Let K be a commutative integral domain and let S = K[x1, x2, . . . , xn] denote the polynomial ring over K in the n variables x1, x2, . . . , xn. If H is a group of automorphisms of the free K-module V = Kx1 +Kx2 + · · ·+Kxn, that is, if H is a subgroup of GL(V ) ∼= GLn(K), then the linear action of H on V extends uniquely to a K-algebra action on S. The relationship between S, H, the H-stable ...

Multiplicative Number Theory 638 27.2 Functions . . . . . . . . . . . . . . . . . 638 27.3 Multiplicative Properties . . . . . . . . . 640 27.4 Euler Products and Dirichlet Series . . . 640 27.5 Inversion Formulas . . . . . . . . . . . . 641 27.6 Divisor Sums . . . . . . . . . . . . . . . 641 27.7 Lambert Series as Generating Functions . 641 27.8 Dirichlet Characters . . . . . . . . . . . . 642...

In this thesis we look at several problems that lie in the intersection between combinatorial and multiplicative number theory. A common theme of many of these problems are estimates for and properties of the smooth numbers, those integers not divisible by any large prime factors. The first is the Ramsey-theoretic problem to determine the maximal size of a subset of the integers containing no 3...

We propose a modiied approach to the abstract convergence theory of the additive and multiplicative Schwarz iterative methods associated with a subspace splitting for solving symmetric positive deenit linear problems. In particular, we obtain a satisfactory bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the c...

The general rationale of analytic number theory is to derive statistical information about a sequence {an} from the analytic behaviour of an appropriate generating function, such as a power series ∑ anz or a Dirichlet series ∑ ann−s . The type of generating function employed depends on the problem being investigated. There are no rigid rules governing the kind of generating function that is app...

We propose a weaker condition for multiplicative unitary operators related to quantum groups, than the condition of manageability introduced by S.L. Woronowicz. We prove that all the main results of the theory of manageable multiplicative unitaries remain true under this weaker condition. We also show that multiplicative unitaries arising naturally in the construction of some recent examples of...

1 Department ofMathematics andComputerScience,EmoryUniversity,Atlanta,GA30322, USA; e-mail: [email protected] 2 Fachbereich Mathematik, Universität Wuppertal, 42097 Wuppertal, Germany; e-mail: [email protected] 3 Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany; e-mail: [email protected] 4 Department of Mathematics, Te...

We consider a slow passage through a point of loss of stability. If the passage is sufficiently slow, the dynamics are controlled by additive random disturbances, even if they are extremely small. We derive expressions for the ‘exit value’ distribution when the parameter is explicitly a function of time and the dynamics are controlled by additive Gaussian noise. We derive a new expression for t...

7 Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate 8 devices between semantics and traditional finitary proof systems, they are commonly found in 9 completeness arguments, automated deduction, verification, etc. However, their proof theory 10 is surprisingly underdeveloped. In particular, very little is known about the computational 11 behavior o...