asymptotic formulas for the eigenvalues of the timoshenko beam abstract asymptotic formulas are derived for the eigenvalues of a free ended timoshenko beam which has variable mass density and constant beam parameters otherwise these asymptotic formulas show how the eigenvalues and hence how the natural frequencies of such a beam depend on the material and geometric parameters which appear as coefficients in the timoshenko differential equations

on bounded approximations of periodicity for computational homogenization of stokes flow in porous media by separation of scales and the homogenization of a flow through porous media a two scale problem arises where a darcy type flow is present on the macroscale and a stokes flow on the subscale in this paper the problem is given as the minimization of a potential additional constraints imposing periodicity in a weak sense are added using lagrange multipliers in particular the upper and lower energy bounds for the corresponding strongly periodic problem are produced quantifying the accuracy of the weakly periodic boundary conditions a numerical example demonstrates the evaluation of energy bounds and the performance of weakly periodic boundary conditions on a representative volume element

eigenvalue asymptotics of layered media and their applications to the inverse problem we compute the asymptotics of the eigenvalues of the classical sturm liouville problem with a piecewise smooth coefficient q this means that q and or its derivatives can have jump discontinuities the boundary conditions are arbitrary our results extend the classical work of h hochstadt see comm pure appl math 14 1961 pp 749 764 and some related formulas discovered by g borg see acta math 78 1946 pp 1 96 then we apply our results to the inverse problem of determining the interfaces in a layered medium from acoustic data since the index of refraction of such a medium can be considered piecewise smooth

discontinuous inverse eigenvalue problems on considere des problemes de sturm liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur on montre que si le potentiel est connu sur la moitie de l intervalle et si une condition limite est donnee alors le potentiel et l autre condition limite sont determines de facon unique par les valeurs propres

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

estimates for periodic and dirichlet eigenvalues of the schrodinger operator with singular potentials abstract in this paper the periodic and the dirichlet problems for the schrodinger operator d2 dx2 v are studied for singular complex valued potentials v in the sobolev space h per 0 1 0 0 2 n 1 2 n n 2 2 v 0 v 2 n v 2 n o n 3 2 1 2 e where v k denote the fourier coefficients of v 2 the dirichlet spectrum consists of a sequence n n 1 of complex eigenvalues satisfying the asymptotics for any e 0 n n 2 2 v 0 v 2 n v 2 n 2 o n 2 1 e

the moving curve ideal and the rees algebra the rees algebra of an ideal in a commutative ring is the quotient of a polynomial ring by its ideal of defining relations for a polynomial ring in two variables this ideal was discovered independently by the geometric modeling community where it is called the moving curve ideal we review some properties of the rees algebra and discuss one result and one conjecture concerning the structure of the moving curve ideal and its relation to adjoint curves some parts of the paper are purely expository

inverse spectral problems for differential pencils on the half line with turning points the inverse spectral problem of recovering pencils of second order differential operators on the half line with turning points is studied we establish properties of the spectral characteristics give a formulation of the inverse problem prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem

effective construction of a class of positive operators in hilbert space which do not admit triangular factorization to introduce the main notions of triangular factorization consider a hilbert space l 2 a b infty leq a b leq infty the orthogonal projectors p xi in l 2 a b are defined by the relations p xi f x f x for a x xi p xi f x 0 for xi x b f epsilon l 2 a b denote the identity operator by i

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

matched filtering for generalized stationary processes the methods for solving optimal filtering problems in the case of the classical stationary processes have been well known since the late 1940s practice often gives rise to what is not a classical stationary process but a generalized one and white noise is one simple example hence it is of interest to describe the system action on the generalized stationary processes and then to carry over filtering methods to them for arbitrary generalized stochastic processes this seems to be a challenging problem in this correspondence we identify a rather general class of s sub j generalized stationary processes for which the desired extension can be done for matched filters this class can be considered as a model of colored noise and it is wide enough to include white noise positive frequencies white noise as well as certain generalized processes occurring in practice namely when the smoothing effect gives rise to the situation in which the distribution of probabilities may not exist at some time instances one advantage of the suggested model is that it connects optimal filter design with inverting of integral operators the methods for the latter can be found in the extensive literature

solution of the inverse spectral problem for an impedance with integrable derivative part ii the inverse eigenvalue problem for a sturm liouville equation in impedance form with dirichlet boundary conditions on a unit interval is solved the solution of this nonlinear problem requires the investigation of a combined map from the logarithmic derivative of the impedance assumed to be square integrable to two sequences of spectral data the first is the sequence of dirichlet eigenvalues shown to be locally bounded with square roots differing from the sequence of integral multiples of by a square summable sequence the second sequence has as first term the mean of the logarithmic derivative of the impedance each term in the remainder of the sequence is the logarithm of the product of the impedance and the derivative of an eigenfunction evaluated at an endpoint it is shown that this second sequence is a locally bounded square summable sequence the combined map is a real analytic isomorphism a uniqueness result follows from the asymptotics and analyticity shown in the first paper and properties of riesz bases the existence is shown by giving a closed form of the solution it is also shown that the same map solves the inverse sturm liouville problem in impedance form with neumann boundary conditions 1993 john wiley sons inc

the inverse sturm liouville problem ii soit q q 0 2 l r 2 0 1 le probleme de sturm liouville y q x y y 0 x 1 y 0 cos y 0 sin 0 y 1 cos y 1 sin 0 a un spectre discret de valeurs propres simples 0 q 1 q avec des fonctions propres h 0 x q h 1 x q on fixe p dans 0 2 l r 2 0 1 et soit m p q 0 2 l r 2 0 1 n q n p n 0 l espace de tous les points q de meme spectre que p on etudie m p comme une sous variete de 0 2 l r 2 0 1

on the equations of the moving curve ideal of a rational algebraic plane curve given a parametrization of a rational plane algebraic curve c some explicit adjoint pencils on c are described in terms of determinants moreover some generators of the rees algebra associated to this parametrization are presented the main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence

rational plane curves parameterizable by conics abstract we introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines also known as monoid curves we show that they are the image of monoid curves via suitable quadratic transformations in projective plane we also describe all the possible proper parameterizations of them and a set of minimal generators of the rees algebra associated to these parameterizations extending well known results for curves parameterizable by lines

computing variational bounds for flow through random aggregates of spheres known formulas for variational bounds on darcy s constant for slow flow through porous media depend on two point and three point spatial correlation functions certain bounds due to prager and doi depending only a two point correlation functions have been calculated for the first time for random aggregates of spheres with packing fractions up to 0 64 three radial distribution functions for hard spheres were tested for up to 0 49 1 the uniform distribution or well stirred approximation 2 the percus yevick approximation and 3 the semi empirical distribution of verlet and weis the empirical radial distribution functions of bennett and finney were used for packing fractions near the random close packing limit rcp 0 64 an accurate multidimensional monte carlo integration method vegas developed by lepage was used to compute the required two point correlation functions the results show that doi s bounds are preferred for 0 10 the upper bounds computed using the well stirred approximation actually become negative which is physically impossible as increases indicating the very limited value of this approximation the other two choices of radial distribution function give reasonable results for up to 0 49 however these bounds do not decrease with as fast as expected for large it is concluded that variational bounds dependent on three point correlation functions are required to obtain more accurate bounds on darcy s constant for large

inverse spectral problems for sturm liouville operators in impedance form abstract the spectral properties of sturm liouville operators with an impedance in w p 1 0 1 for some p 1 are studied in particular a complete solution of the inverse spectral problem is provided

eigenvalue asymptotics for sturm liouville operators with singular potentials abstract we derive eigenvalue asymptotics for sturm liouville operators with singular complex valued potentials from the space w 2 1 0 1 0 1 and dirichlet or neumann dirichlet boundary conditions we also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra

an inverse eigenvalue problem for a vibrating string with two dirichlet spectra a classical inverse problem is can you hear the density of a string clamped at both ends the mathematical model gives rise to an inverse sturm liouville problem for the unknown density rho and it is well known that the answer is negative the dirichlet spectrum from the clamped end point conditions is insufficient there are many known ways to add additional information to gain a positive answer and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode the so called norming constants we make the assumption that neither of these changes are possible instead we will add known mass densities to the string in a way we can prescribe and remeasure the dirichlet spectrum we will not be able to answer the uniqueness question in its most general form but will give some insight to what added masses should be chosen and how this can lead to a reconstruction of the original string density

survey on the theory and applications of μ bases for rational curves and surfaces abstract bases are new representations for rational curves and surfaces which serve as a bridge between their parametric forms and implicit forms geometrically bases are represented by moving lines or moving planes while their algebraic counterparts are special syzygies of the parametric equations of rational curves or surfaces bases have been proven to be significant in solving many important problems in geometric modeling such as fast implicitization singularity computation reparametrization as well as providing easy inversion formulas for points we review the state of the art results in bases theory and applications for rational curves and surfaces and raise unsolved problems for future research

riccati map on l02 t and its applications abstract this paper is concerned with the spectral properties of the schrodinger operator l q def d 2 d x 2 q with periodic potential q from the sobolev space h 1 t we obtain asymptotic formulas and a priori estimates for the periodic and dirichlet eigenvalues which generalize known results for the case of potentials q l 2 t the key idea is to reduce the problem to a known one the spectrum of the impedance operator via a nonlinear analytic isomorphism between l 0 2 t and the sobolev space h 0 1 t

on the closed image of a rational map and the implicitization problem in this paper we investigate some topics around the closed image s of a rational map lambda given by some homogeneous elements f 1 ldots f n of the same degree in a graded algebra a we first compute the degree of this closed image in case lambda is generically finite and f 1 ldots f n define isolated base points in proj a we then relate the definition ideal of s to the symmetric and the rees algebras of the ideal i f 1 ldots f n subset a and prove some new acyclicity criteria for the associated approximation complexes finally we use these results to obtain the implicit equation of s in case s is a hypersurface proj a pp n 2 k with k a field and base points are either absent or local complete intersection isolated points

variationally consistent computational homogenization of transient heat flow a framework for variationally consistent homogenization combined with a generalized macrohomogeneity condition is exploited for the analysis of non linear transient heat conduction within this framework the classical approach of model based first order homogenization for stationary problems is extended to transient problems homogenization is then carried out in the spatial domain on representative volume elements rve which are in practice introduced in quadrature points in standard fashion along with the classical averages a higher order conservation quantity is obtained an iterative fe2 algorithm is devised for the case of non linear diffusion and storage coefficients and it is applied to transient heat conduction in a strongly heterogeneous particle composite parametric studies are carried out in particular with respect to the influence of the internal length associated with the second order conservation quantity copyright c 2009 john wiley sons ltd

inverse spectral theory for uniformly open gaps in a weighted sturm liouville problem abstract motivated by a pde existence problem we study the inverse problem for a weighted sturm liouville operator l s associated with the eigenvalue problem y s x y 0 where s is a real valued periodic even function that is bounded from below by a positive constant and belongs to the l 2 based sobolev space h r 0 1 r 1 choosing gap lengths and gap midpoints as coordinates we define a spectral map g that assigns to a coefficient s the structure of the spectrum of l s we find that g is a real analytic isomorphism locally around s 1 which in particular implies the existence of coefficients s h r 0 1 r 3 2 whose spectrum features band structure with all gaps uniformly open around the gap midpoints this result paves the way for the construction of so called breathers in nonlinear wave equations with such coefficients s apart from the novelty of treating the inverse spectral problem for the full banach scale h r 0 1 r 1 the local nature of our result allows more concise and transparent proofs in particular instead of using any preliminary transformations we treat the weighted problem directly by adapting techniques used for schrodinger operators with distribution potentials

inverse spectral problems for bessel operators abstract we study the inverse spectral problem for a class of bessel operators given in l 2 0 1 by the differential expression d d x x v d d x x v with n and a real valued function v l p 0 1 p 1 subject to various boundary conditions we describe completely the spectral data of these operators i e the spectra and corresponding norming constants and give the algorithm of reconstruction of v from the spectral data

rational normal scrolls and the defining equations of rees algebras consider a height two ideal i which is minimally generated by m homogeneous forms of degree d in the polynomial ring r k x y suppose that one column in the homogeneous presenting matrix of i has entries of degree n and all of the other entries of are linear we identify an explicit generating set for the ideal a which defines the rees ring r r it so r s a for the polynomial ring s r t1 tm we resolve r as an s module and is as an r module for all powers s the proof uses a rational normal scroll ring a s h with aa isomorphic to the nth symbolic power of a height one prime ideal k of a the ideal k n is generated by monomials whenever possible we study a k n in place of a aa because the generators of k n are much less complicated then the generators of aa we obtain a filtration of k n in which the factors are polynomial rings hypersurface rings or modules resolved by eagon northcott complexes the generators of i give rise to an algebraic curve c in projective m 1 space the defining equations of the fiber ring r x y r yield a solution of the implicitization problem for c introduction let i be a height two ideal in the polynomial ring r k x y with i minimally generated by m forms of degree d we consider the following questions what is the defining ideal a of the rees ring r r it r it it what is the hilbert function of r the rees ring r is equal to r t1 tm a what is the minimal resolution of r what are the defining equations of the fiber 1supported in part by the national security agency 2supported in part by the national science foundation and the national security agency 3supported in part by the national science foundation 2000 mathematics subject classification primary 14q05 secondary 13c20 13d02 14e05 14m12 68w30

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

the inverse sturn liouville problem iii on considere la theorie spectrale inverse du probleme de sturm liouville y q x y y 0 x 1 avec les conditions aux limites y 0 0 by 1 y 1 0 q l r 2 0 1 et b r

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

syzygies and singularities of tensor product surfaces of bidegree 2 1 let u h op1 p1 2 1 be a basepoint free four dimensional vector space the sections corresponding to u determine a regular map u p1 p1 p3 we study the associated bigraded ideal iu k s t u v from the standpoint of commutative algebra proving that there are exactly six numerical types of possible bigraded minimal free resolution these resolutions play a key role in determining the implicit equation for u p1 p1 via work of buse jouanolou 5 buse chardin 6 botbol 2 and botbol dickensteindohm 3 on the approximation complex z in four of the six cases iu has a linear first syzygy remarkably from this we obtain all differentials in the minimal free resolution in particular this allows us to explicitly describe the implicit equation and singular locus of the image

on the continuum limit of a discrete inverse spectral problem on optimal finite difference grids we consider finite difference approximations of solutions of inverse stur m liouville problems in bounded intervals using three point finite difference sc hemes we discretize the equations on so called optimal grids constructed as follow s for a staggered grid with 2k points we ask that the finite difference operator a k k jacobi matrix and the sturm liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval this requirement determines uniquely the entries in the jacobi matrix which are grid cell averages of the coefficients in the continuum problem if these coefficients are known we can find the grid which we call optimal becau se it gives by design a finite difference operator with a prescribed spectr al measure we focus attention on the inverse problem where neither the coefficients n or the grid are known a key question in inversion is how to parametrize the coefficients i e how to choose the grid it is clear that to be successful this grid must be close to the optimal one which is unknown fortunately as we show here the grid dependence on the unknown coefficients is weak so the inversion can be done on a pr ecomputed grid for an a priori guess of the unknown coefficients this obse rvation leads to a simple yet efficient inversion algorithm which gives coefficien ts that converge pointwise to the true solution as the number k of data points tends to infinity the cornerstone of our convergence proof is showing that op timal grids provide an implicit natural regularization of the inverse problem by givin g reconstructions with uniformly bounded total variation the analysis is based on a novel explicit perturbation analysis of lanczos recursions and on a discrete