combinatorial point for fused loop models integrable loop models associated with higher representations spin l 2 of u q mathfrak sl 2 are investigated at the point q hbox e pm i pi ell 2 the ground state eigenvalue and eigenvectors are described introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries

hermite biehler routh hurwitz and total positivity simple proofs of the hermite biehler and routh hurwitz theorems are presented the total nonnegativity of the hurwitz matrix of a stable real polynomial follows as an immediate corollary

on the variety of projectors in this paper we are going to give an answer to the following two questions let k be a field and a4 the space of n x n square matrices with coefficients in k the first question we want to investigate is what are the equations of the subvariety a of ikf consisting of rank h matrices a such that a2 a by equations we mean of course a set of explicit generators of the prime ideal of x the second question is what are the equations of the varieties yh h n 2 which are the closures of the nilpotent conjugacy classes in h4 e consisting of matrices of rank h whose square is the matrix 0 in this case we shall also prove that these varieties y are normal and cohen macaulay in arbitrary characteristic thus generalizing a result of hesselink 3 to arbitrary characteristic note that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and cohen macaulay by kraft and procesi 4 but our method is not suitable for generalizing their result the method used in this paper is the one of hodge algebras 11 namely we shall exhibit an explicit linear basis for the coordinate rings of the varieties x and y

classical results on the stability of linear time invariant systems and the schwarz form the paper deals with the classical results of routh and hurwitz biehler and kharitonov concerning the stability of a linear time invariant differential equation it is shown that these results follow directly from a schwarz matrix representation of stable systems

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

spectral collocation method for solving fredholm integral equations on the half line the main purpose of this paper is to present a spectral collocation method to approximate the solution of fredholm integral equations of the second kind on the half line in this approach interpolation operator based on scaled laguerre functions is proposed also convergence theorems are established under suitable assumptions finally two numerical examples are presented and commented

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

a spectral method for the vorticity equation on the surface a spectral scheme is proposed for the vorticity equation defined on the spherical surface generalized stability and convergence are proved the approximation results in this paper are also useful for other nonlinear problems

a new generalized laguerre spectral approximation and its applications a new family of generalized laguerre polynomials is introduced various orthogonal projections are investigated some approximation results are established as an example of their important applications the mixed spherical harmonic generalized laguerre approximation is developed a mixed spectral scheme is proposed for a three dimensional model problem its convergence is proved numerical results demonstrate the high accuracy of this new spectral method

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

the routh hurwitz stability criterion revisited the case of multiple poles on imaginary axis this technical note presents a relation between the number of zero rows rows with all zero elements in the routh array and the multiplicity of j axis poles the main result of the technical note is that the existence of more than one zero row in the routh array amounts to instability of the system irrespective of any sign change in the first column

root distribution for the ellipse it is shown that the number of characteristic roots of a given matrix or polynomial inside outside and on the boundary of a given ellipse can be determined by a common half plane test e g routh test the test should be performed on a polynomial which is received from the original characteristic polynomial by a simple transformation root exclusion and root clustering criteria are included as special cases

the negative routh test and its application to the cases of vanishing leading elements and imaginary roots the ordinary routh test furnishes in a straightforward manner the number r of positive roots of a polynomial of degree n a modification that allows one to obtain the number r of negative roots in the same way is presented thus the number r o of purely imaginary roots is immediately found as n r r the modified test is especially useful when there is a first column vanishing element

linked by loops network structure and switch integration in complex dynamical systems abstract simple nonlinear dynamical systems with multiple stable stationary states are often taken as models for switchlike biological systems this paper considers the interaction of multiple such simple multistable systems when they are embedded together into a larger dynamical supersystem attention is focused on the network structure of the resulting set of coupled differential equations and the consequences of this structure on the propensity of the embedded switches to act independently versus cooperatively specifically it is argued that both larger average and larger variance of the node degree distribution lead to increased switch independence given the frequency of empirical observations of high variance degree distributions e g power law in biological networks it is suggested that the results presented here may aid in identifying switch integrating subnetworks as comparatively homogenous low degree substructures potential applications to ecological problems such as the relationship of stability and complexity are also briefly discussed

numerical integration based on laguerre gauss interpolation abstract in this paper we propose two efficient numerical integrators for ordinary differential equations based on modified laguerre gauss interpolations the global convergence of proposed algorithms is proved numerical results demonstrate the spectral accuracy of these new schemes and agree well with the theoretical analysis

complete root distribution with respect to parabolas and some results with respect to hyperbolas and sectors criteria are derived to determine the distribution of the characteristic roots of a byatem with respect to parabolas and hyperbolas in the complex plane which are related to certain physical properties of the system the criteria are given in terms of a classical half plane algorithm e g a routh array carried out on a transformed polynomial of double the degree of the original characteristic polynomial for parabolas left right or shifted the algorithm provides the complete root distribution including criteria for root clustering and root exclusion as bpecial cases for hyperbolas the algorithm provides in all cases criteria for root clustering inside the open or closed left or right branch of the hyperbola and for characteristic polynomials which are hurwitz the algorithm provides the complete root distribution with respect to the left branch of the hyperbola also by an additional classical test sturm the algorithm provides in all cases the exact number of roots on the left boundary of t

grobner geometry of vertex decompositions and of flagged tableaux we relate a classic algebro geometric degeneration technique dating at least to hodge 1941 to the notion of vertex decompositions of simplicial com plexes the good case is when the degeneration is reduced and we call this a geometric vertex decomposition our main example in this paper is the family of vexillary matrix schubert vari eties whose ideals are also known as one sided ladder determinantal ideals us ing a diagonal term order to specify the grobner degeneration we show that these have geometric vertex decompositions into simpler varieties of the same type from this together with the combinatorics of the pipe dreams of fomin kirillov 1996 we derive a new formula for the numerators of their multigraded hilbert series the double grothendieck polynomials in terms of flagged set valued tableaux this unifies work of wachs 1985 on flagged tableaux and buch 2002 on set valued tableaux giving geometric meaning to both this work focuses on diagonal term orders giving results complementary to those of knutson miller 2004 where it was shown that the generating minors form a grobner basis for any antidiagonal term order and any matrix schubert va riety we show here that under a diagonal term order the only matrix schubert varieties for which these minors form grobner bases are the vexillary ones reach ing an end toward which the ladder determinantal literature had been building contents

integration processes of ordinary differential equations based on laguerre radau interpolations in this paper we propose two integration processes for ordinary differential equations based on modified laguerre radau interpolations which are very efficient for long time numerical simulations of dynamical systems the global convergence of proposed algorithms are proved numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis

the spectral method for high order problems with proper simulations of asymptotic behaviors at infinity in this paper we investigate new spectral and multidomain spectral methods for high order problems we introduce a family of new generalized laguerre functions which are mutually orthogonal with the weight function x a d x c d 0 a and c being arbitrary real numbers the corresponding quasi orthogonal approximation and laguerre gauss radau type interpolation are proposed the spectral and multidomain spectral schemes are provided for several model problems which not only fit the mixed inhomogeneous boundary conditions on the fixed boundary exactly but also match the asymptotic behaviors at infinity reasonably numerical results demonstrate the efficiency of suggested algorithms and confirm the analysis well

generalized laguerre pseudospectral method based laguerre interpolation abstract in this paper we develop the generalized laguerre pseudospectral method for the second order ordinary differential equation based on the modified generalized laguerre approximation which is very efficient for the long time numerical simulations of dynamical systems the numerical solution possesses the spectral accuracy the global convergence of the proposed algorithm is proved numerical results demonstrate the effectiveness of the proposed method and coincide well with theoretical analysis

mixed laguerre legendre pseudospectral method for incompressible fluid flow in an infinite strip in this paper we investigate the mixed laguerre legendre interpolation approximation and its application some approximation results are established a mixed laguerre legendre pseudospectral scheme is constructed for incompressible fluid flow in an infinite strip its stability and convergence are proved numerical results show the efficiency of this new approach

elementary proof of the routh hurwitz test abstract this note presents an elementary proof of the familiar routh hurwitz test the proof is basically one continuity argument it does not rely on sturm chains cauchy index and the principle of the argument and it is fully self contained in the same style an extended routh hurwitz test is derived which finds the inertia of polynomials

root distribution of a polynomial in subregions of complex plane modified processes are proposed for directly treating singularities in the routh array and procedures are developed for determining the respective orders of simple and or repeated roots lying on the imaginary axis the extended sturm test is developed for determining the root distribution on some specified lines moreover to enhance the routh stability theorem the extended routh theorem for finding the numbers of roots of a polynomial lying within some specified regions is proposed

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

jacobi and laguerre quasi orthogonal approximations and related interpolations in this paper we investigate jacobi quasi orthogonal approximation and generalized jacobi gauss lobatto interpolation we also propose laguerre quasi orthogonal approximation and generalized laguerre gauss radau interpolation a series of sharp results on these approximations are established which are applicable to spectral and pseudospectral methods for mixed nonhomogeneous boundary value problems of high order

a generalized laguerre fourier hermite pseudospectral method for computing the dynamics of rotating bose einstein condensates a time splitting generalized laguerre fourier hermite pseudospectral method is proposed for computing the dynamics of rotating bose einstein condensates becs in two and three dimensions the new numerical method is based on the following i the use of a time splitting technique for decoupling the nonlinearity ii the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient and iii the construction of eigenfunctions for the linear operator by properly scaling the generalized laguerre fourier and hermite functions the new method enjoys the following properties i it is explicit time reversible and time transverse invariant ii it conserves the position density and is spectrally accurate in space and second order or fourth order accurate in time and iii it solves the problem in the original whole space instead of in a truncated artificial computational domain the method is also extend

mixed spectral method for three dimensional exterior problems using spherical harmonic and generalized laguerre functions in this paper we develop the mixed spectral method for three dimensional exterior problems using spherical harmonic and generalized laguerre functions some basic approximation results are established the mixed spectral schemes are proposed for two model problems their convergences are proved numerical results demonstrate the efficiency of this new approach

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

the laguerre spectral method for solving neumann boundary value problems in this paper we propose a laguerre spectral method for solving neumann boundary value problems this approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly moreover a tridiagonal matrix is employed instead of the full stiffness matrix encountered in the classical variational formulation of such problems for analyzing the numerical errors some basic results on laguerre approximations are established the convergence is proved the numerical results demonstrate the efficiency of this approach

inverse routh table construction and stability of delay equations abstract based on an inversion of the routh table construction a unimodular characterization of all hurwitz polynomials is obtained in parameter space the hurwitz polynomials of degree n correspond to the positive 2 n tant the method is then used to construct classes of stable continuous time delay difference equations and delay differential equations of neutral type by a suitable limiting process

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

generalized laguerre interpolation and pseudospectral method for unbounded domains in this paper error estimates for generalized laguerre gauss type interpolations are derived in nonuniformly weighted sobolev spaces weighted with omega alpha beta x x alpha e beta x alpha 1 beta 0 generalized laguerre pseudospectral methods are analyzed and implemented two model problems are considered the proposed schemes keep spectral accuracy and with suitable choice of basis functions lead to sparse and symmetric linear systems

a scheme related to the brauer loop model abstract we introduce the brauer loop scheme e m m n c m m 0 where is a certain degeneration of the ordinary matrix product its components of top dimension n 2 2 correspond to involutions s n having one or no fixed points in the case n even this scheme contains the upper upper scheme from a knutson some schemes related to the commuting variety j algebraic geom in press math ag 0306275 as a union of n 2 of its components one of those is a degeneration of the commuting variety of pairs of commuting matrices the brauer loop model is an integrable stochastic process studied in j de gier b nienhuis brauer loops and the commuting variety j stat mech 2005 p01006 math ag 0410392 based on earlier related work in m j martins b nienhuis r rietman an intersecting loop model as a solvable super spin chain phys rev lett 81 1998 504 507 cond mat 9709051 and some of the entries of its perron frobenius eigenvector were observed conjecturally to equal the degrees of the components of the upper upper scheme our proof of this equality follows the program outlined in p di francesco p zinn justin inhomogeneous model of crossing loops and multidegrees of some algebraic varieties math ph 0412031 in that paper the entries of the perron frobenius eigenvector were generalized from numbers to polynomials which allowed them to be calculated inductively using divided difference operators we relate these polynomials to the multi degrees of the components of the brauer loop scheme defined using an evident torus action on e as a consequence we obtain a formula for the degree of the commuting variety previously calculated up to 4 4 matrices

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

a general theory for matrix root clustering in subregions of the complex plane we consider the general problem of root clustering of a matrix in the complex plane let a in c n times n and s subset c find the largest class of s and an algebraic criterion which is necessary and sufficient for lambda i a in s i 1 2 n we introduce two types of regions which constitute the largest class of s known to date the criterion is presented both for open regions and closed ones the results are used to define a design methodology for control systems moreover all classical results are shown to be special cases of the present theory

b orbits in solutions to the equation x2 0 in triangular matrices abstract consider in k the set of solutions to the equation x2 0 in n n strictly upper triangular matrices over field k in this paper we construct the bijection from the set of adjoint orbits in in k onto the set of involutions in symmetric group sn for a finite field k we compute the number of points in the orbits

extended table for eliminating the singularities in routh s array a simple procedure for eliminating the classical epsilon method introduced by the routh hurwitz test is given the procedure extends the construction of the routh table in the special case of vanishing leading array elements without introducing complementary multiplications it is shown that it is always possible to associate with any polynomial g an extended routh table the number of sign changes in the first column of the table equals the number of zeros of g with positive real part

addendum to routh hurwitz test under vanishing leading array elements this note serves to supplement the modified routh array construction procedure presented in the paper for the case that entire rows of zero elements are embedded in rows with some leading zeros

inhomogenous model of crossing loops and multidegrees of some algebraic varieties we consider a quantum integrable inhomogeneous model based on the brauer algebra b 1 and discuss the properties of its ground state eigenvector in particular we derive various sum rules and show how some of its entries are related to multidegrees of algebraic varieties