We discuss the cobordism type of spin manifolds with non-negative sectional curvature. We show that in each dimension 4k ⩾ 12, there are infinitely many cobordism types of simply connected and non-negatively curved spin manifolds. Moreover, we raise and analyse a question about possible cobordism obstructions to non-negative curvature.

We show that if $X$ is an $m$-dimensional definable set in $\mathbb {R}_\text {an}^\text{pow}$, the structure of real subanalytic sets with real power maps added, then for any positive integer $r$ there exists a $C^{r}$-parameterization of $X$ consisting of $cr^{m^{3}}$ maps for some constant $c$. Moreover, these maps are real analytic and this bound is uniform for a definable family.

The calculation of the steady state thermal stresses in an isotropic elastic half space or slab with traction free faces has been the subject of several investigations. Steinberg and McDowell (1), using an extension of the Bousinesq-Papkowitch method of isothermal elasticity, first derived the now well-known result that in such a body which contains no heat sources there exists a plane state of stress parallel to the boundary planes. Sneddon and Lockett (2) approached this class of problems by direct solution of the equations of thermoelasticity using a double Fourier integral transform method, the results being transformed to Hankel type integrals in the case of axial symmetry. A further approach due to Nowinski (3) exploits the fact that in steady state thermoelasticity each component of the displacement vector is a biharmonic function which can be expressed as a combination of harmonics. However, possibly the most economical method of solution of this type of problem is that of Williams (4) who expressed the displacement vector in terms of two scalar potential functions, one of which is directly related to the temperature field. The same principle has also been used by Fox (5) in treating thermoelastic distributions in a slab containing a spherical cavity.

For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{\prime}=f_1(x,y)$, $y^{\prime}=f_2(x,y)$, where f1 has degree one, f2 has degree $n\ge 1$ and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{\prime}=y$, $y^{\prime}=g_1(x) +y g_2(x)$ with g1 and g2 polynomials of degrees n and m, respectively.

We study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson system

in ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

We prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × × … × for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].

Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.

When can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.

In the setting of sphere bundles, a certain degree map degB turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set ℱ of fibrewise homotopy classes of fibrewise maps. We introduce an addition on ℱ, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.

We give an algorithm which for a given set of generators can decide whether a stem product of infinite cyclic groups is a one-relator group. We also generalize this to the case of one-relator products of groups.

In comparison with the general plane quartic on the one hand, and the curves having either two or three nodes on the other, the uninodal curve has been neglected. Many of its properties may of course be deduced from those of the general quartic in the limiting case when an oval shrinks to a point or when two branches approach and ultimately unite. The modifications of properties of the bitangents are shewn more clearly by Geiser's method, in which these lines are obtained by projecting the lines of a cubic surface from a point on the surface. As the point moves up to and crosses a line on the surface, the quartic acquires a node and certain pairs of bitangents obviously coincide, viz. those obtained by projecting two lines coplanar with that on which the point lies. A nodal quartic curve and its double tangents may also be obtained by projecting a cubic surface which has a conical point from an arbitrary point on the surface. Each of these three methods leads us to the conclusion that, when a quartic acquires a node, twelve of the double tangents coincide two and two and become six tangents from the node, and the other sixteen remain as genuine bitangents: the twelve which coincide are six pairs of a Steiner complex.

On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma has

and points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integral

which was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.

Let $L$ be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let $f:L\to\{\pm1\}$ be a homomorphism and, for $\alpha=\sum\alpha_\ell\ell\in\mathbb{Z}L$, define $\alpha^f=\sum f(\ell)\alpha_\ell\ell^{-1}$. Call $\alpha$ f-unitary if $\alpha^f=\alpha^{-1}$ or $\alpha^f=-\alpha^{-1}$. In this paper, we identify the RA loops $L$ with the property that all units in $\mathbb{Z}L$ are $f$-unitary. Along the way, we extend a famous theorem of Higman to a case still undecided in group rings.

AMS 2000 Mathematics subject classification: Primary 20N05. Secondary 17D05; 16S34; 16U60

We introduce the notion of tracial equivalence for $C^*$-algebras. Let $A$ and $B$ be two unital separable $C^*$-algebras. If they are tracially equivalent, then there are two sequences of asymptotically multiplicative contractive completely positive linear maps $\phi_n:A\to B$ and $\psi_n:B\to A$ with a tracial condition such that $\{\phi_n\circ\psi_n\}$ and $\{\psi_n\circ\phi_n\}$ are tracially approximately inner. Let $A$ and $B$ be two unital separable simple $C^*$-algebras with tracial topological rank zero. It is proved that $A$ and $B$ are tracially equivalent if and only if $A$ and $B$ have order isomorphic ranges of tracial states. For the Cantor minimal systems $(X_1,\sigma_1)$ and $(X_2,\sigma_2)$, using a result of Giordano, Putnam and Skau, we show that two such dynamical systems are (topological) orbit equivalent if and only if the associated crossed products $C(X_1)\times_{\sigma_1}\mathbb{Z}$ and $C(X_2)\times_{\sigma_2}\mathbb{Z}$ are tracially equivalent.

Let f(z) be an entire function, M(r) the maximum of f(z) on ∣z∣=r, and λ>1. Let Eλ=Eλ(f{z:log∣f(z)≦(1-λ)log(M∣z∣)}, and denote the density of Eλby

where m is planar measure.

Let E be a separated locally convex barrelled space with continuous dual E′ and algebraic dual E* and let M be a subspace of E* with and dim Robertson, Tweddle and Yeomans have recently considered the question of barrelledness under the Mackey topology τ(E,E' + M) when E is given to be barrelled under its original topology τ(E,E') [5], [6], [7].

In recent years the geometry of generic submanifolds of Euclidean space has been theobject of much study. Thorn hinted in [7] that the focal set of such a submanifold couldprofitably be studied by using the family of distance squared functions on thesubmanifold from points of the ambient space. For a generic submanifold the focal set isthe catastrophe or bifurcation set of this family. The key to obtaining results on thelocal structure of this focal set is a transversality theorem of Looijenga [5]; for analternative exposition see [8].