Let Ω be the weakly pseudoconvex domain

and let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

Let S be a simplex in En which is homothetic to a given simplex S*, which contains no point of the integral lattice in its interior, and which has maximal volume V(S). We conjecture that V(S) > nn/n!, and establish the conjecture for n < 3.

O. Kolberg has shown that if the partition generating function is split into five interlocking series, then certain algebraic relations hold among these series. We show that the same phenomenon occurs whenever the number of such series is not a power of two.

This paper introduces certain generalization of the notions of approximate limit, continuity and derivative and of absolute continuity, of real functions, leading to generalized integrals of Perron and Denjoy types comprising the AP-integral of Burkill (1931) and Sonouchi and Utagawa (1949) and the AD-integral of Kubota (1963), respectiely. The generalizations are all substantiated by appropriate examples.

Let X be a weakly complete proper cone contained in a weak space E and h(E) the Riesz space generated by the continuous linear forms on E. A positive conical measure μ on X is a positive linear form on h(E)|x. G. Choquet has proved μ is a Daniell integral on E when E is weakly complete, but μ is not generally a Daniell integral on X. However we give an integration theory for functions on X and compare this theory with the classical Daniell theory. The case where μ is maximal in the sense of G. Choquet is remarkable.

Let St = exp{−tH}, Tt = exp{−tK}, be C0-semigroups on a Banach space . For appropriate f one can define subordinate semigroups Sft = exp{−tf(H)}, Ttf = exp{−tf(K)}, on and examine order properties of the pairs S, T, and Sf, Tf. If , = Lp(X;dv) we define St≽ Tt ≽ 0 if St − Tt and Tt map non-negative functions into non-negative functions. Then for p fixed in the range 1 > p > ∞ we characterize the functions for which St ≽ Tt ≽ 0 implies Sft ≽ Tft ≽ 0 for each Lp(X;dv) and the converse is true for all Lp(X;dv). Further we give irreducibility criteria for the strict ordering of holomorphic semigroups. This extends earlier results for L2-spaces.

The concept of ‘Almost Distributive Lattices’ (ADL) is introduced. This class of ADLs includes almost all the existing ring theoretic generalisations of a Boolean ring (algebra) like regular rings, P-rings, biregular rings, associate rings, P1-rings, triple systems, etc. This class also includes the class of Baer-Stone semigroups. A one-to-one correspondence is exhibited between the class of relatively complemented ADLs and the class of Almost Boolean Rings analogous to the well-known Stone's correspondence. Many concepts in distributive lattices can be extended to the class of ADLs through its principal ideals which from a distributive lattice with 0. Subdirect and Sheaf representations of an ADL are obtained.

We consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

The lateral completion of a completely distributive lattice-ordered permutation group is investigated via various completions, obtained by adjoining permutations which match some elements of the given group in various ways. This makes known results on the lateral completion of a completely distributive lattice-ordered group both transparent and easy.