Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

Modelling ship operational reliability over a mission under regular inspections

A ship is required to operate for a fixed mission period. Should a critical item of equipment fail at sea, the ship is subject to a costly event with potentially high risk to ship and crew. Given warning of a pending defect, the ship can try to return to port under its own power and thus attempt to avoid an at sea failure. Defects which lead to a failure are detected by inspection, and the task is to select the appropriate frequency of inspection to balance the number of occasions that a ship fails at sea and the number of preventive inspection based returns to port during a mission to correct a defect. The modelling entails using the delay time concept. Expressions are established for the expected number of preventive and failure returns over a mission, and an example given of a cost based balance to select an optimal inspection period. Although addressing ship reliability, the model has relevance to the mission reliability of any repairable equipment with remote main repair facilities.     

On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.     

Generalized Solutions of Nonlinear Parabolic Equations and Diffusion Processes

We reduce the construction of a weak solution of the Cauchy problem for a quasilinear parabolic equation to the construction of a solution to a stochastic problem. Namely, we construct a diffusion process that allows us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for a nonlinear PDE.      

On the properties of orthorecursive expansions with respect to subspaces

In a Hilbert space, for orthorecursive expansions with respect to closed subspaces, we establish a criterion for expansions of elements of a certain finite-dimensional subspace with respect to a finite sequence of subspaces to coincide with the expanded elements. This implies a criterion for an element to be equal to its orthorecursive expansion with respect to a finite sequence of subspaces. We also obtain a number of results related to the best approximations of elements by partial sums of their orthorecursive expansions with respect to a sequence of finite-dimensional subspaces.     

Moufang sets and jordan division algebras

We give a simple criterion which determines when a permutation group U and one additional permutation give rise to a Moufang set. We apply this criterion to show that every Jordan division algebra gives rise in a very natural way to a Moufang set, to provide sufficient conditions for a Moufang set to arise from a Jordan division algebra and to give a characterization of the projective Moufang sets over a commutative field of characteristic different from 2. The first author is a Postdoctoral Fellow of the Research Foundation – Flanders (Belgium) (FWO-Vlaanderen).     

Preservation of the existence of fixed and coincidence points under homotopy of mappings of ordered sets

In problems of topology and analysis, well-known theorem on the preservation by any continuous homotopy of the property of a mapping to have a fixed point and the property of a pair of mappings to have a coincidence point are extensively applied. Thus, for contraction mappings and some of their generalizations, Frigon’s results on the preservation of the property to have a fixed point by a homotopy of a special type are known. This paper presents theorems on the preservation by order homotopy of the property of a pair of mappings to have a coincidence point. As a corollary, conditions under which such a homotopy preserves the property of a mapping to have a fixed point are obtained.     

A comparison of two sourcing tactics for a new component

We compare two sourcing tactics for a manufacturer to purchase a new component to be used in a one-time production run of a new product with uncertain and price-elastic demand. One alternative is to issue a request-for-quote (RFQ), which is where the manufacturer requests a price-quantity schedule from suppliers. The manufacturer uses this information to determine a production quantity and the number of components to purchase from each supplier. The other alternative is to post a bid specifying how the manufacturer’s purchase quantity will depend on the supplier’s component price. The suppliers use this information to compete on quantity.We find that relative to RFQ, which is more challenging for the manufacturer to characterize the supplier response due to the possibility of supplier interaction, the benefit to the manufacturer from posting a bid increases with the number of suppliers due to increased intensity of competition. If the new component is from an emerging industry where there is little mutual awareness among candidate suppliers, then regardless of number of suppliers, expected manufacturer profit is higher under RFQ. Posting a bid is more likely to benefit the manufacturer when the new component is from a more established industry with a high degree of awareness among candidate suppliers.     

Simd language design using prescriptive semantics

The goal of this article is to isolate a set of primitives necessary for the construction of SIMD programs and to give a denotational semantics for these primitives. The intent is to devise a language with a simple semantics rather than to propose a language which may be conveniently implemented. The approach taken results in the addition of a synchronous parallel assignment statement and a synchronous communication statement to the familiar sequential programming language control structures of composition, projection and iteration.     

Lifting Property of Direct Sums of Hollow Modules

We give necessary and sufficient conditions for a module M with a decomposition that complements maximal direct summands to be a lifting module. We apply our results to get a necessary and sufficient condition for a direct sum of modules to be a quasi-discrete (discrete) module. We also give some sufficient conditions for a lifting module to be a direct sum of indecomposables.     

A Note On A Loaded Piezoelectric Transformer

Due to the coupling of mechanical and electrical quantities in a piezoceramic material, it is possible to transform a harmonic input voltage to a harmonic output voltage by using a piezoceramic transformer. The advantage of such a transformer is the avoidance of magnetic fields and the very simple assembly. The piezoelectric transformer considered here consists of several parts. Three brass and two piezoceramic rods, that are glued together. The purpose of the system is to transform a determined input voltage to a determined ouput voltage with a high efficiency. Therefore, one of the piezoceramics is excited harmonically with a frequency close to a resonance frequency of the system. According to the electromechanical coupling in piezoceramics, the transformer oscillates with the excitation frequency. Due to the piezoelectric effect an electric displacement or an electric voltage is generated between the electrodes of the second piezoceramic. The amplitude of the output voltage depends on the geometry and the load connected between the electrodes of the second piezoceramic. In this paper a theoretical model to determine the gain and input impedance for such a transformer is derived and the results are compared with experiments.     

Koszul Bicomplexes and Generalized Koszul Complexes in Projective Dimension One

We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex of finite free modules, we assemble these complexes to what we call Koszul bicomplexes. They are used in order to investigate the homology of the Koszul complexes in projective dimension one. As in the case of the classical Koszul complexes, this homology turns out to be grade sensitive. In a special setup, we obtain necessary conditions for a map of free modules to be lengthened to a short complex of free modules.     

On Euler's Formula exp (iθ) = cosθ + i sinθ

In approximating a function with a pole‐like singularity at a non‐rational point a* by a rational polynomial, the approximating package will be robust (i.e. undetectably erroneous to a user) if the rational polynomial includes a factor which cancels out the pole to the full accuracy of the machine in use. Naturally in order to be portable, the approximation a to a* must be to the full machine accuracy of the receiving machine.     

Limit Functions for Convergence Groups and Uniformly Quasiregular Maps

We investigate the limit functions of iterates of a functionbelonging to a convergence group or of a uniformly quasiregularmapping. We show that it is not possible for a subsequence ofiterates to tend to a non-constant limit function, and for anothersubsequence of iterates to tend to a constant limit function.It follows that the closure of the stabiliser of a Siegel domainfor a uniformly quasiregular mapping is a compact abelian Liegroup, which we further conjecture to be infinite. This resultconcerning possible limits of convergent subsequences of iteratesfor holomorphic rational functions on the Riemann sphere isknown, and the only known method of proof involves universalcovering surfaces and Möbius groups. Hence, our methodyields a new and perhaps more elementary proof also in thatcase.     

On Testing for the Number of Components in a Mixed Poisson Model

Poisson mixtures are usually used to describe overdispersed data. Finite Poisson mixtures are used in many practical situations where often it is of interest to determine the number of components in the mixture. Identifying how many components comprise a mixture remains a difficult problem. The likelihood ratio test (LRT) is a general statistical procedure to use. Unfortunately, a number of specific problems arise and the classical theory fails to hold. In this paper a new procedure is proposed that is based on testing whether a new component can be added to a finite Poisson mixture which eventually leads to the number of components in the mixture. It is a sequential testing procedure based on the well known LRT that utilises a resampling technique to construct the distribution of the test statistic. The application of the procedure to real data reveals some interesting features of the distribution of the test statistic.     

A hybrid and integrated approach to evaluate and prevent disasters

Disasters are, by their nature, very complex phenomena. Their modelling using a systematic and logical methodology can help us identify their root causes and may facilitate in allocating appropriate resources to prevent such situations. Although techniques exist to model such phenomena, a single off-the-shelf model is insufficient to provide an effective and realistic analysis to prevent disasters due to its inherent assumptions. In order to overcome these limitations of single methods, this article proposes a hybrid model of four methods to optimise a safety investment. First, a hierarchy is constructed with a problem structuring approach. Second, a new graphical representation, the Crisis Tree Analysis, is introduced to visualise how a combination of basic events may lead to a disaster. Third, the criticality of each event is assessed using the Analytic Hierarchy Process. Finally, a mathematical programming model is proposed to calculate the optimal allocation of available funds in order to avoid the disaster. A case study of the Bhopal disaster is used to illustrate the proposed four-step method.     

A parallel domain reduction method

We relate a particular version of a parallel multigrid method to a domain decomposition method, showing that the parallel multigrid method reduces computation to a small portion of the domain and then extends the solution to the entire domain using the correct reflections to get the exact solution. We extend a particular example to double the parallelism in a nonobvious manner. While the techniques of this paper are applied to twodimensional problems, they can be applied to higher dimensional problems in an obvious manner.     

Perturbation singulière d'un problème de frottement sec non monotone

In this Note we deal with a singularly perturbed system constituted by a differential inclusion which has a unique solution for each value of the perturbation parameter. The associated degenerated problem, that corresponds to a dynamic dry friction problem, has many solutions. We show that perturbed problem solutions converge to a particular solution of the degenerated problem when the perturbation parameter goes to zero. The singular perturbation approach allows an analysis of a criterion used to select a solution of the degenerated problem, and suggests a method to study more elaborated dry friction problems.     

A pair of orthogonal frames

We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Gabor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal wavelet frames.