the bunch kaufman factorization of symmetric matrices signature similar to sign regular matrices abstract the bunch kaufman pivoting strategy is a most commonly used method in practice to factor symmetric indefinite matrices however this method in general may destroy the band structure of banded matrices and its growth factor bound of 2 57 n 1 can almost be attained in this paper we investigate the behavior of the method for factorizing symmetric indefinite matrices that are signature similar to sign regular matrices with signature the growth factor bound depending on flips in the signature is derived for factorizing such banded matrices if the bunch kaufman pivoting strategy is modified as suggested by sorensen and van loan then not only do the same growth factor bounds hold but also the bandwidth is nicely preserved

determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

on continuous dependence for an inverse initial boundary value problem for the wave equation abstract we show that the bounded coefficient q x of the wave equation u tt u qu x t 0 t u x 0 u t x 0 0 x u v x x x t 0 t can be continuously determined by its neumann to dirichlet map in certain function spaces

gaussian elimination is stable for the inverse of a diagonally dominant matrix let b e m n c be a row diagonally dominant matrix i e i b ii b ij i 1 n where 0 i 1 i 1 n with a max 1 i n i we show that no pivoting is necessary when gaussian elimination is applied to a b 1 moreover the growth factor for a does not exceed 1 the same results are true with row diagonal dominance being replaced by column diagonal dominance

three dimensional earth conductivity inversion abstract we have constructed two inverse algorithms using tikhonov regularization one follows the well known method of solving the linearized data model relation subject to some regularizing operator which forces the solution to satisfy some desirable features we refer to the other more novel approach as the inverse scattering currents method this procedure uses the linear relation between data and scattering currents induced in an inhomogeneous structure and the relation between the scattering currents and the unknown conductivity the inverse problem is solved iteratively using these two relations

a linearized inverse boundary value problem for maxwell s equations abstract in this paper we consider the problem of determining the electromagnetic state of a body from measurements made on the surface of the body we study the full set of maxwell s equations that govern time harmonic electric and magnetic fields we seek to reconstruct the magnetic permeability the electric permitivity and the electric conductivity in the interior of the body from measurements made on the surface we exhibit appropriate boundary measurements in the form of a boundary mapping specifically the mapping from the tangential components of the electric field to the tangential components of the magnetic field this data can be used to reconstruct and approximately provided they deviate only slightly from known constants we also estimate the reconstruction errors

k classes for matroids and equivariant localization to every matroid we associate a class in the k theory of the grassmannian we study this class using the method of equivariant localization in particular we provide a geometric interpretation of the tutte polynomial we also extend results of the second author concerning the behavior of such classes under direct sum series and parallel connection and two sum these results were previously only established for realizable matroids and their earlier proofs were more difficult r esum e a chaque matronous associons une classe dans la k thde la grassmannienne nous

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

orbits in pr n and equivariant quantum cohomology abstract we compute the g l r 1 equivariant chow class of the g l r 1 orbit closure of any point x 1 x n p r n in terms of the rank polytope of the matroid represented by x 1 x n p r using these classes and generalizations involving point configurations in higher dimensional projective spaces we define for each d n matrix m an n ary operation m on the small equivariant quantum cohomology ring of p r which is the n ary quantum product when m is an invertible matrix we prove that m m is a valuative matroid polytope association like the quantum product these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting as an application we compute the number of line sections with given moduli of a general degree 2 r 1 hypersurface in p r generalizing the known case of quintic plane curves

the inverse scattering problem for time harmonic acoustic waves in an inhomogeneous medium with complex refraction index abstract the inverse scattering problem of determining the complex refraction index of a compact inhomogeneous medium from a knowledge of the far field patterns of the scattered fields corresponding to many incident time harmonic plane acoustic waves is solved by using the colton monk method again by using the same method the analogue of karp s theorem for the scattering of acoustic waves through an inhomogeneous medium with compact support is proven

multidimensional inverse problems and completeness of the products of solutions to pde a method is given for proving uniqueness theorems for some inverse problems the method is based on a result on completeness of the products of solutions to pde as an example the following uniqueness theorems are proved 1 the scattering amplitude a k known for all s2 and a fixed k 0 determines the compactly supported q x l2 d uniquely 2 the surface data u x y k known for all x y p x x3 0 and a fixed k 0 determine the compactly supported x l2 d d r 3 x x3 0 is arbitrarily small determine aj x j 1 2 uniquely here 2u k2u k2a1x a2 x u x y in r3 a1 l2 d a2 h2 d the same conclusion holds if the surface data are known at two distinct frequencies 4 the surface data u x y k known for all x y p and all k 0 determine v x and h k uniquely here 2 k2 k2 x u x y h k x l2 d h k is fourier transform of a wavelet of compact support 5 the conductivity x w2 2 d x c 0 is uniquely determined by the measurements of u and un on d here n is the outward normal to d d r3 is a bounded domain with a smooth boundary d x u 0 in d 6 necessary and sufficient conditions are given for a function a k s2 k 0 is fixed to be the scattering amplitude corresponding to a local potential from a certain class

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

geometry in grassmannians and a generalization of the dilogarithm abstract in this paper we introduce some polyhedra in grassman manifolds which we call grassmannian simplices we study two aspects of these polyhedra their combinatorial structure section 2 and their relation to harmonic differential forms on the grassmannian section 3 using this we obtain results about some new differential forms one of which is the classical dilogarithm section 1 the results here unite two threads of mathematics that were much studied in the 19th century the analytic one concerning the dilogarithm goes back to leibnitz 1696 and euler 1779 and the geometric one concerning grassmannian simplices can be traced to binet 1811 in section 4 we give some of this history along with some recent related results and open problems in section 0 we give as an introduction an account in geometric terms of the simplest cases

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

equivariant k theory and equivariant cohomology for t an abelian compact lie group we give a description of t equivariant k theory with complex coefficients in terms of equivariant cohomology in the appendix we give applications of this by extending results of chang skjelbred and goresky kottwitz macpherson from equivariant cohomology to equivariant k theory

stability of the solution to 3d fixed energy inverse scattering problem let sup s2 a a 0 is a given number b a r3 ba 0 is a small number a numerical method is given to calculate a stable approximation q to the fourier transform q of q x an estimate of the error of the approximation is given for q qq and q qa l ba

the maximum likelihood degree maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials we study the algebraic degree of the critical equations of this optimization problem this degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces and to the euler characteristic of the complexified complement under suitable hypotheses the maximum likelihood degree equals the top chern class of a sheaf of logarithmic differential forms exact formulae in terms of degrees and newton polytopes are given for polynomials with generic coefficients

numerical recovery of the layered medium from the surface data abstract in this paper an inverse problem is formulated and its numerical solution is given the problem is of interest for example in geophysics numerical results show that the algorithm is efficient for noisy data practical recommendations are given the problem consists of finding n z the refraction coefficient of the medium given the surface data u x k x p x x3 0 and all k 0 k0 where k0 0 is an arbitrary small wave number the governing equation is 2 k2n z u x x x1 x2 z analytical solution to this inverse problem is given a numerical implementation is suggested and the results of the numerical experiments are reported

formal solutions of inverse scattering problems iv error estimates the formal solutions of certain three dimensional inverse scattering problems presented in papers i iii in this series j math phys 10 1819 1969 17 1175 1976 21 2648 1980 are employed here to obtain quantitative estimates on the error resulting from the use of the born approximations in both direct and inverse potential scattering problems these estimates are uniformly valid at all energies and for all sufficiently weak potentials

the number of critical points of a product of powers of linear functions be the complement of a given a complex n vector 2 21 2n e cn consider the multivalued holomorphic function defined on m by studying bethe vectors in statistical mechanics a varchenko 18 conjectured that for generic 2 all critical points of qs are nondegenerate and the number of critical points is equal to the absolute value of the euler characteristic of m m he proved the conjecture for complexified real arrangements in 18 a similar result was known to k aomoto 1 example 1 who stated without proof that for positive 2 the number of critical points of b equals the number of bounded components of the complement of the real arrangement see 2 theorem 4 4 1 1 for details in this paper we prove varchenko s conjecture for all arrangements more generally we give a formula for the number of critical points of the multivalued holomorphic function

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

torus orbits on homogeneous varieties and kac polynomials of quivers in this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the kac polynomials of supernova quivers which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line we also prove that these polynomials can be expressed as a specialization of tutte polynomials of certain graphs providing a new way to compute them we also discuss connections with the gel fand macpherson correspondence

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

behavior of the solutions to exterior boundary value problems at low frequencies abstract a simple method is given to study the low frequency behavior of the solutions to exterior boundary value problems for elliptic equations of second order

higher algebraic k theory for actions of diagonalizable groups we study the k theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces our main result shows how to reconstruct the k theory ring of such an action from the k theory rings of the loci where the stabilizers have constant dimension we apply this to the calculation of the equivariant k theory of toric varieties and give conditions under which the merkurjev spectral sequence degenerates so that the equivariant k theory ring determines the ordinary k theory ring we also prove a very refined localization theorem for actions of this type

uniqueness theorems for multidimensional inverse problems with unbounded coefficients abstract in a g ramm inverse problems 3 1987 l77 l82 a method to prove uniqueness theorems is given the method uses an estimate of the solution to a partial differential equation with l d coefficients here the estimate is established for l2 d coefficients and the resulting uniqueness theorems are formulated

valuative invariants for polymatroids abstract many important invariants for matroids and polymatroids such as the tutte polynomial the billera jia reiner quasi symmetric function and the invariant g introduced by the first author are valuative in this paper we construct the z modules of all z valued valuative functions for labeled matroids and polymatroids on a fixed ground set and their unlabeled counterparts the z modules of valuative invariants we give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes and explicit formulas for their ranks our results confirm a conjecture of the first author that g is universal for valuative invariants

positivity in the grothendieck group of complex flag varieties abstract we prove a conjecture of a s buch concerning the structure constants of the grothendieck ring of a flag variety with respect to its basis of schubert structure sheaves for this we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs equivalently the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of schubert varieties

property c and an inverse problem for a hyperbolic equation let utt u q x t u 0 in d 0 t where d r3 is a bounded domain with a smooth boundary d t d d diam d q x t c 0 t l d suppose that for every u d f x t c1 d 0 t the value un d h s t is known where n is the outer normal to d u solves and and satisfies the initial conditions u ut 0 at t 0 then q x t is uniquely determined by the data f h f c1 d 0 t in the subset s of d 0 t consisting of the lines which make 45 with the t axis and which meet the planes t 0 and t t outside d 0 t provided that q x t is known outside s here d is the closure of d

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

a matroid invariant via the k theory of the grassmannian abstract let g d n denote the grassmannian of d planes in c n and let t be the torus c n diag c which acts on g d n let x be a point of g d n and let t x be the closure of the t orbit through x then the class of the structure sheaf of t x in the k theory of g d n depends only on which plucker coordinates of x are nonzero combinatorial data known as the matroid of x in this paper we will define a certain map of additive groups from k g d n to z t letting g x t denote the image of 1 n dim t x o t x g x behaves nicely under the standard constructions of matroid theory such as direct sum two sum duality and series and parallel extensions we use this invariant to prove bounds on the complexity of kapranov s lie complexes m kapranov chow quotients of grassmannians i adv soviet math 16 2 1993 29 110 hacking keel and tevelev s very stable pairs p hacking s keel e tevelev compactification of the moduli space of hyperplane arrangements j algebraic geom 15 2006 657 680 and the author s tropical linear spaces when they are realizable in characteristic zero d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 namely in characteristic zero a lie complex or the underlying d 1 dimensional scheme of a very stable pair can have at most n i 1 d i n d i i 1 strata of dimensions n i and d i respectively this prove the author s f vector conjecture from d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 in the case of a tropical linear space realizable in characteristic 0

finding conductivity from surface measurements abstract a method is given to determine conductivity of a body from the measurements of the potential and either normal component of the current on the surface of the body or the normal derivative of the potential on and the value of conductivity at a single point on

symmetry properties of scattering amplitudes and applications to inverse problems abstract for the transformations of the potential q q l where l is one of the following transformations rotation scaling or translation the corresponding transformation of the scattering amplitude is given a necessary and sufficient condition on the scattering amplitude for the scatterer to be spherically symmetric is derived as an application of the above results