numerical performance of the matrix pencil algorithm computing the greatest common divisor of polynomials and comparison with other matrix based methodologies this paper presents a new numerical algorithm for the computation of the greatest common divisor gcd of several polynomials based on system theoretic properties the specific algorithm characterizes the gcd as the output decoupling zero polynomial of an appropriate linear system associated with the given polynomial set the computation of the gcd is thus reduced to specifying a nonzero entry of a vector forming the compound matrix of a matrix pencil directly produced from the associated linear system a detailed description of the implementation of the algorithm is presented and analytical proofs of its stability are also developed the matlab code of the algorithm is also described in the appendix

scattering from an impedance object at the edge of a perfectly conducting wedge in this study scattering from impedance bodies positioned at the edge of a perfectly electrically conducting wedge is investigated in the treatment of the problem eigenfunction expansion in terms of spherical vector wave functions is employed a complete dyadic green s function for the spherical impedance boss at the edge is developed it is observed that the scattering is highly enhanced by the edge guided waves additionally using t matrix method the solution is extended to the general case of irregularly shaped scatterers the t matrix solution is verified by applying it to the case of a spherical scatterer and results are compared with the dyadic green s function solution

approximate gcd and its application to ill conditioned algebraic equations abstract we describe two algorithms of approximate gcd greatest common divisor for polynomials with coefficients of floating point numbers one is for univariate polynomials and the other for multivariate polynomials the algorithms are careful extensions of the conventional euclidean algorithm and they are applied to solving ill conditioned algebraic equations ill conditioned univariate algebraic equations have multiple or close roots and we can ill conditioned system of algebraic equations which have an approprimate common divisor conventional numeric root finding methods such as newton s method give no satisfactory result for such ill conditioned systems by using an approximate gcd algorithm for multivariate polynomials the system is transformed into a well conditioned problems by a combination of algebraic and numeric methods we call such algorithms hybrid algorithms this paper shows the importance and possible fruitfulness of the hybrid algorithm

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

an equivalence classes model of fuzzy relational databases abstract the technique of employing sets of values for tuple components to express imprecision in relational databases was proposed by buckles and petry in their classic works on fuzzy relational databases in addition to providing an intuitively appealing scheme for representing fuzzy information the model of buckles and petry possesses several key properties of the classical relational model by employing finite scalar domains with similarity relations and special fuzzy number domains buckles and petry have demonstrated that the classical properties of uniqueness of tuple interpretations and well definedness of the relational algebra can be retained in the fuzzy relational database model the key to the preservation of these properties is the fact that scalar domains with similarity relations and the fuzzy number domains can be partitioned into equivalence classes however since equivalence classes can be constructed without assuming the existence of similarity relations or special fuzzy number domains it is desirable to generalize the fuzzy relational database model to one based only on equivalence classes in this work we show that the important properties of classical relational databases and of fuzzy relational databases are preserved in a generalized model built on equivalence relations on finite database domains further we generalize the notion of a functional dependency to the fuzzy relational model

system theoretic based characterisation and computation of the least common multiple of a set of polynomials abstract the paper provides a system theoretic characterisation of the least common multiple lcm m s of a given set of polynomials p which leads to an efficient numerical procedure for the computation of lcm that avoids root finding and use of greatest common divisor gcd procedures the procedure that is presented also leads to the computation of the associated set of multipliers of p with respect to lcm the basis of the new characterisation and computational procedure are the controllability properties of a natural realization s a b c associated with the set p it is shown that the coefficients of the lcm are defined by the properties of the controllable subspace of the pair a b which also leads to the characterisation of associated multipliers an algorithmic procedure that exploits the companion structure of a is formulated and its numerical properties are investigated the new method provides a robust procedure for the computation of lcm and enables the computation of approximate values when the original data are innacurate

approximate least common multiple of several polynomials using the eres division algorithm in this paper a numerical method for the computation of the approximate least common multiple alcm of a set of several univariate real polynomials is presented the most important characteristic of the proposed method is that it avoids root finding procedures and computations involving the greatest common divisor gcd conversely it is based on the algebraic construction of a special matrix which contains key data from the original set of polynomials and leads to the formulation of a linear system which provides the degree and the coefficients of the alcm using low rank approximation techniques and numerical optimization tools particularly in the presence of inaccurate data the numerical stability and complexity of the method are analysed and a comparison with other methods is provided

almost zeros of a set of polynomials of r s the notion of an exact zero of a sot of polynomials p of r s is extended to that of an almost zero the almost zeros are defined as minima of a norm function of p and their invariance under a certain equivalence relation defined on p is established the region of the complex plane which contains the prime almost zero global minimum is defined and necessary and sufficient conditions characterizing the set of almost zeros are given the role of almost zeros of p in the zero assignment problem of the combinants of p is investigated under the assumption of non assignability for p it is shown that the almost zeros act as nearly fixed zeros of the combinants every almost zero z is the centre of a disc d z r which contains at least one zero of all combinants of p the size of the radius r of this disc provides a measure of the ability of the almost zero to attract the zeros of the combinants of p a number of results for estimating upper bounds for the radius r associated with subfamilies of the ge

structured singular values and stability analysis of uncertain polynomials part 1 the generalized m abstract one of the primary goals in this two part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis to achieve these goal we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework part 1 of this series is devoted entirely to the generalized structured singular values specifically for the case when a certain matrix representing the nominal system is of rank one an analytical expression for the generalized notion is derived in this case which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable in particular when the general framework is specialized to that of the standard structured singular value the expression is solved explicitly also for several additional important cases explicit solutions are obtained the framework as well as results will then be used in part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties we demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches for several problems of interest we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of kharitonov theorem can be derive from one another and hence establish a link between two drastically different type of results

matrix pencil methodologies for computing the greatest common divisor of polynomials hybrid algorithms and their performance the computation of the greatest common divisor gcd of several polynomials is a problem that emerges in many fields of applications the gcd computation has a non generic nature and thus its numerical computation is a hard problem in this paper we examine the family of matrix pencil methods for gcd computation and investigate their performance as far as their complexity error analysis and their effectiveness for evaluating approximate solutions the relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree the developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations the combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm furthermore it emphasizes the significance of hybrid computations in complex problems such as the computation of gcd all m

time domain uniform geometrical theory of diffraction for a curved wedge a time domain version of the uniform geometrical theory of diffraction td utd is developed to describe in closed form the transient electromagnetic scattering from a perfectly conducting arbitrarily curved wedge excited by a general time impulsive astigmatic wavefront this td utd impulse response is obtained by a fourier inversion of the corresponding frequency domain utd solution an analytic signal representation of the transient fields is used because it provides a very simple procedure to avoid the difficulties that result when inverting frequency domain utd fields associated with rays that traverse line or smooth caustics the td utd response to a more general transient wave excitation of the wedge may be found via convolution a very useful representation for modeling a general pulsed astigmatic wave excitation is also developed which in particular allows its convolution with the td utd impulse response to be done in closed form some numerical examples illustrating the utility of these developments are presented

numerical evaluation of the derivative of the parabolic cylinder function we have presented numerically useful expressions for calculating the derivative of the parabolic cylinder function d sub p z for the argument z xe sup i spl pi 4 and order p 1 2 iy with x and y real this case is needed for calculating the currents excited on a perfectly conducting parabolic cylinder by a te plane wave and for calculating the scattered far field pattern

variable projection methods for approximate greatest common divisor computations we consider the problem of finding for a given n tuple of polynomials real or complex the closest n tuple that has a common divisor of degree at least d extended weighted euclidean seminorm of the coefficients is used as a measure of closeness two equivalent representations of the problem are considered i direct parameterization over the common divisors and quotients image representation and ii sylvester low rank approximation kernel representation we use the duality between least squares and least norm problems to show that i and ii are closely related to mosaic hankel low rank approximation this allows us to apply to the approximate common divisor problem recent results on complexity and accuracy of computations for mosaic hankel low rank approximation we develop optimization methods based on the variable projection principle both for image and kernel representation these methods have linear complexity in the degrees of the polynomials for small and large d we provide a software implementation of the developed methods which is based on a software package for structured low rank approximation

a hybrid method combining the multitemporal resolution time domain method of moments with the time domain geometrical theory of diffraction for thin wire antenna problems in this paper a hybrid method combining a multitemporal resolution mtr enhanced time domain method of moments td mom with the td geometrical theory of diffraction gtd is presented which allows to efficiently calculate the transient fields radiated by antennas in presence of large objects the mtr scheme tailors the time step size and thus implicitly the duration of the temporal basis function on the basis of the distance between source and test element additionally the voltage induced by source elements far away from the test element is interpolated in time the hybrid method is applied to calculate the radiation properties of thin wire antennas in presence of perfectly conducting flat scatterers to demonstrate its basic features and advantages

grassmann invariants almost zeros and the determinantal zero pole assignment problems of linear multivariable systems the strucxpects of the rationxr spaces x c x fwhere x c is the columxof the transfer function matrix g s and x f is the space associated with right matrix fracxcriptions of g s are investigated for x f gcanonixqb s grassmgesentax g x c and g x f arxd and are shown xmplete basis free invariants for x c and x f respectively the almost zeros az and almost decoupling zeros adz of g s gdefinex local minima of a normgn defix g x c and g x f respectively the computation and certain aspects of the distribution in the complex plane of azs and adzs are examined the role of azs and adzs in the determinantal zero and pole assignment problems respectively is examined next two important families of systems are defined the strongly zero non assignable szna and the strongly pole non assignable spna systems for szna and spna systems minimal radius discs dem e z r em e z and dem f zb centred at an az and adz respectively are defined it is shown that dem e r em e z contains at least one zer

incremental length diffraction coefficients for the shadow boundary of a convex cylinder incremental length diffraction coefficients ildcs are obtained for the shadow boundaries of perfectly electrically conducting pec convex cylinders of general cross section a two step procedure is used first the nonuniform nu current in the vicinity of the shadow boundary is approximated using fock 1965 functions the product of the approximated current and the free space green s function is then integrated on a differential strip of the cylinder surface transverse to the shadow boundary to obtain the ildcs this integration is performed in closed form by employing quadratic polynomial approximations for the amplitude and unwrapped phase of the integrand examples are given of both the current approximations and the integration procedure finally as an example the scattered far field of a pec sphere is obtained by adding the integral of the nu ildcs of a circular cylinder along the shadow boundary of the sphere to the physical optics po far field of the sphere this correction to the po field is shown to significantly improve upon the accuracy of the po far field approximation to the total scattered field of the sphere

matrix representation of the shifting operation and numerical properties of the eres method for computing the greatest common divisor of sets of many polynomials the extended row equivalence and shifting eres method is a matrix based method developed for the computation of the greatest common divisor gcd of sets of many polynomials in this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the eres method by introducing its complete algebraic representation then we analyse in depth its overall numerical stability in finite precision arithmetic numerical examples and comparison with other methods are also presented

qr factorization method for computing the greatest common divisor of polynomials with inexact coefficients this paper presents a novel means of computing the greatest common divisor gcd of two polynomials with real valued coefficients that have been perturbed by noise the method involves the qr factorization of a near to toeplitz matrix derived from the sylvester matrix of the two polynomials it turns out that the gcd to within a constant factor is contained in the last nonzero row of the upper triangular matrix r in the qr factorization of the near to toeplitz matrix the qr factorization is efficiently performed by an algorithm due to chun et al 1987 a condition number estimator due to bischof 1990 and an algorithm for rank estimation due to zarowski 1998 are employed to account for the effects of noise

fast iterative image reconstruction for mri in the presence of field inhomogeneities in magnetic resonance imaging magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast fourier transform fft methods several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities but these methods assume that the field map that characterizes the off resonance frequencies is spatially smooth recently iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off resonance effects however straightforward implementations of such iterative methods suffer from inconveniently long computation times this paper describes a tool for accelerating iterative reconstruction of field corrected mr images a novel time segmented approximation to the mr signal equation we use a min max formulation to derive the temporal interpolator speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast fourier transform with normalized root mean squared approximation errors of 0 07 the proposed method provides fast accurate field corrected image reconstruction even when the field map is not smooth

on the zeros of blocked time invariant systems abstract this paper studies the zero properties of blocked linear systems resulting from blocking of linear time invariant systems the main idea is to establish a relation between the zero properties of blocked systems and the zero properties of their corresponding unblocked systems in particular it is shown that the blocked system has a zero if and only if the associated unblocked system has a zero furthermore the zero properties of blocked systems under a generic setting i e a setting which parameter matrices a b c d assume generic values are examined it is demonstrated that nonsquare blocked systems i e blocked systems with number of outputs unequal to the number of inputs generically have no zeros however square blocked systems i e blocked systems with equal number of inputs and outputs generically have only finite zeros and these finite zeros have geometric multiplicity one

uniform physical theory of diffraction equivalent edge currents for truncated wedge strips new uniform closed form expressions for physical theory of diffraction equivalent edge currents are derived for truncated incremental wedge strips in contrast to previously reported expressions the new expressions are well behaved for all directions of incidence and observation and take a finite value for zero strip length this means that the expressions are well suited for implementation in general computer codes the new expressions are expressed as the difference between two terms the first term is obtained by integrating the exact fringe wave current on a wedge along an untruncated incremental strip extending from the leading edge of the structure under consideration the second term is calculated from an integration of the asymptotic fringe wave fw current along another untruncated incremental strip extending from the trailing edge of the structure the new expressions are tested numerically on a triangular cylinder and the results are compared with those obtained using the method of moments and the previously reported expressions

time domain diffraction by a right angled penetrable wedge the electromagnetic diffraction by a right angled dielectric wedge normally illuminated by a plane wave is considered in the time domain the proposed approach exploits the knowledge of uniform asymptotic diffraction coefficients in the frequency domain and uses the inverse laplace transform for obtaining the response to an impulse plane wave then the response to an arbitrary function plane wave is determined in the dielectric material and the surrounding space via a convolution integral comparisons with fdtd data prove the validity of the approach at best authors knowledge time domain solutions are not available to date for the considered diffraction problem

extending the fuzzy database with fuzzy numbers abstract the fuzzy relational database model originated by the authors permits fuzzy domain values from a discrete finite universe the model is extended here by demonstrating that fuzzy numbers may be employed as domain values without loss of consistency with respect to representation or the relational algebra where equivalence is required in an ordinary relational database similarity is employed in a fuzzy relational database for discrete finite universes similarity between atomic elements is described via a fuzzy similarity relation with max min transitivity two or more fuzzy numbers are defined to be similar if their union forms a continuous level set over the real line this convention effects the partitioning of fuzzy number domains that is necessary to assure the well definedness of the fuzzy relational algebra

the computation of multiple roots of a polynomial this paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise and whose exact form contains multiple roots the addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots but the algorithms presented in this paper preserve the multiplicities of the roots in particular even though the given polynomial is corrupted by noise and all computations are performed on these inexact coefficients the algorithms sew together the simple roots that originate from the same multiple root thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial the algorithms described in this paper do not require that the noise level imposed on the coefficients be known and all parameters are calculated from the given inexact coefficients examples that demonstrate the theory are presented

fuzzy loss less decompositions in databases in this paper we deal with the problem of defining projection and join operators in the framework of a fuzzy relational database we study the properties of such operators relating to the concept of fuzzy functional dependency there are two main properties to achieve on the one hand if a relation r satisfies a dependency between two sets of attributes x and y then we want to preserve this dependency in the fuzzy projection over xy of r on the other hand we want to test a fuzzy dependency just by looking at this projection therefore such a projection operator must allow us to store the information conveyed in r into a separate relation with fewer tuples in order to define the criterion for fuzzy dependency we shall need to extend a resemblance relation defined for crisp data to the fuzzy case finally we shall introduce a fuzzy join operator which will allow us to recover the information given in a relation from its projections

generalised resultants dynamic polynomial combinants and the minimal design problem the theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems dap which provides the unifying description of the dynamic as well as static pole and zero dynamic assignment problems in linear systems the assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original dap this paper demonstrates the origin of dynamic polynomial combinants from linear systems examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree and examines their spectral assignment central to this study is the link of dynamic combinants to the theory of generalised resultants which provide the matrix representation of the dynamic combinants the paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above an

time domain equivalent edge currents for transient scattering time domain equivalent edge currents td eec are developed for the transient scattering analysis the development is based an the fourier inversion of frequency domain equivalent edge current expressions the time domain diffracted fields are expressed in terms of a contour integral along the diffracting edges for any arbitrary input pulse shape thereby yielding finite results at the caustics of diffracted rays the approach also eliminates the need for the evaluation of a convolution integral in the time domain geometrical theory of diffraction gtd analysis the results are compared with the first order gtd results for the transient scattering analysis for a circular disk

multiple time domain diffraction for uwb signals the time domain td solution of the two dimensional multiple diffraction case is investigated the proposed td solution is based on the representation of the inverse fourier transform of the corresponding frequency domain fd solution in closed form as it is given by the uniform theory of diffraction utd and it incorporates the td representation of the higher order diffraction coefficients an algorithm to predict the td diffracted field after an arbitrary number of objects is also presented in the proposed algorithm different types of objects along the propagation path can be applied as well such as absorbing knife edges and metallic or nonperfectly conducting wedges the comparison between the td solution and the numerical inverse fast fourier transform of the fd solution proves the validity of the proposed solution

numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using fdtd classical theories such as the uniform geometrical theory of diffraction utd utilize analytical expressions for diffraction coefficient for canonical problems such as the infinite perfectly conducting wedge we present a numerical approach to this problem using the finite difference time domain fdtd method we present results for the diffraction coefficient of the two dimensional 2 d infinite perfect electrical conductor pec wedge the 2 d infinite lossless dielectric wedge and the 2 d infinite lossy dielectric wedge for incident tm and te polarization and a 90 spl deg wedge angle we compare our fdtd results in the far field region for the infinite pec wedge to the well known analytical solutions obtained using the utd there is very good agreement between the fdtd and utd results the power of this approach using fdtd goes well beyond the simple problems dealt with in this paper it can in principle be extended to calculate the diffraction coefficients for a variety of shape and material discontinuities even in three dimensions

stability criteria of matrix polynomials abstractin this paper the stability of matrix polynomials is investigated first upper and lower bounds are derived for the eigenvalues of a matrix polynomial the bounds are based on the spectra

computation of the gcd of polynomials using gaussian transformations and shifting abstract a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s pm d of maximal degree d is presented this method is based on a recently developed theoretical algorithm karcanias 1987 that uses elementary transformations and shifting operations the present algorithm takes into account the non generic nature of gcd and thus uses steps which minimize the introduction of additional errors and defines the gcd in an approximate sense for a given set pm d with a basis matrix pm the method defines first the most orthogonal uncorrupted base pr from the rows of pm where r rank pm m by applying successively gaussian transformations and shifting on the basis matrix pr e rr d 1 we produce each time a new basis matrix pz with z rank pz r the method terminates when the rank of pz is approximately equal to 1 the coefficient vector of the gcd is then defined as a row of the unit rank matrix pz the method defines the exact degree of t

an efficient ray tracing algorithm for multiple straight wedge diffraction we present an efficient algorithm for the tracing of multiply edge diffracted rays the algorithm assumes a given sequence of infinite edges and complete visibility among them the ray tracing problem is formulated as the minimization of the ray total path length since such a cost function is strictly convex except for coplanar edges in the plane wave far field regime then the problem admits a unique global minimum and allows the use of a modified newton search algorithm which exhibits a very high converging rate we also propose a convenient starting point to effectively initialize the minimization algorithm the proposed algorithm is tested by some numerical examples that show its efficiency and effectiveness

numerical evaluation of parabolic cylinder functions the numerical evaluation of the parabolic cylinder functions d sub p z in two cases is described case i is for the argument z xe sup i pi 4 with x real and the order p 1 2 iy with y real case ii is for z arbitrary but p an integer these cases are of special importance in the analysis of wave scattering from a parabolic cylinder expressions for d sub p z are presented which are numerically accurate and efficient

algebraic decomposition of fat and water in mri the decomposition of magnetic resonance imaging mri data to generate water and fat images has several applications in medical imaging including fat suppression and quantification of visceral fat we introduce a novel algorithm to overcome some of the problems associated with current analytical and iterative decomposition schemes in contrast to traditional analytical schemes our approach is general enough to accommodate any uniform echo shift pattern any number of metabolites and signal samples in contrast to region growing method that use a smooth field map initialization to resolve the ambiguities with the ideal algorithm we propose to use an explicit smoothness constraint on the final fieldmap estimate towards this end we estimate the number of feasible solutions at all the voxels prior to the evaluation of the roots this approach enables the algorithm to evaluate all the feasible roots thus avoiding the convergence to the wrong solution the estimation procedure is based on a modification of the harmonic retrieval hr framework to account for the chemical shift dependence in the frequencies in contrast to the standard linear hr framework we obtain the frequency shift as the common root of a set of quadratic equations on most of the pixels with multiple feasible solutions the correct solution can be identified by a simple sorting of the solutions we use a region merging algorithm to resolve the remaining ambiguity and phase wrapping experimental results indicate that the proposed algebraic scheme eliminates most of the difficulties with the current schemes without compromising the noise performance moreover the proposed algorithm is also computationally more efficient

td utd solutions for the transient radiation and surface fields of pulsed antennas placed on pec smooth convex surfaces a time domain formulation of the uniform geometrical theory of diffraction td utd is developed for predicting the transient radiation and surface fields of elemental pulsed antennas placed directly on a smooth perfectly conducting arbitrary convex surface the td utd solution is obtained by employing an analytic time transform att for inverting into time the corresponding frequency domain utd fd utd solution an elemental antenna on the convex surface is excited by a step function in time and a td utd solution is obtained first the td utd response to a more general pulsed excitation of the elemental current is then found via an efficient convolution of the td utd solution for the step function excitation with the time derivative of the general pulsed excitation in particular this convolution integral is essentially evaluated in closed form after representing the time derivative of the general pulsed excitation by a small sum of simple signals whose frequency domain description is a sum of complex exponential functions some numerical examples are presented to illustrate the utility of these td utd solutions for pulsed antennas on a convex body

necessary and sufficient conditions for zero assignment by constant squaring down the problem of zero assignment by constant squaring down czap is studied for minimal systems described by a transfer function g s rm l s m l using tools from exterior algebra and algebraic geometry conditions for its solvability are given in terms of the numbers m l the forney dynamical order and the rank of the plucker matrix p for the cases l 1 or l m 1 it is shown that g s is completely zero assignable cza if and only if 1 for the cases l 1 m 1 it is proved that g s is cza or generically zero assignable gza under a complex squaring down if and only if l m l 1 and 1 the latter conditions also provide necessary conditions for the existence of a real solution for a generic g s sufficient conditions for solvability of czap by a real squaring down are that l m l 1 and that the number g a0 al 1 1 a0 al 1 i j ai aj is odd for some set a0 al 1 satisfying 1 l 1i 0ai l l 1 2 and 0 a0 a1 al 1 m 1 apart from the existence results the present approach also allows the computation of the solutions whenever they exist finally it is proved that the only fixed zeros of czap are the zeros of g s and a procedure for computing the solutions of czap whenever such solutions exist based on an optimization problem is given

time domain version of the physical theory of diffraction a time domain version of the equivalent edge current eec formulation of the physical theory of diffraction is derived the time domain eecs td eecs apply to the far field analysis of diffraction by edges of perfectly conducting three dimensional 3 d structures with planar faces illuminated by a time domain plane wave by adding the field predicted by the td eecs to the time domain physical optics td po field a significant improvement is obtained compared to what can be achieved by using td po alone the td eecs are expressed as the integral of the time domain fringe wave current the exact current minus the td po current on the canonical wedge along truncated incremental strips closed form expressions for the td eecs are obtained in the half plane case by analytically carrying out the integration along the truncated incremental strip directly in the time domain in the general wedge case closed form expressions for the td eecs are obtained by transforming the corresponding frequency domain eecs to the time domain the td eecs are tested numerically on the triangular cylinder and the results are compared with those obtained using the method of moments in combination with the inverse fast fourier transform

computation of approximate polynomial gcds and an extension computation of approximate polynomial greatest common divisors gcds is important both theoretically and due to its applications to control linear systems network theory and computer aided design we study two approaches to the solution so far omitted by the researchers despite intensive recent work in this area correlation to numerical pade approximation enabled us to improve computations for both problems gcds and pade reduction to the approximation of polynomial zeros enabled us to obtain a new insight into the gcd problem and to devise effective solution algorithms in particular unlike the known algorithms we estimate the degree of approximate gcds at a low computational cost and this enables us to obtain certified correct solution for a large class of input polynomials we also restate the problem in terms of the norm of the perturbation of the zeros rather than the coefficients of the input polynomials which leads us to the fast certified solution for any pair of input polynomials via the computation of their roots and the maximum matchings or connected components in the associated bipartite graph

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

a discrete time uniform geometrical theory of diffraction for the fast transient analysis of scattering from curved wedges in this paper a discrete time dt representation of maxwell s equations is employed to describe the time domain td maxwell s equations in terms of matrix equations analogous to time harmonic maxwell s equations which allows one to conveniently develop many td solutions based on their frequency domain fd formulations it was then employed to develop a td version of the uniform geometrical theory of diffraction utd referred as dt utd for the efficient electromagnetic transient analysis of scattering from perfectly conducting curved wedges this dt utd retains the advantages of its corresponding fd utd solution in several aspects including the form ray physical picture and definitions of ray parameters furthermore the transformation of the fd utd to the dt utd formulation can be easily achieved via a simple interpretation of notation numerical examples are presented to validate and illustrate the utilizations of this dt utd

normal factorisation of polynomials and computational issues abstract this paper introduces the notion of normal factorisation of polynomials and then presents a new numerical algorithm for the computation of this factorisation this procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials in this paper the general aspects of the algorithm are discussed and a symbolic implementation is given the theoretical algorithm provides the basis for a numerical procedure the general aspects of which are also discussed here the advantage of such a factorisation is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots for such polynomials robust numerical techniques based on finding roots may then be used if it is desired to work out the usual irreducible factorisation a detailed description of the implementation of the algorithm is presented the problem considered here is an integral part of computations for algebraic control problems

a subspace method for the computation of the gcd of polynomials for the problem of computing the greatest common divisor gcd of a set of polynomials a subspace ss technique is introduced and compared with the matrix pencil mp method proposed by karcanias and mitrouli it is observed that the ss method is easier to implement and more robust to noise than the mp method

global asymptotic linearisation of the pole placement map a closed form solution for the constant output feedback problem abstract the problem of pole assignment by constant output feedback controllers is studied for minimal systems described by a proper transfer function matrix g s r m p s with mcmillan degree n a new method is presented based on asymptotic linearisation around a degenerate point of the pole placement map related to the problem the essence of the present approach is to reduce the multilinear nature of the problem to one of solving a linear set of equations and this is achieved without losing any of the degrees of freedom in the controller the solution is given in closed form in terms of a one parameter family of static feedback compensators for which the closed loop poles approach the required ones as 0 conditions for the generic as well as exact solvability of the arbitrary pole placement problem are given in terms of the numbers m p n and the rank of a new system invariant the d restricted plucker matrix it is shown that the method works generically when mp n which along with the boundary case mp n is the best possible condition as far as the number of states of the open loop system is concerned for achieving arbitrary pole placement via constant output feedback

pulsed field diffraction by a perfectly conducting wedge a spectral theory of transients analysis the canonical problem of pulsed field diffraction by a perfectly conducting wedge is analyzed via the spectral theory of transients stt in this approach the field is expressed directly in the time domain as a spectral integral of pulsed plane waves closed form expressions are obtained by analytic evaluation of this integral thereby explaining explicitly in the time domain how spectral contributions add up to construct the field for impulsive excitation the final results are identical with those obtained previously via time harmonic spectral integral techniques via the stt the authors also derive new solutions for a finite i e nonimpulsive incident pulse approximate uniform diffraction functions are derived to explain the field structure near the wavefront and in various transition zones they are the time domain counterparts of the diffraction coefficients of the geometrical theory of diffraction gtd and the uniform theory of diffraction utd an important feature of the stt technique is that it can be extended to solve the problem of wedge diffraction of pulsed beam fields i e space time wavepackets

a time domain formulation of the uniform geometrical theory of diffraction for scattering from a smooth convex surface a time domain version of the uniform geometrical theory of diffraction td utd is developed to describe in closed form the transient electromagnetic scattering from a smooth convex surface excited by a general time impulsive astigmatic ray field this td utd impulse response is obtained by employing an analytic time transform att for the inversion in time of an available and accurate corresponding frequency domain utd fd utd solution the att is employed because it overcomes the difficulties that occur when inverting fd utd fields associated with rays that traverse line or smooth caustics furthermore the td utd response to a general pulsed astigmatic wave excitation may be found by a convolution of the general excitation with the td utd impulse response which can be performed in closed form some numerical examples illustrating the utility of this td utd are presented

an extended version of the fuzzy relational database model abstract the fuzzy relational database model proposed by buckles and petry is a formal method for organizing and using fuzzy information in relational databases the model possesses two key properties that hold for classical relational databases no two tuples have identical interpretations and each relational algebra operation has a unique result the original fuzzy relational database model was based on similarity relations which were defined on finite scalar domains buckles and petry later extended the model to incorporate fuzzy number domains this extension was done without loss of consistency with the representation or the relational algebra in a recent paper we have extended the original model to proximity relations which generalize similarity relations the key properties of buckles and petry s model are preserved in the extension in this work we demonstrate that the existence of partitions on finite scalar domains in the fuzzy relational model is the key to preserving the important properties of the classical relational model we explain how proximity relations defined on finite scalar domains are used to partition the domains further we show that normal fuzzy sets can be employed as domain values these fuzzy sets become members of fi nite scalar domains and their characteristic functions are employed in generating the proximity relations of the fuzzy relational database model

invariance properties and characterization of the greatest common divisor of a set of polynomials on m sets of polynomials from r s of degree d x1d4c5 m d the notion of extended r equivalence emd under x1d4c5 m d is characterized by a complete invariant and a canonical form it is shown that the greatest common divisor g c d of x1d4c5 m d is an invariant of x1d4c5 m d for proper sets x1d4c5 m d it is also shown that the g c d is invariant under the combined action of e r e transformations and shifting operations finally a systematic procedure for the computation of the g c d of x1d4c5 mdis given this procedure is based on the use of elementary row operations on real matrices and shifting operations

improved theory of acoustic elementary edge waves the high frequency asymptotic theory of acoustic edge waves p ya ufimtsev j acoust soc am 86 463 74 1989 is well suited for investigation of backscattering from perfectly reflecting soft or hard three dimensional objects with edges however it needs to be improved for calculation of forward scattering especially in the directions grazing to the edge faces where it predicts infinite values the present paper removes this singularity by the appropriate choice of the so called uniform component of the surface field it is defined here as the field induced on the half plane tangential to the illuminated face of the scattering edge and to the edge itself an improved theory of elementary edge waves is proposed which is valid for all directions of scattering including the forward grazing directions

a uniform asymptotic solution for the diffraction by a right angled dielectric wedge we propose a uniform asymptotic solution for the field diffracted by a lossless right angled dielectric wedge illuminated by a plane wave at normal incidence the diffraction problem is solved by splitting the observation domain in the inner region of the wedge and the surrounding free space the scattered electric field in each region is assumed to be originated by a set of equivalent electric and magnetic surface currents involved in the well known radiation integral such currents are localized on the interior and exterior faces of the wedge and expressed in terms of the corresponding geometrical optics field useful analytical manipulations and asymptotic evaluations of the resulting integrals allow one to obtain the diffraction coefficients in terms of the fresnel s reflection and transmission coefficients of the structure and the transition function of the uniform geometrical theory of diffraction the related diffracted field compensates the discontinuities of the geometrical optics field and gives total field levels in good agreement with finite difference time domain results

transient image theory for 2 d and 3 d conducting wedge problems in the present paper two and three dimensional transient scattering from a perfectly conducting wedge is studied the exciting sources analyzed are an infinite impulsive line current parallel to edge of the conductor and an arbitrarily oriented electric dipole the analysis presented is based on the time harmonic image solution found for the scattered field the frequency domain image current corresponding to the field contribution from the wedge is an exact function expressed in terms of simple trigonometric functions it is dependent only on the coordinates and the amplitude of the source but it is independent of the observation point it is shown that the transient scattering can be interpreted as arising from an image source by generalizing the time domain green s function to complex source points

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

approximate greatest common divisor of many polynomials generalised resultants and strength of approximation the computation of the greatest common divisor gcd of many polynomials is a nongeneric problem techniques defining approximate gcd solutions have been defined but the proper definition of the approximate gcd and the way we can measure the strength of the approximation has remained open this paper uses recent results on the representation of the gcd of many polynomials in terms of factorisation of generalised resultants to define the notion of approximate gcd and define the strength of any given approximation by solving an optimisation problem the newly established framework is used to evaluate the performance of alternative procedures which have been used for defining approximate gcds

nearest common root of a set of polynomials a structured singular value approach abstract the paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients in particular we seek the minimum magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root it is shown that the problem is equivalent to the solution of a structured singular value problem arising in robust control for which numerous techniques are available it is also shown that the method can be extended to the calculation of an approximate gcd of fixed degree by introducing the notion of the generalized structured singular value of a matrix the work generalizes previous results by the authors involving the calculation of the approximate gcd of two polynomials although the general case considered here is considerably harder and relies on a matrix dilation approach and several preliminary transformations

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

computing the discrete time analytic signal via fft starting with a real valued n point discrete time signal frequency domain algorithms are provided for computing 1 the complex valued standard n point discrete time analytic signal of the same sample rate 2 the complex valued decimated n 2 point discrete time analytic signal of half the original sample rate and 3 the complex valued interpolated nm point discrete time analytic signal of m times the original sample rate special adjustment of the transform end points are shown to be necessary in order to generate proper discrete time analytic signals

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

on structured singular values of reciprocal matrices computing the structured singular value is a bottleneck of the robustness analysis and synthesis of control systems in this paper it is shown that the structured singular value is identical to the maximum singular value for an important class of matrices called reciprocal which reflects the intrinsic symmetry of physical world

direct time domain analysis of an uwb pulse distortion by convex objects with the slope diffraction included we consider time domain td modelling of diffraction of a signal caused by 2d conducting convex objects for soft polarization case two models of the amplitude term of the time domain convex obstacle diffraction coefficient are proposed and compared one model is derived using an analytical numerical method and the other is obtained in a purely analytical way both methods use uniform geometrical theory of diffraction utd formulated in the frequency domain fd the models are used to examine an ultrawideband uwb pulse distortion the accuracy and calculation complexity of both models are compared the slope term of the convex obstacle diffraction coefficient is also considered for the case when slope diffraction is not negligible the uwb pulse distortion caused by overall convex obstacle diffraction is also analysed

fuzzy information in extended fuzzy relational databases abstract the queries on type 2 fuzzy relational databases may get non reasonable answers and as pointed out by buckles a tuple s membership value is not static but is a measure of the appropriateness of that tuple to a given query here we use two supplementary attributes r and i e to model the concepts of fuzziness and uncertainty respectively we also use these two attribute values to compute the appropriateness of a tuple to a given query in this paper the proposed fuzzy relational model is defined by first order logical in our proposed model the classical properties of the database which are 1 database interpretation being redundant free being uniquely determined and 2 the well definedness of relational algebra are preserved for the fuzzy relational algebra therefore the proposed fuzzy relational model can capture more meaning than the conventional approaches and is more suitable to handle the real life application

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

a fast iterated conditional modes algorithm for water fat decomposition in mri decomposition of water and fat in magnetic resonance imaging mri is important for biomedical research and clinical applications in this paper we propose a two phased approach for the three point water fat decomposition problem our contribution consists of two components 1 a background masked markov random field mrf energy model to formulate the local smoothness of field inhomogeneity 2 a new iterated conditional modes icm algorithm accounting for high performance optimization of the mrf energy model the mrf energy model is integrated with background masking to prevent error propagation of background estimates as well as improve efficiency the central component of our new icm algorithm is the stability tracking st mechanism intended to dynamically track iterative stability on pixels so that computation per iteration is performed only on instable pixels the st mechanism significantly improves the efficiency of icm we also develop a median based initialization algorithm to provide good initial guesses for icm iterations and an adaptive gradient based scheme for parametric configuration of the mrf model we evaluate the robust of our approach with high resolution mouse datasets acquired from 7t mri

a matrix pencil based numerical method for the computation of the gcd of polynomials the paper presents a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s p sub m d of maximal degree d it is based on a previously proposed theoretical procedure karcanias 1989 that characterizes the gcd of p sub m d as the output decoupling zero polynomial of a linear system s a spl circ c spl circ that may be associated with p sub m d the computation of the gcd is thus reduced to finding the finite zeros of the pencil sw aw where w is the unobservable subspace of s a spl circ c spl circ if k dim w the gcd is determined as any nonzero entry of the kth compound c sub k sw a spl circ w the method defines the exact degree of gcd works satisfactorily with any number of polynomials and evaluates successfully approximate solutions

resultant properties of gcd of many polynomials and a factorization representation of gcd the properties of the greatest common divisor gcd of a set of polynomials of may be investigated using the sylvester resultant new properties of the sylvester resultant linked to gcd are established and these lead to canonical factorizations of resultants expressing the extraction of common divisors from the elements of the original set these results lead to a new representation of the gcd introduced in terms of a canonical factorization of the sylvester resultant into a reduced sylvester resultant and a toeplitz matrix representing the gcd the use of the sylvester resultant allows a simplification of the eres and matrix pencils computational procedures for gcd computation and provides the means for formulating simpler and robust computational procedures

proximity relations in the fuzzy relational database model abstract the fuzzy relational model of buckles and petry is a rigorous scheme for incorporating non ideal or fuzzy information in a relational database in addition to providing a consistent scheme for representing fuzzy information in the relational structure the model possesses two critical properties that hold for classical relational databases these properties are that no two tuples have identical interpretations and each relational operation has a unique result the fuzzy relational model relies on similarity relations for each scalar domain in the fuzzy database these relations are reflexive symmetric and max min transitive in addition to introducing fuzziness into the relational model each similarity relation induces equivalence classes in its domain it is the existence of these equivalence classes that provides the model with the important properties possessed by classical relational databases in this paper we extend the fuzzy relational database model of buckles and petry to deal with proximity relations for scalar domains since reflexivity and symmetry are the only constraints placed on proximity relations they generalize the notion of similarity relations we show that it is possible to induce equivalence classes from proximity relations thus the nice properties of the fuzzy relational model of buckles and petry are preserved furthermore the removal of the max min transitivity restriction also provides database users with more freedom to express their value structures

on the gap between the complex structured singular value and its convex upper bound the gap between the complex structured singular value of a complex matrix m and its convex upper bound is considered new necessary and sufficient conditions for the existence of the gap are derived it is shown that determining whether there exists such a gap is as difficult as evaluating a structured singular value of a reduced rank matrix whose rank is equal to the multiplicity of the largest singular value of m furthermore if an upper bound on this reduced rank problem can be obtained it is shown that this provides an upper bound on the original problem that is lower than the convex relaxation upper bound an example that illustrates our procedure is given we also give the solution of several structured approximation problems of independent interest

gefred a generalized model of fuzzy relational databases abstract in this paper we present a fuzzy relational databases model whose main characteristics are the integration of previous models in the same framework representation capabilities for a wide series of fuzzy information and a coherent and flexible handling of it this model aims to solve each problem of representation and handling of fuzzy information taking into account its specific nature and hence it allows the user to choose the comparison operator and the fuzzy compatibility measure to be used in a query besides it permits the user to specify the precision with which the conditions involved in a query are satisfied

rank and null space calculations using matrix decomposition without column interchanges abstract the most widely used stable methods for numerical determination of the rank of a matrix a are the singular value decomposition and the qr algorithm with column interchanges here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges one algorithm makes use of the condition estimator of cline moler stewart and wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices the second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix

noise attenuation by a hard wedge shaped barrier this paper is concerned with the problem of sound screening by a wedge like barrier the sound source is assumed to be point like and the receiver is located in the shadow of the source sound field so that according to geometrical optics only the field diffracted by the edge of the barrier is considered first for the hard wedge in space three models are used for calculating the amplitude of the edge diffracted field these are the uniform theory of diffraction utd the hadden pierce model both in the frequency domain and the biot tolstoy theory of diffraction which is a time domain formulation it is first shown that even at relatively low frequencies the frequency domain models perform quite satisfactorily as compared to the exact time domain theory hence and due to its relative simplicity the utd is proposed as an accurate calculation scheme for solving problems with edge diffraction by hard wedges it is also proved from theoretical calculations that the amplitude of the edge diffracted field increases for an increasing angle of the wedge and consequently the hard half plane gives the lowest field amplitude in the shadow zone some applications are then considered for evaluating the performance of a barrier on a flat ground either completely hard or with mixed homogeneous boundary conditions an improvement of the scheme for calculating the sound field in the all hard case is achieved through considering the multiple diffraction in this case only to the second order between the top of the wedge barrier and its base the results show that for usually occurring situations increasing the angle of the hard wedge barrier affects negatively its efficiency through diminishing its insertion loss these conclusions are also supported by the results of some experimental measurements conducted at a scale model level

properties of the mixed spl mu problem and its bounds upper and lower bounds for the mixed spl mu problem have recently been developed and here we examine the relationship of these bounds to each other and to spl mu a number of interesting properties are developed and the implications of these properties for the robustness analysis of linear systems and the development of practical computation schemes are discussed in particular we find that current techniques can only guarantee easy computation for large problems when spl mu equals its upper bound and computational complexity results prohibit this possibility for general problems in this context we present some special cases where computation is easy and make some direct comparisons between mixed spl mu and kharitonov type analysis methods

time domain line integral representations of physical optics scattered fields a time domain td line integral li representation of the physical optics po radiation integral is developed for the scattering from a perfectly conducting surface illuminated by an electric hertzian dipole the proposed li representation which is valid and singularity free for all near field observers is derived directly in the td by introducing dyad potentials this td representation can be further expressed in terms of geometric optics and boundary wave components of the scattering mechanisms as in the conventional frequency domain li representations using the proposed li representation the computational complexity is reduced by an order of the scatterer s linear electric size compared with the straightforward numerical quadrature of the po surface integral simple and complicated numerical examples are presented to demonstrate the efficiency and accuracy of the proposed td li algorithm

time domain utd po solution for the multiple diffraction of spherical waves for uwb signals the time domain td solution for the multiple diffraction of spherical waves after an arbitrary number of obstacles with different shapes such as knife edges and wedges is presented the proposed td solution is based on the representation of the inverse laplace transform of the corresponding frequency domain solution in closed form as it is given by a hybrid of the uniform theory of diffraction utd physic optics po solution the proposed formulation validated with the results from technical literature does not need to incorporate the td version of the higher order diffraction coefficients due to a recursive relation thus significantly reducing both the mathematical complexity and computation time of the formulation when compared with other td solutions