the effects of rounding errors in the nodes on barycentric interpolation we analyze the effects of rounding errors in the nodes on polynomial barycentric interpolation these errors are particularly relevant for the first barycentric formula with the chebyshev points of the second kind here we propose a method for reducing them

the horn problem and planar networks abstract the problem of determining the set of possible eigenvalues of 3 hermitian matrices that sum up to zero is known as the horn problem the answer is a polyhedral cone which following knutson and tao can be described as the projection of a simpler cone in the space of triangular tableaux or hives to the boundary nodes of the tableau in this paper we introduce a combinatorial problem defined in terms of certain weighted planar graphs giving rise to exactly the same polyhedral cone in our framework the values at the inner nodes of the triangular tableaux receive a natural interpretation other problems of linear algebra fit into the same scheme among them the gelfand zeitlin problem our approach is motivated by the works of fomin and zelevinsky on total positivity and by the ideas of tropicalization

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

geometric invariant theory and the generalized eigenvalue problem let g be a connected reductive subgroup of a complex connected reductive group hat g fix maximal tori and borel subgroups of g and hat g consider the cone mathcal lr g hat g generated by the pairs nu hat nu of dominant characters such that v nu is a submodule of v hat nu with usual notation here we give a minimal set of inequalities describing mathcal lr g hat g as a part of the dominant chamber in other words we describe the facets of mathcal lr g hat g which intersect the interior of the dominant chamber we also describe smaller faces finally we are interested in some classical redundant inequalities

a study of saturated tensor cone for symmetrizable kac moody algebras let mathfrak g be a symmetrizable kac moody lie algebra with the standard cartan subalgebra mathfrak h and the weyl group w let p be the set of dominant integral weights for lambda in p let l lambda be the integrable highest weight irreducible representation of mathfrak g with highest weight lambda for a positive integer s define the saturated tensor semigroup as begin aligned gamma s lambda 1 dots lambda s mu in p s 1 exists n ge 1 text with l n mu subset l n lambda 1 otimes dots otimes l n lambda s end aligned the aim of this paper is to begin a systematic study of gamma s in the infinite dimensional symmetrizable kac moody case in this paper we produce a set of necessary inequalities satisfied by gamma s these inequalities are indexed by products in h g mathrm min b mathbb z for b the standard borel subgroup where g mathrm min is the minimal kac moody group with lie algebra mathfrak g the proof relies on the kac moody analogue of the borel weil theorem and geometric invariant theory specifically the hilbert mumford index in the case that mathfrak g is affine of rank 2 we show that these inequalities are necessary and sufficient we further prove that any integer d 0 is a saturation factor for a 1 1 and 4 is a saturation factor for a 2 2

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

degenerate two phase incompressible flow iii sharp error estimates this is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic parabolic partial differential system which describes the flow of two incompressible immiscible fluids in porous media in this paper we consider a finite element approximation for this system the elliptic equation for the pressure and velocity is approximated by a mixed finite element method while the degenerate parabolic equation for the saturation is approximated by a galerkin finite element method a fully discrete approximation is analyzed sharp error estimates in energy norms are obtained for this approximation the error analysis does not use any regularization of the saturation equation the error estimates are derived directly from the degenerate equation also the analysis does not impose any restriction on the nature of degeneracy finally it respects the minimal regularity on the solution of the differential system

git cones and quivers in this work we improve results of ressayre in geometric invariant theory and generalized eigenvalue problem ii pp 1 25 2008 ressayre in ann inst fourier 180 389 441 2010 on git cones associated to the action of a reductive group g on a projective variety x these results are applied to give a short proof of the derksen weyman theorem that parametrizes bijectively the faces of a rational cone associated to any quiver without oriented cycles an important example of such a cone is the horn cone

a generalization of fulton s conjecture for arbitrary groups we prove a generalization of fulton s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory

the stability of barycentric interpolation at the chebyshev points of the second kind we present a new analysis of the stability of the first and second barycentric formulae for interpolation at the chebyshev points of the second kind our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice we also extend our current understanding regarding the accuracy problems of the first barycentric formula

on the numerical stability of newton s formula for lagrange interpolation abstract it was previously noticed in the literature that the computation of newton s formula can be very sensitive to the ordering of the input data and it may be numerically unstable even for interpolation at the chebyshev nodes nevertheless to the best of our knowledge the effects of rounding errors in the computation of newton s formula have not been properly quantized yet in this note we make a full stability analysis of the propagation of rounding errors in the computation of newton s formula for lagrange interpolation we derive sharp upper bounds for the numerical error in the computations of three algorithms for computing polynomial interpolants in newton form that can be used to understand the instabilities mentioned above we throw light on and give a practical meaning to the condition number defined in carnicer et al 2019 by showing that it is the right thermometer for measuring the effects of rounding errors in the computation of newton s formula

degenerate two phase incompressible flow i existence uniqueness and regularity of a weak solution abstract this is the first paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic parabolic partial differential system which describes the flow of two incompressible immiscible fluids in porous media in this paper we first show that this system possesses a weak solution under physically reasonable hypotheses on the data then we prove that this weak solution is unique finally we establish regularity on the weak solution which is needed in the uniqueness proof in particular the holder continuity of the saturation in space and time and the lipschitz continuity of the pressure in space are obtained

root theoretic young diagrams and schubert calculus planarity and the adjoint varieties abstract we study root theoretic young diagrams to investigate the existence of a lie type uniform and nonnegative combinatorial rule for schubert calculus we provide formulas for co adjoint varieties of classical lie type this is a simplest case after the co minuscule family where a rule has been proved by h thomas and the second author using work of r proctor our results build on earlier pieri type rules of p pragacz j ratajski and of a buch a kresch h tamvakis specifically our formula for og 2 2 n is the first complete rule for a case where diagrams are non planar yet the formulas possess both uniform and non uniform features using these classical type rules as well as results of p e chaput n perrin in the exceptional types we suggest a connection between polytopality of the set of nonzero schubert structure constants and planarity of the diagrams this is an addition to work of a klyachko and a knutson t tao on the grassmannian and of k purbhoo f sottile on cominuscule varieties where the diagrams are always planar

a combinatorial rule for co minuscule schubert calculus we prove a root system uniform concise combinatorial rule for schubert calculus of minuscule and cominuscule flag manifolds g p the latter are also known as compact hermitian symmetric spaces we connect this geometry to the poset combinatorics of proctor thereby giving a generalization of schutzenberger s jeu de taquin formulation of the littlewood richardson rule that computes the intersection numbers of grassmannian schubert varieties our proof introduces cominuscule recursions a general technique to relate the numbers for different lie types

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

the stability of extended floater hormann interpolants we present a new analysis of the stability of extended floater hormann interpolants in which both noisy data and rounding errors are considered contrary to what is claimed in the current literature we show that the lebesgue constant of these interpolants can grow exponentially with the parameters that define them and we emphasize the importance of using the proper interpretation of the lebesgue constant in order to estimate correctly the effects of noise and rounding errors we also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants our experiments show that extended interpolants mentioned in the literature satisfy this condition and therefore the formula used to implement them is not backward stable finally we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation

the recursive nature of cominuscule schubert calculus the necessary and sufficient horn inequalities which determine the non vanishing littlewood richardson coefficients in the cohomology of a grassmannian are recursive in that they are naturally indexed by non vanishing littlewood richardson coefficients on smaller grassmannians we show how non vanishing in the schubert calculus for cominuscule flag varieties is similarly recursive for these varieties the non vanishing of products of schubert classes is controlled by the non vanishing products on smaller cominuscule flag varieties in particular we show that the lists of schubert classes whose product is non zero naturally correspond to the integer points in the feasibility polytope which is defined by inequalities coming from non vanishing products of schubert classes on smaller cominuscule flag varieties while the grassmannian is cominuscule our necessary and sufficient inequalities are different than the classical horn inequalities

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

the existence of weak solutions to single porosity and simple dual porosity models of two phase incompressible flow abstract it is shown that there exists a weak solution to a degenerate and singular elliptic parabolic partial integro differential system of equations these equations model two phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium the full model is of dual porosity type though the single porosity case is covered by setting the matrixto fracture flow terms to zero this matrix to fracture flow is modeled simply in terms of fracture quantities that is no distinct matrix equations arise the equations are considered in a global pressure formulation that is justified by appealing to a physical relation between the degeneracy of the wetting fluid s mobility and the singularity of the capillary pressure function in this formulation the elliptic and parabolic parts of the system are separated hence it is natural to consider various boundary conditions including mixed nonhomogeneous saturation dependent ones of the first three types a weak solution is obtained as a limit of solutions to discrete time problems the proof makes no use of the corresponding regularized system the hypotheses required for various earlier results on single porosity systems are weakened so that only physically relevant assumptions are made in particular the results cover the cases of a singular capillary pressure function a pure neumann boundary condition and an arbitrary initial condition

on the lebesgue constant of berrut s rational interpolant at equidistant nodes it is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting in this paper we confirm this observation by proving that the lebesgue constant of berrut s rational interpolant grows only logarithmically in the number of interpolation nodes moreover the numerical results suggest that the lebesgue constant behaves similarly for interpolation at chebyshev as well as logarithmically distributed nodes

a generalization of cachazo douglas seiberg witten conjecture for symmetric spaces we extend the original cachazo douglas seiberg witten conjecture on the structure of the chiral ring of classical supersymmetric yang mills theory to symmetric spaces

extremal rays in the hermitian eigenvalue problem the hermitian eigenvalue problem asks for the possible eigenvalues of a sum of n times n hermitian matrices given the eigenvalues of the summands the regular faces of the cones gamma n s controlling this problem have been characterized in terms of classical schubert calculus by the work of several authors we determine extremal rays of gamma n s which are never regular faces by relating them to the geometry of flag varieties the extremal rays either arise from modular intersection loci or by induction from extremal rays of smaller groups explicit formulas are given for both the extremal rays coming from such intersection loci and for the induction maps

on the cachazo douglas seiberg witten conjecture for simple lie algebras we prove a part of the cachazo douglas seiberg witten conjecture uniformly for any simple lie algebra the main ingredients in the proof are garland s result on the lie algebra cohomology of kostant s result on the diagonal cohomolgy of and its connection with abelian ideals in a borel subalgebra of and a certain deformation of the singular cohomology of the infinite grassmannian introduced by belkale kumar

littlewood richardson coefficients for reflection groups abstract in this paper we explicitly compute all littlewood richardson coefficients for semisimple and kac moody groups g that is the structure constants also known as the schubert structure constants of the cohomology algebra h g p c where p is a parabolic subgroup of g these coefficients are of importance in enumerative geometry algebraic combinatorics and representation theory our formula for the littlewood richardson coefficients is purely combinatorial and is given in terms of the cartan matrix and the weyl group of g however if some off diagonal entries of the cartan matrix are 0 or 1 the formula may contain negative summands on the other hand if the cartan matrix satisfies a i j a j i 4 for all i j then each summand in our formula is nonnegative that implies nonnegativity of all littlewood richardson coefficients we extend this and other results to the structure coefficients of the t equivariant cohomology of flag varieties g p and bott samelson varieties i g

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

meshfree approach for solving multi dimensional systems of fredholm integral equations via barycentric lagrange interpolation abstract a highly accurate meshfree approach based on the barycentric lagrange basis functions is reported for solving the linear and nonlinear multi dimensional systems of fredholm integral equations fies of the second kind the method is an improved lagrange interpolation technique which possesses high precision and inexpensive procedure the systems of fies are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations the stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric lagrange formula which is dominated by barycentric weights the convergence analysis and error estimation are included some computational results indicate that the method is simple and efficient

frontiers of reality in schubert calculus the theorem of mukhin tarasov and varchenko formerly the shapiro conjecture for grassmannians asserts that all a priori complex solutions to certain geometric problems in the schubert calculus are actually real their proof is quite remarkable using ideas from integrable systems fuchsian differential equations and representation theory there is now a second proof of this result and it has ramifica tions in other areas of mathematics from curves to control theory to combinatorics despite this work the original shapiro conjecture is not yet settled while it is false as stated it has several interesting and not quite understood modifications and gen eralizations that are likely true and the strongest and most subtle version of the shapiro conjecture for grassmannians remains open

flow of oil and water in a porous medium abstract the flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations in this paper the existence of a weak solution by regularization is shown

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat