nullity invariance for pivot and the interlace polynomial we show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given square matrix is described by the symmetric difference operator for sets we consider its consequences for graphs and in particular generalize the recursive relation of the interlace polynomial and simplify its proof

fuzzy linear programming and applications abstract this paper presents a survey on methods for solving fuzzy linear programs first lp models with soft constraints are discussed then lp problems in which coefficients of constraints and or of the objective function may be fuzzy are outlined pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives in addition to the commonly applied extended addition based on the min operator and used for the aggregation of the left hand sides of fuzzy constraints and fuzzy objectives a more flexible procedure based on yager s parametrized t norm t p is presented finally practical applications of fuzzy linear programs are listed

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

rank preserving diagonal completions of a matrix abstract suppose a is an n by n matrix over a field f we prove that it is possible to complete the diagonal entries of a so that the resulting rank of a is as small as possible when n 3r where r is the off diagonal rank of a and n r 3 1

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

error calculation of the hosvd based rule base reduction in hierarchical fuzzy systems abstract in real time evaluation systems the decision should be made in time to avoid the serious consequences in the literature there are several techniques which aim to reduce the computational complexity of these kinds of systems among others the higher order singular value decomposition hosvd based reduction method is a useful tool for handling this problem in the case when only non exact reduction can be performed the error calculation is also essential because the reduction degree is based on the calculated error in these systems in this paper the authors present the hosvd reduction error calculation in hierarchical fuzzy systems using mamdani type inference and define a general formula which includes the propagated error from the previous levels of the hierarchy in which the error bound calculated by different ways can be used and takes into account the case when not all the inputs contain a propagated error but only a few of them furthermore a greedy algorithm is presented for reduction optimization

structures preserved by generalized inversion and schur complementation in this paper we investigate the inheritance of certain structures under generalized matrix inversion these structures contain the case of rank structures and the case of displacement structures we do this in an intertwined way in the sense that we develop an argument that can be used for deriving the results for displacement structures from thoses for rank structures we pay particular attention to the moore penrose generalized inverse showing that for the cases of most interest the ranks of the structure satisfied by the moore penrose inverse can at most double with respect to the original ranks we consider also the case of inheritance of structure by generalized schur complements

nullities of submatrices of the moore penrose inverse let a be a complex m n matrix of rank r and with moore penrose inverse a if a a11a21a12a22 a b11b21b12b22 are partitioned so that aii is mi ni and bii is ni mi with m m1 m2 n n1 n 2 and if the row nullity of a matrix is denoted by then m r b22 a11 n r

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

generalized inverses of morphisms with kernels abstract let x y be a morphism with kernel k x in an additive category with an involution then has a moore penrose inverse with respect to iff is invertible in this case 1 if x y then has a group inverse iff has a cokernel x k and 2 is invertible in this case 2 1

more on structure ranks of matrices abstract the structure rank of a matrix is the largest rank of a submatrix which lies within a specified structure we characterize those structures avoiding the main diagonal for which a matrix and its inverse have the same structure rank thereby generalizing some results of m fiedler

completing a symmetric 2 2 block matrix and its inverse abstract we consider the following completion problems suppose n1 n2 are nonnegative integers such that n 1 n 2 n 0 let a11 a12 a21 b22 be matrices with dimensions n1 n1 n1 n2 n2 n1 and n2 n2 respectively we determine necessary and sufficient conditions so that there exists an n2 n2 matrix a22 such that a a 11 a 21 a 12 a 22 and i a is nonsingular and symmetric and b22 is the lower right block of a partitioning of a 1 ii a is symmetric positive definite and b22 is the lower right block of a partitioning of a 1

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

optimization under fuzzy if then rules abstract the aim of this paper is to introduce a novel statement of fuzzy mathematical programming problems and to provide a method for finding a fair solution to these problems suppose we are given a mathematical programming problem in which the functional relationship between the decision variables and the objective function is not completely known our knowledge base consists of a block of fuzzy if then rules where the antecedent part of the rules contains some linguistic values of the decision variables and the consequence part consists of a linguistic value of the objective function we suggest the use of tsukamoto s fuzzy reasoning method to determine the crisp functional relationship between the objective function and the decision variables and solve the resulting usually nonlinear programming problem to find a fair optimal solution to the original fuzzy problem

generalized oscillatory matrices we define a new class of generalized oscillatory matrices shortly go matrices over a noncommutative ring with identity and a positive part similarly to the classical case this class consists of square generalized totally nonnegative matrices gtn matrices of which some power is generalized totally positive using the previously defined ordering of invertible gtn matrices we study in particular so called basic go matrices which form in a sense atoms of this ordering

a vertical horizontal splitting algorithm for three dimensional tidal and storm surge computations an algorithm is described for the computation of the three dimensional velocity fields due to tides and storm surges the surface elevation and depth averaged velocity components are first computed from the shallow water equations these equations are then used as part of the input to the second part of the algorithm in which the velocity profiles are computed from the momentum equations the nonlinear terms in the momentum equations and both the advective terms and the bottom friction terms in the shallow water equations are fully included the shallow water equations are solved by a finite difference scheme that achieves third order local truncation errors in all the dominant terms and that permits correct parallel flow on coastal boundaries of any orientation two alternative algorithms are discussed for computation of the velocity profiles one based on finite elements in the vertical coordinate direction the other based on a generalized crank nicolson scheme the complete algorithm has been tested on several model problems and has been found to be accurate and fast

a factorization of totally nonsingular matrices over a ring with identity abstract we say that a rectangular matrix over a ring with identity is totally nonsingular tns if for all k all its relevant submatrices either having k consecutive rows and the first k columns or k consecutive columns and the first k rows are invertible we prove that a matrix is tns if and only if it admits a certain factorization with bidiagonal type factors and certain invertible entries this approach generalizes the loewner neville factorization usually applied to totally positive matrices

a class of oscillatory matrices with exponent n 1 abstract a matrix a is called an oscillatory matrix if it is totally nonnegative and there exists a positive integer k such that a k is totally positive in this paper we describe the class of n n oscillatory matrices which have maximal exponent equal to n 1

the nullity theorem for principal pivot transform abstract we generalize the nullity theorem of gustafson 1984 8 from matrix inversion to principal pivot transform several special cases of the obtained result are known in the literature such as a result concerning local complementation on graphs as an application we show that a particular matrix polynomial the so called nullity polynomial is invariant under principal pivot transform

hybrid ikebe newton s iteration for inverting general nonsingular hessenberg matrices after a concise survey the expanded ikebe algorithm for inverting the lower half plus the superdiagonal of an n n unreduced upper hessenberg matrix h is extended to general nonsingular upper hessenberg matrices by computing in the reduced case a block diagonal form of the factor matrix hl in the inverse factorization h 1 h l u 1 this factorization enables us to propose hybrid and accurate nongaussian procedures for computing h 1 thus hl is computed directly in the aim to be used as a fine initial guess for newton s iteration which converges to h 1 in a suitable number of iterations

two results on basic oscillatory matrices this paper continues the research of the authors on totally nonnegative and oscillatory matrices first the result of the second author stating that the hadamard product of oscillatory tridiagonal matrices of the same order is again an oscillatory tridiagonal matrix is extended to the class of basic oscillatory matrices introduced recently then it is shown that every oscillatory matrix contains a basic oscillatory matrix as a factor this explains the role of the class of basic oscillatory matrices within the class of oscillatory matrices

degrees of freedom of interference channels with comp transmission and reception we study the degrees of freedom dof of the k user interference channel with coordinated multipoint comp transmission and reception each message is jointly transmitted by m t successive transmitters and is jointly received by m r successive receivers we refer to this channel as the comp channel with a transmit cooperation order of m t and receive cooperation order of m r since the channel has a total of k transmit antennas and k receive antennas the maximum possible dof is equal to k we show that the comp channel has k dof if and only if m t m r geq k 1 the key idea is that the zero forcing of the interference corresponding to the i rm th message at the decoder of the j rm th message where j ne i can be viewed as a shared responsibility between the m t transmitters carrying the i rm th message and the m r receivers decoding the j rm th message for the general case we derive an outer bound that states that the dof is bounded above by left lceil k m t m r 2 2 right rceil for the special case with only comp transmission i e m r 1 we propose a scheme that can achieve k m t 1 2 dof for all k and conjecture that the result holds true for all k in the proposed coding scheme the m t transmitters carrying each message are used to cancel the interference introduced by this message at the first m t 1 receivers thereby allowing each of these receivers to enjoy 1 dof and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space the achievability proofs are based on the notion of algebraic independence from algebraic geometry

generalized hessenberg matrices we define and study generalized hessenberg matrices i e square matrices which have subdiagonal rank one here subdiagonal rank means the maximum order of a nonsingular submatrix all of whose entries are in the subdiagonal part we prove that the property of being generalized hessenberg matrix is preserved by post and premultiplication by a nonsingular upper triangular matrix by inversion for invertible matrices etc we also study a special kind of generalized hessenberg matrices

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

relationships between goals in multiple attribute decision making abstract a novel way of analysis of the decision situation based on fuzzy relationships between goals is presented in contrast to other approaches in multiple attribute decision making the interactive structure of goals for each decision problem is inferred and represented explicitly this interactive structure reflects existing fuzzy relationships between goals and provides for a better understanding of the decision situation

mapped transient infinite elements for heat transfer problems in infinite media abstract based on the fundamental solutions of transient heat transfer problems in infinite media a mapped transient infinite element is presented and the so called heat transfer function of the element has been derived in this paper in order to express the property matrices of the present mapped transient infinite element a mapping technique in which the infinite element in a global coordinate system is mapped into a strip element of finite width and infinite length in a local coordinate system has also been developed using the variable substitution technique the general integral involved in the evaluation of the property matrices of the element has been changed into a standard one for which the conventional gauss legendre integration scheme can be employed since the heat transfer function of the present mapped transient infinite element is dependent on both space and time variables the mechanism of heat transfer problems in infinite media can be rigorously simulated because the property matrices of the element are evaluated at any time of interest in the analysis finally the accuracy and efficiency of the present mapped transient infinite element have been verified through the analysis of a 1d heat conduction problem in a semi infinite medium and a 2d transient heat transfer problem in a full plane

extended tsukamoto s inference method for solving multi objective linguistic optimization problems abstract multi objective linguistic optimization problems molops are quite useful in modeling a number of real life problems the decision variables used to define the problem constraints and the objective functions take linguistic values in molops in those molops where the values of objective functions are unknown at all the points of the variable space the link between the values of decision variables and objective functions is defined using if then rules molops formulated in this way were often solved using the tsukamoto s inference method however the tsukamoto s method can solve molops if the consequents in the if then rules are defined as monotonic functions also the solution obtained from tsukamoto s method is a precise number so in the present work we propose a novel extension of tsukamoto s inference method the extended tsukamoto s inference method can overcome both the above limitations that is it can solve molops where the consequents are defined as non monotonic functions as well as monotonic functions also the extended tsukamoto s method generates intervals instead of unique numbers in the solution from these intervals we then generate type 1 fuzzy set membership functions and compare them using two of their fuzzy set uncertainty measures viz centroid and fuzziness furthermore we generate linguistic recommendations for these generated type 1 fuzzy set membership functions as humans understand linguistic information naturally we have also demonstrated the applicability of the proposed extended tsukamoto s inference method to the case study for student s performance evaluation we strongly believe that our proposed method is a novel one and will provide directions to future research

direct approach to the band completion problem we study the problem of completion of a matrix with a specified band in such a way that the inverse be a band matrix with the same width for the general block matrix case a new approach to solve this and related problems is proposed the relations with the class of quasiseparable matrices play an important role

completion of an orthogonal projector and a related inverse mapping problem we study the completion problem for an orthogonal projector a of r n from knowledge of one or several of its columns as well as the related matrix inverse problem of finding an orthogonal projector a with ax y for two given n by k matrices x and y using geometric properties of orthogonal projections and of the singular value decomposition we derive theoretical and numerical solutions for these problems we give algorithms and test their matlab implementations

generalized totally nonnegative matrices we define a new class of generalized totally nonnegative matrices shortly gtn matrices over a noncommutative ring with identity and a positive part the class of gtn matrices is closed under multiplication every submatrix of a gtn matrix is a gtn matrix as well we also define a standard form of a gtn matrix we then study the invertible gtn matrices of a fixed order and classify them the classes form a partially ordered set can be noncommutatively multiplied and multiplication preserves ordering

a qz method based on semiseparable matrices this manuscript focusses on an alternative method for computing the eigenvalues of a pencil of two matrices based on semiseparable matrices an effective reduction of a matrix pair to lower semiseparable upper triangular form will be presented as well as a qz iteration for this matrix pair important to remark is that this reduction procedure also inherits a kind of nested subspace iteration as was the case when solving the standard eigenvalue problem with semiseparable matrices it will also be shown that the qz iteration for a semiseparable triangular matrix pair is closely related to the qz iteration for a hessenberg triangular matrix pair

on the structure rank of a class of structure matrices abstract we introduce a class of structure matrices called transpose complementary structure matrices we give a characterization of the n extendable transpose complementary structure matrices with trace k where k 0 2 we also prove that a nonsingular matrix and its inverse have the same s rank for this class of structure matrices s

an algorithm for three dimensional convection and diffusion with very different horizontal and vertical length scales an algorithm is developed and tested for the numerical solution of the three dimensional convection diffusion equation appropriate for the marine environment in which the horizontal and vertical scales differ by one or two orders of magnitude the algorithm splits the horizontal and vertical parts of the process treating the horizontal convection and diffusion explicitly and the vertical convection and diffusion by an implicit finite element method that is unconditionally stable the overall stability conditions on the algorithm are discussed and its accuracy is verified through a number of test problems whose exact solutions are known

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

a cholesky lr algorithm for the positive definite symmetric diagonal plus semiseparable eigenproblem abstract we present a cholesky lr algorithm with laguerre s shift for computing the eigenvalues of a positive definite symmetric diagonal plus semiseparable matrix by exploiting the semiseparable structure each step of the method can be performed in linear time

accurate petrov galerkin methods for transient convective diffusion problems the petrov galerkin method is applied to transient convective diffusion problems with convection dominance weighting functions are used which differ from the shape functions by both symmetric and non symmetric modifications it is proved that the symmetric modifications largely influence the accuracy by a one dimensional and a two dimensional example it is shown how to choose this modification to maximize accuracy

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

interdependence in fuzzy multiple objective programming abstract in multiple objective programs mop application functions are establised to measure the degree of fulfillment of the decision maker s requirements achievement of goals nearness to an ideal point satisfaction etc about the objective functions see e g delgado et al 1990 zimmermann 1978 and are extensively used in the process of finding good compromise solutions we have earlier demonstrated that the use of interdependences among objectives of a mop in the definition of the application functions provides for more correct solutions and faster convergence in this paper generalizing the principle of application functions to fuzzy multiple objective programs fmop with interdependent objectives we define a large family of application functions for fmop and illustrate our ideas by a simple three objective program

multiobjective linguistic optimization generalizing our earlier results on optimization with linguistic variables we introduce a novel statement of fuzzy multiobjective mathematical programming problems and provide a method for finding a fair solution to these problems suppose we are given a multiobjective mathematical programming problem in which the functional relationship between the decision variables and the objective functions is not completely known our knowledge base consists of a block of fuzzy if then rules where the antecedent part of the rules contains some linguistic values of the decision variables and the consequence part consists of a linguistic value of the objective functions we suggest the use of tsukamoto s fuzzy reasoning method to determine the crisp functional relationship between the decision variables and objective functions we model the anding of the objective functions by a t norm and solve the resulting usually nonlinear programming problem to find a fair optimal solution to the original fuzzy multiobjective problem

consecutive column and row properties of matrices and the loewner neville factorization abstract we say that a square matrix over a ring with identity has the consecutive column property if for all k all its relevant submatrices having k consecutive rows and the first k columns are invertible similarly the consecutive row cr is defined we show that analogously to the commutative case for totally positive matrices a matrix has both cc and cr properties if and only if it admits a certain loewner neville type factorization with invertible entries this factorization is unique since the result is proved for matrices in such generality it holds also for block matrices over a field with all blocks square explicit both ways formulae are found between two sets of parameters the loewner neville coefficients in the factorization and the schur complements of relevant submatrices in relevant submatrices larger by one we also show that for lower triangular matrices the cc property is preserved by inversion

complementary basic matrices abstract we show that an n n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions except the first and last completely between the subdiagonal and superdiagonal part then this matrix can be factorized into a product of n 1 matrices each consisting of a 2 2 principal submatrix in two consecutive rows and columns in all possible of the n 1 positions and completed by ones along the diagonal the converse is also true it is shown that the spectrum does not depend on the order of the factors

suborthogonality and orthocentricity of matrices we present some results on submatrices of orthogonal and unitary matrices and their relation to so called orthocentric matrices these are then completely characterized

intrinsic products and factorizations of matrices we say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition we present several examples together with important applications these applications include companion matrices and sign nonsingular matrices

some inheritance properties for complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices m fiedler a note on companion matrices linear algebra appl 372 2003 325 331 9 3 the multiplication of the g i s is intrinsic we explore connections between the 2 2 matrices c k and the resulting cb matrix g k in particular which properties are inherited from the c k to the g k we consider two situations for the ordinary real cb matrices and for the corresponding sign pattern matrices

a note on generalized hessenberg matrices two different generalizations of the concept of hessenberg matrices have appeared recently in the literature in this note we study their connections in more detail and give a general structure theorem for these classes of matrices

a remark on oscillatory matrices in this note we revisit a classical criterion obtained by gantmacher and krein for determining when a totally nonnegative matrix is actually oscillatory a new proof of this criterion is presented by incorporating bidiagonal factorizations of invertible totally nonnegative matrices and utilizing certain associated weighted planar diagrams

on the existence of unimodular matrices with a prescribed submatrix abstract we study the existence of unimodular matrices with a prescribed submatrix over a ring r in particular when either r is hermite and dedekind finite or r has stable range one we give necessary and sufficient conditions for the existence of these unimodular completions

fuzzy reasoning for solving fuzzy mathematical programming problems abstract we interpret fuzzy linear programming flp problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes mfr where the antecedents of the scheme correspond to the constraints of the flp problem and the fact of the scheme is the objective of the flp problem then the solution process consists of two steps first for every decision variable x r n we compute the fuzzy value of the objective function max x via sup min convolution of the antecedents constraints and the fact objective then an optimal solution to flp problem is any point which produces a maximal element of the set max x x r n in the sense of the given inequality relation we show that our solution process for a classical crisp lp problem results in a solution in the classical sense and under well chosen inequality relations and objective function coincides with those suggested by buckley fuzzy sets and systems 31 1989 329 341 delgado et al control and cybernetics 16 1987 114 121 fuzzy sets and systems 26 1988 49 62 negoita fuzzy sets and systems 6 1981 261 269 ramik and rimanek fuzzy sets and systems 16 1985 123 138 verdegay fuzzy information and decision processes north holland amsterdam 1982 control and cybernetics 13 1984 230 239 and zimmermann int j general systems 2 1976 209 215 furthermore we show how to extend the proposed solution principle to non linear programming problems with fuzzy coefficients we illustrate our approach by some simple examples

a note on the nullity theorem in this paper we take a closer look at the nullity theorem as formulated by markham and fiedler in 1986 the theorem is a valuable tool in the computations with structured rank matrices it connects ranks of subblocks of an invertible matrix a with ranks of other subblocks in his inverse a 1 a little earlier barrett and feinsilver 1981 proved a theorem very close to the nullity theorem but restricted to semiseparable and tridiagonal matrices which are each others inverses we will adapt the ideas of barrett and feinsilver to come to a new alternative proof of the nullity theorem based on determinantal formulas in the second part of the paper we extend the nullity theorem to make it suitable for two types of decompositions namely the lu and the qr decomposition these theorems relate the ranks of subblocks of the factors l u and q to the ranks of subblocks of the factored matrix it is shown that a combination of the nullity theorem and his extended versions is suitable to predict in an easy manner the structure of decompositions and or of inverses of structured rank matrices e g higher order band higher order semiseparable hessenberg and many other types of matrices as examples to show the power of the nullity theorem and the related theorems we apply them to semiseparable and related matrices

generalized totally positive matrices abstract we say that a rectangular matrix over a in general noncommutative ring with identity having a positive part is generalized totally positive gtp if in all nested sequences of so called relevant submatrices the schur complements are positive here a relevant submatrix is such either having k consecutive rows and the first k columns or k consecutive columns and the first k rows this notion generalizes the usual totally positive matrices we prove e g that a square matrix is gtp if and only if it admits a certain factorization with bidiagonal type factors and certain invertible entries also the product of square gtp matrices of the same order is again a gtp matrix and its inverse has the checkerboard sign property

structured eigenvalue condition numbers for parameterized quasiseparable matrices the development of fast algorithms for performing computations with n times n n n low rank structured matrices has been a very active area of research during the last two decades as a consequence of the numerous applications where these matrices arise the key ideas behind these fast algorithms are that low rank structured matrices can be described in terms of o n parameters and that these algorithms operate on the parameters instead on the matrix entries therefore the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer since this determines the maximum accuracy of a given fast computation in other words it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low rank structured matrices but as far as we know this has not yet been accomplished in any case in this paper we derive structured relative eigenvalue condition numbers for the important class of low rank structured matrices known as 1 1 1 1 quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the givens vector representations of these matrices and we provide fast algorithms for computing them comparisons among the new structured condition numbers and the unstructured one are also presented as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge in addition the approach presented in this paper is general and may be extended to other problems and classes of low rank structured matrices

an operator splitting algorithm for two dimensional convection dispersion reaction problems an operator splitting algorithm for the two dimensional convection dispersion reaction equation is developed the flow domain is discretized into triangular elements which are fixed in time the governing equation is split into three successive initial value problems a pure convection problem a pure dispersion problem and a pure reaction problem for the pure convection problem solutions are found by the method of characteristics the solution algorithm involves tracing the characteristic lines backwards in time from a vertex of an element to an interior point a cubic polynomial is used to interpolate the concentration and its derivatives on an element for the pure dispersion problem an explicit finite element algorithm is employed analytical solutions are obtained for the pure reaction problem the treatment of the boundary conditions is also discussed several numerical examples are presented numerical results agree well with analytical solutions because cubic polynomials are used in the interpolation very little numerical damping and oscillation are introduced even for the pure convection problem

the interplay of ranks of submatrices a banded invertible matrix t has a remarkable inverse all upper and lower submatrices of t 1 have low rank depending on the bandwidth in t the exact rank condition is known and it allows fast multiplication by full matrices that arise in the boundary element method we look for the right proof of this property of t 1 ultimately it reduces to a fact that deserves to be better known complementary submatrices of any t and t 1 have the same nullity the last figure in the paper when t is tridiagonal shows that on and above the diagonal of t 1 all rows are proportional