adding algebraic rewriting to the untyped lambda calculus abstract we investigate the system obtained by adding an algebraic rewriting system r to an untyped lambda calculus in which terms are formed using the function symbols from r as constants on certain classes of terms called here stable we prove that the resulting calculus is confluent if r is confluent and is terminating if r is terminating the termination result has the corresponding theorems for several typed calculi as corollaries the proof of the confluence result suggests a general method for proving confluence of typed reduction plus rewriting we sketch the application to the polymorphic lambda calculus

rewriting systems over similarity and generalized pseudometric spaces and their properties we present a study of confluence and related properties of fuzzy relations defined over similarity spaces the ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems this paper is a continuation of our previous paper where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees leaving the ordinary notions a particular case when the underlying structure is the two valued boolean algebra in this paper we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation we also present definitions of confluence and related properties of relations on a generalized pseudometric spaces using the well known link between generalized pseudometrics and similarities we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces the introduced notions are further illustrated by example of rewriting based on graded if then rules

the condition of vandermonde like matrices involving orthogonal polynomials the condition number relative to the frobenius norm of the n x n matrix p pi ltxj ly j l is investigated where p p dx are orthogonal polynomials with respect to some weight distribution dx and xi are pairwise distinct real numbers if the nodes x j are the zeros of p the condition number is either expressed or estimated from below and above in terms of the christoffel numbers for dh depending on whether the p are normalized or not for arbitrary real xi and normalized p a lower bound of the condition number is obtained in terms of the christoffel function evaluated at the nodes numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions dx

graph theoretic methods for matrix completion problems abstract a pattern is a list of positions in an n n real matrix a matrix completion problem for the class of matrices asks whether every partial matrix whose specified entries are exactly the positions of the pattern can be completed to a matrix we survey the current state of research on matrix completion problems for many subclasses of p 0 matrices including positive definite matrices m matrices inverse m matrices p matrices and matrices defined by various sign symmetry and positivity conditions on p 0 and p matrices graph theoretic techniques used to study completion problems are discussed several new results are also presented including the solution to the m 0 matrix completion problem and the sign symmetric p 0 matrix completion problem

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

polymorphic higher order recursive path orderings this article extends the termination proof techniques based on reduction orderings to a higher order setting by defining a family of recursive path orderings for terms of a typed lambda calculus generated by a signature of polymorphic higher order function symbols these relations can be generated from two given well founded orderings on the function symbols and on the type constructors the obtained orderings on terms are well founded monotonic stable under substitution and include reductions they can be used to prove the strong normalization property of higher order calculi in which constants can be defined by higher order rewrite rules using first order pattern matching for example the polymorphic version of godel s recursor for the natural numbers is easily oriented and indeed our ordering is polymorphic in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule many nontrivial examples are given that exemplify the expressive power of these orderings all have been checked by our implementation this article is an extended and improved version of jouannaud and rubio 1999 polymorphic algebras have been made more expressive than in our previous framework the intuitive notion of a polymorphic higher order ordering has now been made precise the higher order recursive path ordering itself has been made much more powerful by replacing the congruence on types used there by an ordering on types satisfying some abstract properties besides using a restriction of dershowitz s recursive path ordering for comparing types we can integrate both orderings into a single one operating uniformly on both terms and types

on inverse positivity of sub direct sums of matrices abstract in this note the authors consider the problem of inverse positivity of k subdirect sum of matrices the main results provide a solution to an open problem posed recently

spectre premier de oq mn k image canonique et separation normale abstract given any commutative field k denote r oq mn k the coordinate ring of quantum n n matrices over k and assume q is a nonzero element in k which is not a root of unity recall that r is generated by n2 variables xi i 1 n 2 subject only to the following relations if x y z t is any 2 2 sub matrix of x x i then a yx q 1xy zx q 1xz tz q 1zt ty q 1yt zy yz b tx xt q q 1 yz denote r the k algebra generated by the same variables xi subject to the same relations except relations b which are replaced by c tx xt so that r is just the algebra of regular functions on some quantum affine space of dimension n2 over k the theory of derivative elimination defines a natural embedding spec r spec r and asserts that the canonical image spec r is a union of strata spec w r in the sense of goodearl letzter in cms conf proc vol 22 1998 39 58 where w describes some subset w of p 1 n 2 the sets spec w r 1 spec w r w w define the goodearl letzter h stratification of spec r in the sense of goodearl letzter trans amer math soc 352 2000 1381 1403 in this paper we give the precise description of the set w and we compute its cardinality using that description and the derivative elimination algorithm we can verify theorems 6 3 1 6 3 2 that h spec r has an h normal separation in the sense of goodearl in lecture notes in pure and appl math 210 2000 205 237 so that spec r has normal separation in the sense of brown goodearl trans amer math soc 348 1996 2465 2502 this property was conjectured by k brown and k goodearl since r is auslander regular and cohen macaulay this implies by goodearl lenagan j pure appl algebra 111 1996 123 142 that r is catenary and satisfies the tauvel s height formula

total nonnegativity of matrices related to polynomial roots and poles of rational functions abstract in this paper totally nonnegative positive matrices are considered which are matrices having all their minors nonnegative positive the almost totally positive matrices form a class between the totally nonnegative matrices and the totally positive ones an efficient determinantal test based on the cauchon algorithm for checking a given matrix for falling in one of these three classes of matrices is applied to matrices which are related to roots of polynomials and poles of rational functions specifically the hankel matrix associated with the laurent series at infinity of a rational function and matrices of hurwitz type associated with polynomials in both cases it is concluded from properties of one or two finite sections of the infinite matrix that the infinite matrix itself has these or related properties then the results are applied to derive a sufficient condition for the hurwitz stability of an interval family of polynomials finally interval problems for a subclass of the rational functions viz r functions are investigated these problems include invariance of exclusively positive poles and exclusively negative roots in the presence of variation of the coefficients of the polynomials within given intervals

linear complexity parallel algorithms for linear systems of equations with recursive structure abstract a method of derivation of parallel algorithms for n 1 n 1 matrices with recursive structure is presented and applied to toeplitz hankel and other toeplitz like matrices the derived algorithms executed on o n parallel processors require o n arithmetic operations per processor

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

parallel reductions in λ calculus the notion of parallel reduction is extracted from the simple proof of the church rosser theorem by tait and martin lof intuitively this means to reduce a number of redexes existing in a term simultaneously thus in the case of s reduction the effect of a parallel reduction is same as that of a complete development which is defined by using residuals of s redexes a nice feature of parallel reduction however is that it can be defined directly by induction on the structure of terms without referring to residuals or other auxiliary notions and the inductive definition provides us exactly what we need in proving the theorem inductively moreover the notion can be easily extended to other reduction systems such as girard s second order system f and godel s system t in this paper after reevaluating the significance of the notion of parallel reduction in tait and martin lof type proofs of the church rosser theorems we show that the notion of parallel reduction is also useful in giving short and direct proofs of some other fundamental theorems in reduction theory of calculus among others we give such simple proofs of the standardization theorem for s reduction a special case of which is known as the leftmost reduction theorem for s reduction the quasi leftmost reduction theorem for s reduction the postponement theorem of reduction in s reduction and the leftmost reduction theorem for s reduction

confluent polynomial vandermonde like matrices displacement structures inversion formulas and fast algorithm in the present paper we use the displacement structure approach to introduce a new class of what are called confluent polynomial vandermonde like matrices which generalize the ordinary simple nodes polynomial vandermonde matrices studied earlier by different authors the displacement structure approach leads to the explicit inversion formulas for confluent polynomial vandermonde like matrices and fast algorithms for inversion and for solving the associated linear systems

on the inertia set of a signed graph with loops abstract a signed graph is a pair g where g v e is a graph in which parallel edges and loops are permitted with v 1 n and e the edges in are called odd edges and the other edges of e even by s g we denote the set of all n n real symmetric matrices a a i j such that if a i j 0 then among the edges connecting i and j there must be at least one even edge if a i j 0 then among the edges connecting i and j there must be at least one odd edge and if a i j 0 then either there must be at least one odd edge and at least one even edge connecting i and j or there are no edges connecting i and j here we allow i j for a real symmetric matrix a the partial inertia of a is the pair p q where p and q are the number of positive and negative eigenvalues of a respectively if g is a signed graph we define the inertia set of g as the set of the partial inertias of all matrices a s g by mr g we denote max rank a a s g we say that a signed graph g satisfies the northeast property if for each p q with p q mr g in the inertia set of g also p 1 q p q 1 belong to the inertia set of g in this paper we show that if g is a signed graph where g is a tree with possibly loops attached at some of the vertices then g satisfies the northeast property furthermore we present a formula for calculating the inertia set of a signed graph g where g is a tree with possibly loops attached at some of the vertices

pnl to hol from the logic of nominal sets to the logic of higher order functions permissive nominal logic pnl extends first order predicate logic with term formers that can bind names in their arguments it takes a semantics in permissive nominal sets in pnl the quantifier or l binder are just term formers satisfying axioms and their denotation is functions on nominal atoms abstraction then we have higher order logic hol and its models in ordinary i e zermelo fraenkel sets the denotation of or l is functions on full or partial function spaces this raises the following question how are these two models of binding connected what translation is possible between pnl and hol and between nominal sets and functions we exhibit a translation of pnl into hol and from models of pnl to certain models of hol it is natural but also partial we translate a restricted subsystem of full pnl to hol the extra part which does not translate is the symmetry properties of nominal sets with respect to permutations to use a little nominal jargon we can translate names and binding but not their nominal equivariance properties this seems reasonable since hol and ordinary sets are not equivariant thus viewed through this translation pnl and hol and their models do different things but they enjoy non trivial and rich subsystems which are isomorphic

invariant prime ideals in quantizations of nilpotent lie algebras de concini kac and procesi defined a family of subalgebras u of a quantized universal enveloping algebra uq g associated to the elements of the corresponding weyl group w they are deformations of the universal enveloping algebras u n adw n where n are the nilradicals of a pair of dual borel subalgebras based on results of gorelik and joseph and an interpretation of u as quantized algebras of functions on schubert cells we construct explicitly the h invariant prime ideals of each u and show that the corresponding poset is isomorphic to w where h is the group of group like elements of uq g moreover for each h prime of u w we construct a generating set in terms of demazure modules related to fundamental representations using results of ramanathan and kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related poisson structures on schubert cells in flag varieties

termination of rewrite relations on λ terms based on girard s notion of reducibility in this paper we show how to extend the notion of reducibility introduced by girard for proving the termination of reduction in the polymorphic calculus to prove the termination of various kinds of rewrite relations on terms including rewriting modulo some equational theory and rewriting with matching modulo by using the notion of computability closure this provides a powerful termination criterion for various higher order rewriting frameworks including klop s combinatory reductions systems with simple types and nipkow s higher order rewrite systems

sub direct sums and positivity classes of matrices abstract it is well known that a direct sum is positive semidefinite if and only if each of the direct summands is positive semidefinite in fact it is also known that this statement remains true if positive semidefinite is replaced with doubly nonnegative completely positive totally nonnegative m matrix and p matrix etc for each of these classes we consider corresponding questions for a more general sum of two matrices of which the direct sum and ordinary sum are special cases

les ideaux premiers invariants de oq mm p c abstract for q generic we give a positive answer to a conjecture of goodearl and lenagan the h invariant prime ideals of o q m m p c are generated by quantum minors

intervals of totally nonnegative matrices totally nonnegative matrices i e matrices having all their minors nonnegative and matrix intervals with respect to the checkerboard ordering are considered it is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative and in addition all their zero minors are identical then all matrices from this interval are also nonsingular and totally nonnegative with identical zero minors

torus invariant prime ideals in quantum matrices totally nonnegative cells and symplectic leaves the algebra of quantum matrices of a given size supports a rational torus action by automorphisms it follows from work of letzter and the first named author that to understand the prime and primitive spectra of this algebra the first step is to understand the prime ideals that are invariant under the torus action in this paper we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size as a corollary we obtain explicit generating sets of quantum minors for the torus invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over mathbb q

inverses of generalized vandermonde matrices on demontre plusieurs equations matricielles comprenant des matrices de vandermonde generalisees qui donnent une information algebrique explicite sur les matrices inverses

some results on b matrices and doubly b matrices abstract a real matrix with positive row sums and all its off diagonal elements bounded above by their corresponding row means was called in 4 a b matrix in 5 the class of doubly b matrices was introduced as a generalization of the previous class we present several characterizations and properties of these matrices and for the class of b matrices we consider corresponding questions for subdirect sums of two matrices a general sum of matrices introduced in 1 by s m fallat and c r johnson of which the direct sum and ordinary sum are special cases for the hadamard product of two matrices and for the kronecker product and sum of two matrices

infinitary combinatory reduction systems we define infinitary combinatory reduction systems icrss thus providing the first notion of infinitary higher order rewriting the systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary l calculus are special cases furthermore we generalise a number of known results from first order infinitary rewriting and infinitary l calculus to icrss in particular for fully extended left linear icrss we prove the well known compression property and for orthogonal icrss we prove that 1 if a set of redexes u has a complete development then all complete developments of u end in the same term and that 2 any tiling diagram involving strongly convergent reductions s and t can be completed iff at least one of s t and t s is strongly convergent we also prove an ancillary result of independent interest a set of redexes in an orthogonal icrs has a complete development iff the set has the so called finite jumps property

signal flow graph approach to inversion of h m quasiseparable vandermonde matrices and new filter structures abstract we use the language of signal flow graph representation of digital filter structures to solve three purely mathematical problems including fast inversion of certain polynomial vandermonde matrices deriving an analogue of the horner and clenshaw rules for polynomial evaluation in a h m quasiseparable basis and computation of eigenvectors of h m quasiseparable classes of matrices while algebraic derivations are possible using elementary operations specifically flow reversal on signal flow graphs provides a unified derivation reveals connections with systems theory etc

displacement structure approach to polynomial vandermonde and related matrices abstract we introduce a new class of what we call polynomial vandermonde like matrices this class generalizes the polynomial vandermonde matrices studied earlier by various authors who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems a displacement structure approach allows us to carry over all these results to the wider class of polynomial vandermonde like matrices

computations with quasiseparable polynomials and matrices in this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials quasiseparable matrices generalize two classical matrix classes jacobi tridiagonal matrices and unitary hessenberg matrices that are known to correspond to real orthogonal polynomials and szego polynomials respectively the latter two polynomial families arise in a wide variety of applications and their short recurrence relations are the basis for a number of efficient algorithms for historical reasons algorithm development is more advanced for real orthogonal polynomials recent variations of these algorithms tend to be valid only for the szego polynomials they are analogues and not generalizations of the original algorithms herein we survey several recent results for the superclass of quasiseparable matrices which includes both jacobi and unitary hessenberg matrices as special cases the interplay between quasiseparable matrices and their associated polynomial sequences which contain both real orthogonal and szego polynomials allows one to obtain true generalizations of several algorithms specifically we discuss the bjorck pereyra algorithm the traub algorithm certain new digital filter structures as well as qr and divide and conquer eigenvalue algorithms

the inertia sets of graphs with a 2 separation abstract for a graph g v e with v 1 2 n let s g be the set of all symmetric real n n matrices a a i j with a i j 0 i j if and only if i j e the inertia set of a graph g is the set of all possible inertias of matrices in s g in this paper we give a formula that expresses the inertia set of a graph with a 2 separation in terms of the inertia sets of subgraphs of g this formula follows from an extension to arbitrary fields with characteristic not equal to two

perpetuality and uniform normalization in orthogonal rewrite systems we study perpetuality of reduction steps as well as perpetuality of redexes in orthogonal rewrite systems a perpetual step is a reduction step which retains the possibility of infinite reductions a perpetual redex is a redex which when put into an arbitrary context yields a perpetual step we generalize and refine existing criteria for the perpetuality of reduction steps and redexes in orthogonal term rewriting systems and the calculus due to bergstra and klop and others we first introduce context sensitive conditional expression reduction systems ccerss and define a concept of orthogonality which implies confluence for them in particular several important calculi and their extensions and restrictions can naturally be embedded into orthogonal ccerss we then define a perpetual reduction strategy which enables one to construct minimal w r t levy s permutation ordering on reductions infinite reductions in orthogonal fully extended ccerss using the properties of the minimal perpetual strategy we prove 1 perpetuality of any reduction step that does not erase potentially infinite arguments which are arguments that may become via substitution infinite after a number of outside steps and 2 perpetuality in every context of any safe redex which is a redex whose substitution instances may discard infinite arguments only when the corresponding contracta remain infinite we prove both these perpetuality criteria for orthogonal fully extended ccerss and then specialize and apply them to restricted calculi demonstrating their usefulness in particular we prove the equivalence of weak and strong normalization whose equivalence is here called uniform normalization for various restricted calculi most of which cannot be derived from previously known perpetuality criteria

curry howard for incomplete first order logic derivations using one and a half level terms the curry howard correspondence connects derivations in natural deduction with the lambda calculus predicates are types derivations are terms this supports reasoning from assumptions to conclusions but we may want to reason backwards from the desired conclusion towards the assumptions at intermediate stages we may have a partial derivation with holes this is natural in informal practice but it can be difficult to formalise the informal act of filling holes in a partial derivation suggests a capturing substitution since holes may occur in the scope of quantifier introduction rules as other authors have observed this is not immediately supported by the lambda calculus also universal quantification requires a fresh name and it is not immediately obvious what formal meaning to assign to this notion if derivations are incomplete further issues arise with proof normalisation this corresponds with lambda calculus reduction which can require alpha conversion to avoid capture when beta reducing and it is not immediately clear how to alpha convert a name in an incomplete derivation we apply a one and a half level technique based on nominal terms to construct a curry howard correspondence for first order logic this features two levels of variable but with no lambda abstraction at the second level predicates are types derivations are terms proof normalisation is reduction and the two levels of variable are respectively the assumptions and the holes of an incomplete derivation we give notions of proof term typing alpha conversion and beta reduction for our syntax we prove confluence we exhibit several admissible rules including a proof that instantiation of level two variables is type safe this corresponds with the act of filling holes in an incomplete derivation and can be viewed as a form of cut rule and we explore the connection with traditional curry howard in the case that the derivation is in fact complete our techniques are not specifically tailored to first order logic and the same ideas should be applicable without any essential new difficulties to similar logical systems

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

a fast algorithm for generalized hankel matrices arising in finite moment problems abstract consider an n n lower triangular matrix l whose i 1 st row is defined by the coefficients of the real polynomial p i x of degree i such that p i x is a set of orthogonal polynomials satisfying a standard three term recurrence relation if h is an n n real hankel matrix with nonsingular leading principal submatrices then h lhl t will be referred to as a strongly nonsingular hankel matrix with respect to the orthogonal polynomial basis p i x in this paper we develop an efficient o n 2 algorithm for the solution of a system of linear equations with a real symmetric coefficient matrix h which is a hankel matrix with respect to a suitable orthogonal polynomial basis this leads to an efficient method for computing polynomial approximations of an unknown function given its modified moments

polynomial bezoutian matrix with respect to a general basis we introduce a class of so called polynomial bezoutian matrices the operator representation relative to a pair of dual bases and the generalized barnett factorization formula for this kind of matrix are derived an intertwining relation and generalized bezoutian reduction via polynomial vandermonde matrix are presented

the minimum rank of a sign pattern matrix with a 1 separation abstract a sign pattern matrix is a matrix whose entries are from the set 0 if a is an m n sign pattern matrix the qualitative class of a denoted q a is the set of all real m n matrices b b i j with b i j positive respectively negative zero if a i j is respectively 0 the minimum rank of a sign pattern matrix a denoted mr a is the minimum of the ranks of the real matrices in q a determination of the minimum rank of a sign pattern matrix is a longstanding open problem for the case that the sign pattern matrix has a 1 separation we present a formula to compute the minimum rank of a sign pattern matrix using the minimum ranks of certain generalized sign pattern matrices associated with the 1 separation

a bjorck pereyra type algorithm for szego vandermonde matrices based on properties of unitary hessenberg matrices abstract in this paper we carry over the bjorck pereyra algorithm for solving vandermonde linear systems to what we suggest to call szego vandermonde systems v x i e polynomial vandermonde systems where the corresponding polynomial system is the szego polynomials the properties of the corresponding unitary hessenberg matrix allow us to derive a fast o n 2 computational procedure we present numerical experiments that indicate that for ill conditioned matrices the new algorithm yields better forward accuracy than gaussian elimination

criteria for sign regularity of sets of matrices abstract we consider matrix intervals with respect to a certain checkerboard partial ordering we show that every real matrix contained in a matrix interval is sign regular if two special matrices taken from that matrix interval are sign regular

on a class of nonsingular matrices containing b matrices abstract a real matrix with positive row sums and all its off diagonal elements bounded above by their corresponding row means was called in 12 a b matrix the classes of doubly b matrices and of b r matrices were introduced in 13 and 11 as generalizations of the previous class we present characterizations and properties of b r matrices and we consider the problem of when a subdirect sum of two matrices in this class lies in the same class

displacement structures and fast inversion formulas for confluent polynomial vandermonde like matrices in the present paper confluent polynomial vandermonde like matrices with general recurrence structure are introduced three kinds of displacement structure equations and two kinds of fast inversion formulas for this class of matrices are derived by using displacement structure matrix method a relationship between confluent polynomial vandermonde like matrices and confluent cauchy like matrices is pointed out

combinatorics of h primes in quantum matrices abstract for q c transcendental over q we give an algorithmic construction of an order isomorphism between the set of h primes of o q m n c and the sub poset s of the reverse bruhat order of the symmetric group s 2 n consisting of those permutations that move any integer by no more than n positions further we describe the permutations that correspond via this bijection to rank t h primes more precisely we establish the following result imagine that there is a barrier between positions n and n 1 then a 2n permutation s corresponds to a rank t h invariant prime ideal of o q m n c if and only if the number of integers that are moved by from the right to the left of this barrier is exactly n t the existence of such an order isomorphism was conjectured by goodearl and lenagan

the undecidability of the second order unification problem abstract it is shown that there is no effective procedure for determining whether or not two terms of the language of second order logic have a common instance

complete subsets of a diffraction pattern a positive atomic density r j 1nnj r rj in a d dimensional space can be exactly reconstructed from an appropriate finite subset complete set of its fourier series coefficients uh hd or even limited to the support rj j 1 n from a finite subset of moduli uh it is necessary first to determine a complete set of fourier coefficients uh possibly inside the unavoidable high resolution cut off h l and then by these coefficients to determine the unknown density we report some procedures which are able to single out complete sets they are based on a property of goedkoop s vector lattice ah hd defined so that ah ak uk h the property states that if the vector with index h h1 hd is a linear combination of the vectors relevant to a set of indices with a particular shape then all the vectors relevant to the hyperquadrant h h h h 1 d are linear combinations of the vectors relevant to 0 h moreover the determination of from a complete set passes through the solution of a system of polynomial equations in d variables whose roots determine the position vectors we show how to convert this problem into the simpler problem of sequentially solving a set of polynomial equations in one variable

displacement structure approach to q adic chebyshev vandermonde like matrices abstract in this paper a new class of so called q adic chebyshev vandermonde like matrices over an arbitrary non algebraically closed field is introduced this class generalizes both the ordinary chebyshev vandermonde like matrices over the complex field studied earlier by kailath and olshevsky t kailath v olshevsky displacement structure approach to chebyshev vandermonde and related matrices integral equations operator theory 22 1995 65 92 and the classical q adic vandermonde like matrices with respect to power basis by yang and hu z h yang y j hu displacement structure and fast inversion formulas for q adic vandermonde like matrices j comput appl math 176 2005 1 14 three kinds of displacement structures and consequently three kinds of fast inversion formulas are presented for this class of matrices by using displacement structure theory method which generalize the corresponding results for chebyshev vandermonde like and q adic vandermonde like matrices

invariance of total nonnegativity of a matrix under entry wise perturbation and subdirect sum of totally nonnegative matrices abstract a real matrix is called totally nonnegative if all of its minors are nonnegative in this paper the minors are determined from which the maximum allowable entry perturbation of a totally nonnegative matrix can be found such that the perturbed matrix remains totally nonnegative also the total nonnegativity of the first and second subdirect sum of two totally nonnegative matrices is considered

higher order substitutions the calculus is a concrete calculus of explicit substitutions designed for reasoning about implementations of calculi higher order abstract syntax is an approach to metaprogramming that explicitly captures the variable binding aspects of programming language constructs a new calculus of explicit substitutions for higher order abstract syntax is introduced allowing a high level description of variable binding in object languages while also providing substitutions as explicit programmer manipulable data objects the new calculus is termed the s0 calculus since it makes essential use of an extension of s0 unification described in another paper termination and confluence are verified for the s0 calculus similarly to that for the calculus and an efficient implementation is given in terms of first order renaming substitutions the verification of confluence makes use of a verified adaptation of nipkow s higher order critical pairs lemma to the forms of rewrite rules required for the statement of the s0 calculus

polymorphic rewriting conserves algebraic confluence we study combinations of many sorted algebraic term rewriting systems and polymorphic lambda term rewriting algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus as higher order constants we show that if a many sorted algebraic rewrite system r has the church rosser property is confluent then r s type s type rewriting of mixed terms has the church rosser property too reduction does not commute with algebraic reduction in general however using long normal forms we show that if r is canonical confluent and strongly normalizing then equational provability from r s type s type is still decidable

fast inversion of chebyshev vandermonde matrices this paper contains two fast algorithms for inversion of chebyshev vandermonde matrices of the first and second kind they are based on special representations of the bezoutians of chebyshev polynomials of both kinds the paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster but also more stable than other algorithms available it is also efficient to use the above two algorithms for solving chebyshev vandermode systems of equations with preprocessing

the totally positive completion problem abstract an n n real matrix is said to be totally positive if every minor is non negative in this paper we are interested in totally positive completion problems that is does a partial totally positive matrix have a totally positive matrix completion this problem has in general a negative answer therefore we analyze the question for which labeled graphs g does every partial totally positive matrix whose associated graph is g have a totally positive completion when g is a path or a cycle we give necessary and sufficient conditions that guarantee the existence of the desired completion

the lambda context calculus extended version we present the lambda context calculus this simple lambda calculus features variables arranged in a hierarchy of strengths such that substitution of a strong variable does not avoid capture with respect to abstraction by a weaker variable this allows the calculus to express both capture avoiding and capturing substitution instantiation the reduction rules extend the vanilla lambda calculus in a simple and modular way and preserve the look and feel of a standard lambda calculus with explicit substitutions good properties of the lambda calculus are preserved the lamcc is confluent and a natural injection into the lamcc of the untyped lambda calculus exists and preserves strong normalisation we discuss the calculus and its design with full proofs in the presence of the hierarchy of variables functional binding splits into a functional abstraction l lambda and a name binder new we investigate how the components of this calculus interact with each other and with the reduction rules with examples in two more extended case studies we demonstrate how global state can be expressed and how contexts and contextual equivalence can be naturally internalised using function application

intersection type assignment systems with higher order algebraic rewriting abstract general computational models obtained by integrating different specific models have recently become a stimulating and promising research subject in particular extensions of various calculi with algebraic rewriting either typed or untyped have been deeply investigated in the present paper this subject is addressed from the point of view of type assignment a powerful type assignment system the one based on intersection types is extended with first and higher order algebraic rewriting and relevant properties like strong normalization and completeness are proved

intervals of almost totally positive matrices we consider the class of the totally nonnegative matrices i e the matrices having all their minors nonnegative and intervals of matrices with respect to the chequerboard partial ordering which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum for these intervals in 1982 we stated in this journal the following conjecture if the left and right endpoints of an interval are nonsingular and totally nonnegative then all matrices taken from the interval are nonsingular and totally nonnegative in this paper we show that this conjecture is true if we restrict ourselves to the subset of the almost totally positive matrices

poisson structures on affine spaces and flag varieties ii the standard poisson structures on the flag varieties g p of a complex reductive algebraic group g are investigated it is shown that the orbits of symplectic leaves in g p under a fixed maximal torus of g are smooth irreducible locally closed subvarieties of g p isomorphic to intersections of dual schubert cells in the full flag variety g b of g and their zariski closures are explicitly computed two different proofs of the former result are presented the first is in the framework of poisson homogeneous spaces and the second one uses an idea of weak splittings of surjective poisson submersions based on the notion of poisson dirac submanifolds for a parabolic subgroup p with abelian unipotent radical in which case g p is a hermitian symmetric space of compact type it is shown that all orbits of the standard levi factor l of p on g p are complete poisson subvarieties which are quotients of l equipped with the standard poisson structure moreover it is proved that the poisson structure on g p vanishes at all special base points for the l orbits on g p constructed by richardson rohrle and steinberg

a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type in this note a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type is introduced the operator representation relative to a pair of dual bases and the generalized barnett type factorization formula are derived an intertwining relation and a bezoutian reduction to a block diagonal form by congruence via a generalized vandermonde matrix are presented fujiwara hermite and routh hurwitz criteria in terms of this generalized bezout matrix are obtained

p matrix completions under weak symmetry assumptions an n by n matrix is called a matrix if it is one of weakly sign symmetric positive nonnegative p matrix weakly sign symmetric positive nonnegative p0 1 matrix or fischer or koteljanskii matrix in this paper we are interested in matrix completion problems that is when a partial matrix has a matrix completion here we prove that a combinatorially symmetric partial positive p matrix has a positive p matrix completion if the graph of its specified entries is an n cycle in general a combinatorially symmetric partial matrix has a matrix completion if the graph of its specified entries is a 1 chordal graph this condition is also necessary for weakly sign symmetric p0 or p0 1 matrices

bernstein bezoutian matrices several computational and structural properties of bezoutian matrices expressed with respect to the bernstein polynomial basis are shown the exploitation of such properties allows the design of fast algorithms for the solution of bernstein bezoutian linear systems without never making use of potentially ill conditioned reductions to the monomial basis in particular we devise an algorithm for the computation of the greatest common divisor gcd of two polynomials in bernstein form a series of numerical tests are reported and discussed which indicate that bernstein bezoutian matrices are much less sensitive to perturbations of the coefficients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the bernstein and the power basis

notions of computation and monads abstract the calculus is considered a useful mathematical tool in the study of programming languages since programs can be identified with terms however if one goes further and uses conversion to prove equivalence of programs then a gross simplification is introduced programs are identified with total functions from values to values that may jeopardise the applicability of theoretical results in this paper we introduce calculi based on a categorical semantics for computations that provide a correct basis for proving equivalence of programs for a wide range of notions of computation

further applications of the cauchon algorithm to rank determination and bidiagonal factorization abstract for a class of matrices connected with cauchon diagrams cauchon matrices and the cauchon algorithm a method for determining the rank and for checking a set of consecutive row or column vectors for linear independence is presented cauchon diagrams are also linked to the elementary bidiagonal factorization of a matrix and to certain types of rank conditions associated with submatrices called descending rank conditions

higher order rewrite systems and their confluence abstract we study higher order rewrite systems hrss which extend term rewriting to terms hrss can describe computations over terms with bound variables we show that rewriting with hrss is closely related to undirected equational reasoning we define pattern rewrite systems prss as a special case of hrss and extend three confluence results from term rewriting to prss the critical pair lemma by knuth and bendix confluence of rewriting modulo equations a la huet and confluence of orthogonal prss

nominal rewriting nominal rewriting is based on the observation that if we add support for a equivalence to first order syntax using the nominal set approach then systems with binding including higher order reduction schemes such as l calculus beta reduction can be smoothly represented nominal rewriting maintains a strict distinction between variables of the object language atoms and of the meta language variables or unknowns atoms may be bound by a special abstraction operation but variables cannot be bound giving the framework a pronounced first order character since substitution of terms for variables is not capture avoiding we show how good properties of first order rewriting survive the extension by giving an efficient rewriting algorithm a critical pair lemma and a confluence theorem for orthogonal systems

a quantum analogue of the dihedral action on grassmannians in recent work launois and lenagan have shown how to construct a cocycle twisting of the quantum grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the n cycle c 1 2 cdots n up to a power of q this twisting is needed because c does not naturally induce an automorphism of the quantum grassmannian as it does classically and semi classically we extend this construction to give a quantum analogue of the action on the grassmannian of the dihedral subgroup of s n generated by c and w 0 the longest element and this analogue takes the form of a groupoid we show that there is an induced action of this subgroup on the torus invariant prime ideals of the quantum grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive grassmannians then we see that this dihedral subgroup action exists classically semi classically by poisson automorphisms and anti automorphisms a result of yakimov and in the quantum and nonnegative settings

some results about inverse positive matrices abstract a nonsingular real matrix a is said to be inverse positive if all the elements of its inverse are nonnegative this class of matrices contains the m matrices from which inherit some of their properties and applications especially in economy and in the description of iterative methods for solving nonlinear systems in this paper we present some new characterizations for inverse positive matrices and we analyze when this concept is preserved by the sub direct sum of matrices

subdirect sums of p p0 matrices and totally nonnegative matrices abstract in this paper the problem of when the sub direct sum of two strictly diagonally dominant p matrices is a strictly diagonally dominant p matrix is studied in particular it is shown that the subdirect sum of overlapping principal submatrices of strictly diagonally dominant p matrices is a strictly diagonally dominant p matrix it is also established that the 2 subdirect sum of two totally nonnegative matrices is a totally nonnegative matrix under some conditions it is obtained that a partial totally nonnegative matrix whose graph of the specified entries is a monotonically labeled 2 chordal graph has a totally nonnegative completion finally a positive answer to the question iv in fallat and johnson shaun m fallat c r johnson j r torregrosa a m urbano p matrix completions under weak symmetry assumptions linear algebra appl 312 2000 73 91 is given for p 0 matrices

pivoting and backward stability of fast algorithms for solving cauchy linear equations abstract three fast o n2 algorithms for solving cauchy linear systems of equations are proposed a rounding error analysis indicates that the backward stability of these new cauchy solvers is similar to that of gaussian elimination thus suggesting to employ various pivoting techniques to achieve a favorable backward stability it is shown that cauchy structure allows one to achieve in o n2 operations partial pivoting ordering of the rows and several other judicious orderings in advance without actually performing the elimination the analysis also shows that for the important class of totally positive cauchy matrices it is advantageous to avoid pivoting which yields a remarkable backward stability of the suggested algorithms it is shown that vandermonde and chebyshev vandermonde matrices can be efficiently transformed into cauchy matrices using discrete fourier cosine or sine transforms this allows us to use the proposed algorithms for cauchy matrices for rapid and accurate solution of vandermonde and chebyshev vandermonde linear systems the analytical results are illustrated by computed examples

a typed context calculus this paper develops a typed calculus for contexts i e lambda terms with holes in addition to ordinary lambda terms the calculus contains labeled holes hole abstraction and context application for manipulating first class contexts the primary operation for contexts is hole filling which captures free variables this operation conflicts with substitution of the lambda calculus and a straightforward mixture of the two results in an inconsistent system we solve this problem by defining a type system that precisely specifies the variable capturing nature of contexts and that keeps track of bound variable renaming these mechanisms enable us to define a reduction system that properly integratesb reduction and hole filling the resulting calculus is church rosser and the type system has the subject reduction property we believe that the context calculus will serve as a basis for developing a programming language with advanced features that call for manipulation of open terms copyright 2001 elsevier science b v

the completion problem for m matrices and inverse m matrices abstract the completion problems for m matrices and inverse m matrices are discussed the former has a simple solution but is not at all analogous to the positive definite completion problem the latter is more subtle and is analogous to the positive definite problem a graph theoretic completion result as well as a determinant maximizing completion that is characterized by having a zero entry in the inverse in every position in which the partial matrix has an unspecified entry is given for the inverse m matrix problem these mirror to some extent the positive definite case but the methodology is necessarily different

eigenvector computation for almost unitary hessenberg matrices and inversion of szegő vandermonde matrices via discrete transmission lines abstract in this paper we use a discrete transmission line model known to geophysicists as a layered earth model to derive several computationally efficient solutions for the following three problems i as is well known a hessenberg matrix capturing recurrence relations for szeg polynomials differs from unitary only by its last column hence the first problem is how to rapidly evaluate the eigenvectors of this almost unitary hessenberg matrix ii the second problem is to design a fast o n 2 algorithm for inversion of szeg vandermonde matrices generalizing the well known traub algorithm for inversion of the usual vandermonde matrices iii finally the third problem is to extend the well known horner rule to evaluate a polynomial represented in the basis of szeg polynomials as we shall see all three problems are closely related and their solutions can be computed by the same family of fast algorithms although all the results can be derived algebraically here we reveal a connection to system theory to deduce these algorithms via elementary operations on signal flow graphs for digital filter structures including the celebrated markel gray filter widely used in speech processing and certain other filter structures this choice not only clarifies the derivation and suggests a variety of possible computational schemes but it also makes an interesting connection to many other results related to szeg polynomials which have already been interpreted via signal flow graphs for generalized lattice filter structures including the formulas of the gohberg semencul type for inversion of toeplitz like matrices schur type and levinson type algorithms etc for example this connection allows us to show that moment matrices corresponding to what we called horner szeg polynomials though not toeplitz are quasi toeplitz i e they have a certain shift invariance property