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.

Any representation of a group G on a vector space V extends uniquely to a representation of G on the free metabelian Lie algebra on V. In this paper we study such representations and make some group-theoretic applications.

We state best approximation and fixed point theorems in modular spaces endowed with an H-space structure given by the modular topology. We consider both the cases of single valued functions and multifunctions. These theorems extend some previous results due to Ky Fan.

The second dual of the vector-valued function space C0(S, A) is characterized in terms of generalized functions in the case where A* and A** have the Radon-Nikodým property. As an application we present a simple proof that C0 (S, A) is Arens regular if and only if A is Arens regular in this case. A representation theorem of the measure μh is given, where , h ∈ L∞ (|μ;|, A**) and μh is defined by the Arens product.

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.

The general properties of lattice-perfect measures are discussed. The relationship between countable compactness and measure perfectness, and the relationship between lattice-measure tightness and lattice-measure perfectness are investigated and several applications in topological measure theory are given.

The Kasparov groups are extended to the setting of inverse limits of G-C*-algebras, where G is assumed to be a locally compact group. The K K-product and other important features of the theory are generalized to this setting.

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.

An example of two disjoint special classes whose upper radicals coincide is presented. It is shown that the left hereditary subradical of the hereditary idempotent radical is right hereditary. An example of a hereditary and principally left hereditary radical which is not left hereditary is constructed.

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 consider coactions of a locally compact group G on a C*-algebra A, and the associated crossed product C*-algebra A× G. Given a normal subgroup N of G, we seek to decompose A× G as an iterated crossed product (A× G/ N) × N, and introduce notions of twisted coaction and twisted crossed product which make this possible. We then prove a duality theorem for these twisted crossed products, and discuss how our results might be used, especially when N is abelian.

The free product *CRSi of an arbitrary family of disjoint completely simple semigroups {Si}i∈i, within the variety CR of completely regular semigroups, is described by means of a theorem generalizing that of Kaďourek and Polák for free completely regular semigroups. A notable consequence of the description is that all maximal subgroups of *CRSi are free, except for those in the factors Si themselves. The general theorem simplifies in the case of free CR-products of groups and, in particular, free idempotent-generated completely regular semigroups.

Let M be an invariant subspace of L2 (T2) on the bidisc. V1 and V2 denote the multiplication operators on M by coordinate functions z and ω, respectively. In this paper we study the relation between M and the commutator of V1 and , For example, M is studied when the commutator is self-adjoint or of finite rank.

Let Δ be a thick building of type Ã2, and let be its set of vertices. We study a commutative algebra of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C*-algebra induced by on l2, find its spectrum Σ, prove positive definiteness of a kernel kz for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings ΔJ arising from the groups ΓJ introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of ΓJ.

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.

It is proved that on certain kinds of homogeneous spaces, the only Lp function, 1≤ p < ∞, satisfying the mean value property is the zero function.

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.

The operators K, k, T and t are defined on the lattice of congruences on a Rees matrix semigroup S as follows. For ρ ∈ (S), ρK and ρk (ρT and ρt) are the greatest and the least congruences with the same kernel (trace) as ρ, respectively. We determine the semigroup generated by the operators K, k, T and t as they act on all completely simple semigroups. We also determine the network of congruences associated with a congruence ρ ∈ (S) and the lattice generated by it. The latter is then represented by generators and relations.

We give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

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.