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

modified numerov s method for inverse sturm liouville problems in this paper we propose a new modified numerov s method for recovering from eigenvalues a symmetric potential of a sturm liouville operator with dirichlet boundary conditions we use interpolation to refine the mesh sufficiently for numerov s method to be effective even without the asymptotic correction technique of andrew and paine accuracy and stability of the method are investigated convergence of the method is established our method is extended to deal with natural boundary conditions numerical experiments confirm its competitiveness

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

reconstruction techniques for classical inverse sturm liouville problems this paper gives constructive algorithms for the classical inverse sturm liouville problem it is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator two methods of solving this latter problem are then provided and numerical examples are presented

on the uncertainty in the regularized solution regularization is a fundamental technique in the processing of measurement data these data are typically noisy and sparse and in some cases only a projection of the data on a low dimensional subspace is available very often direct analysis of measured data yields unphysical results due to sensitivity to noise and neglect of the underlying distribution of the measured physical phenomena therefore regularization is extensively applied in physics one of the most common regularization methods was developed by a few researchers most notably tikhonov and phillips during the 1940s 1960s in the early 1980s geman and geman reformulated regularization in a bayesian framework in which every candidate function f x is assigned a probability with the expectation of the random variable f x0 equal to the value of the regularized interpolating function at x0 taking this probabilistic approach to second order the variance of this random variable which reflects the amount of uncertainty inherent in the solution is computed when estimating physical phenomena not only the estimated value is important but also the uncertainty associated with it for example more measurements should yield a more certain solution this review provides a formal definition of this uncertainty explores some numerical issues with its computation and extends it to compute the uncertainties of the integral and derivative with the intriguing result that in some cases the uncertainty of the function and the derivative are highly uncorrelated for example there are cases in which the point at which the function is least reliable is the one at which the derivative is most reliable lastly a related definition of optimal sampling is discussed the relevance of optimal sampling to physics is that it offers a method to determine the location of the measuring devices which yields the most reliable restoration of the measured physical phenomenon

perron eigenvector of the tsetlin matrix abstract we consider the move to position k linear search scheme where the sequence of record requests is a markov chain formulas are derived for the stationary distribution of the permutation chain for k 1 2 n 1 and n where n is the number of records certain identities for the perron complement are established in the process

on the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues abstract consider the inverse acoustic scattering problem for spherically symmetric inhomogeneity of compact support define the corresponding homogeneous and inhomogeneous interior transmission problems we study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem these transmission eigenvalues are shown to be zeros of the scattering amplitude and also the set of eigenvalues of a special sturm liouville problem a uniqueness theorem for the potential of the derived sturm liouville problem is proved when the data are the given spectra and partial knowledge of the potential a corollary of this theorem is a uniqueness theorem for the original inverse acoustic scattering problem

bounds for the perron root of nonnegative matrices and spectral radius of iteration matrices abstract new bounds for the perron root a of a nonnegative matrix a are proposed we prove that min 1 i n r i a b r i b a max 1 i n r i a b r i b where b is an arbitrary matrix with row sums r i b k b i k 0 i 1 n the bounds of h minc 1 and shulin liu 6 are both special cases of this result and based on this result we also get some bounds for the spectral radius of iteration matrices

stability for the multifrequency inverse medium problem abstract the solution of a multi frequency 1d inverse medium problem consists of recovering the refractive index of a medium from measurements of the scattered waves for multiple frequencies in this paper rigorous stability estimates are derived when the frequency takes value in a bounded interval it is showed that the ill posedness of the inverse medium problem decreases as the width of the frequency interval becomes larger more precisely under certain regularity assumptions on the refractive index the estimates indicate that the power in holder stability is an increasing function of the largest value in the frequency band finally a lipschitz stability estimate is obtained for the observable part of the medium function defined through a truncated trace formula

the uniqueness for inverse discrete transmission eigenvalue problems abstract the inverse problem for the discrete analog of the transmission eigenvalue problem with a spherically symmetric index of refraction is considered some uniqueness results are provided which imply that can be completely determined if only partial entries are given on together with partial transmission eigenvalues furthermore some similar results are presented for a related schrodinger equation

two component eigenfunction expansion for open systems described by the wave equation i completeness of expansion the concept of eigenfunction expansions for the wave equation is generalized to open systems in which waves escape to the outside these non conservative systems are non hermitian in the usual sense it is shown that the natural framework is an eigenfunction expansion within a two component formalism that treats the wavefunction and its conjugate momentum together provided the system approaches spatial infinity rapidly without tails and possesses spatial discontinuities the expansion in terms of the eigenfunctions which are now quasinormal modes is shown to be valid

inverse spectral analysis with partial information on the potential ii the case of discrete spectrum we discuss results where the discrete spectrum or partial information on the discrete spectrum and partial information on the potential q of a one dimensional schrodinger operator h d 2 dx 2 q determine the potential completely included are theorems for finite intervals and for the whole line in particular we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of h on a finite interval and knowledge of q over a corresponding fraction of the interval the methods employed rest on weyl m function techniques and densities of zeros of a class of entire functions

convergence of numerov s method for inverse sturm liouville problems in this paper we discuss the convergence of numerov s method in andrew 2005 2006 8 9 for computing sturm liouville potentials from the given eigenvalues by using the asymptotic estimate for the eigenvalue of the sturm liouville problem and the error in the finite difference eigenvalue convergence of numerov s method for symmetric potentials is proved based on the method of symmetric extension we establish a convergence result of numerov s method for the nonsymmetric potential from two spectra numerical examples are reported to confirm the theoretically predicted convergence

a polynomial invariant of graphs on orientable surfaces we consider cyclic graphs that is graphs with cyclic orders at the vertices corresponding to 2 cell embeddings of graphs into orientable surfaces or combinatorial maps we construct a three variable polynomial invariant of these objects the cyclic graph polynomial which has many of the useful properties of the tutte polynomial although the cyclic graph polynomial generalizes the tutte polynomial its definition is very different and it depends on the embedding in an essential way 2000 mathematical subject classification 05c10

an inverse spectral problem for a nonsymmetric differential operator reconstruction of eigenvalue problem in this paper an algorithm is established to reconstruct an eigenvalue problem from the given data satisfying certain conditions these conditions are proved to be not only necessary but also sufficient for the given data to coincide with the spectral characteristics corresponding to the reconstructed eigenvalue problem

boundary value method for inverse sturm liouville problems in this paper we present a method to recover symmetric and non symmetric potential functions of inverse sturm liouville problems from the knowledge of eigenvalues the linear multistep method coupled with suitable boundary conditions known as boundary value method bvm is the main tool to approximate the eigenvalues in each iteration step of the used newton method the bvm was extended to work for neumann neumann boundary conditions moreover a suitable approximation for the asymptotic correction of the eigenvalues is given numerical results demonstrate that the method is able to give good results for both symmetric and non symmetric potentials

a comparison of two methods for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium abstract a comparison is made of two methods for solving the inverse scattering problem for acoustic waves in an inhomogeneous absorbing medium these methods were recently introduced by colton kirsch and monk and are based on considering a weighted average of the far field data instead of the far field data itself the comparison is based on numerical examples for the special case of spherically stratified media

a second degree method for nonlinear inverse problems the paper is concerned with the solution of nonlinear ill posed problems by methods that utilize the second derivative a general predictor corrector approach is developed one which avoids solving quadratic equations during the iteration process combining regularization of each iteration step with an adequate stopping condition leads to a general regularization scheme for nonlinear equations possible implementations and discussion of the performance of this method are illustrated by applications to some well known inverse problems

two component eigenfunction expansion for open systems described by the wave equation ii linear space structure for a broad class of open systems described by the wave equation the eigenfunctions which are quasinormal modes provide a complete basis for simultaneously expanding outgoing wavefunctions in this paper the linear space structure associated with this expansion is developed under a modified inner product the time evolution operator is self adjoint even though energy is not conserved for the system alone thus the eigenfunctions are mutually orthogonal consequently the usual tools of eigenfunction expansions can be transcribed to these open systems as an example the time independent perturbation theory is developed in straightforward analogy with quantum mechanics giving the shift in both the real part and the imaginary part of the eigenvalues

graphs determined by polynomial invariants many polynomials have been defined associated to graphs like the characteristic matchings chromatic and tutte polynomials besides their intrinsic interest they encode useful combinatorial information about the given graph it is natural then to ask to what extent any of these polynomials determines a graph and in particular whether one can find graphs that can be uniquely determined by a given polynomial in this paper we survey known results in this area and at the same time we present some new results

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

equivalence of inverse sturm liouville problems with boundary conditions rationally dependent on the eigenparameter abstract three inverse problems for a sturm liouville boundary value problem y qy y y 0 cos y 0 sin and y 1 f y 1 are considered for rational f it is shown that the weyl m function uniquely determines f and q and is in turn uniquely determined by either two spectra from different values of or by the prufer angle for this it is necessary to produce direct results of independent interest on asymptotics and oscillation

reconstruction of a radially symmetric potential from two spectral sequences we consider the problem of determining a radially symmetric potential in the three dimensional schrodinger equation from eigenvalues associated with two different angular momentum quantum numbers this leads to a singular eigenvalue problem for which there are no known general uniqueness results for the most reasonable conjectures we are able to give strong evidence for uniqueness in some cases and we discuss a computational solution method together with some supporting analysis one application we have in mind is the determination of certain physical parameters in the standard model of the sun constructed from eigenvalue data

stability in the inverse nodal solution for the interior transmission problem abstract the inverse nodal problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from zeros for eigenfunctions corresponding to the transmission eigenvalues it is shown that the space of potential functions q which correspond to interior transmission problems characterized by q q l 2 0 1 under a certain metric is homeomorphic to the partition set of the space of quasinodal sequences as a consequence the inverse nodal problem defined on the partition set of admissible sequence is stable

uncoupling the perron eigenvector problem abstract for a nonnegative irreducible matrix a with spectral radius this paper is concerned with the determination of the unique normalized perron vector which satisfies a 62 0 j j 1 it is explained how to uncouple a large matrix a into two or more smaller matrices say p 11 p 22 p kk such that this sequence of smaller matrices has the following properties 1 each p ii is also nonnegative and irreducible so that each pii has a unique perron vector i 2 each pii has the same spectral radius as a 3 it is possible to determine the i s completely independently of each other so that one can execute the computation of the i s parallel 4 it is easy to couple the smaller perron vectors i back together in order to produce the perron vector for the original matrix a

determination of a finite range potential from discrete phase shift data by inverse scattering a practical method of inverse scattering is proposed to determine a finite range scattering potential from phase shift data evaluated at a set of discrete energies by using the asymptotic behaviour of the phase shifts and the connection between short and long wavelength scales a relatively few input data are found to be adequate the method works well for repulsive and moderately attractive potentials useful in many applications

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

the recovery of even polynomial potentials we consider the problem of reconstructing an even polynomial potential from one set of spectral data of a sturm liouville problem we show that we can recover an even polynomial of degree 2m from m 1 given taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum the idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function we then compute these coefficients by solving a nonlinear algebraic system and provide numerical examples at the end because of its algebraic nature the method applies also to non self adjoint problems

inverse problem with transmission eigenvalues for the discrete schrodinger equation the discrete schrodinger equation with the dirichlet boundary condition is considered on a half line lattice when the potential is real valued and compactly supported the inverse problem of recovery of the potential from the corresponding transmission eigenvalues is analyzed the marchenko method and the gel fand levitan method are used to solve the inverse problem uniquely except in one unusual case where the sum of the transmission eigenvalues is equal to a certain integer related to the support of the potential it is shown that in the unusual case there may be a unique potential corresponding to a given set of transmission eigenvalues there may be a finite number of distinct potentials for a given set of transmission eigenvalues or there may be infinitely many potentials for a given set of transmission eigenvalues the theory presented is illustrated with several explicit examples

reconstruction of a potential on the line that is a priori known on the half line in this paper the authors study the inverse schrodinger scattering on the real line a method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies in particular no information on the forming constants is required the method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the schrodinger equation the deift trubowitz approach to inverse scattering is followed the main new point is the way in which bound states are treated in addition to its mathematical interest the case when the potential is a priori known on the half line is particularly interesting in many applications one can consider for example a potential that has compact support or that it is zero on a half line

inverse scattering with partial information on the potential the one dimensional schrodinger equation is considered when the potential is real valued and integrable and has a finite first moment the recovery of such a potential is analyzed in terms of the scattering data consisting of a reflection coefficient all the bound state energies knowledge of the potential on a finite interval and all of the bound state norming constants except one it is shown that a missing norming constant in the data can cause at most a double nonuniqueness in the recovery in the particular case when the missing norming constant in the data corresponds to the lowest energy bound state the necessary and sufficient conditions are obtained for the nonuniqueness and the two norming constants and the corresponding potentials are determined some explicit examples are provided to illustrate the nonuniqueness

numerical solution of the goursat problem on a triangular domain with mixed boundary conditions abstract for the goursat problem we consider a triangular domain with mixed dirichlet and impedance boundary conditions imposed on it we develop an algorithm for its numerical solution mainly based on runge kutta method and trapezoidal formula iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition error estimates are derived examples are presented to illustrate the effectiveness of the method

inverse nodal problems for the sturm liouville operator with a constant delay we study inverse nodal problems for non self adjoint second order differential operator with a constant delay the uniqueness theorem is proved for this inverse problem and a constructive procedure for the solution of the inverse nodal problems is provided we show that the space of delay differential operators characterized by q l1 a 0 2 a 0 under a certain metric is homeomorphic to the partition set of the space of quasinodal sequences which is all admissible sequences x xnj n 2 j 1 n 1 which form sequences that converge to q and individually as a consequence the inverse nodal problem when defined on the partition set of admissible sequence induced by the same equivalence relation is stable

on the choice of the regularization parameter in nonlinear inverse problems this paper focuses on regularization techniques for nonlinear ill posed inverse problems tikhonov regularization and regularization due to the use of norm constraints are analyzed a model function technique is proposed to iteratively determine an optimal regularization parameter or the parameter characterizing the norm constraint and to estimate the error in the data if it is not known a priori

jordan blocks and generalized bi orthogonal bases realizations in open wave systems dissipation can sometimes be described by a non hermitian hamiltonian h whose left and right eigenvectors f j fj form a bi orthogonal basis bb for waves in a class of open systems this is known to lead to exact complete bb expansions if f j fj 0 for all j if not normalization seems impossible and many familiar formulae fail examples are given the problem is related to the merging of eigenmodes so that h can only be diagonalized to jordan blocks the resolution involves a generalized bb containing extra vectors whose dynamics are modified by polynomials in the time t the splitting of merged modes under a perturbation is also treated one thus obtains a non trivial extension of the bb formalism for open systems

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

inverse scattering problems where the potential is not absolutely continuous on the known interior subinterval the inverse scattering problem for the schrodinger operators on the line is considered when the potential is real valued and integrable and has a finite first moment it is shown that the potential on the line is uniquely determined by the left or right reflection coefficient alone provided that the potential is known on a finite interval and it is not absolutely continuous almost everywhere on this known interval

phaseless inverse scattering and the phase problem in optics two related problems are considered i the inverse scattering problem for a potential v x supported on the half line x 0 when the given data is r k the amplitude of the reflection coefficient and ii determination of a function g t supported on the half line t 0 when the given data is g k the amplitude of the fourier transform of g under certain conditions on v or g uniqueness theorems are proved and computational methods are developed a numerical example of recovery of v x from r k is given

numerical technique for the inverse resonance problem motivated by the work of regge nuovo cimento 8 1958 671 9 1958 491 we are interested in the problem of recovering a radial potential in r3 from its resonance parameters which are zeros of the appropriately defined jost function for a potential of compact support these may in turn be identified as the complex eigenvalues of a nonselfadjoint sturm liouville problem with an eigenparameter dependent boundary condition in this paper we propose and study a particular computational technique for this problem based on a moment problem for a function g t which is related to the boundary values of the corresponding gelfand levitan kernel

recovery of multiple coefficients in a reaction diffusion equation abstract this paper considers the inverse problem of recovering both the unknown spatially dependent conductivity a x and the potential q x in a parabolic equation from overposed data consisting of the value of solution profiles taken at a later time t we show both uniqueness results and the convergence of an iteration scheme designed to recover these coefficients we also allow a more general setting in particular when the usual time derivative is replaced by one of fractional order and when the potential term is coupled with a known nonlinearity f of the form q x f u

recovering the density of a string from only lowest frequency data a classical inverse problem is can you hear the density of a string clamped at both ends the answer as narrowly stated is negative one has to give an additional data sequence such as the energy in each mode norming constants or information on the string slope at the endpoints or alternatively change the boundary conditions and obtain the new frequencies what if neither of these options is possible one solution recently proposed was to add a known mass density and remeasure the dirichlet spectrum w rundell and p e sacks siam j appl math 73 2013 pp 1020 1037 in this work we take a related approach we are able to add any number of known mass densities but we can only measure the fundamental frequency that is the lowest eigenvalue in each case we will show that our problem can be closely approximated by a related problem for which there is uniqueness and we discuss a suite of reconstruction methods

an inverse sturm liouville problem with a fractional derivative in this paper we numerically investigate an inverse problem of recovering the potential term in a fractional sturm liouville problem from one spectrum the qualitative behaviors of the eigenvalues and eigenfunctions are discussed and numerical reconstructions of the potential with a newton method from finite spectral data are presented surprisingly it allows very satisfactory reconstructions for both smooth and discontinuous potentials provided that the order a 1 2 of fractional derivative is sufficiently away from 2

schur complements and its applications to symmetric nonnegative and z matrices in linear algebra appl 177 1992 137 smith proved that if h is a hermitian semidefinite matrix and a is a nonsingular principal submatrix then the eigenvalues of the schur complement h a interlace those of h in this paper we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve perron complements for an irreducible symmetric nonnegative matrix we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order we apply our results to an irreducible symmetric z matrix and to the adjacency matrix or the general laplacian matrix of a connected weighted graph the equality cases for the bounds for spectral radii or least eigenvalues are also examined

an inverse problem in coupled mode theory we study an inverse problem for the zakharov shabat system which is motivated by an application to the design of co directional couplers with prescribed response properties

on the spectral characterization of t shape trees a graph g is said to be determined by its spectrum if any graph having the same spectrum as g is isomorphic to g a t shape is a tree with exactly one of its vertices having maximal degree 3 in this paper we show that all t shape trees are determined by their spectra except for a few well defined cases

boundary value methods for the reconstruction of sturm liouville potentials abstract in this paper we present numerical procedures for solving the two inverse sturm liouville problems known in the literature as the two spectra and the half inverse problems the method proposed looks for a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension in order to compute such an approximation a sequence of direct problems has to be solved this is done by applying one of the boundary value methods generalizing the classical numerov scheme recently introduced by the authors numerical results confirming the effectiveness of the approach proposed are also reported

recovery of a potential from the ratio of reflection and transmission coefficients for the one dimensional schrodinger equation the analysis is provided to recover the potential from the data consisting of the ratio of a reflection coefficient to the transmission coefficient it is investigated whether such data uniquely constructs a reflection coefficient the number of bound states bound state energies bound state norming constants and a corresponding potential in all three cases when there is no knowledge of the support of the potential the support of the potential is confined to a half line and the support is confined to a finite interval various uniqueness and nonuniqueness results are established the precise criteria are provided for the uniqueness and the nonuniqueness and the degree of nonuniqueness and the recovery is illustrated with some explicit examples

a symmetric function generalization of the chromatic polynomial of a graph abstract for a finite graph g with d vertices we define a homogeneous symmetric function xg of degree d in the variables x1 x2 if we set x1 xn 1 and all other xi 0 then we obtain g n the chromatic polynomial of g evaluated at n we consider the expansion of xg in terms of various symmetric function bases the coefficients in these expansions are related to partitions of the vertices into stable subsets the mobius function of the lattice of contractions of g and the structure of the acyclic orientations of g the coefficients which arise when xg is expanded in terms of elementary symmetric functions are particularly interesting and for certain graphs are related to the theory of hecke algebras and kazhdan lusztig polynomials

waves in open systems via a biorthogonal basis dissipative quantum systems are sometimes phenomenologically described in terms of a non hermitian hamiltonian h with different left and right eigenvectors forming a biorthogonal basis it is shown that the dynamics of waves in open systems can be cast exactly into this form thus providing a well founded realization of the phenomenological description and at the same time placing these open systems into a well known framework the formalism leads to a generalization of norms and inner products for open systems which in contrast to earlier works is finite without the need for regularization the inner product allows transcription of much of the formalism for conservative systems including perturbation theory and second quantization

on the bivariate permanent polynomials of graphs abstract in 1980 balasubramanian and parthasarathy introduced the bivariate permanent polynomials of graphs and conjectured that this polynomial is a graph characterizing polynomial that is any two graphs with the same bivariate permanent polynomial are isomorphic in this paper we give counterexamples to this conjecture by a computer search furthermore we show that several well known families of graphs are determined by the bivariate permanent polynomial complete graphs complete bipartite graphs regular complete multipartite graphs cycles and their complements

inverses of perron complements of inverse m matrices the concept of the perron complement of a nonnegative and irreducible matrix was introduced by meyer in 1989 and it was used by him to construct an algorithm for computing the stationary distribution vector for markov chains here we consider properties of the perron complement of an n n matrix k which is an inverse of an irreducible m matrix we first show that the perron complements of k are inverses of m matrices and that the inverses of associated principal submatrices of k are sandwiched between the inverses of the perron complements of k and the inverses of the corresponding schur complements of k we then investigate the directed graph of the inverse of the perron complements of such matrices k

numerical solution of the inverse spectral problem for bessel operators we consider some inverse spectral problems associated with the singular sturm liouville equation u q x 1 x 2 u lu0

localization of perron roots abstract this paper is concerned with the localization of the perron root of a nonnegative irreducible matrix a a new localization method that utilizes the relationship between the perron root of a nonnegative matrix and the estimates of the row sums of its generalized perron complement is presented the method is efficient because it gives the bounds on a only by computing the estimates of the row sums of the generalized perron complement rather than the generalized perron complement itself several numerical examples are given to illustrate the effectiveness of our method

multifrequency inverse problem for the reduced wave equation with sparse data the inverse problem for the reduced wave equation u k2n2 x u 0 is considered where the quantity to be determined is the value of n x in a compact domain d in r3 the data consists of a finite set of measurements of the scattered field produced by different incident fields the measurements are made at various points exterior to d and at possibly different frequencies the mathematical problem involves solving a system of nonlinear functional equations conditions are developed which indicate when the measured body is close to a particular comparison body a linearized perturbation is valid a new higher order nonlinear iterative procedure is developed for the full nonlinear problem which reduces to the usual solution in the linearized region the method is illustrated by computational results for the one dimensional case

reconstruction of a spherically symmetric speed of sound consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises among other places in nondestructive testing define the corresponding homogeneous and inhomogeneous interior transmission problems see e g d colton and p monk quart j mech math 41 1988 pp 97 125 here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem it is shown in mclaughlin and polyakov j differential equations to appear that these eigenvalues are the zeros of an average of the scattering amplitude and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data in the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented the method given here completely determines the sound

a modified algorithm for the perron root of a nonnegative matrix abstract an algorithm of diagonal transformation for the perron root of nonnegative matrices is proposed by duan and zhang f duan k zhang an algorithm of diagonal transformation for perron root of nonnegative irreducible matrices appl math comput 175 2006 762 772 this method can be used for all nonnegative irreducible matrices in this paper an improved algorithm which is based on this method is proposed the new algorithm inherits all the above mentioned advantages of the original algorithm and has higher efficiency it is testified by numerical testing that the efficiency of the new algorithm is improved greatly

two dimensional monomer dimer systems are computationally intractable the classic problem of counting monomer dimer arrangements on a two dimensional lattice is analyzed using techniques from theoretical computer science under a certain assumption made precise in the text it can be shown that the general problem is computationally intractable this negative result contrasts with the special case of a system with monomer density zero for which efficient solutions have been known for some time a second much easier result obtained under the same assumption is that the partition function of a three dimensional ising system is computationally intractable again the negative result contrasts with known efficient techniques for evaluating the partition function of a two dimensional system

numerical algorithm for the determination of the potential of a conservative system from its normal mode spectra an efficient numerical algorithm is developed for determining the potential of a conservative system described by a klein gordon or schrodinger equation from its normal mode spectra the algorithm is based on first solving the linear problem of perturbative inversion by means of tikhonov regularization and then using a series of perturbative steps to approach the full potential by newton s method the algorithm is stable and convergence is rapid

a computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

transmission eigenvalues and a problem of hans lewy abstract subject to certain assumptions on the matrix n n x we show that except for a discrete set of values of k called transmission eigenvalues the only solution of n u k 2 u 0 v k 2 v 0 in d u v u v on d where d r n is a domain with smooth boundary d having unit outward normal is the trivial solution u v 0 this problem has important applications to the inverse scattering problem for anisotropic media and is in a class of problems first considered by lewy bull amer math soc 65 1959 37 58

perron complement and perron root for a nonnegative irreducible matrix a this paper is concerned with the estimation and determination of the unique perron root or spectral radius of a we present a new method that utilizes the relation between perron roots of the nonnegative matrix and its generalized perron complement several numerical examples are given to show that our method is effective at least for some classes of nonnegative matrices

multifrequency inverse problem for the reduced wave equation resolution cell and stability the multifrequency inverse problem associated with the reduced wave equation u k2n2 x u 0 x r3 is examined for the case where the data set is sparse the resolution cell or solution set is examined in detail and is shown to be an infinite dimensional manifold the concept of stability is introduced it is shown that the intrinsic condition of structural stability to the inverse process selects out a preferred set of solutions from the solution set the structural stability of various iterative schemes used in the inverse process are examined

bound states and inverse scattering for the schrodinger equation in one dimension the one dimensional schrodinger equation is considered when the potential and its first moment are absolutely integrable when the potential has support contained on the left right half line it is uniquely constructed by using only the reflection coefficient from the right left the bound state norming constants determine whether the potential has support contained on a half line or on the full line the bound state energies and the unique set of norming constants yielding the potential with support contained on the left right half line are completely determined by the reflection coefficient from the right left an explicit example is provided

bvms for computing sturm liouville symmetric potentials abstract the paper deals with the numerical solution of inverse sturm liouville problems with unknown potential symmetric over the interval 0 the proposed method is based on the use of a family of boundary value methods obtained as a generalization of the numerov scheme aimed to the computation of an approximation of the potential belonging to a suitable function space of finite dimension the accuracy and stability properties of the resulting procedure for particular choices of such function space are investigated the reported numerical experiments put into evidence the competitiveness of the new method

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

on the inverse resonance problem for schrodinger operators we consider schrodinger operators on 0 with compactly supported possibly complex valued potentials in l1 0 it is known at least in the case of a real valued potential that the location of eigenvalues and resonances determines the potential uniquely from the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data in this paper we prove the validity of this statement i e we show conditional stability for finite data as a by product we also obtain a uniqueness result for the inverse resonance problem for complex valued potentials

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

reconstruction of potential energy profiles from multiple rupture time distributions we explore the mathematical and numerical aspects of reconstructing a potential energy profile of a molecular bond from its rupture time distribution while reliable reconstruction of gross attributes such as the height and the width of an energy barrier can be easily extracted from a single first passage time fpt distribution the reconstruction of finer structure is ill conditioned more careful analysis shows the existence of optimal bond potential amplitudes represented by an effective peclet number and initial bond configurations that yield the most efficient numerical reconstruction of simple potentials furthermore we show that reconstruction of more complex potentials containing multiple minima can be achieved by simultaneously using two or more measured fpt distributions obtained under different physical conditions for example by changing the effective potential energy surface by known amounts additional measured fpt distributions improve the reconstruction we demonstrate the possibility of reconstructing potentials with multiple minima motivate heuristic rules ofthumb for optimizing the reconstruction and discuss further applications and extensions

determination of partially known sturm liouville potentials abstract it is well known that the determination of a general potential function employs two sets of information one or both may be a sequence of eigenvalues in this work a partially known potential and one finite sequence of eigenvalues are given in order to approximate a non symmetric potential we utilize not only the finite difference method but also numerov s method to approximate the necessary eigenvalues in each iteration step a complete set of eigenvalues is the main information to approximate a potential in this paper a method recovering a missing eigenvalue in the sequence is given

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

uniqueness of the interior transmission problem with partial information on the potential and eigenvalues abstract the inverse spectral problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b a uniqueness theorem for the potential q of the derived sturm liouville problem b q is presented from the data involving fractions of the eigenvalues of the problem b q on a finite interval and knowledge of q over a corresponding fraction of the interval the methods employed rest on weyl function techniques and properties of zeros of a class of entire functions

inverse spectral problems for the sturm liouville operator with discontinuity abstract in this work we consider the sturm liouville operator on a finite interval 0 1 with discontinuous conditions at 1 2 we prove that if the potential is known a priori on a subinterval b 1 with b 1 2 then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions for the case b 1 2 parts of either one or two spectra can uniquely determine the potential and a part of parameters

on the determination of radially dependent lame coefficients in this paper we prove that one can determine radially dependent lame coefficients lambda and mu of an isotropic elastic ball by choosing only two correspondences in the dirichlet to neumann map i e twice making measurements of displacement and stress on the surface of the ball this is a one dimensional multiunknown undetermined coefficient problem for a system of partial differential equations we use a matrix form of the transmutation mapping for solving this problem the procedure for proving the uniqueness based on transmutation can be viewed as a computational algorithm also the method we develop has the potential of solving other one dimensional multiunknown inverse problems

the lower and upper bounds on perron root of nonnegative irreducible matrices let a be an nxn nonnegative irreducible matrix let a a be the principal submatrix of a based on the nonempty ordered subset a of 1 2 n and define the generalized perron complement of a a by p t a a a i e p t a a a a b a b a ti a a 1a a b t r a a this paper gives the upper and lower bounds on the perron root of a an upper bound on perron root is derived from the maximum of the given parameter t 0 and the maximum of the row sums of p t 0 a a a synchronously a lower bound on perron root is expressed by the minimum of the given parameter t 0 and the minimum of the row sums of p t 0 a a a it is also shown how to choose the parameter t after a to get tighter upper and lower bounds of r a several numerical examples are presented to show that our method compared with the methods in l z lu m k ng locations of perron roots linear algebra appl 392 2004 103 117 is more effective

sturm liouville problems with boundary conditions rationally dependent on the eigenparameter ii necessary and sufficient conditions are given for two sequences n and n to be the eigenvalues and norming constants of a sturm liouville boundary value problem y qy y y 0 cos y 0 sin and y 1 f y 1 where f is a rational function of herglotz nevanlinna type it is also proved that q and f are uniquely determined by the sequences n and n

on the determination of potentials without bound state data abstract we investigate the feasibility of recovering a compactly supported potential from the s matrix for general potentials this is known to be insufficient information on the bound states being essential we prove uniqueness results and derive reconstruction algorithms for potential recovery using several different combinations of s matrix data the basic device is the transformation of the scattering data into boundary conditions for a related hyperbolic boundary value problem some generalizations to the case of step like potentials are proved