The Mumford class κ1 on Mg,0 was shown to be proportional to the cohomology class [ωWP ] of the Weil-Petersson form by Wolpert in [WO]. Furthermore he showed that the restriction of this class to any component of the compactyfying divisor coincides with the corresponding Weil-Petersson class. Arbarello and Cornalba introduced classes κ1 on Mg,n, proved a similar restriction property for these a...

Let k-l,ml,..~+k denote non-negative integers, and suppose the greatest common divisor of ml,...,mk is 1 . We show that if '1, "',sk are sufficiently long blocks of consecutive integers, then the set mlSl+ . ..+mkSk contains a sizable block of consecutive integers. For example; if m and n are relatively prime natural numbers, and U, U ? Vt V are integers with U-u 2 n-l , V-v 2 m-l 1 then the se...

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...

A general subresultant method is introduced to compute elements of a given ideal with few terms and bounded coefficients. This subresultant method is applied to solve over-determined polynomial systems by either finding a triangular representation of the solution set or by reducing the problem to eigenvalue computation. One of the ingredients of the subresultant method is the computation of a m...

Let S = {x1,x2, . . . ,xn} be a set of positive integers, and let f be an arithmetical function. The matrices (S) f = [ f (gcd(xi,xj))] and [S] f = [ f (lcm[xi,xj])] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f , respectively. In this paper, we assume that the elements of the matrices (S) f and [S] f are integers and st...

1 Division 3 1.1 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Greatest common divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . ....