Volume 25, Issue 6

Abstract. Let A and B be complete normed algebras with a unit such that B is simple and power-associative, and let be a dense-range homomorphism. We prove that if θ is irreducible (that is, , for every closed proper ideal I of A ), then θ is continuous. The continuity of non-irreducible homomorphisms is also obtained provided that the set of «spectrally rare »elements in the range algebra is not dense in B . These results extend the classical Rickart's dense-range homomorphism theorem to the non-associative setting.

Abstract. The connection problem associated with a Selberg type integral is solved. The connection coefficients are given in terms of the q -Racah polynomials. As an application of the explicit expression of the connection coefficients, examples of the monodromy-invariant Hermitian form of non-diagonal type are presented. It is noteworthy that such Hermitian forms are intimately related with the correlation functions of non-diagonal type in -conformal field theory.

Abstract. We show that the space of all holomorphic maps of degree one from the Riemann sphere into a Grassmann manifold is a sphere bundle over a flag manifold. Using the notions of “kernel” and “span” of a map, we completely identify the space of unparameterized maps as well. The illustrative case of maps into the quadric Grassmann manifold is discussed in detail and the homology of the corresponding spaces is computed.

Abstract. We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces with and .

Abstract. In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating vectors with as many component as the Frobenius number.

Abstract. Let be an arbitrary infinite sequence of nontrivial finite Abelian transitive groups such that the topological rank ρ of the infinite Cartesian product of these groups is finite. We consider the corresponding inverse limit of iterated permutational wreath products. By using the geometric language of automorphisms of a certain rooted tree we show that the group is topologically generated by the union of two sets each containing ρ elements and such that both the group generated by S and the group generated by are free Abelian groups of rank ρ. Moreover, the group G generated by the above union decomposes as a semidirect product of the group generated by one of these sets and the normal closure of the other set, and the semigroup generated by this union is a free product of semigroups generated by S and , respectively. We derive other algebraic properties of the group G . In particular, the condition for the groups to be finitely generated as profinite groups and some nontrivial results concerning topological ranks of their subgroups are given in this way. For example, for every our construction gives an explicit and naturally defined finitely generated subgroup of an inverse limit of iterated wreath products of finite Abelian groups such that the rank of this subgroup and the topological rank of its topological closure differ by n .

Abstract. We study fully nonlinear partial differential equations of Monge–Ampère type involving the derivatives with respect to a family of vector fields. The main result is a comparison principle among viscosity subsolutions, convex with respect to , and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to the Dirichlet problem. Its assumptions include the equation of prescribed horizontal Gauss curvature in Carnot groups. By the Perron method we also prove the existence of a solution either under a growth condition of the nonlinearity with respect to the gradient of the solution, or assuming the existence of a subsolution attaining continuously the boundary data, therefore generalizing some classical result for Euclidean Monge–Ampère equations.