a trace formula for canonical differential expressions we prove a trace formula for pairs of self adjoint operators associated to canonical differential expressions an important role is played by the associated weyl function

on the subgroup structure of classical groups in this paper we establish a result on the subgroup structure of classical groups over algebraically closed elds and use this to give a new proof of a fundamental theorem of m aschbacher on subgroups of nite classical groups let v be a nite dimensional vector space over an algebraically closed eld k and let g be one of the classical algebraic groups sl v sp v or so v our result is a reduction theorem concerning the subgroups of g we de ne a certain collection c of natural proper subgroups of g and prove that any closed nite or in nite subgroup of g either lies in a member of c or is roughly speaking a simple group acting irreducibly on v aschbacher s result is an analogous reduction theorem for subgroups of nite classical groups we obtain this as a relatively easy consequence of our main result by taking xed points under the action of a frobenius morphism using a standard process involving lang s theorem the proof of the main result uses elementary linear algebra together with a few basic facts from the theory of algebraic groups various complications which arise in the nite group setting in as become much more straightforward in the algebraic group setting in particular questions involving extension elds do not occur and issues of conjugacy are easily settled when we descend to nite invent math 134 427 453 1998

on the cohen macaulay and buchsbaum property for unions of planes in affine space the coordinate ring a of a union of planes in affine space is studied and it is asked when a is a cohen macaulay or buchsbaum ring these properties are related to the position of the planes via the notion of seminormality it is shown that a is cohen macaulay iff a is connected in codimension 2 and seminormal in an appropriate sense consideration of the cohen macaulification then yields a simple criterion for a to be buchsbaum methods with several examples are illustrated

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

applications of homological mirror symmetry to hypergeometric systems duality conjectures abstract homological mirror symmetry for crepant resolutions of gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of pdes we explain these conjectures and verify them in some cases

adaptive thinning algorithm of nonsingular triangulation this paper studies adaptive thinning strategies for the non singular triangulation of scattered data by c 1 rational spline function given a set of points in r 2 luo liu and chen have presented a triangulation algorithm which ensures the non singularity of s 2 1 and s 3 1 spaces in this paper we improve the algorithm to reduce the number of knots of the triangulation within a given tolerance while the non singularity of s 2 1 and s 3 1 spaces is ensured our strategies presented here depend on both the locations of the data points in the plane and the data values at these points we give the definition of discrete norm for c 1 rational spline function by using its coefficients then a weight is assigned to each knot which is a measure of the importance of knot in the representation of spline when the weight of the knot is less than the given tolerance its influence is regarded negligible then it can be removed it s a discrete method in the end of this paper several numerical examples are presented to show the feasibility and validity of our algorithm

discrete skew selfadjoint canonical systems and the isotropic heisenberg magnet model abstract a discrete analog of a skew self adjoint canonical zakharov shabat or akns system with a pseudo exponential potential is introduced for the corresponding weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices as an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic heisenberg magnet model state space techniques from mathematical system theory play an important role in the proofs

on the spectral side of arthur s trace formula combinatorial setup in arthur s trace formula a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the coxeter fan g m families these include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side we introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes in a companion paper the results are used to study the spectral side of the trace formula

splines with boundary conditions this paper explores spaces of splines satisfying boundary conditions using the long exact sequence for relative homology we relate them to the boundary complex of d where d is a planar simplicial complex which triangulates a pseudomanifold in this case there is a natural relationship between the hilbert polynomials which measure the dimension of c k r d for k 0 of the spline modules

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

fat points inverse systems and piecewise polynomial functions abstract we explore the connection between ideals of fat points which correspond to subschemes of p nobtained by intersecting mixed powers of ideals of points and piecewise polynomial functions splines on ad dimensional simplicial complex embedded inrd using the inverse system approach introduced by macaulay 11 we give a complete characterization of the free resolutions possible for ideals ink x y generated by powers of homogeneous linear forms we allow the powers to differ we show how ideals generated by powers of homogeneous linear forms are related to the question of determining for some fixed the dimension of the vector space of splines on of degree less than or equal tok we use this relationship and the results above to derive a formula which gives the number of planar mixed splines in sufficiently high degree

algorithms for the quillen suslin theorem abstract we give an algorithm for computing a free basis of a projective c x1 xn module which is presented as the image kernel or cokernel of a polynomial matrix our method can be implemented using grobner bases and it provides an elementary constructive proof of the quillen suslin theorem

cayley bacharach theorem of piecewise algebraic curves the piecewise algebraic curve determined by a bivariate spline function is a generalization of the classical algebraic curve in this paper by using bezout s theorem and nother type theorem of piecewise algebraic curves the cayley bacharach theorem and hilbert function of c0 piecewise algebraic curves are presented

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

lagrange interpolation by bivariate splines on cross cut partitions the piecewise algebraic curve is a generalization of the classical algebraic curve in this paper we give the nother type theorem of piecewise algebraic curves on cross cut partitions by using interpolation along a piecewise algebraic curve we also present a method to construct lagrange interpolation sets for bivariate spline spaces on cross cut partitions

inverse spectral problem for differential operators with rational scattering matrix functions in this paper we study the inverse scattering problem for linear canonical differential equations of a special type in the case when the scattering function is a rational matrix valued function the main result here is the form of the potential in terms of a realisation of the scattering function the difference between this publication and the previous one 3 consists in the fact that here is used the marchenko method instead of kre n s method for rational scattering functions we also prove the equivalence of the two methods

exact solutions of nonlinear equations and the method of operator identities abstract exact solutions and wave functions of integrable nonlinear equations are constructed by the method of operator identities

an algorithm for constructing a basis for c r spline modules over polynomial rings let be a polyhedral complex embedded in the euclidean space e d and s r r 0 denote the set of all c r splines on then s r is an r module where r e x 1 x d is the ring of polynomials in several variables in this paper we state and prove the existence of an algorithm to write down a free basis for the above r module in terms of obvious linear forms defining common faces of members of this is done for the case when consists of a finite number of parallelopipeds properly joined amongst themselves along the above linear forms

counting positive solutions for polynomial systems with real coefficients based on the subresultant polynomial chain and the method presented by hongguang fu and lu yang et al we propose a practical algorithm for computing the number of sign changes of the leading coefficients and the constant terms of a generalized sturm sequence gss with the principal minors sequence of the discrimination matrix of the polynomial and the number of sign changes of the gss a formula determining the number of positive solutions of a given bivariate polynomial systems with real coefficients with a finite number of complex solutions and its algorithm are presented using the techniques of the b net form of bivariate splines function this algorithm can be used for computing the number of real intersection points of two piecewise algebraic curves and its distributions in all cells when common points are finite a piecewise algebraic curve is defined by a bivariate spline function

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

the correspondence between multivariate spline ideals and piecewise algebraic varieties as a piecewise polynomial with a certain smoothness the spline plays an important role in computational geometry the algebraic variety is the most important subject in classical algebraic geometry as the zero set of multivariate splines the piecewise algebraic variety is a generalization of the algebraic variety in this paper the correspondence between piecewise algebraic varieties and spline ideals is discussed furthermore hilbert s nullstellensatz for the piecewise algebraic variety is also studied

stability of the procedure of explicit recovery of skew selfadjoint dirac systems from rational weyl matrix functions abstract procedures to recover explicitly discrete and continuous skew selfadjoint dirac systems on semi axis from rational weyl matrix functions are considered their stability is shown some new facts on asymptotics of pseudo exponential potentials i e of explicit solutions of inverse problems are proved as well the gbdt version of backlund darboux transformation methods from system theory and results on algebraic riccati equations are used for this purpose

on the compatibility condition for linear systems and a factorization formula for wave functions the well known compatibility condition for linear systems wx gw and wt fw is considered and new results are obtained in this way a factorization formula for wave functions which is basic in the inverse spectral transform approach to initial boundary value problems is proved in greater generality than before some applications follow

the algebra of continuous piecewise polynomials abstract for a triangulated d dimensional region r d we consider the algebra c 0 of all continuous piecewise polynomial functions on we find generators for c 0 as an r algebra and use these to give an isomorphism between c 0 and a quotient of the face ring a of we then study the structure of c 0 as a module over r r y 1 y d the polynomial ring in d indeterminates giving generators for c 0 as an r module these form a free basis when is a shellable complex in general we show that c 0 is a free r module whenever is a disk

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

on structure of bivariate spline space of lower degree over arbitrary triangulation the structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property a new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces the upper bound of the dimension of spline spaces over triangulation given by l l schumaker is slightly improved via the new vertex coding method the structure of multivariate spline spaces s 2 1 and s 3 1 over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by luo and wang a kind of sufficient conditions on judging non singularity of the s 2 1 and s 3 1 spaces over arbitrary triangulation is given which only depends on the topological property of the triangulation from the sufficient conditions a triangulation strategy is presented at the end of the paper the strategy ensures that the constructed triangulation is non singular or generic for s 2 1 and s 3 1

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

pseudo canonical systems with rational weyl functions explicit formulas and applications abstract the paper extends earlier results of the authors for canonical systems with spectral functions of which the absolutely continuous part has a rational derivative to a class of differential systems with skew selfadjoint potentials the corresponding direct and inverse spectral problems are solved explicitly using state space methods from mathematical system theory applications to nonlinear integrable partial differential equations are included

scattering for general type dirac systems on the semi axis reflection coefficients and weyl functions abstract we show that for general type self adjoint and skew self adjoint dirac systems on the semi axis weyl functions are unique analytic extensions of the reflection coefficients new results on the extension of the weyl functions to the real axis and on the existence in the skew self adjoint case of the weyl functions follow important procedures to recover general type dirac systems from the weyl functions are applied to the recovery of dirac systems from the reflection coefficients we explicitly recover dirac systems from the rational reflection coefficients as well

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

the number of homomorphisms from finite groups to classical groups we provide upper and lower bounds for the number of completely reducible homomorphisms from a finite group to general linear and unitary groups over arbitrary finite fields and to orthogonal and symplectic groups over finite fields of odd characteristic