the determinant formula for a matroid bilinear form abstract we introduce a symmetric bilinear form of a weighted matroid and prove that the determinant of the matrix of this form is a product of linear functions of weights this formula is an analog of the formula for the determinant of the shapovalov form in representation theory

quantized knizhnik zamolodchikov equations quantum yang baxter equation and difference equations for q hypergeometric functions thesl2 quantized knizhnik zamolodchikov equations are solved inq hypergeometric functions new difference equations are derived for generalq hypergeometric functions the equations are given in terms of quantum yang baxter matrices and have the form similar to quantum knizhnik zamolodchikov equations for quantum affine algebras introduced by frenkel and reshetikhin

ringel duality for perverse sheaves on hypertoric varieties abstract motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities we study a category of perverse sheaves with coefficients in a field k on any affine unimodular hypertoric variety m our main result is that this is a highest weight category whose ringel dual is the corresponding category for the gale dual hypertoric variety m on the way to proving our main result we confirm a conjecture of finkelberg kubrak in the case of hypertoric varieties we also show that our category is equivalent to representations of a combinatorially defined algebra recently introduced in a related paper

unitarity of the knizhnik zamolodchikov bernard connection and the bethe ansatz for the elliptic hitchin systems we work out finite dimensional integral formulae for the scalar product of genus one states of the groupg chern simons theory with insertions of wilson lines assuming convergence of the integrals we show that unitarity of the elliptic knizhnik zamolodchikov bernard connection with respect to the scalar product of cs states is closely related to the bethe ansatz for the commuting hamiltonians building up the connection and quantizing the quadratic hamiltonians of the elliptic hitchin system

a level set method for imaging salt structures using gravity data abstractwe have developed a level set method for the inverse gravimetry problem of imaging salt structures with density contrast reversal under such a circumstance a part of the salt structure contributes two completely opposite anomalies that counteract with each other making it unobservable to the gravity data as a consequence this amplifies the inherent nonuniqueness of the inverse gravimetry problem so that it is much more challenging to recover the whole salt structure from the gravity data to alleviate the severe nonuniqueness it is reasonable to assume that the density contrast between the salt structure and the surrounding sedimentary host depends upon the depth only and is known a priori consequently the original inverse gravity problem reduces to a domain inverse problem where the supporting domain of the salt body becomes the only unknown we have used a level set function to parametrize the boundary of the salt body so that we reformulated the domain inverse problem into a nonlinear

a selberg integral for the lie algebra an a new q binomial theorem for macdonald polynomials is employed to prove an a n analogue of the celebrated selberg integral this confirms the mathfrak g rm a n case of a conjecture by mukhin and varchenko concerning the existence of a selberg integral for every simple lie algebra mathfrak g

spaces of quasi exponentials and representations of gln we consider the action of the bethe algebra on the weight subspace of weight of the tensor product of k polynomial irreducible modules with highest weights respectively the bethe algebra depends on n complex numbers under the assumption that k1 kn are distinct we prove that the image of in end is isomorphic to the algebra of functions on the intersection of suitable schubert cycles in the grassmannian of n dimensional spaces of quasi exponentials with exponents we also prove that the module is isomorphic to the coregular representation of that algebra of functions we present a bethe ansatz construction identifying the eigenvectors of the bethe algebra with points of that intersection of schubert cycles

bases for cohomology of local systems on hyperplane complements we study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes a the cohomology plays an important role for the theory of generalized hy pergeometric functions we combine several known results to construct explicit bases of logarithmic forms for the only non vanishing cohomol ogy group under some nonresonance conditions on the local system for any arrangement a the bases are determined by a linear ordering of the hyperplanes and are indexed by certain no broken circuits bases of a the basic forms depend on the local system but any two bases con structed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system in certain special cases we show the existence of bases of monomial logarithmic forms

an iterative method for the computation of nonlinear wide angle pulsed acoustic fields of medical diagnostic transducers the development and optimization of medical ultrasound transducers and imaging modalities require a computational method that accurately predicts the nonlinear acoustic pressure field a prospective method should provide the wide angle pulsed field emitted by an arbitrary planar source distribution and propagating in a three dimensional large scale domain holding a nonlinear acoustic medium in this paper a method is presented that is free of any assumed wavefield directionality the nonlinear acoustic wave equation is solved by treating the nonlinear term as a contrast source this formulation leads to an iterative scheme that involves the repetitive solution of a linear wave problem through green s function method it is shown that accurate field predictions may be obtained within a few iterations moreover by employing a dedicated numerical convolution technique the method allows for a discretization down to two points per wavelength or period of the highest frequency of interest the performance of the method is evaluated through a number of nonlinear field predictions for pulsed transducers with various geometries the results demonstrate the directional independence of the method moreover comparison with results from several existing methods shows that the method accurately predicts the nonlinear field for weak to moderate nonlinearity

linear inversion of gravity data for 3 d density distributions we have developed an improved levenburg marquart technique to rapidly invert bouguer gravity data for a 3 d density distribution as a source of the observed field this technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user where such information is not available objective models are generated the technique estimates the density distribution within the source volume using a least squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential householder transformations the source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density this discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution convergence of the solution to the system is tightly controlled by a damping pa

critical points of master functions and integrable hierarchies abstract we consider the population of critical points generated from the trivial critical point of the master function with no variables and associated with the trivial representation of the affine lie algebra sl n we show that the critical points of this population define rational solutions of the equations of the mkdv hierarchy associated with sl n we also construct critical points from suitable n tuples of tau functions the construction is based on a wronskian identity for tau functions in particular we construct critical points from suitable n tuples of schur polynomials and prove a wronskian identity for schur polynomials

unmitigated numerical solution to the diffraction term in the parabolic nonlinear ultrasound wave equation various numerical algorithms have been developed to solve the khokhlov kuznetsov zabolotskaya kzk parabolic nonlinear wave equation in this work a generalized time domain numerical algorithm is proposed to solve the diffraction term of the kzk equation this algorithm solves the transverse laplacian operator of the kzk equation in three dimensional 3d cartesian coordinates using a finite difference method based on the five point implicit backward finite difference and the five point crank nicolson finite difference discretization techniques this leads to a more uniform discretization of the laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term in addition a new empirical algorithm based on the lu decomposition technique is proposed to solve the system of linear equations obtained from this discretization the proposed empirical algorithm improves the calculation speed and memory usage while the order of computational complexity remains linear in calculation of the diffraction term in the kzk equation for evaluating the accuracy of the proposed algorithm two previously published algorithms are used as comparison references the conventional 2d texas code and its generalization for 3d geometries the results show that the accuracy efficiency performance of the proposed algorithm is comparable with the established time domain methods

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

constrained inversion of gravity fields for complex 3 d structures as part of a research program to develop gravity interpretation tools that can be merged with seismic techniques a full 3 d complex structural inversion scheme for possibly multibody polyhedral models has been developed the forward modeling algorithm was adopted from previous work because the inverse problem is generally very ill posed several methods of regularizing the inversion were investigated and a combination of the most useful was adopted the combination includes i a structured matrix formulation for the system equations ii an analytical expression for the jacobian calculation iii first derivative damping iv a choice of damping parameter based on a variation of the trust region method v a weighted scheme for parameter correction and vi complete freezing of degrees of freedom found not to influence the gravity field significantly this combination yields a robust inversion which was successfully demonstrated on data over the galveston island salt dome offshore texas variat

adaptive learning 3d gravity inversion for salt body imaging we have developed an iterative scheme for inverting gravity data produced by salt bodies with density contrasts relative to the sediments varying from positive to negative crossing in this way the nil zone our inversion method estimates a 3d density contrast distribution through a piecewise constant function defined on a user specified grid of cells it consists of two nested iterative loops the outer loop uses an adaptive learning strategy that starts with a coarse grid of cells a set of first guess geometric elements axes and points and the corresponding assigned density contrasts from the second iteration on this strategy refines the grid and automatically creates a new set of geometric elements points only and associated density contrasts each geometric element operates as the first guess skeletal outline of a section of the salt body to be imaged the inner loop estimates the 3d density contrast distribution for the grid of cells and for the set of geometric elements defined in the oute

a filtered convolution method for the computation of acoustic wave fields in very large spatiotemporal domains the full wave computation of transient acoustic fields with sizes in the order of 100x100x100 wavelengths by 100 periods requires a numerical method that is extremely efficient in terms of storage and computation iterative integral equation methods offer a good performance on these points provided that the recurring spatiotemporal convolutions are computed with a coarse sampling and relatively few computational operations this paper describes a method for the numerical evaluation of very large scale four dimensional convolutions that employs a fast fourier transformation and that uses a sampling rate close to or at the limit of two points per wavelength and per period to achieve this the functions involved are systematically filtered windowed and zero padded with respect to all relevant coordinates prior to sampling the method is developed in the context of the neumann iterative solution of the acoustic contrast source problem for an inhomogeneous medium the implementation of the method on a parallel computer is discussed the obtained numerical results have a relative root mean square error of a few percent when sampling at two points per wavelength and per period further the results prove that the method enables the computation of transient fields in the order of the indicated size

the combinatorial laplacian of the tutte complex abstract let m be an ordered matroid and c m be an exterior algebra over its underlying set e graded by both corank and nullity then c 0 m is the simplicial chain complex of in m the simplicial complex whose simplices are indexed by the independent sets of the matroid dually c0 m is the cochain complex of in m we give a combinatorial description of a basis of eigenvectors for the combinatorial laplacian of a family of boundary maps on the double complex extending work by w kook v reiner and d stanton 2000 j amer math soc 13 129 148 on in m the eigenvalues are enumerated by a weighted version of the tutte polynomial using an identity of g etienne and m las vergnas 1998 discrete math 179 111 119 as an application we prove a duality theorem for the cohomology of orlik solomon algebras

imaging geologic surfaces by inverting gravity gradient data with depth horizons the nonuniqueness problem that occurs when inverting potential field data is well known it can however be surmounted by jointly inverting these data with independent data sets incorporating depth information and regularizing the solution the goal is to produce a geologic model that is compatible with all measured quantities does not exceed any prescribed limits and is geologically plausible to achieve this we have developed a spatially based surface inversion algorithm that solves for the geometric interface between geologic bodies the bodies are constructed from grids of rectangular prisms that have their bottom depths adjusted by the algorithm to form the inverted surface to solve large scale inversions approximations are used in the potential field calculations that allow internal matrices to be stored in sparse format with minimal loss of accuracy the impetus for the work came from the need to combine airborne gravity gradient data with depth horizons estimated from interpreted 2d seismic

gauss manin connections for arrangements ii nonresonant weights we study the gauss manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system aomoto and kita determined this gauss manin connection for arrangements in general position we use their results and an algorithm constructed in this paper to determine this gauss manin connection for all arrangements

jackson integrals of jordan pochhammer type and quantum knizhnik zamolodchikov equations we show that theq difference systems satisfied by jackson integrals of jordan pochhammer type give a class of the quantum knizhnik zamolodchikov equation for u q widehat mathfrak s mathfrak l 2 in the sense of frenkel and reshetikhin

modeling of nonlinear ultrasound propagation in tissue from array transducers a computationally efficient model capable of simulating finite amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described it is based on a second order operator splitting approach tavakkoli et al j acoust soc am 104 2061 2072 1998 with a fractional step marching scheme whereby the effects of diffraction attenuation and nonlinearity can be computed independently over incremental steps this approach is an extension to that of christopher and parker j acoust soc am 90 507 521 90 488 499 1991 wherein linear and nonlinear effects are propagated separately over incremental steps and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources results of the model are compared with published data predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented

hypergeometric integrals and arrangements abstract this is an outline of the aomoto gelfand theory of multivariable hypergeometric integrals and varchenko s formula for the determinant of the period matrix of the hypergeometric pairing a significant feature of this work is the use of the theory of arrangements of hyperplanes to transform a problem in analysis into one in combinatorics

toric reduction and a conjecture of batyrev and materov this paper grew out of our efforts to understand the toric residue mirror conjecture formulated by batyrev and materov in 2 this conjecture has its origin in physics and is based on a work by morrison and plesser 14 according to the philosophy of mirror symmetry to every manifold in a certain class one can associate a dual manifold the so called mirror so that the intersection numbers of the moduli spaces of holomorphic curves in one of these manifolds are related to integrals of certain special differential forms on the other while at the moment this mirror manifold is only partially understood there is an explicit construction due to victor batyrev 1 in which the two manifolds are toric varieties whose defining data are related by a natural duality notion for polytopes let us recall the setting of the conjecture of batyrev and materov let t be a d dimensional real vector space endowed with an integral structure a lattice t t of full rank we denote by t the embedded dual lattice v t v z for all t consider two convex polytopes t and t containing the origin in their respective interiors and related by the duality

full wave modeling of therapeutic ultrasound efficient time domain implementation of the frequency power law attenuation for the simulation of therapeutic ultrasound applications a method including frequency dependent attenuation effects directly in the time domain is highly desirable this paper describes an efficient numerical time domain implementation of the power law attenuation model presented by szabo szabo j acoust soc am 96 491 500 1994 simulations of therapeutic ultrasound applications are feasible in conjunction with a previously presented finite differences time domain fdtd algorithm for nonlinear ultrasound propagation ginter et al j acoust soc am 111 2049 2059 2002 szabo implemented the empirical frequency power law attenuation using a causal convolutional operator directly in the time domain equation though a variety of time domain models has been published in recent years no efficient numerical implementation has been presented so far for frequency power law attenuation models solving a convolutional integral with standard time domain techniques requires enormous computational ef

quantum algebra structure of certain jackson integrals theq difference system satisfied by jackson integrals with a configuration ofa type root system is studied we explicitly construct some linear combination of jackson integrals which satisfies the quantum knizhnik zamolodchikov equation for the 2 point correlation function ofq vertex operators introduced by frenkel and reshetikhin for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 the expression of integrands for then point case is conjectured and a set of linear relations for the corresponding jackson integrals is proved

topological representations of the quantum groupuq sl2 we define a topological action of the quantum groupuq sl2 on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc this action commutes with the monodromy action of the braid groupoid which is given by ther matrix ofuq sl2

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 application of aomoto gelfand hypergeometric functions to the su n knizhnik zamolodchikov equation solutions to the knizhnik zamolodchikov equation for verma modules of the lie algebra mathfrak s mathfrak l n 1 mathbb c are explicitly given by certain integrals called aomoto gelfand hypergeometric functions

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

a numerical method for general finite amplitude wave propagation in two dimensions and its application to spark pulses the general equations of finite amplitude acoustics including classical absorption effects and second order nonlinear effects are written in a form suitable for two dimensional numerical solution a finite difference scheme then is applied to numerically solve the equations to demonstrate the method examples are given of spherical free field propagation normal plane reflection from a hard surface and oblique spherical reflection from a hard surface for spark pulses this method has an advantage over burgers equation methods one way wave equation methods and pestorius type algorithms in that it can predict the interaction of multiple finite amplitude acoustic waves at arbitrary propagation angles

on operad structures of moduli spaces and string theory we construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy lie algebras gravity algebras and batalin vilkovisky algebras these algebras appeared recently in the context of string theory and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory

the b and m shapiro conjecture in real algebraic geometry and the bethe ansatz we prove the b and m shapiro conjecture that if the wronskian of a set of polynomials has real roots only then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients this in particular implies the following result if all ramification points of a parametrized rational curve 1 r lie on a circle in the riemann sphere 1 then maps this circle into a suitable real subspace r r the proof is based on the bethe ansatz method in the gaudin model the key observation is that a symmetric linear operator on a euclidean space has real spectrum in appendix a we discuss properties of differential operators associated with bethe vectors in the gaudin model in particular we prove a statement which may be useful in complex algebraic geometry it claims that certain schubert cycles in a grassmannian intersect transversally if the spectrum of the corresponding gaudin hamiltonians is simple in appendix b we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with lie algebras of types a r b r and c r

gauss manin connections for arrangements iii formal connections abstract we study the gauss manin connection for the moduli space of anarrangement of complex hyperplanes in the cohomology of a complex rank onelocal system we de ne formal gauss manin connection matrices in the ao moto complex and prove that for all arrangements and all local systems theseformal connection matrices specialize to gauss manin connection matrices 1 introductionlet a h 1 h n be an arrangement of nordered hyperplanes in c l andlet l be a local system of coe cients on m m a c l s nj 1 h j the comple ment of a the need to calculate the local system cohomology h m l arises invarious contexts for instance local systems may be used to study the milnor berof the non isolated hypersurface singularity at the origin obtained by coning thearrangement see 8 5 in mathematical physics local systems on complements ofarrangements arise in the aomoto gelfand theory of multivariable hypergeometricintegrals 2 12 18 and the representation theory of lie algebras and quantumgroups these considerations lead to solutions of the knizhnik zamolodchikov dif ferential equation from conformal eld theory see 21 23 here a central problemis the determination of the gauss manin connection on h

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

on algebraic equations satisfied by hypergeometric correlators in wzw models ii the paper contains an explicit description of genus 0 conformal block bundles for wess zumino witten models of conformal field theory we prove that an earlier construction due to the second and the third authors gives a map of these bundles to certain de rham cohomology bundles

on the gaudin model associated to lie algebras of classical types we derive explicit formulas for solutions of the bethe ansatz equations of the gaudin model associated to the tensor product of one arbitrary finite dimensional irreducible module and one vector representation for all simple lie algebras of classical type we use this result to show that the bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

on representations and k theory of the braid groups let be the fundamental group of the complement of a k 1 hyperplane arrangement such as artin s pure braid group or more generally a homologically toroidal group as defined below the triviality of bundles arising from orthogonal representations of is characterized completely as follows an orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups furthermore the subgroup of elements in the complex k theory of b which arises from complex unitary representations of is shown to be trivial in the case of real k theory the subgroup of elements which arises from real orthogonal representations of is an elementary abelian 2 group which is characterized completely in terms of the first two stiefel whitney classes of the representation in addition quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the jacobi identity for certain choices of poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized heisenberg groups these cohomology relations correspond to non trivial spin representations of the pure braid groups which give rise to trivial bundles

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

a q deformation of a trivial symmetric group action let a be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system an 1 let b b q be the varchenko matrix for this arrangement with all hyperplane parameters equal to q we show that b is the matrix with rows and columns indexed by permutations with entry equal to qi 1 where i is the number of inversions of equivalently b is the matrix for left multiplication on csn by n q sn q clearly b commutes with the right regular action of sn on csn a general theorem of varchenko applied in this special case shows that b is singular exactly when q is a j j 1 st root of 1 for some j between 2 and n in this paper we prove two results which partially solve the problem originally posed by varchenko of describing the sn module structure of the nullspace of b in the case that b is singular our first result is that ker b indn sn 1 lien 1 lien in the case that q e2 i n n 1 where lien denotes the multilinear part of the free lie algebra with n generators our second result gives an elegant formula for the determinant of b restricted to the virtual sn module p with characteristic the power sum symmetric function p x

computation of nonlinear ultrasound fields using a linearized contrast source method nonlinear ultrasound is important in medical diagnostics because imaging of the higher harmonics improves resolution and reduces scattering artifacts second harmonic imaging is currently standard and higher harmonic imaging is under investigation the efficient development of novel imaging modalities and equipment requires accurate simulations of nonlinear wave fields in large volumes of realistic lossy inhomogeneous media the iterative nonlinear contrast source incs method has been developed to deal with spatiotemporal domains measuring hundreds of wavelengths and periods this full wave method considers the nonlinear term of the westervelt equation as a nonlinear contrast source and solves the equivalent integral equation via the neumann iterative solution recently the method has been extended with a contrast source that accounts for spatially varying attenuation the current paper addresses the problem that the neumann iterative solution converges badly for strong contrast sources the remedy is linearization of the nonlinear contrast source combined with application of more advanced methods for solving the resulting integral equation numerical results show that linearization in combination with a bi conjugate gradient stabilized method allows the incs method to deal with fairly strong inhomogeneous attenuation while the error due to the linearization can be eliminated by restarting the iterative scheme

singular and nonsingular eigenvectors for the gaudin model we present a method to construct a basis of singular and nonsingular common eigenvectors for gaudin hamiltonians in a tensor product module of the lie algebra sl 2 the subset of singular vectors is completely described by analogy with covariant differential operators the relation between singular eigenvectors and the bethe ansatz is discussed in each weight subspace the set of singular eigenvectors is completed to a basis by a family of nonsingular eigenvectors we discuss also the generalization of this method to the case of an arbitrary lie algebra

the varchenko determinant for oriented matroids we generalize the varchenko matrix of a hyperplane arrangement to oriented matroids we show that the celebrated determinant formula for the varchenko matrix first proved by varchenko generalizes to oriented matroids it follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for closed supertopes in oriented matroids we follow a proof strategy for the original varchenko formula first suggested by denham and hanlon besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest we show that a supertope considered as a subposet of the tope poset has a contractible order complex

discrete miura opers and solutions of the bethe ansatz equations solutions of the bethe ansatz equations associated to the xxx model of a simple lie algebra open image in new window come in families called the populations we prove that a population is isomorphic to the flag variety of the langlands dual lie algebra open image in new window the proof is based on the correspondence between the solutions of the bethe ansatz equations and special difference operators which we call the discrete miura opers the notion of a discrete miura oper is one of the main results of the paper