We establish isomorphisms between certain specializations of BMW algebras and the symmetric squares of Temperley–Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an application, we compute the closed images of the irreducible braid group representations factoring over these specialized BMW algebras.

A chain is a configuration in ℝd of segments of length ℓ1, . . ., ℓn−1 consecutively joined to each other such that the resulting broken line connects two given points at a distance ℓn. For a fixed generic set of length parameters the space of all chains in ℝd is a closed smooth manifold of dimension (n − 2)(d − 1) − 1. In this paper we study cohomology algebras of spaces of chains. We give a complete classification of these spaces (up to equivariant diffeomorphism) in terms of linear inequalities of a special kind which are satisfied by the length parameters ℓ1, . . ., ℓn. This result is analogous to the conjecture of K. Walker which concerns the special case d=2.

Consider networks on n vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as n13/10. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order n3/2. Finally, if we require the network length to be O(n) then the mean number of hops grows as order log log n. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense.

Let μ be a finite complex Borel measure supported on the unit circle

.

In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form

.

and for all ξ ∈ , | ξ | = R for some fixed R > 0.

The object of this note is the following theorem together with a table of the (1 + i)n division points for n = 1, 2, 3, 4, 5 on the elliptic curve y2 = x3 − Dx which has complex multiplication by i = √(−1).

Let X be a curve with an involution T which fixes r points. We show that the Weierstrass weight of a fixed point is at least (r−2)(r−4)/8. Our proof is independent of the recent result of Torres.

We consider the case where X=Fn, the nth Fermat curve, and T is any of the involutions of Fn. We find that our bound is equal to the actual weight in all known cases (n[les ]7) and compute then n=8 case to demonstrate that the equality continues to hold.

The problem of the steady motion of a heavy cylinder, of infinite length, over the surface of an elastic half-space has been examined by Craggs and Roberts(1), Clements(2) and Roberts (3). In papers (1) and (2) the surfaces of both the cylinder and the half-space are assumed to be smooth so that the shear stress over the region of contact is zero. With this assumption Craggs and Roberts analysed the motion for an isotropic half-space while Clements considered the case of an anisotropic half-space. In the paper by Roberts (3) various problems concerning the rolling of rigid and non-rigid cylinders over an isotropic elastic half-space are discussed. In this paper we consider a similar class of rolling cylinder problems for an anisotropic half-space. We begin, in section 2, by deriving the relevant basic equations for the stress and displacement while in section 3 some properties of certain constants occurring in the equations of section 2 are derived for use in later sections. The problem of the steady motion of an inflated tyre is discussed in section 4 under the assumptions that the tyre exerts a constant pressure over the contact region which is instantaneously at rest. The solution shows that, with these assumptions, the shear stress is infinite near the ends of the region of contact. This indicates that there will be slipping of the tyre over the half-space at both ends of the contact region. If these zones of slip are large compared with the total contact area then the assumption of a contact region which is instantaneously at rest is invalidated so that the solution may only be regarded as being applicable when the zones of slip are small. An expression to determine the width of the zones of slip is derived and this expression is used, together with the numerical results of section 5, to obtain information about the influence of anisotropy on the width of the zones of slip. In section 6 the problem of the inflated tyre is discussed with allowance made for a zone of slip at both ends of the region of contact. The results of this section could be expected, particularly when the zones of slipping are wide, to more accurately predict the response of the anisotropic half-space to the rolling tyre than the results of section 4. Unfortunately, in order to satisfactorily complete the analysis, it is necessary to restrict the class of anisotropic materials for which the results are applicable. As a consequence, the work of section 4 is in many ways preferable to that of section 6 in indicating the influence which elastic anisotropy has upon the results.

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg’s conjectures.

1. Introduction and statement of results. The most substantial contribution in recent years to the literature on the Galton–Watson embedding problem has been that of Karlin and McGregor (4) and (5), to which the readeris referred for a more complete bibliography of the subject. Their results and methods very much involve the analytic properties of the associated probability generating functions (p.g.f.'s), and most of the results they describe as ‘negative’ in the sense that they are sufficient conditions for non-embeddability. The two results here are similarly ‘negative’ but obtained from more direct probabilistic considerations.

In this paper a periodic solution of the gravitational equations is examined. The solution, which is valid throughout all space, is such that the internal and external fields and their derivatives are continuous at the boundary of a spheroid rotating about an axis other than that of symmetry. At great distances from the system the solution has the same form as the field due to a rotating rod and can be applied to problems like the determination of the velocity of propagation of gravitational waves and the loss of energy of a rotating cohesive system.

For any locally compact Hausdorff topological group G, let M(G) denote the topo-logical semigroup of all probability measures on G, where M (G) is given the weak topology and the multiplication is convolution.

It is shown that the solutions of the field equations generated by the most general effective quadratic Lagrangian aR2 + bRklRkl which are static, singularity-free, of class C4 and asymptotically flat (of rank A3) are (i) conformally flat when 3a + b = 0, (ii) flat when 3a + b ≠ 0 (b ≠ 0), and (iii) have zero scalar curvature when b = 0.

1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.

We prove the following generalized version of the complex Adams conjecture (see Theorem 10·4), as announced in (5).

The Wigner 6j–symbol, written in the form

is shown to be invariant under separate permutations of A, B, C alone, separate permutations of α, β, γ alone and separate change in sign of any pair of α, β, γ; results equivalent to the new symmetry relations of Regge. Alternatively, written in the form

with J0 + J1 + J2 + J3 = K1 + K2 + K3, Jr(r = 0, 1, 2, 3) and K3 (s = 1, 2, 3) integral, it is invariant for separate permutations of the Jr and of the Ks. If Jm = max (J0, J1, J2, J3), then each 6j-symbol with distinct value may be associated with an ordered partition of Jm into 6 integral parts: Jm = n1 + n2 + n3 + p1 + p2 + p3, n1 ≥ n2 ≥ n3; p1 ≥ p2 ≥ n3. The 6j-symbol is proportional to the Saalschützian

of unit argument.

Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequality

for any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.

It is the object of this paper to prove a transversality theorem for homology manifolds which is strong enough to be then able to prove a Thom theorem for the cobordism ring of homology manifolds. The new bundle theory necessary for this follows on naturally from that developed in ‘Homology Cobordism Bundles’(2).

We observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics , non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls have the same first eigenvalue, the same volume and the distance spheres and, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,[7]

There is developed the theory of the propagation of waves through a system of potential barriers. Many properties are derived and discussed, mainly by the use of matrix theory arising from the description of each barrier by a transmission matrix. The extent to which certain previously used approximate methods are applicable and valid is also examined. Two distinct kinds of resonant phenomena are discussed – first when there is no incident wave and secondly when there is no reflected wave. This latter case of perfect transmission is investigated in detail, an examination being given of both why and when it occurs and the time taken to establish such exceptional fields within the barriers.

The statistical theory of long-range interactions between adsorbed particles on a plane lattice is worked out approximately, by treating in detail the distribution of adsorbed particles among a few sites inside and on the boundary of a circular region, and regarding the distribution outside the circle as uniform and continuous with a density Kθ per unit area, where K is the number of lattice points per unit area and θ is the fraction of surface covered by adsorbed particles. The continuous distribution begins at a distance ρ from the centre of the circle, ρ being determined by the condition that the probability of occupation of a first shell site is equal to the probability θ of occupation of the central site. Using this method, general formulae for the adsorption isotherm and the heat of adsorption are obtained. Numerical applications for dipole interactions and for quadratic and hexagonal lattices are worked out in detail and the case in which the dipole moment varies with θ is discussed.