the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

associative and jordan algebras and polynomial time interior point algorithms for symmetric cones we present a general framework whereby analysis of interior point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras in particular such analyses are extendible to the cone of positive semidefinite hermitian matrices with complex and quaternion entries and to the lorentz cone we prove the case of the lorentz cone by using the embedding of its associated jordan algebra in the clifford algebra as an example of such extensions we take monterio s polynomial time complexity analysis of the family of similarly scaled directions introduced by monteiro and zhang 1998 and generalize it to cone lp over all representable symmetric cones

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

the directional instability problem in systems with frictional contacts the research summarized in this paper addresses the directional instability of finite dimensional systems with unilateral frictional contacts conditions for the occurrence of this divergence type instability are discussed complementarity formulations are developed and numerical procedures are proposed for the solution of the corresponding non smooth stability eigenproblems various examples are analytically or numerically solved and discussed namely some finite element examples that have instability modes involving evolution towards slip or stick in different portions of the contact surface

second order conic approximation for disassembly line design with joint probabilistic constraints a problem of profit oriented disassembly line design and balancing with possible partial disassembly and presence of hazardous parts is studied the objective is to design a production line providing a maximal revenue with balanced workload task times are assumed to be random variables with known normal probability distributions the cycle time constraints are to be jointly satisfied with at least a predetermined probability level an and or graph is used to model the precedence relationships among tasks several lower and upper bounding schemes are developed using second order cone programming and convex piecewise linear approximation to show the relevance and applicability of the proposed approach a set of instances from the literature are solved to optimality

generalised linear models for correlated pseudo observations with applications to multi state models will be in a given state at some time is a complex nonlinear function of the intensity regression coefficients we present a technique which models the state probabilities directly this method is based on the pseudo values from a jackknife statistic constructed from simple summary statistic estimates of the state probabilities these pseudo values are then used in a generalised estimating equation to obtain estimates of the model parameters we illustrate how this technique works by studying examples of common regression problems we apply the technique to model acute graft versus host disease in bone marrow transplants

a goodness of fit test for binary regression models based on smoothing methods a new global test statistic for models with continuous covariates and binary response is introduced the test statistic is based on nonparametric kernel methods explicit expressions are given for the mean and variance of the test statistic asymptotic properties are considered and approximate corrections due to parameter estimation are presented properties of the test statistic are studied by simulation the goodness of fit method is illustrated on data from a dutch follow up study on preterm infants recommendations for practitioners are given

product form cholesky factorization in interior point methods for second order cone programming second order cone programming socp problems are typically solved by interior point methods as in linear programming lp interior point methods can in theory solve socps in polynomial time and can in practice exploit sparsity in the problem data specifically when cones of large dimension are present the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an lp here we propose a product form cholesky factorization pfcf approach and show that it is more numerically stable than the alternative sherman morrison woodbury approach we derive several pfcf variants and compare their theoretical perfomance finally we prove that the elements of l in the cholesky factorizations ldlt that arise in interior point methods for socp are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path

estimating a time dependent concordance index for survival prediction models with covariate dependent censoring given a predictive marker and a time to event response variable the proportion of concordant pairs in a data set is called concordance index a specifically useful marker is the risk predicted by a survival regression model this article extends the existing methodology for applications where the length of the follow up period depends on the predictor variables a class of inverse probability of censoring weighted estimators is discussed in which the estimates rely on a working model for the conditional censoring distribution the estimators are consistent for a truncated concordance index if the working model is correctly specified and if the probability of being uncensored at the truncation time is positive in this framework all kinds of prediction models can be assessed and time trends in the discrimination ability of a model can be captured by varying the truncation time point for illustration we re analyze a study on risk prediction for prostate cancer patients the effects of misspecification of the censoring model are studied in simulated data

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

schwarz method for earthquake source dynamics dynamic faulting under slip dependent friction in a linear elastic domain in plane and 3d configurations is considered the use of an implicit time stepping scheme newmark method allows much larger values of the time step than the critical cfl time step and higher accuracy to handle the non smoothness of the interface constitutive law slip weakening friction the finite element form of the quasi variational inequality is solved by a schwarz domain decomposition method by separating the inner nodes of the domain from the nodes on the fault in this way the quasi variational inequality splits into two subproblems the first one is a large linear system of equations and its unknowns are related to the mesh nodes of the first subdomain i e lying inside the domain the unknowns of the second subproblem are the degrees of freedom of the mesh nodes of the second subdomain i e lying on the domain boundary where the conditions of contact and friction are imposed this nonlinear subproblem is solved by the same schwarz algorithm leading to some local nonlinear subproblems of a very small size numerical experiments are performed to illustrate convergence in time and space instability capturing energy dissipation and the influence of normal stress variations we have used the proposed numerical method to compute source dynamics phenomena on complex and realistic 2d fault models branched fault systems

special functions as solutions to discrete painleve equations we present results on special solutions of discrete painleve equations these solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second order discrete equations these linear equations define the discrete analogues of special functions

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

on the convergence of the method of analytic centers when applied to convex quadratic programs this work examines the method of analytic centers of sonnevend when applied to solve generalized convex quadratic programs where also the constraints are given by convex quadratic functions we establish the existence of a two sided ellipsoidal approximation for the set of feasible points around its center and show that a simple zero order algorithm starting from an initial center of the feasible set generates a sequence of strictly feasible points whose objective function values converge to the optimal value concerning the speed of convergence it is shown that an upper bound for the gap in between the objective function value and the optimal value is reduced by a factor ofe with o sqrt m left ln varepsilon right iterations wherem is the number of inequality constraints here each iteration involves the computation of one newton step the bound of o sqrt m left ln varepsilon right newton iterations to guarantee an error reduction by a factor ofe in the objective function is as good as the one currently given forlinear programs however the algorithm considered here is of theoretical interest only full efficiency of the method can only be obtained when accelerating it by some higher order extrapolation scheme see e g the work of jarre sonnevend and stoer

picard and chazy solutions to the painleve vi equation we study the solutions of a particular family of painleve vi equations with parameters beta gamma 0 delta frac 1 2 and 2 alpha 2 mu 1 2 for 2 mu in mathbb z we show that in the case of half integer mu all solutions can be written in terms of known functions and they are of two types a two parameter family of solutions found by picard and a new one parameter family of classical solutions which we call chazy solutions we give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0 1 infty and their nonlinear monodromy we study the structure of analytic continuation of the solutions to the pvi mu equation for any mu such that 2 mu in mathbb z as an application we classify all the algebraic solutions for mu half integer we show that they are in one to one correspondence with regular polygons or star polygons in the plane for mu integer we show that all algebraic solutions belong to a one parameter family of rational solutions

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

critical friction for wedged configurations abstract a wedged configuration with coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads we analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3 d context the critical friction coefficient w is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses for friction coefficients with w the wedged problem has at least a solution and for the wedged problem is stated in a discrete framework using a mixed finite element approach and the discrete critical friction coefficient h w is introduced as the solution of a global minimization problem involving a non differentiable and non convex functional the existence of h the displacement field of a critical wedged state is proved and a specific numerical method based on a genetic algorithm was developed to compute the critical wedged configurations some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

improvements on cross validation the 632 bootstrap method abstract a training set of data has been used to construct a rule for predicting future responses what is the error rate of this rule this is an important question both for comparing models and for assessing a final selected model the traditional answer to this question is given by cross validation the cross validation estimate of prediction error is nearly unbiased but can be highly variable here we discuss bootstrap estimates of prediction error which can be thought of as smoothed versions of cross validation we show that a particular bootstrap method the 632 rule substantially outperforms cross validation in a catalog of 24 simulation experiments besides providing point estimates we also consider estimating the variability of an error rate estimate all of the results here are nonparametric and apply to any possible prediction rule however we study only classification problems with 0 1 loss in detail our simulations include smooth prediction rules like fisher s linear discriminant fun

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

enumeration of all wedged equilibrium configurations in contact problem with coulomb friction abstract for a linear structure subjected to the unilateral contact condition with a fixed obstacle we refer to a nontrivial equilibrium state as a wedged configuration finding a wedged configuration is called a wedging problem this paper discusses theoretical properties of solution set of the finite dimensional wedging problem with the coulomb friction and presents numerical methods for computing all the wedged configurations we propose algorithms for enumerating all the finitely many representative solutions with which we can completely describe the solution set of the wedging problem there exists a positive critical friction coefficient defined as the minimum value of friction coefficient with which at least one wedged configuration exists we also propose an algorithm for computing the critical friction coefficient which is based on the bisection method and the second order cone program

assessing the complete solution set of the planar frictional wedging problem abstract this paper examines the computation of all the infinitely many solutions of the frictional wedging problem in the non coercive context by an algorithm applied to its complementarity formulation our analysis offers insights regarding the mechanical and the geometrical meaning of the solution set we find that the solution structure depends much on the value of the coefficient of friction the evidence indicates that non coercivity implies much longer computation times coefficients of friction at the onset of wedging are computed and related

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

algorithmic method for deriving lax pairs from the invariant painleve analysis of nonlinear partial differential equations given a partial differential equation its painleve analysis will first be performed with a built in invariance under the homographic group acting on the singular manifold function then assuming an order for the underlying lax pair a multicomponent pseudopotential of projective riccati type the components of which are homographically invariant is introduced if the equation admits a classical darboux transformation a very small set of determining equations whose solution yields the lax pair will be generated in the basis of the pseudopotential this new method will be applied to find the yet unpublished lax pair of the scalar hirota satsuma equation

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

on polyhedral approximations of the second order cone we demonstrate that a conic quadratic problem is polynomially reducible to linear programming we demonstrate this by constructing for every an lp program explicitly given in terms of e and the data of cqp with the following properties i the number dimx dimu of variables and the number dimp of constraints in lp do not exceed ii every feasible solutionx to cqp can be extended to a feasible solution x u to lp iii if x u is feasible for lp thenx satisfies the e relaxed constraints of cqp namely

nearest neighbor estimation of a bivariate distribution under random censoring we consider the problem of estimating the bivariate distribution of the random vector x y when y may be subject to random censoring the censoring variable c is allowed to depend on x but it is assumed that y and c are conditionally independent given x x the estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of y given x x over a range of values of x the weak convergence of the centered estimator multiplied by n 1 2 is obtained and a closed form expression for the covariance function of the limiting process is given it is shown that the proposed estimator is optimal in the beran sense this is similar to an optimality property the product limit estimator enjoys using the proposed estimator of the bivariate distribution an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function abstract let xi yi i 1 2 n be independent identically distributed as x y then the rth ordered x variate is denoted by xr n and the associated y variate the concomitant of the rth order statistic by y r n this paper considers statistics of the form and more generally of the form where j is a bounded smooth function and may depend on n under certain regularity conditions the asymptotic normality of these statistics is established these statistics are used to construct consistent estimators of various conditional quantities for example e y x x p y a x x and var y x x

unstable perturbation of the equilibrium under coulomb friction nonlinear eigenvalue analysis abstract equilibrium configurations of elastic structures under coulomb friction are considered we use a nonlinear eigenvalue analysis to see when an equilibrium configuration can be dynamically unstable or when it admits quasi static bifurcations we prove that if the nonlinear eigenvalue problem has a smooth solution then there exists a family of dynamic perturbations of the equilibrium which have an exponential growth in time the above results are obtained under some conditions on the boundary between the free stick and slipping zones of the contact boundary these conditions are specific to the continuous problem and they are not necessary when we reformulate the same result for the discrete finite element problem we propose a specific numerical algorithm to compute the eigenvalue and the corresponding eigenfunction for a mixed finite element approach of the nonlinear eigenvalue problem it makes use of the successive iterates of a nonlinear operator we proved that the algorithm is consistent i e if it is convergent then its limit is a solution but we cannot give a mathematical proof of its convergence we give some numerical experiments to examine the convergence of the proposed algorithm other examples illustrate how from the nonlinear eigenvalue analysis we get information on the stability of the equilibrium configurations

first degree birational transformations of the painleve equations and their contiguity relations we present a consistent truncation allowing us to obtain the first degree birational transformation found by okamoto for the sixth painleve equation the discrete equation arising from its contiguity relation is then just the sum of six simple poles an algebraic solution is presented which is equivalent to but simpler than the umemura solution finally the well known confluence provides a unified picture of all first degree birational transformations for the lower painleve equations ranging them in two distinct sequences

time dependent predictive accuracy in the presence of competing risks competing risks arise naturally in time to event studies in this article we propose time dependent accuracy measures for a marker when we have censored survival times and competing risks time dependent versions of sensitivity or true positive tp fraction naturally correspond to consideration of either cumulative or prevalent cases that accrue over a fixed time period or alternatively to incident cases that are observed among event free subjects at any select time time dependent dynamic specificity 1 false positive fp can be based on the marker distribution among event free subjects we extend these definitions to incorporate cause of failure for competing risks outcomes the proposed estimation for cause specific cumulative tp dynamic fp is based on the nearest neighbor estimation of bivariate distribution function of the marker and the event time on the other hand incident tp dynamic fp can be estimated using a possibly nonproportional hazards cox model for the cause specific hazards and riskset reweighting of the marker distribution the proposed methods extend the time dependent predictive accuracy measures of heagerty lumley and pepe 2000 biometrics 56 337 344 and heagerty and zheng 2005 biometrics 61 92 105

a product form cholesky factorization method for handling dense columns in interior point methods for linear programming cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods ipms for linear programming lp its main advantages being numerical stability and efficiency for sparse systems however in the presence of dense columns there is dramatic loss of efficiency a typical approach to remedy this problem is to apply the sherman morrison woodbury smw update formula to the dense part this approach while being very efficient is not numerically stable here we propose using product form cholesky factorization to handle dense columns the proposed approach is essentially as stable as the original cholesky factorization and nearly as efficient as the smw approach we demonstrate these properties both theoretically and computationally a key part of our theoretical analysis is the proof that the elements of the cholesky factors of the matrices that arise in ipms for lp are uniformly bounded as the duality gap converges to zero

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

the generalized linear complementarity problem and an algorithm to find all its solutions motivated by a number of typical applications a generalization of the classicallinear complementarity problem is presented together with an algorithm to determine the complete solution set the algorithm is based on the double description method for solving linear inequalities and succeeds in describing continuous as well as unbounded solution sets

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

similarity and other spectral relations for symmetric cones abstract the similarity relations that are derived in this paper reduce to well known results in the special case of symmetric matrices in particular for two positive definite matrices x and y the square of the spectral geometric mean is known to be similar to the matrix product xy it is shown in this paper that this property carries over to symmetric cones more elementary similarity relations such as xy 2 x yx 2 y are generalized as well we also extend the result that the eigenvalues of a matrix product xy are less dispersed than the eigenvalues of the jordan product xy yx 2 the paper further contains a number of inequalities on norms and spectral values this type of inequality is often used in the analysis of interior point methods in optimization we also derive an extension of stein s theorem to symmetric cones

calibration plots for risk prediction models in the presence of competing risks a predicted risk of 17 can be called reliable if it can be expected that the event will occur to about 17 of 100 patients who all received a predicted risk of 17 statistical models can predict the absolute risk of an event such as cardiovascular death in the presence of competing risks such as death due to other causes for personalized medicine and patient counseling it is necessary to check that the model is calibrated in the sense that it provides reliable predictions for all subjects there are three often encountered practical problems when the aim is to display or test if a risk prediction model is well calibrated the first is lack of independent validation data the second is right censoring and the third is that when the risk scale is continuous the estimation problem is as difficult as density estimation to deal with these problems we propose to estimate calibration curves for competing risks models based on jackknife pseudo values that are combined with a nearest neighborhood smoother and a cross validation approach to deal with all three problems

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

a mixed finite element method and solution multiplicity for coulomb frictional contact abstract this paper is concerned with the discrete contact problem governed by coulomb s friction law we propose and study a new technique using mixed finite elements with two multipliers in order to determine numerically critical friction coefficients for which multiple solutions to the friction problem exist the framework is based on eigenvalue problems and it allows to exhibit non uniqueness cases involving an infinity of solutions located on a continuous branch the theory is illustrated with several computations which clearly show the accuracy of the proposed method

complete characterizations of stable farkas lemma and cone convex programming duality we establish necessary and sufficient conditions for a stable farkas lemma we then derive necessary and sufficient conditions for a stable duality of a cone convex optimization problem where strong duality holds for each linear perturbation of a given convex objective function as an application we obtain stable duality results for convex semi definite programs and convex second order cone programs

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

a new full nesterov todd step feasible interior point method for convex quadratic symmetric cone optimization in this paper we generalize the classical primal dual logarithmic barrier method for linear optimization to convex quadratic optimization over symmetric cone by using euclidean jordan algebras the symmetrization of the search directions used in this paper is based on the nesterov todd scaling scheme and only full nesterov todd step is used at each iteration we derive the iteration bound that matches the currently best known iteration bound for small update methods namely o r log r e some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm

volumetric barrier decomposition algorithms for stochastic quadratic second order cone programming ariyawansa and zhu 2011 have derived volumetric barrier decomposition algorithms for solving two stage stochastic semidefinite programs and proved polynomial complexity of certain members of the algorithms in this paper we utilize their work to derive volumetric barrier decomposition algorithms for solving two stage stochastic convex quadratic second order cone programming and establish polynomial complexity of certain members of the proposed algorithms

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm