a study of the performance of neville elimination using two kinds of partitioning techniques abstract in this paper a method to compute the solution of a system of linear equations by means of neville elimination is described using two kinds of partitioning techniques block and block striped this type of approach is especially suited to the case of totally positive linear systems which is present in different fields of application although neville elimination carried out more floating point operations than gaussian elimination in some cases in this study we confirm that these advantages disappear when we use multiprocessor systems on the other hand the overall parallel run time of neville elimination is better than gauss time as neville elimination uses a lower cost communication model

on monotone and convex spline interpolation this paper is concerned with the problem of existence of monotone and or convex splines having degree n and order of continuity k which interpolate to a set of data at the knots the interpolating splines are obtained by using bernstein polynomials of suitable continuous piecewise linear functions they satisfy the inequality k n k the theorems presented here are useful in developing algorithms for the construction of shape preserving splines interpolating arbitrary sets of data points earlier results of mcallister passow and roulier can be deduced from those given in this paper

some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

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

isogeometric collocation methods with generalized b splines we introduce isogeometric collocation methods based on generalized b splines and we analyze their performance through numerical examples for univariate and multivariate scalar and vector valued problems in particular advection diffusion and linear elasticity model problems are addressed the resulting method combines the favorable properties of isogeometric collocation and the geometrical and analytical problem oriented advantages of generalized b splines

convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas abstract subdivision schemes are popular iterative processes to build graphs of functions curves and surfaces we analyze the 2 point hermite c 2 subdivision scheme introduced by merrien in 26 for the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme in this paper by an explicit computation of the joint spectral radius of such pairs of matrices we determine necessary and sufficient conditions for the scheme to be c 2 convergent whenever it reproduces cubic polynomials in addition we present two one parameter families of convergent subdivision schemes belonging to the class in 26 possessing interesting properties from the shape control point of view

curve and surface construction using hermite subdivision schemes in this paper we present a very efficient hermite subdivision scheme based on rational functions and outline its potential applications with special emphasis on the construction of cubic like b splines well suited for the design of constrained curves and surfaces

full rank factorization in echelon form of totally nonpositive negative rectangular matrices abstract an n m real matrix a is said to be totally nonpositive negative if every minor is nonpositive negative in this paper we study the full rank factorization in echelon form of a totally nonpositive negative matrix this characterization allows us to significantly reduce the number of minors to be checked in order to decide the total negativity of a matrix

quasi extended chebyshev spaces and weight functions we recently showed that the class of quasi extended chebyshev spaces is the largest class of sufficiently differentiable functions permitting design in previous articles we mentioned a simple procedure to build such spaces by means of both generalised derivatives associated with non vanishing weight functions and two dimensional chebyshev spaces in the present one we prove that conversely on a closed bounded interval any quasi extended chebyshev space can be obtained via the latter procedure we then draw a few interesting consequences from the latter result

which spaces for design we determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms it is the class of all spaces containing constants of which the spaces derived under differentiation are quasi extended chebyshev spaces i e they permit hermite interpolation taylor interpolation excepted it is also the class of all spaces which possess bernstein bases or of all spaces for which any associated spline space does possess a b spline basis note that blossoms guarantee that such bases are normalised totally positive bases they even are the optimal ones

on the zeros of complex van vleck polynomials polynomial solutions to the generalized lame equation the stieltjes polynomials and the associated van vleck polynomials have been studied extensively in the case of real number parameters in the complex case relatively little is known numerical investigations of the location of the zeros of the stieltjes and van vleck polynomials in special cases reveal intriguing patterns in the complex case suggestive of a deeper structure in this article we report on these investigations with the main result being a proof of a theorem confirming that the zeros of the van vleck polynomials lie on special line segments in the case of the complex generalized lame equation having three free parameters furthermore as a result of this proposition we are able to obtain in this case a strengthening of a classical result of heine on the number of possible van vleck polynomials associated with a given stieltjes polynomial

mixed hyperbolic trigonometric spaces for design we investigate the class of five dimensional null spaces of linear differential operators with constant coefficients and odd characteristic polynomials one of the advantages of this class is that it permits to mix trigonometric and hyperbolic functions within the same space and we will more specially focus on this interesting blending whenever necessary we determine the critical lengths for design this yields the largest possible intervals on which existence of bernstein bases is guaranteed such bases being then automatically the optimal normalised totally positive bases this also enables us to show the interest of this class of spaces for geometric design

deriving calculus with cotriples we construct a taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors the tower was inspired by goodwillie s taylor tower for functors of spaces and is related to dold and puppe s stable derived functors and mac lane s q construction we study the layers d n f fiber p n f p n 1 f and the limit of the tower for the latter we determine a condition on the cross effects that guarantees convergence we define differentials for functors and establish chain and product rules for them we conclude by studying exponential functors in this setting and describing their taylor towers

growth factors of pivoting strategies associated with neville elimination neville elimination is a direct method for solving linear systems several pivoting strategies for neville elimination including pairwise pivoting are analyzed bounds for two different kinds of growth factors are provided finally an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed using random matrices

error estimates for the numerical identification of a variable coefficient error estimates are derived for the approximate identification of an unknown transmissivity coefficient in a partial differential equation describing a model problem in groundwater now the approximation scheme considered determines the coefficient by least squares fitting of the observed pressure data 1 introduction in this paper we wish to present an error analysis of a common numerical scheme used in the identification of parameters in distributed systems specifically we shall concern ourselves here with a model problem in groundwater flow the problem is to identify a spatially varying transmissivity coefficient a x from observations of the piezometric head u x in a two dimensional static aquifer s where a and u are related by the equations 1 div avw ins2

on positivity of principal minors of bivariate bezier collocation matrix it is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct thus the corresponding interpolation matrix m is nonsingular schumaker stated the conjecture that all principal submatrices of m are nonsingular too furthermore all of the corresponding determinants the principal minors are conjectured to be positive this result would solve the constrained interpolation problem in this paper the conjecture on minors for polynomial degree 17 and conjecture for some particular configurations of domain points are confirmed

multiresolution analyses originated from nonstationary subdivision schemes we analyze the properties of a new class of totally positive refinable functions obtained from nonstationary subdivision schemes we show that the corresponding system of the integer translates is linearly independent satisfies a whitney schoenberg condition reproduces polynomials up to a certain degree and generates a multiresolution analysis finally pre wavelets and bases on the interval are constructed

brief paper necessary conditions for hadamard factorizations of hurwitz polynomials a property of hurwitz polynomials is related with the hadamard product garloff and wagner proved that hadamard products of hurwitz polynomials are hurwitz polynomials and garloff and shrinivasan shown that there are hurwitz polynomials of degree 4 which do not have a hadamard factorization into two hurwitz polynomials of the same degree 4 in this paper we give necessary conditions for an even degree hurwitz polynomial to have a hadamard factorization into two even degree hurwitz polynomials such conditions are given in terms of the coefficients of the given polynomial alone furthermore we show that if an odd degree hurwitz polynomial has a hadamard factorization then a system of nonlinear inequalities has at least one solution

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

cauchy data requirement of the inverse problem of the mean age equation a classical theoretical inverse problem in groundwater hydraulics is the calibration of a heterogeneous isotropic hydraulic conductivity field using known hydraulic head data at each point in the domain it has been long known that a necessary condition for the uniqueness of the solution to this theoretical inverse problem is the cauchy data i e the hydraulic conductivity values or equivalently flux along a line crossing all streamtubes in the entire domain this result has been cited in identifying the value of boundary flux data in practical inverse problems although in general such data are generally not available in this work we show that groundwater mean age data replace this cauchy data requirement

splitting criteria for homotopy functors of spectra we explore the interaction between the taylor tower and cotower as defined in deriving calculus with cotriples and dual calculus for functors to spectra of functors of spectra this leads to new splitting criteria which generalize the results in dual calculus for functors to spectra

characterizations of rectangular totally and strictly totally positive matrices abstract an n m real matrix a is said to be totally positive strictly totally positive if every minor is nonnegative positive in this paper we study characterizations of these classes of matrices by minors by their full rank factorization and by their thin qr factorization

constructing totally positive piecewise chebyshevian b spline bases abstract we consider piecewise chebyshevian splines in the sense of splines with pieces taken from any different five dimensional extended chebyshev spaces and with connection matrices at the knots in this large context we establish necessary and sufficient conditions for the existence of totally positive refinable b spline bases these conditions are applied in many important special cases e g symmetric cardinal geometrically continuous quartic b spline parametrically continuous mixed l splines the great variety of illustrations provided proves the richness of this class of splines for design this richness can be exploited in various other fields as well

how to build all chebyshevian spline spaces good for geometric design in the present work we determine all chebyshevian spline spaces good for geometric design by chebyshevian spline space we mean a space of splines with sections in different extended chebyshev spaces and with connection matrices at the knots we say that such a spline space is good for design when it possesses blossoms to justify the terminology let us recall that in this general framework existence of blossoms defined on a restricted set of tuples makes it possible to develop all the classical geometric design algorithms for splines furthermore existence of blossoms is equivalent to existence of a b spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots we show that chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions with respect to which the connection matrices are identity matrices many interesting consequences can be drawn from the latter characterisation as an example all chebsyhevian spline spaces good for design can be built by means of integral recurrence relations

elimination technique from extrapolation to totally positive matrices and cagd in this survey we will show some connections between several mathematical problems such as extrapolation linear systems totally positive matrices and computer aided geometric design with elimination techniques as the common tool to deal with all of them

equal entries in totally positive matrices we show that the maximal number of equal entries in a totally positive resp totally nonsingular n by n matrix is n 4 3 resp n 3 2 relationships with point line incidences in the plane bruhat order of permutations and tp completability are also presented we also examine the number and positionings of equal 2 by 2 minors in a 2 by n tp matrix and give a relationship between the location of equal 2 by 2 minors and outerplanar graphs

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

effortless quasi interpolation in hierarchical spaces we present a general and simple procedure to construct quasi interpolants in hierarchical spaces such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement the proposed hierarchical quasi interpolants are described in terms of the so called truncated hierarchical basis assuming a quasi interpolant is selected for each space associated with a particular level in the hierarchy the hierarchical quasi interpolants are obtained without any additional manipulation the main properties like polynomial reproduction of the quasi interpolants selected at each level are locally preserved in the hierarchical construction we show how to construct hierarchical local projectors and the local approximation order of the underling hierarchical space is also investigated the presentation is detailed for the truncated hierarchical b spline basis and we discuss its extension to a more general framework

extremal and optimal properties of b bases collocation matrices totally positive matrices are related with the shape preserving representations of a space of functions the normalized b basis of the space has optimal shape preserving properties bernstein polynomials b splines and rational bernstein bases are examples of normalized b bases it is proven that the minimal eigenvalue and singular value of a collocation matrix of a normalized b basis is bounded below by the minimal eigenvalue and singular value of the corresponding collocation matrix of any normalized totally positive basis of the same space the optimal conditioning for the infty norm of a collocation matrix of a normalized b basis among all the normalized totally positive bases of a space of functions is also shown numerical examples confirm the theoretical results and answer related questions

gb splines of arbitrary order explicit formulae and recurrence relations for the calculation of generalized b splines gb splines of arbitrary order are given we derive main properties of gb splines and their series i e partition of unity shape preserving properties invariance with respect to ane transformations etc it is shown that such splines have the variation diminishing property and are chebyshevian splines c 1999 elsevier science b v all rights reserved

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

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 hyers ulam stability constant for chebyshevian bernstein operators abstract on a closed bounded interval a given extended chebyshev space which possesses a bernstein basis generates infinitely many operators of the bernstein type we show that all these operators share the same hyers ulam stability constant this constant is the maximum in absolute value of the bezier coefficients of the generalised chebyshev polynomial associated with the given space we establish an optimality property of these bernstein operators with respect to the hyers ulam stability constant numerical computations of these constants are investigated in two cases rational and muntz bernstein operators with special emphasis on their behaviour with respect to the concerned interval and to dimension elevation

total positivity of a kind of generalized toric bernstein basis abstract the normalized totally positive bases are widely used in many fields based on the generalized vandermonde determinant the normalized total positivity of a kind of generalized toric bernstein basis is proved which is defined on a set of real points by this result then the progressive iterative approximation property of the generalized toric bezier curve is obtained and some examples are provided to illustrate this property

on a class of weak tchebycheff systems in this paper we study the approximation power the existence of a normalized b basis and the structure of a degree raising process for spaces of the form requiring suitable assumptions on the functions u and v the results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of cagd

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

a note on matrices with maximal growth factor for neville elimination neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations the growth factor is an indicator of the numerical stability of an algorithm in the literature bounds for the growth factor of neville elimination with some pivoting strategies have appeared in this work we determine all the matrices such that the minimal upper bound of the growth factor of neville elimination with those pivoting strategies is reached

a geometric approach for hermite subdivision we present a non stationary non uniform scheme for two point hermite subdivision the novelty of this approach relies on a geometric interpretation of the subdivision steps related to generalized bernstein bases which permits to overcome the usually unavoidable analytical difficulties the main advantages consist in extra smoothness conditions which in turn produce highly regular limit curves and in an elegant structure of the subdivision described by three de casteljau type matrices as a by product the scheme is inherently shape preserving

accurate computations with said ball vandermonde matrices abstract a generalization of the vandermonde matrices which arise when the power basis is replaced by the said ball basis is considered when the nodes are inside the interval 0 1 then those matrices are strictly totally positive an algorithm for computing the bidiagonal decomposition of those said ball vandermonde matrices is presented which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is shown to be fast and to guarantee high relative accuracy some numerical experiments which illustrate the good behaviour of the algorithm are included

isogemetric analysis and symmetric galerkin bem isogeometric approach applied to boundary element methods is an emerging research area see e g simpson et al 2012 33 in this context the aim of the present contribution is that of investigating from a numerical point of view the symmetric galerkin boundary element method sgbem devoted to the solution of 2d boundary value problems for the laplace equation where the boundary and the unknowns on it are both represented by b splines de boor 2001 9 we mainly compare this approach which we call iga sgbem with a curvilinear sgbem aimi et al 1999 2 which operates on any boundary given by explicit parametric representation and where the approximate solution is obtained using lagrangian basis both techniques are further compared with a standard conventional sgbem approach aimi et al 1997 1 where the boundary of the assigned problem is approximated by linear elements and the numerical solution is expressed in terms of lagrangian basis several examples will be presented and discussed underlying benefits and drawbacks of all the above mentioned approaches

the hadamard core of the totally nonnegative matrices abstract an m by n matrix a is called totally nonnegative if every minor of a is nonnegative the hadamard product of two matrices is simply their entry wise product this paper introduces the subclass of totally nonnegative matrices whose hadamard product with any totally nonnegative matrix is again totally nonnegative many properties concerning this class are discussed including a complete characterization for min m n a characterization of the zero nonzero patterns for which all totally nonnegative matrices lie in this class and connections to oppenheim s inequality

blocking neville elimination algorithm for exploiting cache memories neville elimination is a method for solving a linear system of equations that introduces zeros in a matrix column by adding to each row an adequate multiple of the previous one in this paper we explore block algorithms for neville elimination which take into account the memory hierarchies of a computer these algorithms try to manage the memory movements to optimize them thus the matrix of the system is divided following three different strategies blocking by rows columns or submatrices in each case we study the performance of the algorithm according to the ratio of floating point operations to memory references q theoretical estimations show that q depends on data partitioning being submatrix blocks the best choice

understanding recurrence relations for chebyshevian b splines via blossoms the purpose of this article is to show how naturally recurrence relations for most general chebyshevian b splines emerge from blossoms in particular this work gives a new insight into previous results by lyche a recurrence relation for chebyshevian b splines constr approx 1 1985 155 178 the importance of which it underlines

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

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

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

spaces of generalized splines over t meshes we consider a class of non polynomial spaces namely a noteworthy case of extended chebyshev spaces and we generalize the concept of polynomial spline space over t mesh to this non polynomial setting in other words we focus on a class of spaces spanned in each cell of the t mesh both by polynomial and by suitably chosen non polynomial functions which we will refer to as generalized splines over t meshes for such spaces we provide under certain conditions on the regularity of the space a study of the dimension and of the basis based on the notion of minimal determining set as well as some results about the dimension of refined and merged t meshes finally we study the approximation power of the just constructed spline spaces

the direct inverse problem in aquifers solution of the inverse problem for aquifers requires identification of a distributed parameter system in addition to ill posedness the inverse problem may be underdetermined if recharge is treated as a spatially and temporally variable distribution parameter galerkin s method is applied to a weak formulation of the direct inverse problem leading to a possibly nonlinear weighted least squares problem consideration of generic errors suggests that if the estimate for some derivative of head contains a significant error on one element for all time then significant estimation errors are possible for all nodes and for all reservoir parameters in contrast if some derivative of head is small in some spatial region for all time causing the matrix to be inverted in the least squares algorithm to be nearly singular then significant estimation errors can occur in the corresponding reservoir parameters in that spatial region although it is still possible that other reservoir parameters in that region as well as all reservoir parameters elsewhere may be satisfactorily estimated

sign regular matrices and neville elimination abstract a pivoting strategy of o n operations for the neville elimination of n n nonsingular sign regular matrices is introduced among other nice properties it is proved that it preserves sign regularity it is also shown its relationship with scaled partial pivoting strategies for neville elimination

bidiagonal decompositions of oscillating systems of vectors abstract we establish necessary and sufficient conditions in the language of bidiagonal decompositions for a matrix v to be an eigenvector matrix of a totally positive matrix namely this is the case if and only if v and v t are lowerly totally positive these conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of v and v t using these decompositions we give elementary proofs of the oscillating properties of v in particular the fact that the jth column of v has j 1 changes of sign our new results include the fact that the q matrix in a qr decomposition of a totally positive matrix belongs to the above class and thus has the same oscillating properties

on dimension elevation in quasi extended chebyshev spaces via blossoms we analyse the dimension elevation process from mathcal e n p q to mathcal e n 1 p q where mathcal e n p q is spanned over 0 1 by 1 x xn 2 xp 1 x q p q being any convenient real numbers such spaces are not extended chebyshev spaces but quasi extended chebyshev spaces they were recently introduced in cagd for shape preservation purposes costantini in math comp 46 203 214 1986 costantini in advanced course on fairshape pp 87 114 in 1996 costantini in curves and surfaces with applications in cagd pp 85 94 1997 our results give a new insight into the special case p q for which dimension elevation had already been considered first when p q was supposed to be an integer goodman and mazure in j approx theory 109 48 81 2001 then without the latter requirement costantini et al in numer math 101 333 354 2005 the question of dimension elevation in more general quasi extended chebyshev spaces is also addressed

polynomial least squares fitting in the bernstein basis the problem of polynomial least squares fitting in which the usual monomial basis is replaced by the bernstein basis is considered the coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular bernstein vandermonde matrix in order to use the method based on the qr decomposition of a the first stage consists of computing the bidiagonal decomposition of the coefficient matrix a starting from that bidiagonal decomposition an algorithm for obtaining the qr decomposition of a is the applied finally a triangular system is solved by using the bidiagonal decomposition of the r factor of a some numerical experiments showing the behavior of this approach are included

shape preserving interpolants with high smoothness the aim of the present paper is to introduce a new method to construct a c3 function which interpolates a given set of data in a shape preserving way the resulting function is a parametric quintic curve whose shape is controlled by tension parameters which have a direct geometric meaning

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

shape preserving c1 hermite interpolants generated by a gori pitolli subdivision scheme in this paper we present a method for the construction of c 1 hermite interpolants obtained from a particular family of refinable spline functions introduced by gori and pitolli they constitute a one parameter subfamily of the hermite interpolants generated by the general merrien s subdivision scheme we compare this family to the other one parameter subfamily studied by merrien and sablonniere and lyche and merrien on the solution of two points hermite interpolation problems with arbitrary monotonicity or convexity constraints

a classification of taylor towers of functors of spaces and spectra abstract we describe new structure on the goodwillie derivatives of a functor and we show how the full taylor tower of the functor can be recovered from this structure this new structure takes the form of a coalgebra over a certain comonad which we construct and whose precise nature depends on the source and target categories of the functor in question the taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives we get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad for functors with values in the category of spectra we give a rather explicit description of the associated comonads and their coalgebras in particular for functors from based spaces to spectra we interpret this new structure as that of a divided power right module over the operad formed by the derivatives of the identity on based spaces

curve construction based on four αβ bernstein like basis functions a b s t r a c t four new bernstein like basis functions with two exponential shape parameters are constructed in this paper which include the cubic said ball basis functions and the cubic bernstein basis functions within the general framework of quasi extended chebyshev space we prove that the proposed bernstein like basis is an optimal normalized totally positive basis in order to compute the corresponding bezier like curves stably and efficiently a new corner cutting algorithm is developed necessary and sufficient conditions are derived for the planar bezier like curve having single or double inflection points a loop or a cusp or be locally or globally convex in terms of the relative position of its control polygons side vectors based on the new proposed bernstein like basis a class of b spline like basis functions with two local exponential shape parameters is constructed their totally positive property is also proved the associated b splinelike curves have c 2 continuity at single knots and include the cubic non uniform b spline

shape preserving properties of ω q bernstein polynomials abstract in this paper we discuss shape preserving properties of the q bernstein polynomials b n q f x introduced by lewanowicz and wozny in s lewanowicz p wo ny generalized bernstein polynomials bit 44 1 2004 63 78 for q 0 1 when 0 we recover the q bernstein polynomials introduced by phillips g m phillips bernstein polynomials based on the q integers ann numer math 4 1997 511 518 when q 1 we recover the classical bernstein polynomials for q 0 1 we show that the basic q bernstein polynomial basis is a normalized totally positive basis on 0 1 and that the q bernstein operators b n q on c 0 1 are variation diminishing monotonicity preserving and convexity preserving we also show that the q bernstein polynomials of a convex function f in the case q 0 1 are monotonic in the parameters and q for fixed n

bidiagonal factorizations and quasi oscillatory rectangular matrices abstract a real matrix a of size m n is called totally nonnegative totally positive if all its minors are nonnegative positive a variant of the neville elimination process is studied in relation to the existence of a totally nonnegative elementary bidiagonal factorization of a the class of quasi oscillatory rectangular matrices which in the square case contains the oscillatory matrices is introduced and a characterization of this class of matrices by incorporating bidiagonal factorization is showed

q bernstein polynomials and bezier curves we define q bernstein polynomials which generalize the classical bernstein polynomials and show that the difference of two consecutive q bernstein polynomials of a function f can be expressed in terms of second order divided differences of f it is also shown that the approximation to a convex function by its q bernstein polynomials is one sided a parametric curve is represented using a generalized bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve we study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing as for the classical bernstein basis

monotonicity preserving representations of non polynomial surfaces we prove that in contrast to the case for rational surfaces some tensor product representations through spaces containing algebraic trigonometric and hyperbolic polynomials are monotonicity preserving the surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product bezier and tensor product b spline surfaces

weighted lupas q bezier curves this paper is concerned with a new generalization of rational bernstein bezier curves involving q integers as shape parameters a one parameter family of rational bernstein bezier curves weighted lupas q bezier curves is constructed based on a set of lupas q analogue of bernstein functions which is proved to be a normalized totally positive basis the generalized rational bezier curve is investigated from a geometric point of view the investigation provides the geometric meaning of the weights and the representation for conic sections we also obtain degree evaluation and de casteljau algorithms by means of homogeneous coordinates numerical examples show that weighted lupas q bezier curves have more modeling flexibility than classical rational bernstein bezier curves and lupas q bezier curves and meanwhile they provide better approximations to the control polygon than rational phillips q bezier curves

comparing pivoting strategies for almost strictly sign regular matrices abstract in this paper some properties of two determinant pivoting for neville elimination are presented in particular we consider a zero increasing property and we show an optimal normwise growth factor comparisons with other pivoting strategies for neville elimination and with gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed numerical examples are included

a general class of bernstein like bases this paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials bases with optimal shape preserving and stability properties are constructed evaluation and subdivision algorithms for them are provided bases for the corresponding and mixed spline spaces are also constructed some nice properties of these bases and the generated curves are shown

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

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

a collection of examples where neville elimination outperforms gaussian elimination neville elimination is an elimination procedure alternative to gaussian elimination it is very useful when dealing with totally positive matrices for which nice stability results are known here we include examples most of them test matrices used in matlab which are not totally positive matrices where neville elimination outperforms gaussian elimination

neville elimination a study of the efficiency using checkerboard partitioning abstract it is well known that checkerboard partitioning can exploit more concurrency than striped partitioning because the matrix computation can be divided among more processors than in the case of striping in this work we analyze the performance of neville method when a checkerboard partitioning is used focusing on the special case of block cyclic checkerboard partitioning this method is an alternative to gaussian elimination and it has been proved to be very useful for some classes of matrices such as totally positive matrices the performance of this parallel system is measured in terms of the efficiency the fraction of time for which a processor is usefully employed which in our model is close to one when the optimum block size is used also we have executed our algorithms on a parallel pc cluster observing that both efficiencies theoretical and empirical are quite similar

one sided hermite interpolation by piecewise different generalized polynomials piecewise generalized polynomials of different kinds of order n ect splines of order n are constructed from different ect systems of order n via connection matrices which are nonsingular and totally positive a well known zero count for polynomial splines is extended to ect splines it is used to construct ect b splines and to show under which conditions ect splines will solve modified hermite type interpolation problems also conditions are specified such that piecewise generalized polynomials form rect systems and the interpolation problems associated with may be solved recursively

recovery of a variable coefficient in a coastal evolution equation a method is presented for reconstructing an unknown coefficient in a linear diffusion equation from measured data this equation arises in the description of coastline evolution and preliminary results are presented here the unknown term may vary with both space and time although time variation is assumed to be slow inversion is carried out by first expressing the solution of the direct problem formally in terms of the governing operators and making explicit approximations to these expressions using data at two time steps this then allows equations to be derived and solved to give explicit expressions for the required function