Every knot admits a special projection with the property that under the projection discs in the canonical Seifert surface project disjointly. Under an isotopy, such a projection can be turned into a connected sum of what we call inseparable projections. The main result is that if there is no band in an inseparable projection with half-twisting number +1 or −1, then the projection is not a projection of the trivial knot. To prove this a non-cyclic Coxeter group is constructed as a quotient of the knot group. The construction is possibly of interest in itself. The techniques developed are applied to give a criterion to decide when an inseparable projection with 3 discs comes from the trivial knot.

Problems dealing with certain functional calculi for systems of non-commuting operators, and ordered calculi for systems of certain types of pseudo-differential operators, can sometimes be treated via the methods of integration with respect to polymeasures. The polymeasures arising in this fashion (called Radon polymeasures) often have additional structure not available in the general theory. This allows for a more extensive class of “integrable” functions than just the product functions allowed in the abstract theory. The purpose here is to further develop special aspects of integration with respect to Radon polymeasures with a particular emphasis on identifying large classes of “integrable” functions.

Let u(x, t) be a smooth function in the domain Q = Ω × (0, L), Ω in n, let Du be the spatial gradient of u(x, t) and let ∇u = (Du, u1). If u(x, t) satisfies the parabolic equation F(u, Du, D2u) = ut, we define w(x, t) by g(w) = │∇u│−1G(∇u) (g is positive and decreasing, G is concave and homogeneous of degree one) and we prove that w(x, t) attains its maximum value on the parabolic boundary of Q. If u(x, t) satisfies the equation Δu + 2h(q2) uiujuij = ut(q2 = │Du│2, 1 + 2q2h(q2) > 0) we prove that qf (u) takes its maximum value on the parabolic boundary of Q provided f satisfies a suitable condition. If u(x, t) satisfies the parabolic equation aij (Du)uij − b(x, t, u, Du) = ut (b is concave with respect to (x, t, u)) we define C(x, y, t, τ) = u(z, θ) − αu(x, t) − βu(y, τ) (0 < α, 0 < β, α + β = 1, z αx +y, θ = αt + βτ) and we prove that if C(x, y, t, r) ≤0 when x, y, z ∈ Ω2 and one of t, τ = 0, and when t, τ ∈ (0, L], and one of x, y, z, ∈ ∂Ω, then it is C(x, y, t, τ) ≤0 everywhere.

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

We develop a concrete Fourier transform on a compact Lie group by means of a symbol calculus, or *-product, on each integral co-adjoint orbit. These *-products are constructed by means of a moment map defined for each irreducible representation. We derive integral formulae for these algebra structures and discuss the relationship between two naturally occurring inner products on them. A global Kirillov-type character is obtained for each irreducible representation. The case of SU(2) is treated in some detail, where some interesting connections with classical spherical trigonometry are obtained.

In Zermelo-Fraenkel set theory weakened to permit the existence of atoms and without the axiom of choice we investigate the deductive strength of five statements which make assertions about the cardinality of the union of a well-ordered collection of sets. All five of the statements considered are consequences of the axiom of choice.

Recall a closed convex set C is said to have the weak drop property if for every weakly sequentially closed set A disjoint from C there exists x ∈ A such that co({x} ∩ C) ∪ A = {x}. Giles and Kutzarova proved that every bounded closed convex set with the weak drop property is weakly compact. In this article, we show that if C is an unbounded closed convex set of X with the weak drop property, then C has nonempty interior and X is a reflexive space.

Let fn be a sequence of nonnegative integers and let f(x): = Σn≥0 fn xn be its generating function. Assume f(x) has the following properties: it has radius of convergence r, 0 < r < 1, with its only singualarity on the circle of convergence at x = r and f(r) = s; y = f(x) satisfies an analytic identity F(x, y) = 0 near (r, s); for some k ≥ 2 F0.j = 0, 0 ≤ j < k, F0.k ≠ 0 where Fi is the value at (r, s) of the ith partial derivative with respect to x and the jth partial derivative with respect to y of F. These assumptions form the basis of what we call the typical and general cases. In both cases we show how to obtain an asymptotic expansion of fn. We apply our technique to produce several terms in the asymptotic expansion of combinatorial sequences for which previously only the first term was known.