The main purpose of this paper is the following algebraic generalization of the corona theorem for the disc algebra A (D): I f d is a greatest common divisor of the functions f l . . . . . fn cA (D), then there exist functions gl,.",gnEA (D) with d-~flgl q-... q-fngn. This generalization is false for many algebras of holomorphic functions, e. g. in case of the Banach algebra H ~ Under the assum...

We establish a connection between the L norm of sums of dilated functions whose jth Fourier coefficients are O(j−α) for some α ∈ (1/2, 1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L and for the almost everywhere convergence of series of dilated functions.

A number b divides a if the remainder is zero. We denote it by b | a. b a denotes that b does not divide a. If b divides a then a is a multiple of b. Now we can define the greatest common divisor (GCD). The GCD of two numbers a and b is defined as the biggest number which divides both a as well as b. It is also denoted by gcd(a, b). One of the important case is when gcd(a, b) = 1, i.e., there i...

We prove that every projective variety of dimension n over a field of positive characteristic admits a morphism to projective n-space, étale away from the hyperplane H at infinity, which maps a chosen divisor into H and a chosen smooth point not on the divisor to some point not in H .

We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all Deligne–Faltings log structures (in particular simple normal crossings divisor) from the simplest case with characteristic generated by N (essentially the smooth divisor case).

We discuss the uniruledness of various base loci of linear systems related to the canonical divisor. In particular we prove that the stable base locus of the canonical divisor of a smooth projective variety of general type is covered by rational curves.

The purpose of this note is to add some important properties to the results obtained in [2]. Specifically, it is shown that (i) an apportionment for relaxed divisor methods remains unchanged over an interval and (ii) any relaxed divisor method approaches the Webster method as the house size increases.

In this paper, we determine the structures of zero-divisor semigroups whose graph is Kn + 1, the complete graph Kn together with an end vertex. We also present a formula to calculate the number of non-isomorphic zero-divisor semigroups corresponding to the complete graph Kn, for all positive integer n.