efficient sequential designs with binary data abstract a class of sequential designs for estimating the percentiles of a quantal response curve is proposed its updating rule is based on an efficient summary of all of the data available via a parametric model the logit mle version of the proposed designs can be viewed as a natural analog of the robbins monro procedure in the case of binary data it is shown to be asymptotically consistent optimal and nonparametric via its connection with the latter procedure for certain choices of initial designs the proposed method performs very well in a simulation study for sample sizes up to 35 a nonparametric sequential design via the spearman karber estimator for estimating the median is also proposed

a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

the inverse nodal problem and the ambarzumyan problem for the p laplacian we study the issues of the reconstruction and stability of the inverse nodal problem for the one dimensional p laplacian eigenvalue problem a key step is the application of a modified prufer substitution to derive a detailed asymptotic expansion for the eigenvalues and nodal lengths two associated ambarzumyan problems are also solved

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

half inverse problem for diffusion operators on the finite interval abstract the potential function q x in the regular and singular sturm liouville problem can be uniquely determined from two spectra inverse problem for diffusion operator given at the finite interval eigenvalues normal numbers also on two spectra are solved half inverse spectral problem for a sturm liouville operator consists in reconstruction of this operator by its spectrum and half of the potential in this study by using the hochstadt and lieberman s method we show that if q x is prescribed on 2 then only one spectrum is sufficient to determine q x on the interval 0 2 for diffusion operator

existence of a tree of stieltjes strings corresponding to two given spectra transversal vibrations of a plane rooted tree of stieltjes strings are considered with dirichlet boundary conditions at each pendant vertex continuity and kirchhoff conditions are imposed at each interior vertex except the root the dirichlet problem is that with the dirichlet condition at the root and the neumann problem is that with continuity and kirchhoff conditions at the root it is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of neumann and dirichlet problems generated by stieltjes string recurrent relations on any prescribed tree

a proof of the local borg marchenko theorem in 2 5 a local version of a basic uniqueness theorem in inverse spectral theory is given this paper gives a simple proof of this theorem

extension of ambarzumyan s theorem to general boundary conditions abstract we extend the classical ambarzumyan s theorem for the sturm liouville equation which is concerned only with neumann boundary conditions to the general boundary conditions by imposing an additional condition on the potential function our result supplements the poschel trubowitz inverse spectral theory we also have parallel results for vectorial sturm liouville systems

inverse spectral theory for a singular sturm liouville operator on 0 1 xer3 acting on functions in p 1x1 1 vanishing on 1x1 1 we set x 1x1 and assume q x is a real valued function in l 0 11 the closure h of i is a self adjoint operator with compact resolvent hence h has pure point spectrum without finite accumulation points moreover using the mapping u x xu x and expansion in spherical harmonics one shows that h is unitarily equivalent to a direct sum of sturm liouville operators acting on l o 11 cf 11 the first operator in this sum is

optimal sequential designs of case control studies fixed case control studies separately collect a case sample and a control sample with the two sample sizes being fixed prior to studies and sometimes arbitrarily chosen this often results in loss of efficiency of case control designs in terms of cost saving or time saving of the studies we study sequential case control designs and in connection with treatment allocation and stochastic approximation derive a simple sampling rule that leads to optimal case control designs some important issues such as fixed width confidence intervals and sequential tests of hypotheses with possible early stopping to save time or costs which cannot be answered with fixed case control designs are shown to be naturally solved with the derived optimal sequential case control designs

recovering differential pencils on compact graphs abstract we study boundary value problems on compact graphs without circles i e on trees for second order ordinary differential equations with nonlinear dependence on the spectral parameter we establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so called weyl vector which is a generalization of the weyl function m function for the classical sturm liouville operator for this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings

bayesian sample size determination for case control studies case control studies are among the most commonly used means of assessing association between exposure and outcome sample size determination and the optimal control to case ratio are vital to the design of such studies in this article we investigate bayesian sample size determination and the control to case ratio for case control studies when interval estimation is the goal of the eventual statistical analysis in certain cases we are able to derive approximate closed form sample size formulas we also describe two monte carlo methods each of which provides a unified approach to the sample size problem because they may be applied to a wide range of interval based criteria we compare the accuracy of the different methods we also extend our methods to include cross sectional designs and designs for gene environment interaction studies

the free dirac operator on compact and noncompact graphs the free dirac operator defined on composite one dimensional structures consisting of finitely many half lines and intervals is investigated the influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self adjoint extensions of the dirac operator general relations between the parameters of the extensions and the eigenvalues resp the scattering coefficients are derived and then applied to the cases of a bundle of half lines a point defect a branching line and an eye shaped structure

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

on multiplicity of a quantum graph spectrum the sturm liouville equations on the edges of a metric connected graph together with the boundary and matching conditions at the vertices generate a spectral problem for a self adjoint operator it is shown that if the graph is not cyclically connected then the maximal multiplicity of an eigenvalue of such an operator is gt ptin where is the cyclomatic number of the graph and gt and ptin are the number of edges and the number of interior vertices respectively for the tree obtained by contracting all the cycles of the graph into vertices if the graph is cyclically connected then the maximal multiplicity of an eigenvalue is 1

inverse spectral problems for the sturm liouville operator on a d star graph abstract in this work we consider inverse spectral problems for the sturm liouville differential operator on a d star type graph with standard matching conditions in the internal vertex where the integer d 2 by using the yurko s method yurko 2008 27 yurko 2009 28 we show that 1 if the potential function q j x on a fixed edge e j is prescribed on the interval 2 then the reciprocal of d of the spectrum suffices to determine q j x on the whole interval 0 2 the 2 over d of the spectrum suffices to determine q j x on a fixed edge e j

on local borg marchenko uniqueness results we provide a new short proof of the following fact first proved by one of us in 1998 if two weyl titchmarsh m functions m j z of two schrodinger operators h j d 2 dx 2 q j j 1 2 in l 2 o r o r are exponentially close that is m 1 z m 2 z z o e 2 im z 1 2 a o r then q 1 q 2 a e on o the result applies to any boundary conditions at x o and x r and should be considered a local version of the celebrated borg marchenko uniqueness result which is quickly recovered as a corollary to our proof moreover we extend the local uniqueness result to matrix valued schrodinger operators

visibility of quantum graph spectrum from the vertices we investigate the relation between the eigenvalues of the laplacian with kirchhoff vertex conditions on a finite metric graph and a corresponding titchmarsh weyl function a parameter dependent ne

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

corrigendum to extension of ambarzumyan s theorem to general boundary conditions j math anal appl 263 2001 333 342

inverse problems for the differential operator on a graph with cycles abstract this work deals with inverse problems of the sturm liouville operator on a graph with multi cycles endowed with standard matching conditions in the internal vertex and with neumann boundary conditions at the boundary vertices we show that the potential on a graph with multi cycles including the potential on cycles can be constructed by the dense nodal points on the interval considered moreover we investigate the so called incomplete inverse problems of recovering the potential on a fixed edge from a subset of nodal points situated only on a part of the edge

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

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

a generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system abstract a generalized eigenvalue algorithm for a certain class of tridiagonal matrix pencils is presented the algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil the convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system

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

determining the resistors in a network in this paper it is shown that the resistors in a rectangular network can be determined by measurements at the boundary of the voltages generated by imposed currents an algorithm for using the boundary measurements to compute the resistances is also given

the inverse spectral problem for radial schrodinger operators on 0 1 we consider an inverse spectral problem for a class of singular sturm liouville operators on the unit interval with explicit singularity a a 1 x2 a n related to the schrodinger operator with a radially symmetric potential the purpose of this paper is the global parametrization of potentials by the spectral data a and some norming constants a for a 0 or 1 a a is already known to be a global coordinate system on lr2 0 1 using some transformation operators we show that this result holds for any non negative integer a moreover we give a description of the isospectral sets

a local borg marchenko theorem for complex potentials the borg marchenko theorem states that the weyl titchmarsh m function of the differential expression d2 dx2 q with a real valued potential q determines this potential uniquely we investigate the validity of the borg marchenko theorem and its local version for complex valued potentials

simultaneous band reduction of two symmetric matrices abstract in this paper we consider simultaneous band reduction of two dense symmetric matrices by congruent transformations the ideas of simultaneous tridiagonalization are generalized to propose an efficient algorithm for the simultaneous band reduction in contrast to the algorithms of simultaneous tridiagonalization which are mainly based on matrix vector operations the proposed algorithm of simultaneous band reduction has the advantage that matrix matrix operations can be fully used to achieve better performance on modern computer architecture numerical results are presented to illustrate the effectiveness of our proposed algorithm

a borg levinson theorem for trees we prove that the dirichelet to neumann map for a schrodinger operator on a finite simply connected tree determines uniquely the potential on that tree

a tikhonov regularization for the inverse nodal problem for p laplacian abstract we apply the tikhonov regularization method to reconstruct potentials of a p laplacian eigenvalue problem using only zeros of one eigenfunction with explicit error bounds this is an inverse problem on a class of nonlinear equations when p 2 the study is motivated by our previous work on sturm liouville operators chen et al 2011 6

ambarzumyan type theorems on graphs with loops and double edges abstract in this work we consider inverse spectral problems for two types of graphs with loops and or double edges it is shown that analogs of ambarzumyan s theorem are true for sturm liouville problems on graphs with neumann boundary conditions at the pendant vertices and kirchhoff s conditions at the interior vertices

an inverse spectral problem for a star graph of krein strings we solve an inverse spectral problem for a star graph of krein strings where the known spectral data comprises the spectrum associated with the whole graph the spectra associated with the individual edges as well as so called coupling matrices in particular we show that these spectral quantities uniquely determine the weight within the class of borel measures on the graph which give rise to trace class resolvents furthermore we obtain a concise characterization of all possible spectral data for this class of weights

consistency and asymptotic efficiency of slope estimates in stochastic approximation schemes adaptive stochastic approximation schemes for choosing the levels of x at which y is to be observed are useful in applications of the following nature suppose that in 1 1 x is the dosage level of a drug given to the i th patient who turns up for treatment and that yi is the response of the patient suppose also that an optimal response value y is desired without loss of generality we shall replacing y by y y if necessary assume that y 0 if 0 were known then the dosage levels should all be set at 0 since 0 is usually unknown how can the dosage levels xi be chosen so as to approach the

some inverse scattering problems on star shaped graphs having in mind applications to the fault detection diagnosis of lossless electrical networks here we consider some inverse scattering problems for schrodinger operators over star shaped graphs we restrict ourselves to the case of minimal experimental setup consisting in measuring at most two reflection coefficients when an infinite homogeneous potential less branch is added to the central node first by studying the asymptotic behavior of only one reflection coefficient in the high frequency limit we prove the identiability of the geometry of this star shaped graph the number of edges and their lengths next we study the potential identification problem by inverse scattering noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings potentials with bounded h1 norms the main result states that under some assumptions on the geometry of the graph the measurement of two reflection coefficients associated to two different sets of boundary conditions at the external vertices of the tree determines uniquely the potentials it can be seen as a generalization of the theorem of the two boundary spectra on an interval

schrodinger operators on graphs and geometry i essentially bounded potentials the inverse spectral problem for schrodinger operators on finite compact metric graphs is investigated the relations between the spectral asymptotics and geometric properties of the underlying graph are studied it is proven that the euler characteristic of the graph can be calculated from the spectrum of the schrodinger operator in the case of essentially bounded real potentials and standard boundary conditions at the vertices several generalizations of the presented results are discussed

asymptotically isospectral quantum graphs and generalised trigonometric polynomials abstract the theory of almost periodic functions is used to investigate spectral properties of schrodinger operators on metric graphs also known as quantum graphs in particular we prove that two schrodinger operators may have asymptotically close spectra if and only if the corresponding reference laplacians are isospectral our result implies that a schrodinger operator is isospectral to the standard laplacian on a may be different metric graph only if the potential is identically equal to zero

a nodal inverse problem for a quasi linear ordinary differential equation in the half line abstract in this paper we study an inverse problem for a quasi linear ordinary differential equation with a monotonic weight in the half line first we find the asymptotic behavior of the singular eigenvalues and we obtain a weyl type asymptotics imposing an appropriate integrability condition on the weight then we investigate the inverse problem of recovering the coefficients from nodal data we show that any dense subset of nodes of the eigenfunctions is enough to recover the weight

simultaneous tridiagonalization of two symmetric matrices we show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations no assumptions are made on the non singularity or definiteness of the two matrices the reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via householder reflectors two algorithms are proposed one using non orthogonal rank one modifications of the identity matrix and the other more costly but more stable using a combination of householder reflectors and non orthogonal rank one modifications of the identity matrix with minimal condition numbers each of these tridiagonalization processes requires o n 3 arithmetic operations and respects the symmetry of the problem we illustrate and compare the two algorithms with some numerical experiments

determination of a differential pencil from interior spectral data abstract in this paper inverse spectra problems for a differential pencil are studied by using the approach similar to those in hochstadt and lieberman 1978 14 and ramm 2000 26 we prove that 1 if p x or q x is full given on the interval 0 then a set of values of eigenfunctions at the mid point of the interval 0 and one spectrum suffice to determine q x or p x on the interval 0 and all parameters in the boundary conditions 2 if p x or q x is full given on the interval 0 then some information on eigenfunctions at some internal point b 2 and parts of two spectra suffice to determine q x or p x on the interval 0 and all parameters in the boundary conditions

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

the dirichlet to neumann map for a resistor network relations that characterize the dirichlet to neumann map for a square network of resistors are given using these relations a parametrization of the set of dirichlet to neumann maps is given

fixed size confidence regions for parameters of a logistic regression model let mathbf x i y i be independent identically distributed observations that satisfy a logistic regression model that is for each i log lbrack p y i 1 mathbf x i p y i 0 mathbf x i rbrack mathbf x t i beta 0 where y i in 0 1 mathbf x i in mathbf r p and beta 0 in mathbf b p is the unknown parameter vector of the model the marginal distribution of the covariate vectors mathbf x i is assumed to be unknown sequential procedures for constructing fixed size and fixed proportional accuracy confidence regions for beta 0 are proposed and shown to be asymptotically efficient as the size of the region becomes small

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

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

scattering on graphs and one dimensional approximations to n dimensional schrodinger operators in the present article we develop the spectral analysis of schrodinger operators on lattice type graphs for the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices using an explicit expression for the resolvent of lattice type operator we develop in the second section appropriate lippmann schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix it serves as a base for the approximation of the multi dimensional schrodinger operator by a one dimensional operator on the graph in the third section of the paper for given n dimensional schrodinger operators with rapidly decreasing potential we construct a lattice type operator on a cubic graph embedded into rn and show that the original n dimensional scattering problem can be approximated in a p

dirichlet neumann inverse spectral problem for a star graph of stieltjes strings we solve two inverse spectral problems for star graphs of stieltjes strings with dirichlet and neumann boundary conditions respectively at a selected vertex called root the root is either the central vertex or in the more challenging problem a pendant vertex of the star graph at all other pendant vertices dirichlet conditions are imposed at the central vertex at which a mass may be placed continuity and kirchhoff conditions are assumed we derive conditions on two sets of real numbers to be the spectra of the above dirichlet and neumann problems our solution for the inverse problems is constructive we establish algorithms to recover the mass distribution on the star graph i e the point masses and lengths of subintervals between them from these two spectra and from the lengths of the separate strings if the root is a pendant vertex the two spectra uniquely determine the parameters on the main string i e the string incident to the root if the length of the main string is known the mass distribution on the other edges need not be unique the reason for this is the non uniqueness caused by the non strict interlacing of the given data in the case when the root is the central vertex finally we relate of our results to tree patterned matrix inverse problems

inverse scattering for lasso graph the inverse problem for the magnetic schrodinger operator on the lasso graph with different matching conditions at the vertex is investigated it is proven that the titchmarsh weyl function known f

the m matrix inverse problem for the sturm liouville equation on graphs abstractwe consider an inverse spectral problemfor sturm liouville boundaryvalue problemson a graph with formally self adjoint boundary conditions at the nodes where the giveninformation is the m matrix based on the results found in s currie b a watson m matrix asymptotics for sturm liouville problems on graphs j com appl math doi 10 1016 j cam 2007 11 019 using the green s function we prove that the poles ofthe m matrix are at the eigenvalues of the associated boundary value problem and aresimple located on the real axis and that the residue at a pole is a negative semi de nitematrix with rank equal to the multiplicity of the eigenvalue we de ne the so callednorming constants and relate them to the spectral measure and the m matrix thisenables us to recover from the m matrix the boundary conditions and the potential up to a unitary equivalence for co normal boundary conditions keywords inverse problem di erential operators on graphs m function sturm liouville 2000 msc 34a55 34b45 34b20 34l05 34b27

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

on the simultaneous tridiagonalization of two symmetric matrices we discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal tridiagonal form under the very mild assumption that the matrix pencil is regular we outline the general principles and propose a unified framework for the problem this allows us to gain new insights leading to an economical approach that only uses gauss transformations and orthogonal householder transformations numerical experiments show that the approach is numerically robust and competitive