Syntomic regulators andp-adic integration II:K 2 of curves

We show that the ad hoc regulator of Coleman and de Shalit for elements ofK ₂ of curves evaluated on a holomorphic differential is the same as the syntomic regulator of the same elements cup produced with the same differential. Combined with the results of Coleman and de Shalit this gives a relation between syntomic regulators and special values ofp-adicL-functions. The main technical innovation is the notion of a local index — a kind of generalized residue.     

Syntomic regulators and special values of p-adic L-functions

In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's p-adic class number formula, and express special values of p-adic L-functions in terms of orders of K-groups and higher p-adic regulators. The approach uses syntomic regulator maps, which are the p-adic equivalent of the Beilinson regulator maps. They can be compared with étale regulators via the Fontaine-Messing map, and computations of Bloch-Kato in the case that p is unramified in F lead to results about generalized Coates-Wiles homomorphisms and cyclotomic characters. Oblatum 14-V-96 & 9-X-97     

Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem – an inverse problem in fractal construction – is proposed. Dedicated to Walter Gautschi.     

Synthesis of multivariable regulators: The internal model principle

For the multivariable control system described by \(\dot x = Ax + Bu,y = Cx,z = Dx\) (wherey is the measured output andz the output to be regulated) conditions are given for the existence of a controller which preserves output regulation and loop stability, in the presence of small parameter variations in controller and plant. Under mild conditions, such a “strong” synthesis is shown to exist if and only if the regulator problem with internal stability (RPIS) is well-posed. Synthesis is achieved by means of a feedback configuration which in general incorporates an invariant, and suitably redundant, copy of the dynamic model adopted for the exogenous disturbance and reference signals which the system is required to process.     

Diophantine geometry over groups III: Rigid and solid solutions

This paper is the third in a series on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the third paper we analyze exceptional families of solutions to a parametric system of equations. The structure of the exceptional solutions, and the global bound on the number of families of exceptional solutions we obtain, play an essential role in our approach towards quantifier elimination in the elementary theory of a free group presented in the next papers of this series. The argument used for proving the global bound is a key in proving the termination of the quantifier elimination procedure presented in the sixth paper of the series. Partially supported by an Israel Academy of Sciences Fellowship.     

Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'s

Panovsky and Richardson [A family of implicit Chebyshev methods for the numerical integration of second-order differential equations, J. Comput. Appl. Math. 23 (1988) 35–51] presented a method based on Chebyshev approximations for numerically solving the problem y^″=f(x,y)$y^{″}=f(x,y)$, being the steplength constant. Coleman and Booth [Analysis of a Family of Chebyshev Methods for y^″=f(x,y)$y^{″}=f(x,y)$, J. Comput. Appl. Math. 44 (1992) 95–114] made an analysis of the above method and suggested the convenience to design a variable steplength implementation. As far as we know this goal has not been achieved until now. Later on we extended the above method (this journal, 2003), and obtained a scheme for numerically solving the equation y^″-2gy^′+(g²+w²)=f(x,y)$y^{″}-2{\mathit{gy}}^{\prime }+(g^2+w^2)=f(x,y)$. The question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper.     

Generalized Rigid Groups: Definitions,Basic Properties,and Problems

We find a natural generalization of the concept of rigid group. The generalized rigid groups are also called r-groups. The terms of the corresponding rigid series of every r-group can be characterized by both ?-formulas and ?-formulas. We find a recursive system of axioms for the class of r-groups of fixed solubility length. We define divisible r-groups and give an appropriate system of axioms. Several fundamental problems are stated.     

On Optimum Propulsion by Means of Small Periodic Motions of a Rigid Profile. I. Properties of Optimum Motions

The problem of optimum thrust generation by means of a rigid profile performing small arbitrarily periodic motions in an inviscid incompressible fluid is studied. The motions considered have to generate a prescribed mean value of thrust and must be such that the contribution to this mean thrust by the suction at the leading edge does not exceed a certain given value. Furthermore, the motions are in general subjected to a maximum type constraint on their amplitude. For this infinite dimensional, nonconvex and nonsmooth optimization problem, a generalized Lagrange multiplier rule is derived. In case the constraint on the amplitude is omitted, the optimum motions are calculated analytically; for the general case a number of properties of the solutions are derived from the Lagrange multiplier rule.     

Measure and integration: the basic extension theorems

The present article returns to the new foundations of measure and integration due to the author. In this development the basic extension procedures lead from the so-called outer and inner premeasures to their unique maximal extensions. The initial version was for extended real valued set functions. In the sequel we want to achieve a major simplification, in that we develop the procedures—with no loss in the essentials—in the traditional frame of nonnegative set functions. The final section then will obtain an important extension theorem in the inner theories.     

Cognitive map: KBDSS integration in transportation planning

This paper discusses how to use an integrated cognitive map-knowledge based decision support model to help the transportation planners in their selection of the most suitable alternative for a water crossing infrastructure. It also illustrates the evolution of this research since 1994. In this latest version, the knowledge is acquired from 19 experts on the subject and then an expert system is used to evaluate the alternatives based on their centrality, namely relative importance, shown by the aggregated cognitive map. Finally, this version of the model is applied to solve the water crossing traffic congestion of the Bosphorus at Istanbul and the validity of the results is discussed.     

Romberg integration: the importance of being analytical

The problem of reflection of water waves by a nearly vertical wall is studied. A simplified perturbational analysis followed by Havelock's expansion of water wave potential is employed to tackle the problem. Assuming some particular shapes of the nearly vertical wall, first‐order correction to the reflection coefficient is calculated for deep water as well as for uniform finite depth of water.     

Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type -- Part I: RK-methods

Summary. For implicit RK-methods applied to singularly perturbed systems of ODEs it is shown that the resulting discrete systems preserve the geometric properties of the underlying ODE. This invariant manifold result is used to derive sharp bounds on the global error of RK-solutions. Received August 26, 1993 / Revised version received May 10, 1994