abrikosov lattices in finite domains in 1957 abrikosov published his work on periodic solutions to the linearized ginzburg landau equations abrikosov s analysis assumes periodic boundary conditions which are very different from the natural boundary conditions the minimizer of the ginzburg landau energy functional should satisfy in the present work we prove that the global minimizer of the fully non linear functional can be approximated in every rectangular subset of the domain by one of the periodic solution to the linearized ginzburg landau equations in the plane furthermore we prove that the energy of this solution is close to the minimum of the energy over all abrikosov s solutions in that rectangle

fitting regression models to case control data by maximum likelihood summary we consider fitting categorical regression models to data obtained by either stratified or nonstratified case control or response selective sampling from a finite population with known population totals in each response category with certain models such as the logistic with appropriate constant terms a method variously known as conditional maximum likelihood breslow cain 1988 or pseudo conditional likelihood wild 1991 which involves the prospective fitting of a pseudo model results in maximum likelihood estimates of case control data we extend these results by showing the maximum likelihood estimates for any model can be found by iterating this process with a simple updating of offset parameters attention is also paid to estimation of the asymptotic covariance matrix one benefit of the results of this paper is the ability to obtain maximum likelihood estimates of the parameters of logistic models for stratified case control studies compare breslow cain 1988 scott wild 1991 using an ordinary logistic regression program even when the stratum constants are modelled

mixed normal superconducting states in the presence of strong electric currents we study the ginzburg landau equations in the presence of large electric currents that are smaller than the critical current where the normal state loses its stability for steady state solutions in the large kappa limit we prove that the superconductivity order parameter is exponentially small in a significant part of the domain and simply small in the rest of it similar results are obtained for the time dependent problem in continuation of the paper by the two first authors 3 we conclude by obtaining some weaker results albeit similar for steady state solutions in the large domain limit

a simplicial approach to the determination of an integer point of a simplex we propose an algorithm to find an integer point of a simplex the algorithm is based on an integer labeling rule and a triangulation of the space starting at an integer point the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension within a finite number of iterations the algorithm either yields an integer point of the simplex or proves that no such point exists

the 2 n 1 2 ray algorithm a new simplicial algorithm to compute economic equilibria a new variable dimension simplicial restart algorithm is introduced to compute economic equilibria the number of rays along which the algorithm can leave the starting point differs from the thusfar known algorithms more precisely the new algorithm has one ray to each of the 2n 1 2 faces of then dimensional price simplex whereas the existing algorithms haven 1 rays either to each facet or to each vertex of the unit simplex the path of points followed by the algorithm can be interpreted as a globally and universally convergent price adjustment process the process is also economically meaningful and therefore it is a good alternative for the well known walras tatonnement process computational results show that the algorithm is competitive with the most efficient simplicial algorithms developed thusfar

variable dimension algorithms basic theory interpretations and extensions of some existing methods in this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by van der laan and talman we introduce a new concept primal dual pair of subdivided manifolds and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms our basic model furnishes interpretations to several existing methods lemke s algorithm for the linear complementarity problem its extension to the nonlinear complementarity problem a variable dimension algorithm on conical subdivisions and merrill s algorithm we shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us a method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed

newton type minimization via the lanczos method this paper discusses the use of the linear conjugate gradient method developed via the lanczos method in the solution of large scale unconstrained minimization problems at each iteration of a newton type method the direction of search is defined as the solution of a quadratic subproblem when the number of variables is very large this subproblem may be solved using the linear conjugate gradient method of hestenes and stiefel we show how the equivalent lanczos characterization of the linear conjugate gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive definite hessian matrices at all points of the region of interest this derivation also makes it possible to compute a negative curvature direction at a stationary point the idea of a truncated newton method is to perform only a limited number of iterations of the quadratic subproblem this effectively gives a search direction that interpolates between the steepest de

a variable rate refining triangulation a new variable rate refining subdivisionb4 for the solution of equations with piecewise linear homotopies and restart homotopies is described the essential virtue of the subdivisionb4 is that it offers vast latitude to the user in particular the parameters ofb4 can be set so one restriction ofb4 is thatj3 triangulation and another restriction is the octahedral subdivision more generally full flexibility is available in the placement of local focal points and perhaps most importantly b4 permits the new capability of refining coordinates at variable andindependent rates

surface superconductivity in 3 dimensions we study the ginzburg landau system for a superconductor occupying a 3 dimensional bounded domain and improve the estimate of the upper critical field h c3 obtained by k lu and x pan in j diff eqns 168 2000 386 452 we also analyze the behavior of the order parameters we show that under an applied magnetic field lying below and not far from h c3 order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field and exponentially decay both in the normal and tangential directions away from the sheath in the l 2 sense as the applied field decreases further but keeps in between and away from h c2 and h c3 the superconducting sheath expands but does not cover the entire surface and superconductivity at the surface portion orthogonal to the applied field is always very weak this phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by x pan in comm math phys 228 2002 327 370 where under an axial applied field lying in between h c2 and h c3 the entire surface is in the superconducting state

a simplicial algorithm for stationary point problems on polytopes a simplicial variable dimension restart algorithm for the stationary point problem or variational inequality problem on a polytope is proposed given a polytope c in rn and a continuous function f c rn find a point x in c such that fx x fx x for any point x in c starting from an arbitrary point v in c the algorithm generates a piecewise linear path of points in c this path is followed by alternating linear programming pivot steps to follow a linear piece of the path and replacement steps in a simplicial subdivision of c within a finite number of function evaluations and linear programming pivot steps the algorithm finds an approximate stationary point the algorithm leaves the starting point v along a ray pointing to one of the vertices w of c the vertex w is obtained from the optimum solution of the linear programming problem maximize fv x subject to x c

non linear surface superconductivity for type ii superconductors in the large domain limit the ginzburg landau model for superconductivity is considered in two dimensions we show for smooth bounded domains that superconductivity remains concentrated near the surface when the applied magnetic field is decreased below hc3 as long as it is greater than hc2 we demonstrate this result in the large domain limit i e when the domain s size tends to infinity additionally we prove that for applied fields greater than hc2 the only solution in 2 satisfying normal state conditions at infinity is the normal state the above results have been proved in the past for the linear case here we prove them for non linear problems

computing an integer point of a simplex with an arbitrary starting homotopy like simplicial algorithm abstract an arbitrary starting homotopy like simplicial algorithm is developed for computing an integer point of an n dimensional simplex the algorithm is derived from the use of an integer labeling rule and a triangulation of r n 0 1 and consists of two interchanging phases one phase of the algorithm constitutes a homotopy simplicial algorithm which generates n 1 dimensional simplices in r n 0 1 and the other phase of the algorithm constitutes a pivoting procedure which generates n dimensional simplices in either r n 0 or r n 1 the algorithm varies from one phase to the other when the matrix defining the simplex is in the so called canonical form starting at an arbitrary integer point in r n 0 the algorithm within a finite number of iterations either yields an integer point of the simplex or proves that no such point exists

regression with missing x s a review abstract the literature of regression analysis with missing values of the independent variables is reviewed six classes of procedures are distinguished complete case analysis available case methods least squares on imputed data maximum likelihood bayesian methods and multiple imputation methods are compared and illustrated when missing data are confined to one independent variable and extensions to more general patterns are indicated attention is paid to the performance of methods when the missing data are not missing completely at random least squares methods that fill in missing x s using only data on the x s are contrasted with likelihood based methods that use data on the x s and y the latter approach is preferred and provides methods for elaboration of the basic normal linear regression model it is suggested that more widely distributed software is needed that advances beyond complete case analysis available case analysis and naive imputation methods bayesian simulation methods and mu

spectral modified polak ribiere polyak projection conjugate gradient method for solving monotone systems of nonlinear equations abstract in this paper we present a modification of polak ribi e re polyak prp conjugate gradient method for solving system of monotone nonlinear equations which is a combination of spectral conjugate gradient method and the hyperplane projection technique the method is based on two methods for unconstrained optimization proposed by wan et al 2011 and sun 2015 we obtained a new search direction by the use of a different formula for the conjugate gradient parameter the search direction satisfies the sufficient descent condition and the global convergence of the method is established under some assumptions preliminary numerical comparison with some existing methods shows the efficiency of the proposed method

implementation of a continuation method for normal maps this paper presents an implementation of a nonsmooth continuation method of which the idea was originally put forward by alexander et al we show how the method can be computationally implemented and present numerical results for variational inequality problems in up to 96 variables

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

emergence of the triangular lattice near a flat wall the bifurcation of periodic solutions near a flat wall for applied magnetic fields which are slightly weaker than hc2 is considered for a reduced ginzburg landau model obtained in the large limit we formally demonstrate that following the bifurcation of the first mode when the applied magnetic field is further decreased there is a second bifurcation after which the solution develops continuously into the well known triangular lattice

weighted semiparametric estimation in regression analysis with missing covariate data abstract this article investigates estimation of the regression coefficients in an assumed mean function when covariates on some subjects are missing we examine the performance of a horvitz and thompson 1952 type weighted estimator by using different estimates of the selection probabilities which may be treated as nuisance parameters or a nuisance function in particular we investigate the properties of the estimate of the regression parameters when the selection probabilities are estimated by kernel smoothers we present large sample theory for the new estimator and conduct simulation studies comparing the proposed estimator to the maximum likelihood estimator and multiple imputation under various model assumptions and different missingness mechanisms in addition we provide two real examples that motivate this investigation

the octahedral algorithm a new simplicial fixed point algorithm a new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented it uses the restrart technique of merrill to improve the accuracy of the solution the algorithm is shown to converge quadratically under certain conditions the algorithm should be efficient and relatively easy to implement

a class of indefinite dogleg path methods for unconstrained minimization in this paper we propose a convenient curvilinear search method to solve trust region problems arising from unconstrained optimization problems the curvilinear paths we set forth are dogleg paths generated mainly by employing bunch parlett factorization for general symmetric matrices that may be indefinite this method is easy to implement and globally convergent it is proved that the method satisfies the first and second order stationary point convergence properties and that the convergence rate is quadratic under commonly used conditions on functions numerical experiments are conducted to compare this method with some existing methods

on the computation of fixed points in the product space of unit simplices and an application to noncooperative n person games in this paper an algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space s of unit simplices into itself the algorithm is a modification of an algorithm introduced in an earlier paper the main feature is that it starts with an arbitrary chosen point in s and that the triangulation of s depends on the starting point moreover the algorithm can terminate with a non full dimensional subsimplex yielding a good approximation an application is given for non cooperative n person games where s is the strategy space some computational experiences are given

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results

on the bifurcation and stability of periodic solutions of the ginzburg landau equations in the plane the linear bifurcation and stability of periodic solutions to the ginzburg landau equations in the plane are investigated in particular we find new infinite families of solutions which include the few solutions previously reported in the literature then the vortex structure of these new solutions is examined in addition the energy of a large class of solutions is approximated in the limit case for which the fundamental cell is a very thin and long rectangle in that limit we find that the energy of the solution representing the well known triangular lattice is the lowest finally we examine the stability of one infinite family of solutions including both the triangular and square lattices in an infinite dimensional space of perturbations in contrast to a previous work in which stability was examined only in a finite dimensional space we find that in addition to the triangular lattice other solutions are stable as well

the d 2 triangulation for simplicial homotopy algorithms for computing solutions of nonlinear equations a triangulation of arbitrary refinement of grid sizes of 0 1 n is proposed for simplicial homotopy algorithms for computing solutions of nonlinear equations on each level the new triangulation called thed2 triangulation subdivides n into simplices according to thed1 triangulation we prove that thed2 triangulation is superior to thek2 triangulation andj2 triangulation in the number of simplices numerical tests show that the simplicial homotopy algorithm based on thed2 triangulation indeed is much more efficient

semiparametric methods in the proportional odds model for ordinal response data with missing covariates we consider the estimation problem of a proportional odds model with missing covariates based on the validation and nonvalidation data sets we propose a joint conditional method that is an extension of wang et al 2002 statistica sinica 12 555 574 the proposed method is semiparametric since it requires neither an additional model for the missingness mechanism nor the specification of the conditional distribution of missing covariates given observed variables under the assumption that the observed covariates and the surrogate variable are categorical we derived the large sample property the simulation studies show that in various situations the joint conditional method is more efficient than the conditional estimation method and weighted method we also use a real data set that came from a survey of cable tv satisfaction to illustrate the approaches

a continuous deformation algorithm for variational inequality problems on polytopes a continuous deformation algorithm is proposed for solving a variational inequality problem on a polytopek the algorithm embeds the polytopek intok 0 and starts by applying a variable dimension algorithm onk 0 until an approximate solution is found onk 0 then by tracing the path of solutions of a system of equations the algorithm virtually follows a path of approximate solution ink 0 when the path ink 0 returns to level 0 i e k 0 the variable dimension algorithm is again used until a new approximate solution is found onk 0 the setk 0 is triangulated so that the approximate solution on the path improves the accuracy as the level increases a contrivance for a practical implementation of the algorithm is proposed and tested for some test problems

efficiency and implementation of simplicial zero point algorithms in this paper we compare the efficiency of several simplicial variable dimension algorithms to do so we first treat the issues of degeneracy and accelerating we present a device for solving degeneracy furthermore we compare several accelerating techniques the technique of iterated quasi newton steps after each major cycle of the simplicial algorithm is implemented in a computer code which is used to compare the efficiency of the n 1 ray 2n ray 2n ray and 3n 1 ray algorithms except for the n 1 ray algorithm the number of function evaluations does not differ very much between the various algorithms it appeared however that the 2n algorithm needs considerably less computation time

a simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron a path following algorithm is proposed for finding a solution to the nonlinear stationary point problem on an unbounded convex and pointed polyhedron the algorithm can start at an arbitrary point of the polyhedron the path to be followed by the algorithm is described as the path of zeros of some piecewise continuously differentiable function defined on an appropriate subdivided manifold this manifold is induced by a generalized primal dual pair of subdivided manifolds the path of zeros can be approximately followed by dividing the polyhedron into simplices and replacing the original function by its piecewise linear approximation with respect to this subdivision the piecewise linear path of this function can be generated by alternating replacement steps and linear programming pivot steps a condition under which the path of zeros converges to a solution is also stated and a description of how the algorithm operates when the problem is linear or when the polyhedron is the cartesian product of a poly

simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds ideas of a simplicial variable dimension restart algorithm to approximate zero points onrn developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds the algorithm can be considered as a modification of the2n ray zero point finding algorithm onrn it appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n ray algorithm applied to an equivalent zero point problem this is caused by the fact that the algorithm utilizes the complementarity conditions on the variables

a fully polynomial time approximation algorithm for computing a stationary point of the general linear complementarity problem we apply a potential reduction algorithm to solve the general linear complementarity problem glcp minimize xty subject to ax by cz q and x y z 0 we show that the algorithm is a fully polynomial time approximation scheme fptas for computing an e approximate stationary point of the glcp note that there are some glcps in which every stationary point is a solution xty 0 these include the lcps with row sufficient matrices we also show that the algorithm is a polynomial time algorithm for a special class of glcps

a three terms polak ribiere polyak conjugate gradient algorithm for large scale nonlinear equations in this paper a conjugate gradient algorithm for systems of large scale nonlinear equations is designed by the following steps i a three terms conjugate gradient direction d k is presented where the direction possesses the sufficient descent property and the trust region property independent of line search technique ii a backtracking line search technique along the direction is proposed to get the step length k and construct a point iii if the point satisfies the given condition then it is the next point otherwise the projection proximal technique is used and get the next point both the direction and the line search technique are the derivative free approaches then the large scale nonlinear equations are successfully solved 100 000 variables the global convergence of the given algorithm is established under suitable conditions numerical results show that the proposed method is efficient for large scale problems

stable nucleation for the ginzburg landau system with an applied magnetic field we consider the ginzburg landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder of radius overline r and the applied field is parallel to the axis it is shown that there is an upper critical value overline h such that if the modulus of the applied field is greater than overline h the normal nonsuperconducting state in which the order parameter is identically zero is stable and if the modulus of the applied field is slightly below oh the normal state is unstable in addition it is shown that there is a positive lower critical value underline h leq overline h such that the normal state is unstable if the modulus of the applied field is less than underline h and stable if the modulus is slightly above underline h in the case of type ii materials for whic h the ginzburg landau constant is large it is shown that there is a discrete set of radii such that if overline r notin cal b kappa and kappa overline r is sufficiently large then for each applied field of modulus slightly less than overline h or slightly more than underline h there is precisely one small superconducting solution up to a gauge transformation which is stable moreover for this solution the complex valued order parameter is zero only on the axis of the cylinder and its winding number is proportional to the product of 2 and the cross sectional area of the cylinder in addition the solution exhibits surface superconductivity as predicted by the physicists de gennes and st james

a new variable dimension algorithm for the fixed point problem we propose a new variable dimension algorithm for solving a system ofn equations inn variables the algorithm solves a sequence of subproblems consisting of the first several equations the dimension of subproblems does not however increase monotonically a convergence condition is also presented

an efficient implementation of merrill s method for sparse or partially separable systems of nonlinear equations when merrill s method without an extra dimension is applied to solving sparse or partially separable systems of nonlinear equations the computational efficiency can be further improved i e a larger piece of linearity can be traversed in one step by using a suitable linear system one of the linear systems is updated and the corresponding technique is shown to update information about the linear system and the large piece in the implementation of the method some numerical results of the method support the claim that it is efficient also some mistakes from a previous paper in which the main technique exploiting sparsity was proposed are corrected

a unifying model based on retraction for fixed point algorithms we present a unifying model based on retraction for several restart fixed point algorithms the model embraces the interpretation of the algorithms in terms of stationary point problem by van der laan and talman and fully explains the 2 ray method

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

boundary concentration for eigenvalue problems related to the onset of superconductivity we examine the asymptotic behavior of the eigenvalue h and corresponding eigenfunction associated with the variational problem in the regime h 1 here a is any vector field with curl equal to 1 the problem arises within the ginzburg landau model for superconductivity with the function h yielding the relationship between the critical temperature vs applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross section omega subset r 2 we first carry out a rigorous analysis of the associated problem on a half plane and then rigorously justify some of the formal arguments of bs obtaining an expansion for while also proving that the first eigenfunction decays to zero somewhere along the sample boundary when is not a disc for interior decay we demonstrate that the rate is exponential

a class of simplicial restart fixed point algorithms without an extra dimension in an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices ideas of that paper are used to construct a class of triangulations ofrn more precisely for somek 1 k n and positive integersm1 mk with sumn a triangulation ofrn is obtained by triangulating the cells which are formed by taking the product of given triangulations ofrmj j 1 k the triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inrn being the starting point of the algorithm fork n we obtain thek triangulation originally due to todd each element of the class can be used to find a simplex which approximates a fixed point of a mapping onrn by generating a unique path of adjacent simplices of variable dimension starting with the pointv

eigenvalue problems of ginzburg landau operator in bounded domains in this paper we study the eigenvalue problems for the ginzburg landau operator with a large parameter in bounded domains in r2 under gauge invariant boundary conditions the estimates for the eigenvalues are obtained and the asymptotic behavior of the associated eigenfunctions is discussed these results play a key role in estimating the critical magnetic field in the mathematical theory of superconductivity

a modified integer labeling for complementarity algorithms nonlinear complementary problems can be solved by algorithms based on the principle of simplicial approximation we present such an algorithm that allows the choice of an arbitrary starting point without using an artificial labeling or an extra dimension furthermore the relation between the grid size of the triangulation and the accuracy of an approximate solution is investigated and it is shown that a combination of the algorithm with newton steps yields better accuracy finally the algorithm is adapted for solving systems of equations

a conjugate gradient projection method for solving equations with convex constraints abstract in this paper we propose a derivative free iterative method for a class of equations with convex constraints appearing in a variety of the practical problems such as compressing sensing fluid mechanics plasma physics nonlinear optics and solid state in the iteration our search direction can be viewed as an extension of a modified three term cg method by an appropriate line search and the projection step our method is convergent to the solution our method inherits the advantages of cg method and projection method and thus is suitable for solving large scale non smooth problem under the assumption that f is lipschitz continuous and satisfies a weaker condition of monotonicity our method is globally convergent numerical results show the efficiency of the proposed method