Let D be the factor of the enveloping algebra of a semisimple Lie algebra by its minimal primitive ideal with trival central character. We give a geometric description of the Chern character ch: K0(D)→HC0(D) and the state (of the maximal ideal m) s: K0(D)→K0(D/m) = ℤ in terms of the Euler characteristic χ:K0()→ℤ, where is the associated flag variety.

In this article we prove that if a completely positive linear map Φ of a unital C*-algebra A into another B with only finite dimensional irreducible representations is pure, then we have NΦ = Φker + kerΦ, where NΦ={x∈A|Φ(x) = 0}, Φker = {x∈A|Φ(x*x) = 0}, and kerΦ={x∈A|Φ(xx*) = 0}. We also prove that for every unital strongly positive and n-positive linear map Φ of a C*-algebra A onto another B with n≧2, if NΦ = Φker + kerΦ, then Φ is extreme in Pn(A, B, IB). By this null-kernel condition, many new extreme n-positive linear maps are identified. A general procedure for constructing extreme n-positive linear maps is suggested and discussed.

In this paper we consider a nonlinear periodic parabolic boundary value problem with a discontinuous nonmonotone nonlinearity. Using a lifting result for operators of type (S+), a general surjectivity theorem for operators of monotone type and an auxiliary problem defined by truncation and penalization we prove the existence of a solution in the order interval formed by an upper and lower solution. Moreover we show that the set of all such solutions is compact in Lp(T, (Z)).

The 6Ψ6 summation theorem was first proved by Bailey1, who deduced it indirectly from a transformation of a well-poised 8Φ7 series into two 4Φ3 series. No direct proof of the theorem has been published, and, since it has interesting applications in the proofs of various identities which occur in combinatory analysis, for example the A series of Rogers2 and some elegant identities due to Ramanujan3, we give two new proofs of the theorem in this paper.

Let be a covering of a topological space X and ℱ a sheaf of abelian groups over X. By a well known result of Leray, (3) II theorems 5.2.4. and 5.4.1., if is open, or closed and locally finite, there exists a spectral sequence {Er} satisfying isomorphisms and for some filtration of the graded group H*(X, ℱ). ℋq(ℱ) denotes the system of coefficients over : s→Hq(| s |, ℱ).

The term Geometrography is new to mathematical science, and it may be defined, in the words of its inventor, as “the art of geometrical constructions.”

The name Pfaffian was given by Cayley to the rational function which is the square root of a zero-axial skew determinant of even order

We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.

§ 1. Extensions of Pascal's theorem are already known if we look at the matter from certain particular points of view – an extension of the theorem from the more general point of view is still a desideratum.