A polynomial system is a real autonomous system of ordinary differential equations on the plane with polynomial nonlinearities:

with aij, bij ∈ ℝ and where x = x(t) and y = y(t) are real-valued functions.

The problem of analysing limit cycles (isolated periodic solutions) in polynomial systems was first discussed by Poincaré[16]. Then, in the famous list of 23 mathematical problems stated in 1900, Hilbert[9] asked in the second part of the 16th problem for an upper bound for the number of limit cycles for nth degree polynomial systems, in terms of n. Recently, it has been proved that, given a particular choice of coefficients for a system of form (1·1), the number of limit cycles is finite. This result is known as Dulac's theorem, see Ecalle[8] or Il'yashenko[10]. However, it is unknown whether or not there exists an upper bound for the number of limit cycles in system (1·1) in terms of n. Even for quadratic systems (i.e. polynomial systems with quadratic nonlinearities) this remains an open question.

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

Let K be a hyper-cube of side 2 in n dimensions where n is any positive integer, and H an arbitrary, fixed (n − 1)-dimensional hyper-plane through the centre of K.

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

A band is a semigroup in which every element is idempotent. In this note we give an explicit description of the Jacobson radical of the semigroup ring of a band over a ring with unity. It is shown, further, that this radical is nil if and only if the Jacobson radical of the coefficient ring is nil. For the particular case of a normal band (see below for the definition) the Jacobson radical of the band ring is nilpotent if and only if the Jacobson radical of the coefficient ring is nilpotent; but this result does not extend to arbitrary bands.

Alexandrov (1), Zeeman (3) and others after them have investigated at length the problem of characterizing as Lorentz transformations those mappings f of Minkowski space M which ‘preserve cones’, i.e. if C(a) denotes the null-cone in M with vertex a ∈ M, then

for all a ∈ M.

Let G be an arbitrary group and Zn(G) denote the group algebra of G over the integers modulo n. If δi(G) denotes ith power of the augmentation ideal δ(G) of Zn(G), then

is easily seen to be a normal subgroup of G. It is denoted by Di, n(G) and is called ith dimension subgroup of G modulo n. It can be shown that these dimension subgroups are determined by the dimension subgroups modulo a power of a prime p. Hence we shall restrict our attention to these dimension subgroups.

The Frattini subgroup φ(G) of a group G is the intersection of G and all its maximal subgroups. The following results for finite groups are well known:

THEOREM A0. If G is a finite group, then the following three conditions are equivalent:

(i) G is nilpotent,

(ii) G/φ(G) is nilpotent,

(iii) φ(G) ≥ G′.

1. Introductory. The present paper is mainly concerned with some forms for the reciprocal of the square matrix u of which the typical element is

and k is not an integer within the range ± (n − 1) inclusive but is otherwise arbitrary; thus

Let $\mathcal{C}$ be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the category C($\mathcal{C}$) of unbounded chain complexes in $\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

The possibility of the propagation of disturbances along a free surface with a velocity exceeding the velocity of sound in the liquid is examined. Simple necessary conditions for the occurrence of such ultrasonic propagation are found in terms of the boundary condition. It is shown that the conditions are not satisfied in the case of gravity waves, and a detailed examination is made of the pressure and surface waves due to a localized disturbance. Capillary waves are shown to satisfy the conditions for ultrasonic propagation, the existance of a ‘faster-than-sound’ wave is proved, and its main properties are examined.

The set of critical values of a map of the plane (of corank 1) can be regarded as the apparent contour of a smooth surface. It is a classical result of Whitney[13] that generically the apparent contour is a smooth fold curve with isolated cusps and transverse fold crossings. More recent classifications of smooth map-germs of the plane of low codimension (occurring in generic 2- or 3-parameter families), e.g. in [1, 5, 7], were motivated by a question in differential geometry: given any smooth surface which is generically embedded in ℝ3, produce a list of all possible (orthogonal or central) projections of such a surface.

A direct method of obtaining the limitations of crystal symmetry on physical phenomena is described and discussed in greater detail for the case of diamagnetic magnetostriction. Formulae are obtained for the variation of the longitudinal and transverse effects with crystal orientation in the case of bismuth.

We obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.

Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.

The new concept of a constitutive surface is introduced into inviscid fluid mechanics, with special reference to compressible gas dynamics and to shallow water theory. The detailed shape of such surfaces is calculated, including the manner of their transition across singularities where the Mach or Froude number passes through unity. The calculation draws upon recent work describing the transition of a Legendre transformation through its singularity. For example, mass flow Q, total energy h and flow stress P are always related by part of a ‘swallowtail’ surface, regardless of the particular motion.

The addition of dynamical conditions defines particle history tracks which always lie on constitutive surfaces even for unsteady flow, except that they may jump from one part to another of such a surface when shock waves or hydraulic jumps are passed through.

Illustrations given include the steady flow of a general gas through a standing normal shock, general shallow water theory, and flow along a sloping-sided channel. Connections with existing literature are described.

There is a sense in which the homology group HA of a free Abelian chain complex A may be said to be a ‘complete system of invariants’ of A, to within chain equivalence; certainly any graded Abelian group G is isomorphic to HA for a suitable A, and if HA and HB are isomorphic then A and B are chain equivalent. Such a result is useful in showing that it is fruitless to seek other homotopy invariants of A; whatever depends only on the homotopy class of A depends only on HA, so that we can, for instance, predict the existence of a formula giving H(A ⊗ G), to within isomorphism, in terms of HA and G. The theorem on the existence and uniqueness to within chain equivalence of projective resolutions of modules is a variant of the above theorem, more general in one direction and more special in another.

In this paper the solution is found of certain boundary value problems appropriate to the flow of an invisoid, compressible fluid past a wedge-shaped profile. The most general hodograph equation in the form

is used. Expressions are found for the position coordinates and the drag. The resulting solutions are then discussed for functions k(σ) appropriate to Chaplygin's equation, Tricomi's equation and the equation of Tomotika and Tamada. The case of Helmholtz flow past a wedge with sonic velocity upstream is treated in detail, and the resulting expressions for the drag enable an estimate to be made of the error which results in using approximations to Chaplygin's equation.

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.

We give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

Let ℕ be the natural numbers object in , the effective topos. It was shown in [1] that the maps ℕ → ℕ are (internally or externally) just the total recursive functions. Now a subset A ⊆ ω of natural numbers corresponds to a ¬¬-closed subobject A ↣ ℕ; let kA be the least topology forcing A to be decidable, and let ℕA be the sheafification of ℕ with respect to this topology. Then one would expect the maps ℕ → ℕA to be the total functions recursive in A.