristic 0 which satisfies the identical relation xn = 0 satisfies also the relation x1x2 … xN = 0, where N is an integer depending only on n. He remarked further that it is a corollary that the result remains true if the ground field is of prime characteristic p, provided that p is large enough compared with n; and he conjectured that the obviously necessary condition p > n is in fact sufficient. The object of this note is to prove Nagata's conjecture. To do this, we give a new proof of his theorem, and as a by-product we obtain a rather better bound for N than his, showing, namely, that we can take N = 2n − 1. The determination of the best possible value of N, or even of its order of magnitude, seems not to be easy; at any rate, the best I have been able to do in the opposite direction is to show that for large n we cannot take N as small as n2/e2, where e is the base of natural logarithms.

In the second part of this paper I pointed out how the presence of any singular point or points of the kind considered in the first part, on a rational surface, corresponds to a subgroup of the group of symmetry of the polytope which represents the properties of the system of base points in the plane representation of the surface. The subgroup is generated by reflexions, and may be the direct product of one or more factors (all the primes of symmetry in one factor being perpendicular to all those in any other factor). Each factor corresponds to a singular point on the surface, namely (in Coxeter's notation), a factor [ ] to a conic node C2, a factor [3n] to a binode Bn+2 (n ≥ 1), a factor [3n,1,1] to a unode Un+5, (n ≥ 1), and finally a factor [3n,2,1] to a unode The possible subgroups in the finite groups that arise have been enumerated by Coxeter; and we shall find that every subgroup generated by reflexions in the group of symmetry of the polytope in ε dimensions which represents ε base points corresponds to a possible configuration of the base points, in which just those rational curves of grade − 2 are actual which correspond to primes of symmetry belonging to the subgroup; without exception, for ε ≤ 6; with one exception—the subgroup [ ]7—for ε = 7; and with three exceptions—the subgroups [ ]7, [ ]8, and [31,1,1 × [ ]4—for ε = 8. Since moreover the system |k| of cubics passing simply through all the base points is in all these cases an actually existing system, for which all the rational curves of grade − 2 are fundamental, its projective model (or in the case ε = 8, in which |k| is only a pencil, the projective model of the system |2k|) provides a rational surface on which all the sets of curves corresponding to the subgroups in question actually appear as singular points.

Introduction and summary: In the distribution of the sum of n random variablesfor which we suppose , the tail probability will be called an extreme tail probability if the normal approximationis poor, when is asymptotically normal and n is large. This paper concerns the specialization of (1.1) in whichwith a random permutation of nλ ones and n − nλ zeros, that is, in which T is the total for a simple random sample from the population with sampling fraction λ. In this case, we will always write T as Tλ. For brevity, we may write {c1,…, cn} for .

The object of this paper is to make more precise the results given by Enriques-Campedelli for canonical surfaces, that is, surfaces whose prime sections are the canonical curves. We consider the case of regular surfaces in space of three dimensions (pg = pn = 4), and of regular multiple planes (pg = pn = 3).

1. Making use of the familiar abbreviationlet us adopt a contracted notation for the generalized hypergeometric function AFB(x) and writewhere (a) denotes the sequence of parameters a1, a2,…, aA. It will be assumed throughout the present paper that there are A of the ‘a’ parameters, A′ of the ‘a′’ parameters, and so on. Thus [(a)]m is to be interpreted aswith similar interpretations for [(a′)]m, etc.

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A. A reduction of a is an ideal b contained in a which satisfies ar+1 = arb for all sufficiently large r. This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m, and that a is m-primary. It is well known (Samuel (10)) that the length of the ideal an is, for large values of n equal to Pa(n) where Pa(n) is a polynomial in n whose degree d is equal to the dimension of A. If we write the coefficient of nd in Pa(n) in the form e(a)/d!, e(a) is a positive integer termed the multiplicity of a. If now b is a reduction of a, then b is also m-primary, and e(b) = e(a).

Several papers have recently been published which include proofs of the following:Theorem 1. Those trisecant planes of a rational normal quartic curve Cu which meet a second rational normal quartic curve Cv having six points in common with Cu also meet a third rational normal quartic curve Cw through these same six points. The three suffixes may be permuted in any way, the three curves forming a symmetrical set.

Some recent work by the authors (1) on the distribution of the residues of a cubic polynomial modulo an odd prime p led to the conjecture that, in general, two cubic polynomials with integer coefficients possessing the same residues modulo p (not necessarily occurring to the same multiplicity) are equivalent, that is are related by a linear transformation modulo p. The purpose of the present paper is to prove this conjecture. We establish the following theorem.

The object of this paper is two-fold: first to collect together the known facts about combinatorial cobordism in general, and then to calculate the groups for the first 8 dimensions. As in (29), we shall denote the unoriented and oriented cobordism groups in dimensionnbyand Ωn, and will distinguish the combinatorial from the differential case by affixesc, d.

The object of the present paper is to evaluate m(k), the invariant measure (defined in §2) of the set of real matrices with determinant 1 and norm bounded by a positive real number k. The norm is that induced on the linear transformations of Euclidean n-space by the usual Euclidean norm in that space, and is also defined in § 2. We were led to this problem as a result of work arising from the Geometry of Numbers. A knowledge of the behaviour of m(k) was necessary in order to generalize a certain mean value formula (Macbeath and Rogers (9)) from Riemann to Lebesgue integrable functions.

The incitement, or provocation, to assemble the following pages was the recognition in a recent paper (5) of a geometrical property characterizing those pairs, in a pencil of quadric surfaces, such that the quadrics of a pair are both outpolar and inpolar to each other: symmetrically apolar let it be said. That there are three such pairs was known (8), but their identification by means of the principal chords of their common curve had apparently not been remarked upon. The question naturally arose as to the number and identification of such pairs of quadrics of a pencil in n-dimensional projective space [n]; it is easy to show that the number is ½n(n–1), but geometrical identification seems quite another matter and no success whatever has yet been achieved even when n is 4.

AK-space (E, τ) is a linear spaceEof sequences with a locally convex topology τ for which the inclusion map: (E, τ) → (ω, product topology) is continuous. In (2) topological properties ofK-spaces were determined directly from properties of the spaceEand the topology τ. It is, however, very natural to consider duality properties ofK-spaces and the purpose of this paper is to determine some of these properties.

Suppose that t is a group homomorphism from a topological group E into a topological group F. When is it true, that the closure of t−1(V) in E is a neighbourhood of the identity in E for every neighbourhood V of the identity in F? This question arises naturally in the study of the closed graph theorem in the context of topological groups; for example, see (1) and ((3), p. 213). The concept of a g-ultrabarrelled space introduced in this paper is the result of an investigation aimed at answering this question.

The following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequencesare uniformly distributed on the circle, for almost all x in G.

This third and last part of the paper is concerned with the interpretation of the Schläfli symbol {k1, k2, …, km−1} when the k's are unrestricted. It is shown that, whenever the k's are integers (greater than 2), the symbol represents a polytope in a generalized space having time-like as well as space-like dimensions.

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.

Let (E, τ) be a complete, semi-metrizable topological vector space. Let p be a pseudo-norm (not to be confused with a semi-norm, cf. (8), p. 18) inducing the topology τ. For each positive real number r, letLet f be a continuous linear function from E into a topological vector space F. The open mapping theorem of Banach may be stated as follows: If f is nearly open, that is, if the closure of each f(Vr) is a neighbourhood of O in F then whenever β > α > O; in particular, each f(Vr) is a neighbourhood of O. We note that f, identifying with its graph, is a closed linear subspace of the product space E × F. In this paper, we shall employ techniques developed by Kelley (6) and Baker (1) to extend the theorem to the case where f is taken to be a closed cone in E × F. The generalized theorem throws some light onto the duality theory of ordered spaces. In particular, the theorem of Andô–Ellis is generalized to (not assumed, a priori to be complete) normed vector spaces.

The present paper is concerned with particular cases, obtained by suitably restricting the spaces involved, of the following general problem.Given a topological space X, we ask whether there exist integers n ≥ 2 and non-contractible spaces X1, …, Xn such that X has the homotopy type of the Cartesian product X1, × … × Xn or of the union X1, v … v Xn.

In this paper it is shown how all the regular polytopes (including the Kepler-Poinsot polyhedra and Hess's analogous star polytopes in four dimensions) arise as rational solutions of trigonometrical equations.

In this paper we have evaluated an integral involving Fox's H-function and Jacobi polynomial and employed its particular cases to establish two expansion formulae for the H-function involving Jacobi polynomials.

Introduction. In his paper ((1)), Baumslag has shown that the wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group, the prime p being the same for both groups. Liebeck ((3)) has obtained the exact nilpotency class of A wr B when A and B are Abelian. Let A be an Abelian p -group of exponent pn and let B be a direct product of cyclic groups, whose orders are pβ1, …, pβn, with β1 ≤ β2 ≤ … βn. Then A wr B has nilpotency class .

In two classical papers (1, 2) J. M. Whittaker introduced the study of integral functions bounded at the lattice points m + in(m, n = 0, ± 1, …,). He succeeded in showing (cf. also G. Polya(3)) that an integral function of at most the minimum type of order 2 uniformly bounded at the lattice points was necessarily constant. This result was improved almost simultaneously by A. Pflüger(5) and V. Ganapathy Iyer(11), who showed that the result was true also for functions of type Kz) function shows that theirs is a best possible result in this direction.

Introduction. Two previous papers (1,2) have dealt with additive processes defined on finite Markov chains. Such a process in discrete time may be treated as a bivariate Markov process {R(k), X(k)}. The process R(k) is an irreducible Markov chain on states r = 1, 2, …, R governed by a stochastic transition matrix B0 with components brs. The marginal process X(k) ‘defined’ on the chain R(k) is a sum of random increments ξ(i) dependent on the chain, i.e. if the ith transition takes the chain from state r to state s, ξ,(i) is chosen from a distribution function Drs(x) indexed by r and s. The distribution of the bivariate process may be represented by a vector F(x, k) with componentsThese are generated recursively by the relationwhere the increment matrix distribution B(x) has components brsDrs(x). We denote the nth moment of B(x) by Bn = ∫xndB(x), so that B0 = B(∞).

In a recent paper (3) Diananda has given an extension of the central limit theorem to stationary sequences of random variables {Yt} (t = 1,2,3,…), which are m–dependent, i.e. are such that m + 1 is the smallest integer r having the property that two sets of the Y's are independent whenever the suffix of any member of one set differs from that of any member of the other set by at least r. As Diananda points out, this can be used to show that for linear autoregressive processes with m–dependent residuals, the joint distribution of any finite number of the forms which occur in Bartlett and Diananda's goodness of fit test (see (1)) is asymptotically multivariate normal, provided also that the fourth moment of the residuals is finite. These forms are linear functions of the sample serial correlations rt, and if the autoregressive process is defined bythey may be denoted by , whereEt being the usual shift operator such that for any function ft defined for integral t, In particular, when , as for a moving average process, the joint distribution of any finite number of the rt, is asymptotically multivariate normal.

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) isso thatUsing this relation, Churchhouse has shown thatandwherewithMoreoverand b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,andThe object of this note is to prove these conjectures.

This paper is a sequel to one recently published, in which the regularity of surfaces was investigated by means of certain criteria suggested by Comessatti. We suppose throughout that the surface F under consideration is of general type in Sr, and that it has curve sections C of order n and genus π. It may be recalled that in the previous work the two most useful criteria were the following theorems:If π < ½ (n + 2), then F is regular or referable. (I)If πr (n) ≤ π ≤ n, where πr (n) is the maximum genus of a Cn in Sr, then F is regular or referable. (II)

A graph G is a pair (V{G), E(G)); V(G) is the set of vertices of G, and E(G) the set of edges of G; E(G) is a subset of the Cartesian product V(G) × V(G) (all pairs (υi, υj) where υi∈ V(G), υj ∈ V(G)). If υ is a vertex of G we shall write υ ∈ G. The graph-theoretic terminology used in this paper is that of Berge (1). We shall consider only graphs which are finite, undirected, and without loops.

The following note was suggested by an interesting paper written by F. P. White, where many references are given. It refers to a theorem given by W. F. Meyer, by whom the proof is indicated as possible by generalization of an intricate analytical proof given by him for a simple case. His result is that if on the rational curve of orderr, in space [r], say the curvecr[r], there be an involution ∞k, of sets ofmpoints, expressed, suppose, by an equationthen the spaces [r− 1], formed fromrpoints of any one of the polyhedra ofmpoints, are an aggregate ∞kof primes of this space [r], which is of class (m−k,m−r), the notation (p, q) meaning the binomial coefficientp! /q!(p − q)!. By Meyer, the conditionsk

In this paper upper bounds on the Engel length of a group G of prime-power exponent are obtained, under varying conditions on a normal subgroup H and the quotient G/H. Examples are constructed to show that the bounds are best possible when the nilpotency class of H is less than the relevant prime.

It is known that, for n ≤3, ‘ought to have’ can truthfully be replaced by ‘has’ (see (4), and (5), Cor. 6·6). In this paper, this conjecture will be proved for n ≥ 5. The only unsolved case then will be in dimension four.

Let Q1,…, Q5 be five fixed points (no four coplanar) of the real projective space S3: let s be a variable quadric surface through these points. The set of all such quadrics can be represented by the points of a real S4, in which there is a quartic primal that represents cones. The geometry of this threefold is well known in the complex case, but has hardly been considered at all in the real case: and one object of the present paper is to describe the real threefold and determine its homology groups.

1. Discussion of results1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.

1. Letbe a numerical series. If for sufficiently small h > 0 the seriesis convergent, we can form the upper and lower limits of J (h) as h → 0. These limits are called respectively the upper and lower sums (R, 1) of the series (1). For the purposes of the present paper it will be convenient to consider a more extended definition of these upper and lower sums. We shall suppose that for sufficiently small h the series J (h) is summable by Poisson's method. We denote the Poisson sum by PJ (h). The upper and lower limits of PJ (h) as h → 0 will be called the upper and lower sums (R, 1) of the series (1).

The problem of evaluating the non-homogeneous critical determinant of the plane region |y − x2| ≤ 1 is solved as a particular case of a result concerning the non-homogeneous properties of convex regions proved in the author's thesis (not yet published). The search for an alternative method of proof led to a proof of the existence of an infinite sequence of minima analogous to the Markoff chain. This is the subject of the present paper.

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled‘Properties ofp(n) and τ(n) defined by the relationsand was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties ofp(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.

An important problem in geometric topology is to find a homology 3-sphere M with Rohlin invariant μ(M) = ½ such that M # M bounds an acyclic 4-manifold. If such an M exists then, for instance, all closed topological manifolds of dimension ≥ 5 support a polyhedral structure(3, 4). One way of producing such examples is to find a homology 3-sphere M with μ(M) = ½ such that M admits an orientation-reversing diffeomorphism h, for then M # M would be diffeomorphic to M # − M which bounds an acyclic manifold. In this paper we observe by elementary means that if h is further assumed to be an involution, then this is not possible, namely:Theorem. Let M be a homology 3-sphere which admits an orientation reversing involution. Then M bounds a parallelizable rational 4-ball. In particular, μ(M) = 0.

In this paper I shall show how certain bounds on the possible diagrams presenting a given oriented knot or link K can be found from its two-variable polynomial PK defined in [3]. The inequalities regarding exponent sum and braid index of possible representations of K by a closed braid which are proved in [5] and [2] follow as a special case.

The aim of this paper is to prove the following result:Theorem. Let G be the group defined by the presentationwhere each ui is a word in the xj and , and suppose that there is a homomorphism σ such that the orders of the elements respectively. Let If G is finite, then α−d + 1 > 0 and .

This paper extends the fundamental identity of Wald in the theory of sequential analysis to the case where the variables are Markovian rather than independent.