a criterion for the existence of common invariant subspaces of matrices it is shown that square matrices a and b have a common invariant subspace w of dimension k 1 if and only if for some scalar s a c si and b c si are invertible and their kth compounds have a common eigenvector which is a grassmann representative for w t he applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated 2001 elsevier science inc all rights reserved

lattice sums for off axis electromagnetic scattering by gratings we consider both the spatial domain and spectral domain forms of the green s function appropriate in the electromagnetic diffraction of a plane wave incident at a general angle in the xy plane on a singly periodic structure or grating oriented along the x axis we equate the spatial and spectral forms of the green s function and so establish expressions from which grating lattice sums can be evaluated for oblique incidence we also obtain a set of identities among the lattice sums we use these lattice sums in an expression for the green s function which we show to be computationally fast if knowledge of this function at several points is required for small values of y

on probability possibility transformations abstract a probability possibility transformation is a purely mechanical transformation of probabilistic support to possibilistic support and vice versa that is a conversion of the measure of support of one theory into that of the other that is independent of the problem domain in this paper it is shown that any bijective mapping between probability distributions and normalized possibility distributions that is symmetric and expansible cannot preserve the properties of projection noninteraction or conditionalization

neumann series and lattice sums we consider sums over the square lattice which depend only on radial distance and provide formulas which enable sums of functions with neumann series to be reexpressed as combinations of hypergeometric series we illustrate the procedure using trigonometric sums previously studied by borwein and borwein sums combining logarithms bessel functions j and powers of distance and sums of neumann functions we also exhibit sums which may be evaluated analytically and recurrence formulas linking sums

problems in formal temporal reasoning abstract ever since its introduction by mccarthy and hayes in 1969 the so called frame problem has been the object of much fascination and debate although it was defined in the narrow context of the situation calculus a specific temporal formalism it was clear from the start that it is in fact a manifestation of some fundamental problem in temporal reasoning our aim in this informal paper is to identify the general form of certain classes of problems that arise in formal temporal reasoning we argue that problems such as the frame problem arise from the conflicting desires to reason both rigorously and efficiently about the future this conflict does not depend on the particular underlying temporal formalism in particular we identify two formalism independent problems called the qualification problem and the extended prediction problem which subsume the frame problem to illustrate the fact that these problems are indeed inherent to the prediction task and not to a particular formalism we show that they arise in two distinct frameworks classical mechanics and hayes histories notation

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

evaluation and regularization of phase modulated eisenstein series and application to double schlomilch type sums in the study of periodic media conditionally convergent series are frequently encountered and their regularization is crucial for applications we derive an identity that regularizes two dimensional phase modulated eisenstein series for all bravais lattices yielding physically meaningful values we also obtain explicit forms for the phase modulated series in terms of holomorphic eisenstein series enabling their closed form evaluation for important high symmetry lattices results are then used to obtain representations for the related double schlomilch type sums which are also given for all bravais lattices finally we treat displaced lattices of high symmetry expressing them in terms of origin centered lattices via geometric multi set identities these identities apply to all classes of two dimensional sums allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions in the study of periodic media conditionally convergent series are frequently encountered and their regularization is crucial for applications we derive an identity that regularizes two dimensional phase modulated eisenstein series for all bravais lattices yielding physically meaningful values we also obtain explicit forms for the phase modulated series in terms of holomorphic eisenstein series enabling their closed form evaluation for important high symmetry lattices results are then used to obtain representations for the related double schlomilch type sums which are also given for all bravais lattices finally we treat displaced lattices of high symmetry expressing them in terms of origin centered lattices via geometric multi set identities these identities apply to all classes of two dimensional sums allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions

measures of uncertainty in expert systems abstract this paper compares four measures that have been advocated as models for uncertainty in expert systems the measures are additive probabilities used in the bayesian theory coherent lower or upper previsions belief functions used in the dempster shafer theory and possibility measures fuzzy logic special emphasis is given to the theory of coherent lower previsions in which upper and lower probabilities expectations and conditional probabilities are constructed from initial assessments through a technique of natural extension mathematically all the measures can be regarded as types of coherent lower or upper previsions and this perspective gives some insight into the properties of belief functions and possibility measures the measures are evaluated according to six criteria clarity of interpretation ability to model partial information and imprecise assessments especially judgements expressed in natural language rules for combining and updating uncertainty and their justification consistency of models and inferences feasibility of assessment and feasibility of computations each of the four measures seems to be useful in special kinds of problems but only lower and upper previsions appear to be sufficiently general to model the most common types of uncertainty

fast algorithms for helmholtz green s functions the formal representation of the quasi periodic helmholtz green s function obtained by the method of images is only conditionally convergent and thus requires an appropriate summation convention for its evaluation instead of using this formal sum we derive a candidate green s function as a sum of two rapidly convergent series one to be applied in the spatial domain and the other in the fourier domain as in ewald s method we prove that this representation of green s function satisfies the helmholtz equation with the quasi periodic condition and furthermore leads to a fast algorithm for its application as an operator we approximate the spatial series by a short sum of separable functions given by gaussians in each variable for the series in the fourier domain we exploit the exponential decay of its terms to truncate it we use fast and accurate algorithms for convolving functions with this approximation of the quasi periodic green s function the resulting method yields a fast solver for the helmholtz equation with the quasi periodic boundary condition the algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when green s function is applied to discontinuous functions or potentials with singularities we also construct helmholtz green s functions with dirichlet neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi periodic green s function to apply them the complexity in dimension d 2 of these algorithms is graphic 1 d log c log 1 d where is the desired accuracy is proportional to the number of wavelengths contained in the computational domain and c is a constant we illustrate our approach with examples 1 embed inline graphic 1 gif

lattice sums for the helmholtz equation a survey of different representations for lattice sums for the helmholtz equation is made these sums arise naturally when dealing with wave scattering by periodic structures one of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d lambda lattice sums are related to and can be calculated from the quasi periodic green s function and this object serves as the starting point of the analysis

properties of measures of information in evidence and possibility theories abstract an overview of information measures recently introduced by several authors in the settting of shafer s theory of evidence is proposed new results pertaining to additivity and monotonicity properties of these measures of information are presented the interpretation of each measure of information as opposed to others is discussed the potential usefulness of measures of specificity or imprecision is suggested and a principle of minimal specificity is stated for the purpose of reconstructing a body of evidence from incomplete knowledge

the split ring resonator we obtain band diagrams for a two dimensional periodic structure consisting of an infinite square array of infinitely thin concentric circles split rings with narrow gaps our approach exploits the narrowness of the gaps and yields algebraic equations relating the frequency to the bloch wavenumber and geometric properties of the array further asymptotic analysis indicates that the gravest mode has a frequency that scales in an inverse logarithmic fashion with the size of the gap and that exhibits anomalous dispersion near the origin of the brillouin zone this acoustic mode is dispersionless numerical solution of the eigenvalue problem in the single gap case confirms these conclusions the two lowest modes of the split ring can be interpreted as a splitting of the gravest propagating rayleigh mode

common invariant subspaces of two matrices abstract a finite rational procedure of the shemesh type is proposed to check whether given complex n n matrices a and b have a common invariant subspace of dimension k 2 k n the only restriction is that at least one of a and b must have distinct eigenvalues this fact can also be verified by a finite rational computation

a new integral representation for quasi periodic fields and its application to two dimensional band structure calculations in this paper we consider band structure calculations governed by the helmholtz or maxwell equations in piecewise homogeneous periodic materials methods based on boundary integral equations are natural in this context since they discretize the interface alone and can achieve high order accuracy in complicated geometries in order to handle the quasi periodic conditions which are imposed on the unit cell the free space green s function is typically replaced by its quasi periodic cousin unfortunately the quasi periodic green s function diverges for families of parameter values that correspond to resonances of the empty unit cell here we bypass this problem by means of a new integral representation that relies on the free space green s function alone adding auxiliary layer potentials on the boundary of the unit cell itself an important aspect of our method is that by carefully including a few neighboring images the densities may be kept smooth and convergence rapid this framework results in an integral equation of the second kind avoids spurious resonances and achieves spectral accuracy because of our image structure inclusions which intersect the unit cell walls may be handled easily and automatically our approach is compatible with fast multipole acceleration generalizes easily to three dimensions and avoids the complication of divergent lattice sums

temporal similarity by measuring possibilistic uncertainty in cbr similarity is an essential concept in case based reasoning cbr in domains in which time plays a relevant role cbr systems require good temporal similarity measures to compare cases temporal cases are traditionally represented by a set of temporal features defining time series and temporal event sequences in the particular situation where these features are not homogeneous i e combination of qualitative and quantitative information systems find difficulties in performing the cbr cycle furthermore temporal similarity measures cannot directly apply the efficient time series techniques requiring new approaches to deal with these heterogeneous sequences to this end recent proposals are focused on direct matching between pairs of features within sequences mainly based on classical distances however three limitations to the traditional approaches have been identified 1 they do not consider the implicit temporal relations amongst all features of the sequence ignoring a large amount of temporal information 2 they ignore the uncertainty produced in any process of analogy 3 they are designed to compare pairs of sequences limiting their use to basic aspects of the retrieval step of cbr no benefits on other cbr steps temporal constraint networks have proved to be useful tools for temporal representation and reasoning and can be easily extended to manage imprecision and uncertainty an approach to solve similarity problems could be the transformation of these heterogeneous sequences into uncertain temporal relations obtaining a temporal constraint network the overall uncertainty of this network can be considered as an effective indicator of the sequences similarity therefore this paper proposes a non classical approach to measure temporal similarity of cases which are heterogeneous temporal event sequences given two or more sequences the temporal similarity is measured by describing a unique temporal scenario of possibilistic temporal relations and calculating the uncertainty produced

loss and gain functions for cbr retrieval the method described in this article evaluates case similarity in the retrieval stage of case based reasoning cbr it thus plays a key role in deciding which case to select and therefore in deciding which solution will be eventually applied in cbr there are many retrieval techniques one feature shared by most is that case retrieval is based on attribute similarity and importance however there are other crucial factors that should be considered such as the possible consequences of a given solution in other words its potential loss and gain as their name clearly implies these concepts are defined as functions measuring loss and gain when a given retrieval case solution is applied moreover these functions help the user to choose the best solution so that when a mistake is made the resulting loss is minimal in this way the highest benefit is always obtained

sums of spherical waves for lattices layers and lines we consider the connections between sums of spherical wave functions over lattices layers and lines the differences between sums over lattices and those over a doubly periodic constituent layer are expressed in terms of series with exponential convergence correspondingly sums over the layer can be regarded as composed of a sum over a central line and another sum over displaced lines exhibiting exponential convergence we exhibit formulas which can be used to calculate accurately and efficiently sums of spherical waves over lattices layers and lines which in turn may be used to construct quasiperiodic green s functions for the helmholtz equation of use in scattering problems for layers and lines of spheres and for finding the bloch modes of lattices of spheres we illustrate the numerical accuracy of our expressions

integral representations of harmonic lattice sums we present new integral formulas for coulombic lattice sums that arise in solid state physics material science and complex analysis unlike ewald summation which can yield approximate integral representations the formulas described here are exact simple quadrature rules with modest numbers of nodes yield highly accurate results

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

quasi periodic green s functions of the helmholtz and laplace equations a classical problem of free space green s function g0 representations of the helmholtz equation is studied in various quasi periodic cases i e when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space exponentially convergent series for the free space quasi periodic g0 and for the expansion coefficients dl of g0 in the basis of regular cylindrical in two dimensions and spherical in three dimension 3d waves or lattice sums are reviewed and new results for the case of a one dimensional 1d periodicity in 3d are derived from a mathematical point of view a derivation of exponentially convergent representations for schlomilch series of cylindrical and spherical hankel functions of any integer order is accomplished exponentially convergent series for g0 and lattice sums dl hold for any value of the bloch momentum and allow g0 to be efficiently evaluated also in the periodicity plane the quasi periodic green s functions of the laplace equation are obtained from the corresponding representations of g0 of the helmholtz equation by taking the limit of the wave vector magnitude going to zero the derivation of relevant results in the case of a 1d periodicity in 3d highlights the common part which is universally applicable to any of remaining quasi periodic cases the results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics remote sensing of periodic surfaces periodic gratings and infinite arrays of resonators coupled to a waveguide in many contexts of simulating systems of charged particles in molecular dynamics for the description of quasi periodic arrays of point interactions in quantum mechanics and in various ab initio first principle multiple scattering theories for the analysis of diffraction of classical and quantum waves

efficient calculation of lattice sums for free space periodic green s function an efficient method to calculate the lattice sums is presented for a one dimensional 1 d periodic array of line sources the method is based on the recurrence relations for hankel functions and the fourier integral representation of the zeroth order hankel function the lattice sums of arbitrary high order are then expressed by an integral of elementary functions which is easily computed using a simple scheme of numerical integration the calculated lattice sums are used to evaluate the free space periodic green s function the numerical results show that the proposed method provides a highly accurate evaluation of the green s function with far less computation time even when the observation point is located near the plane of the array

one and two dimensional lattice sums for the three dimensional helmholtz equation the accurate and efficient computation of lattice sums for the three dimensional helmholtz equation is considered for the cases where the underlying lattice is one or two dimensional we demonstrate using careful numerical computations that the reduction method in which the sums for a two dimensional lattice are expressed as a sum of one dimensional lattice sums leads to an order of magnitude improvement in performance over the well known ewald method in the process we clarify and improve on a number of results originally formulated by twersky in the 1970s

the common invariant subspace problem an approach via grobner bases let a be an n n matrix it is a relatively simple process to construct a homogeneous ideal generated by quadrics whose associated projective variety parametrizes the one dimensional invariant subspaces of a given a finite collection of n n matrices one can similarly construct a homogeneous ideal again generated by quadrics whose associated projective variety parametrizes the one dimensional subspaces which are invariant subspaces for every member of the collection grobner basis techniques then provide a finite rational algorithm to determine how many points are on this variety in other words a finite rational algorithm is given to determine both the existence and quantity of common one dimensional invariant subspaces to a set of matrices this is then extended for each d to an algorithm to determine both the existence and quantity of common d dimensional invariant subspaces to a set of matrices 2004 published by elsevier inc

lattice sums for gratings and arrays lattice sums arising in quasiperiodic green s functions for the helmholtz equation over general two dimensional arrays are investigated the array sums are related to those over a single quasiperiodic line of sources and their difference is be expressed in terms of exponentially convergent series it is shown that our expressions can be used to generate the sums pertaining to the case of photonic gap states associated with complex quasiperiodicity bloch vectors the accuracy and computational speed of our expressions are illustrated

a fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions we present a new integral equation method for the calculation of two dimensional scattering from periodic structures involving triple points multiple materials meeting at a single point the combination of a robust and high order accurate integral representation and a fast direct solver permits the efficient simulation of scattering from fixed structures at multiple angles of incidence we demonstrate the performance of the scheme with several numerical examples

fuzzy constraint networks for signal pattern recognition this paper deals with representation and reasoning on information concerning the evolution of a physical parameter by means of a model based on the fuzzy constraint satisfaction problem formalism and with which it is possible to define what we call fuzzy temporal profiles ftp based on fundamentally linguistic information this model allows the integration of knowledge on the evolution of a set of parameters into a knowledge representation scheme in which time plays a fundamental role the ftp model describes the behaviour of a physical parameter on the basis of a set of signal events and which allows the evolution of the parameter between each pair of events to be modelled as signal episodes given the fundamentally linguistic nature of the information represented the consistency analysis of this information is an essential task nevertheless the obtention of the minimal representation of the network that defines an ftp is an np hard problem in spite of this we supply algorithms guaranteeing local levels of consistency that allow to correct a large proportion of the errors committed by a human expert in the linguistic description of the profile furthermore we propose a new topology whose consistency can be guaranteed in polynomial time we also study the applicability of this model in the recognition of signal patterns