On Affine Geometry of Purley Real Submanifolds

We propose to study n-dimensional purely real submanifolds of the standard affine complex space Cⁿ from an affine point of view. Such submanifolds are naturally endowed with a unique transversal bundle. Fundamental theorems are given as well as a theorem of Cartan–Norden type. Examples illustrating particular affine properties of such submanifolds are provided.     

Delay performance in stochastic processing networks with priority service

We evaluate the delay performance of an open multi-class stochastic processing network of multi-server resources with preemptive-resume priority service. We show that the stationary distribution of aggregate queue lengths has product form. For each service class we derive explicit expressions for the following stationary performance measures: The mean and, under feedforward routing, the Laplace transform of the delay distribution at each resource. We show that these measures are the same as if the resources were operating in isolation.     

Error estimates for finite element method solution of the Stokes problem in the primitive variables

Summary In this paper we derive error estimates for a class of finite element approximation of the Stokes equation. These elements, popular among engineers, are conforming lagrangian both in velocity and pressure and therefore based on a mixed variational principle. The error estimates are established from a new Brezzi-type inequality for this kind of mixed formulation. The results are true in 2 or 3 dimensions.     

Efficient parallel algorithms for shortest paths in planar digraphs

Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graphG with real-valued edge costs but no negative cycles. We assume that a planar embedding ofG is given, together with a set ofq faces that cover all the vertices. Then our algorithm runs inO(log² n) time and employsO(nq+M(q)) processors (whereM(t) is the number of processors required to multiply twot×t matrices inO(logt) time). Let us note here that wheneverq<n then our processor bound is better than the best previous one (M(n)).O(log² n) time,n-processor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directedouterplanar graph. Our work is based on the fundamental hammock-decomposition technique of G. Frederickson. We achieve this decomposition inO(logn log*n) parallel time by usingO(n) processors. The hammock-decomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms.This work was partially supported by the EEC ESPRIT Basic Research Action No. 3075 (ALCOM) and by the Ministry of Industry, Energy and Technology of Greece.     

On semidefinite representations of non-closed sets

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinitely representable. So far, all results focus on the case of closed sets.In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinitely representable set is shown to be semidefinitely representable. More general, one can remove faces of a semidefinitely representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.     

Mathematical justification of a heuristic for statistical correlation of real-life time series

Many of the analyses of time series that arise in real-life situations require the adoption of various simplifying assumptions so as to cope with the complexity of the phenomena under consideration. Whilst accepting that these simplifications lead to heuristics providing less accurate processing of information compared to the solution of analytical equations, the intelligent choice of the simplifications coupled with the empirical verification of the resulting heuristic has proven itself to be a powerful systems modelling paradigm. In this study, we look at the theoretical underpinning of a successful heuristic for estimation of urban travel times from lane occupancy measurements. We show that by interpreting time series as statistical processes with a known distribution it is possible to estimate travel time as a limit value of an appropriately defined statistical process. The proof of the theorem asserting the above, supports the conclusion that it is possible to design a heuristic that eliminates the adverse effect of spurious readings without loosing temporal resolution of data (as implied by the standard method of data averaging). The original contribution of the paper concerning the link between the analytical modelling and the design of heuristics is general and relevant to a broad spectrum of applications.     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.     

A general extrapolation algorithm

Summary In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied.     

Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

Summary The multidimensional (partial) difference equation with periodical coefficients is transformed into an equation for a vector sequence. Integral formulae for the vector fundamental solution are developed and some results about its asymptotic properties are explained. As an example, the results are used for a simple difference equation on a hexagonal grid.     

On the fast solutions of nonlinear elliptic equations

Summary In a recent paper we described a multi-grid algorithm for the numerical solution of Fredholm's integral equation of the second kind. This multi-grid iteration of the second kind has important applications to elliptic boundary value problems. Here we study the treatment of nonlinear boundary value problems. The required amount of computational work is proportional to the work needed for a sequence of linear equations. No derivatives are required since these linear problems are not the linearized equations.     

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide. Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies. We then turn to expansions of lattices with additional operations, and address the question of which equational properties of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras. For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation properties the primitive operations satisfy, the more equations are preserved. Received August 21, 2005; accepted in final form October 17, 2006.     

Fehleranalyse für die Gauß-Elimination zur Berechnung der Lösung minimaler Länge

Summary Presented is a realistic, elementwise analysis for the rounding errors of a generalization of Gauss elimination for solving the linear best least squares problem without pivoting. The bounds are suitable to determine the class of well-posed problems to the given method. A mixed error analysis is given and then the effects of errors in the input data are studied. Numerical examples demonstrate the efficiency.

     

Quivers of finite mutation type and skew-symmetric matrices

Quivers of finite mutation type are certain directed graphs that first arised in Fomin-Zelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations of surfaces. In this paper, we study structural properties of finite mutation type quivers in relation with the corresponding skew-symmetric matrices. We obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes.     

Integration over a simplex,truncated cubes,and Eulerian numbers

Summary A method of integrating a function over a simplex is described in which (i) the simplex is first transformed into a right-angled isosceles simplex; (ii) this simplex is dissected into small cubes and truncated cubes; (iii) the integration over the truncated cubes is performed by the centroid method or by Stroud's method, and this requires the use of formulae for the moments of a truncated cube. These formulae are developed and are expressed in terms of Eulerian numbers. In the special case when the truncated cube is itself a right-angled isoceles simplex a new algorithm is given, depending on the discrete Fourier transform, for calculating the moments as polynomials inn wheren is the dimensionality.     

The derivation of cubic splines with obstacles by methods of optimization and optimal control

Summary Interpolating splines which are restricted in their movement by the presence of obstacles are investigated. For simplicity we mainly treat cubic splines which are required to be non-negative. The extension to splines of higher order and to certain other forms of obstacles is straightforward. Methods of optimization and of optimal control are used to obtain necessary optimality criteria. These criteria are applied to derive an algorithm to compute splines which are restricted to constant lower or upper bounds. There is a numerical example which illustrates the method presented.Dedicated to Günter Meinardus on the occasion of his 60th birthday     

A gentle guide to the basics of two projections theory

This paper is a survey of the basics of the theory of two projections. It contains in particular the theorem by Halmos on two orthogonal projections and Roch, Silbermann, Gohberg, and Krupnik’s theorem on two idempotents in Banach algebras. These two theorems, which deliver the desired results usually very quickly and comfortably, are missing or wrongly cited in many recent publications on the topic, The paper is intended as a gentle guide to the field. The basic theorems are precisely stated, some of them are accompanied by full proofs, others not, but precise references are given in each case, and many examples illustrate how to work with the theorems.     

Singularly perturbed finite element methods

Summary This paper introduces a new piecewise linear finite element, which is designed to handle singularly perturbed ordinary differential equations. Both pointwise and global estimates (which are independent of the perturbation parameter) are obtained.     

Chord techniques and Newton's method for discrete bifurcation problems

Summary The numerical treatment of discrete bifurcation problems (2) with chord methods or Newton's method is a question of constructing appropriate initial approximations to prevent the sequence from converging to the trivial solution. This problem is being discussed under conditions which are satisfied for quite a few examples arising in applications (see Sect. 3).     

On the efficient solution of nonlinear finite element equations. II

Summary We consider nonlinear variational inequalities corresponding to a locally convex minimization problem with linear constraints of obstacle type. An efficient method for the solution of the discretized problem is obtained by combining a slightly modified projected SOR-Newton method with the projected version of thec g-accelerated relaxation method presented in a preceding paper. The first algorithm is used to approximately reach in relatively few steps the proper subspace of active constraints. In the second phase a Kuhn-Tucker point is found to prescribed accuracy. Global convergence is proved and some numerical results are presented.