information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

sample average approximation for the continuous type principal agent problem abstract we develop a method for finding approximate solutions to the continuous agent type principal agent problem when analytical methods are not available the solution is calculated by solving a discrete agent type version of the problem using sample average approximation and bootstrapping we show how a solution to the approximate problem can be used to derive a lower bound and expected upper bound for the optimal objective function and evaluate the error associated with the approximation numerical examples illustrate convergence in the approximate solution to the true solution as the number of samples increases this works yields a method for obtaining some tractability in continuous type principal agent problems where solutions were previously unavailable

stability and finite element approximation of phase change models for natural convection in porous media abstract in this paper we study a phase change problem for non isothermal incompressible viscous flows the underlying continuum is modelled as a viscous newtonian fluid where the change of phase is either encoded in the viscosity itself or in the brinkman boussinesq approximation where the solidification process influences the drag directly we address these and other modelling assumptions and their consequences in the simulation of differentially heated cavity flows of diverse type a second order finite element method for the primal formulation of the problem in terms of velocity temperature and pressure is constructed and we provide conditions for its stability we finally present several numerical tests in 2d and 3d corroborating the accuracy of the numerical scheme as well as illustrating key properties of the model

computational geometry of positive definiteness abstract in matrix computations such as in factoring matrices hermitian and preferably positive definite elements are occasionally required related problems can often be cast as those of existence of respective elements in a matrix subspace for two dimensional matrix subspaces first results in this regard are due to finsler to assess positive definiteness in larger dimensional cases the task becomes computational geometric for the joint numerical range in a natural way the hermitian element of the frobenius norm one with the maximal least eigenvalue is found to this end extreme eigenvalue computations are combined with ellipsoid and perceptron algorithms

stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization this paper introduces an algorithm for minimizing a single variable locally lipschitz function subject to a like function being nonpositive the method combines polyhedral and quadratic approximation a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point the convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases the algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms

integrability analyticity isochrony equilibria small oscillations and diophantine relations results from the stationary korteweg de vries hierarchy the isochronous variant is exhibited of the dynamical system corresponding to the mth ordinary differential equation of the stationary korteweg de vries kdv hierarchy new diophantine relations are thereby obtained in the guise of matrices of arbitrary order having integer eigenvalues or equivalently of polynomials of arbitrary degree having integer zeros generalizations of these formulas to relations among rational functions are also obtained the basic idea to arrive at such relations is not new but the specific application reported in this paper is new and it is likely to open the way to several analogous new findings

properties of the zeros of generalized basic hypergeometric polynomials we define the generalized basic hypergeometric polynomial of degree n in terms of the generalized basic hypergeometric function by choosing one of its parameters to allow the termination of the series after a finite number of summands in this paper we obtain a set of nonlinear algebraic equations satisfied by the n zeros of the polynomial moreover we obtain an n n matrix m defined in terms of the zeros of the polynomial which in turn depend on the parameters of the polynomial the eigenvalues of this remarkable matrix m are given by neat expressions that depend only on some of the parameters of the polynomial that is the matrix m is isospectral moreover in case the parameters that appear in the expressions for the eigenvalues of m are rational the matrix m has rational eigenvalues a diophantine property

a computational comparison of the ellipsoid algorithm with several nonlinear programming algorithms a computational comparison of several general purpose nonlinear programming algorithms is presented this study was motivated by the preliminary results in 12 which show that the recently developed ellipsoid algorithm is competitive with a widely used augmented lagrangian algorithm to provide a better perspective on the value of ellipsoid algorithms in nonlinear programming the present study includes some of the most highly regarded nonlinear programming algorithms and is a much more comprehensive study than 12 the algorithms considered here are chosen from four distinct classes and 50 well known test problems are used the algorithms used represent augmented lagrangian ellipsoid generalized reduced gradient and iterative quadratic programming methods results regarding robustness and relative efficiency are presented

the neatest many body problem amenable to exact treatments a goldfish various formulations findings and conjectures are reviewed which relate to a many body problem that is arguably the neatest nontrivial such model amenable to exact treatments

equilibrium positions shape invariance and askey wilson polynomials we show that the equilibrium positions of the ruijsenaars schneider van diejen systems with the trigonometric potential are given by the zeros of the askey wilson polynomials with five parameters the corresponding single particle quantum version which is a typical example of discrete quantum mechanical systems with a q shift type kinetic term is shape invariant and the eigenfunctions are the askey wilson polynomials this is an extension of our previous study 1 2 which established the discrete analogue of the well known fact the equilibrium positions of the calogero systems are described by the hermite and laguerre polynomials whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the hermite and laguerre polynomials

proof of certain diophantine conjectures and identification of remarkable classes of orthogonal polynomials certain diophantine conjectures are proven and to do so certain remarkable classes of orthogonal polynomials are identified yielding additional diophantine findings

solving nonlinear principal agent problems using bilevel programming abstract while significant progress has been made analytic research on principal agent problems that seek closed form solutions faces limitations due to tractability issues that arise because of the mathematical complexity of the problem the principal must maximize expected utility subject to the agent s participation and incentive compatibility constraints linearity of performance measures is often assumed and the linear exponential normal len model is often used to deal with this complexity these assumptions may be too restrictive for researchers to explore the variety of relationships between compensation contracts offered by the principal and the effort of the agent in this paper we show how to numerically solve principal agent problems with nonlinear contracts in our procedure we deal directly with the agent s incentive compatibility constraint we illustrate our solution procedure with numerical examples and use optimization methods to make the problem tractable without using the simplifying assumptions of a len model we also show that using linear contracts to approximate nonlinear contracts leads to solutions that are far from the optimal solutions obtained using nonlinear contracts a principal agent problem is a special instance of a bilevel nonlinear programming problem we show how to solve principal agent problems by solving bilevel programming problems using the ellipsoid algorithm the approach we present can give researchers new insights into the relationships between nonlinear compensation schemes and employee effort

a basis change strategy for the reduced gradient method and the optimum design of large structures this paper proposes a basis change strategy within the reduced gradient method for optimization under linear constraints it ensures a non singular basis matrix at every iteration the same strategy can reliably be used within the generalized reduced gradient method for optimization under non linear constraints this method is applied to the minimum weight design of large structures under displacement and stress constraints exploiting the sparsity of the constraint jacobian matrix

analysis and mixed primal finite element discretisations for stress assisted diffusion problems abstract we analyse the solvability of a static coupled system of pdes describing the diffusion of a solute into an elastic material where the process is affected by the stresses exerted in the solid the problem is formulated in terms of solid stress rotation tensor solid displacement and concentration of the solute existence and uniqueness of weak solutions follow from adapting a fixed point strategy decoupling linear elasticity from a generalised poisson equation we then construct mixed primal and augmented mixed primal galerkin schemes based on adequate finite element spaces for which we rigorously derive a priori error bounds the convergence of these methods is confirmed through a set of computational tests in 2d and 3d

a new conjugate gradient method with guaranteed descent and an efficient line search a new nonlinear conjugate gradient method and an associated implementation based on an inexact line search are proposed and analyzed with exact line search our method reduces to a nonlinear version of the hestenes stiefel conjugate gradient scheme for any inexact line search our scheme satisfies the descent condition bf g k sf t bf d k le frac 7 8 bf g k 2 moreover a global convergence result is established when the line search fulfills the wolfe conditions a new line search scheme is developed that is efficient and highly accurate efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer high accuracy is achieved by using a convergence criterion which we call the approximate wolfe conditions obtained by replacing the sufficient decrease criterion in the wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion numerical comparisons are given with both l bfgs and conjugate gradient methods using the unconstrained optimization problems in the cute library

optimal contract design in the joint economic lot size problem with multi dimensional asymmetric information abstract previous work has studied the classical joint economic lot size model as an adverse selection problem with asymmetric cost information solving this problem is challenging due to the presence of countervailing incentives and two dimensional information asymmetry under which the classical single crossing condition does not need to hold in the present work we advance the existing knowledge about the problem on hand by conducting its optimality analysis which leads to a better informed and an easier problem solution first we refine the existing closed form solution which simplifies problem solving and its analysis second we prove that karush kuhn tucker conditions are necessary for optimality and demonstrate that the problem may in general possess non optimal stationary points due to non convexity third we prove that certain types of stationary points are always dominated which eases the analytical solution of the problem fourth we derive a simple optimality condition stating that a weak pareto efficiency of the buyer s possible cost structures implies optimality of any stationary point it simplifies the analytical solution approach and ensures a successful solution of the problem by means of conventional numerical techniques e g with a general purpose solver we further establish properties of optimal solutions and indicate how these are related with the classical results on adverse selection

tridiagonal matrices orthogonal polynomials and diophantine relations i it is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three term recursion relations and being therefore members of an orthogonal set a class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers in the process certain neat formulae are also obtained which perhaps deserve to be included in standard compilations since they involve classical polynomials such as the jacobi polynomials and other named polynomials

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

a new implementation of the force method and the optimum design of large trusses an implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system the matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices no computational procedure is required to generate the compatibility conditions or the self stress matrix and no partitioning of the force vector is made into a basic set and a redundant set unlike the conventional force method a slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns this is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions a sparse generalized reduced gradient package is used as the optimizer a class of test problems involving large truss structures is solved the results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures

evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping abstract bilevel optimization problems are a class of challenging optimization problems which contain two levels of optimization tasks in these problems the optimal solutions to the lower level problem become possible feasible candidates to the upper level problem such a requirement makes the optimization problem difficult to solve and has kept the researchers busy towards devising methodologies which can efficiently handle the problem despite the efforts there hardly exists any effective methodology which is capable of handling a complex bilevel problem in this paper we introduce bilevel evolutionary algorithm based on quadratic approximations bleaq of optimal lower level variables with respect to the upper level variables the approach is capable of handling bilevel problems with different kinds of complexities in relatively smaller number of function evaluations ideas from classical optimization have been hybridized with evolutionary methods to generate an efficient optimization algorithm for a wide class of bilevel problems the performance of the algorithm has been evaluated on two sets of test problems the first set is a recently proposed smd test set which contains problems with controllable complexities and the second set contains standard test problems collected from the literature the proposed method has been compared against three benchmarks and the performance gain is observed to be significant the codes related to the paper may be accessed from the website http bilevel org

primal mixed formulations for reaction diffusion systems on deforming domains we propose a finite element formulation for a coupled elasticity reaction diffusion system written in a fully lagrangian form and governing the spatio temporal interaction of species inside an elastic or hyper elastic body a primal weak formulation is the baseline model for the reaction diffusion system written in the deformed domain and a finite element method with piecewise linear approximations is employed for its spatial discretization on the other hand the strain is introduced as mixed variable in the equations of elastodynamics which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction diffusion system written in a deformed domain the discrete mechanical problem yields a mixed finite element scheme based on row wise raviart thomas elements for stresses brezzi douglas marini elements for displacements and piecewise constant pressure approximations the application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed and some illustrative examples are provided and extensively analyzed

on isochronous bruschi ragnisco ruijsenaars toda lattices equilibrium configurations behaviour in their neighbourhood diophantine relations and conjectures isochronous versions of the bruschi ragnisco ruijsenaars toda lattice and of some of its also integrable variants are introduced their equilibrium configurations are found when they exist and by investigating the motions of these systems near equilibrium some diophantine relations are obtained as well as some insight into the solution of those of these integrable models whose solutions are not yet known

integrability analyticity isochrony equilibria small oscillations and diophantine relations results from the stationary burgers hierarchy an isochronous system is introduced by modifying the nth ode of the stationary burgers hierarchy and then by investigating its behaviour near its equilibria neat diophantine relations are identified involving well known polynomials of arbitrary degree having integer zeros or equivalently matrices the determinants of which yield such polynomials the basic idea to arrive at such relations is not new but the specific application reported in this paper is new and it is likely to open the way to several analogous new findings

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

a derivative based bracketing scheme for univariate minimization and the conjugate gradient method abstract a derivative based scheme for univariate minimization is developed this scheme has a quadratic convergence rate and requires an evaluation of the function and its derivative in each iteration an application involving the conjugate gradient method is presented

using block norms for location modeling in formulating a continuous location model with facilities represented as points in rn e g typically in the plane one must characterize the distance between two points as a function of their coordinates two criteria in selecting a distance function are 1 to obtain good approximations of actual distances and 2 to obtain a mathematical model of the location problem that is easy to solve in this paper we show how a class of norms with polygonal contours called block norms can yield attractive choices as distance functions with respect to these criteria in particular we consider the following relevant properties of block norms they generalize the concepts of rectilinear or city block travel they are dense in the set of all norms they have interesting travel interpretations in the plane they can be expressed as a sum of the absolute values of linear functions they often give better approximations to actual highway distances than the most frequently used family of norms the lp norms and finally they yield linear programming formulations of certain facility location problems i e the weber problem and the rawls problem

a class of rank two ellipsoid algorithms for convex programming in this paper we develop a method for constructing minimum volume ellipsoids containing a wedge shaped subset of a given ellipsoid this construction yields a class of ellipsoid algorithms for convex programming that use rank two update formulae

bagging binary and quantile predictors for time series abstract bootstrap aggregating or bagging introduced by breiman 1996a bagging predictors machine learning 24 123 140 has been proved to be effective to improve on unstable forecast theoretical and empirical works using classification regression trees variable selection in linear and non linear regression have shown that bagging can generate substantial prediction gain however most of the existing literature on bagging has been limited to the cross sectional circumstances with symmetric cost functions in this paper we extend the application of bagging to time series settings with asymmetric cost functions particularly for predicting signs and quantiles we use quantile predictions to construct a binary predictor and the majority voted bagging binary prediction we show that bagging may improve the binary prediction in small sample but it does not improve in large sample various bagging forecast combination weights are used such as equal weighted and bayesian model averaging bma weighted combinations for demonstration we present results from monte carlo experiments and from empirical applications using monthly s p500 and nasdaq stock index returns

the staircase method integrals for periodic reductions of integrable lattice equations we show in full generality that the staircase method papageorgiou et al 1990 phys lett a 147 106 14 quispel et al 1991 physica a 173 243 66 provides integrals for mappings and correspondences obtained as traveling wave reductions of systems of integrable partial difference equations we apply the staircase method to a variety of equations including the korteweg de vries equation the five point bruschi calogero droghei equation the quotient difference qd algorithm and the boussinesq system we show that in all these cases if the staircase method provides r integrals for an n dimensional mapping with then one can introduce q 2r variables which reduce the dimension of the mapping from n to q these dimension reducing variables are obtained as joint invariants of k symmetries of the mappings our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable we also study reductions on other quad graphs than the regular lattice and we prove linear growth of the multi valuedness of iterates of high dimensional correspondences obtained as reductions of the qd algorithm

a linear second order ode with only polynomial solutions abstract a linear second order ode featuring an arbitrary number of free parameters is identified all solutions of which are polynomials

a numerical investigation of rank two ellipsoid algorithms for nonlinear programming we study the performance of some rank two ellipsoid algorithms when used to solve nonlinear programming problems experiments are reported which show that the rank two algorithms studied are slightly less efficient than the usual rank one center cut algorithm some results are also presented concerning the growth of ellipsoid asphericity in rank one and rank two algorithms

a study of mathematical programmingmethods for structural optimization part ii numerical results various mathematical programming methods for structural optimization are studied in a companion paper these methods have been studied based on certain theoretical considerations in this paper the methods are studied based on solving a set of test problems the methods that are studied include recursive qp feasible directions gradient projection sumt and multiplier methods various computer codes have been developed and are studied together with some existing programs such as conmin and optdyn the test problems considered have 3 47 design variables and 3 252 constraints the evaluation criteria consist of studying the accuracy reliability and efficiency of a code it turns out that globally convergent algorithms multiplier methods in particular are very reliable but not efficient primal algorithms like conmin which are not proved to be globally convergent are efficient but not reliable

can the general solution of the second order ode characterizing jacobi polynomials be polynomial the answer to the question posed in the title is affirmative indeed while the second order ode 1 x2 p 2 x p n n 1 p 0 for generic values of the two parameters and and n a non negative integer identifies uniquely up to an overall multiplicative constant characterizing its normalization the jacobi polynomial p n x as its polynomial solution if instead n m and n n with m and n being two non negative integers satisfying the restriction m n n then the general solution of this ode is a polynomial of degree n depending in addition of course to the three parameters n m and n on two arbitrary parameters i e on an additional one besides the overall multiplicative parameter characterizing its normalization

properties of the zeros of the polynomials belonging to the q askey scheme in this paper we provide properties which are to the best of our knowledge new of the zeros of the polynomials belonging to the q askey scheme these findings include diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted

diophantine properties of the zeros of certain laguerre and para jacobi polynomials matrices and of arbitrary rank n given by simple expressions in terms of the n zeros of certain laguerre or para jacobi polynomials of degree n feature a diophantine property in the laguerre case this property states that the 2n zeros of the polynomial are all integers indeed we conjecture that the results in the para jacobi case are somewhat analogous they refer to the zeros of the general solution of the ode generally characterizing jacobi polynomials in the special case in which this solution is a para jacobi polynomial featuring an additional arbitrary parameter

convergence rates of the ellipsoid method on general convex functions the ellipsoid method is applied to the unconstrained minimization of a general convex function the method converges at a geometric rate which depends only upon the dimension of the space but not on the actual function this rate can be improved somewhat if the function satisfies some lipschitz type condition or if the minimum set has dimension greater than zero if the ellipsoid entirely contains the optimal set equating the steiner polynomial associated to the optimal set and the volume of the ellipsoid at a given iteration will give an upper bound on the minimum recorded function value

an augmented mixed finite element method for 3d linear elasticity problems in this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3d our approach is an extension of a technique developed recently for plane elasticity which is based on the introduction of consistent terms of galerkin least squares type we consider non homogeneous and homogeneous dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms in this way the associated galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces in particular raviart thomas spaces of order 0 for the stress tensor continuous piecewise linear elements for the displacement and piecewise constants for the rotation can be utilized moreover we show that in this case the number of unknowns behaves approximately as 9 5 times the number of elements tetrahedrons of the triangulation which is cheaper by a factor of 3 than the classical peers in 3d several numerical results illustrating the good performance of the augmented schemes are provided

descent algorithms on oblique manifold for source adaptive ica contrast a riemannian manifold optimization strategy is proposed to facilitate the relaxation of the orthonormality constraint in a more natural way in the course of performing independent component analysis ica that employs a mutual information based source adaptive contrast function despite the extensive development of manifold techniques catering to the orthonormality constraint only a limited number of works have been dedicated to oblique manifold ob algorithms to intrinsically handle the normality constraint which has been empirically shown to be superior to other riemannian and euclidean approaches imposing the normality constraint implicitly in line with the ica definition essentially guarantees a substantial improvement in the solution accuracy by way of increased degrees of freedom while searching for an optimal unmixing ica matrix in contrast with the orthonormality constraint designs of the steepest descent conjugate gradient with hager zhang or a hybrid update parameter quasi newton and cost effective quasi newton methods intended for ob are presented in this paper their performance is validated using natural images and systematically compared with the popular state of the art approaches in order to assess the performance effects of the choice of algorithm and the use of a riemannian rather than euclidean framework we surmount the computational challenge associated with the direct estimation of the source densities using the improved fast gauss transform in the evaluation of the contrast function and its gradient the proposed ob schemes may find applications in the offline image signal analysis wherein on one hand the computational overhead can be tolerated and on the other the solution quality holds paramount interest

a deep cut ellipsoid algorithm for convex programming theory and applications this paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems in each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided moreover for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed finally some computational results of the deep and central cut version of the algorithm applied to a min max stochastic queue location problem are reported

properties of the zeros of generalized hypergeometric polynomials abstract we define the generalized hypergeometric polynomial of degree n as follows p n 1 p 1 q z m 0 n n m 1 m p m z n m m 1 m q m z n f q p 1 n 1 p 1 q 1 z here n is an arbitrary positive integer p and q are arbitrary nonnegative integers the p q parameters j and k are arbitrary generic possibly complex numbers m is the pochhammer symbol and f q p 1 0 1 p 1 q z is the generalized hypergeometric function in this paper we obtain a set of n nonlinear algebraic equations satisfied by the n zeros n of this polynomial we moreover manufacture an n n matrix l in terms of the 1 p q parameters n j k characterizing this polynomial and of its n zeros n and we show that it features the n eigenvalues m m k 1 q k 1 m m 1 n these n eigenvalues depend only on the q parameters k implying that the n n matrix l is isospectral for variations of the p parameters j and they clearly are integer or rational numbers if the q parameters k are themselves integer or rational numbers a nontrivial diophantine property

finite dimensional representations of difference operators and the identification of remarkable matrices two square matrices of arbitrary order n are introduced they are defined in terms of n arbitrary numbers zn and of an arbitrary additional parameter a respectively q and provide finite dimensional representations of the two operators acting on a function f z as follows f z a f z a respectively f qz f z q 1 z these representations are exact in a sense explained in the paper when the function f z is a polynomial in z of degree less than n this formalism allows to transform difference equations valid in the space of polynomials of degree less than n into corresponding matrix vector equations as an application of this technique several remarkable square matrices of order n are identified which feature explicitly n arbitrary numbers zn or the n zeros of polynomials belonging to the askey and q askey schemes several of these findings have a diophantine character

quasi maximum likelihood estimation for conditional quantiles in this paper we derive the asymptotic distribution of a new class of quasi maximum likelihood estimators qmle based on a tick exponential family of densities we show that the tick exponential assumption is a necessary and sufficient condition for a qmle to be consistent for the parameters of a correctly specified model of a given conditional quantile hence the role of this family of densities in the conditional quantile estimation is analog to the role of the linear exponential family in the conditional mean estimation the tick exponential qmles are shown to be asymptotically normal with an asymptotic covariance matrix that has a novel form not seen in earlier work and which accounts for possible model misspecification for practical purposes we show that the maximization of the tick exponential quasi log likelihood can conveniently be carried out by using standard gradient based optimization techniques more importantly we provide a consistent estimator for the asymptotic covariance matrix based on the scores of the log likelihood which allows us to compute the conditional quantile confidence intervals

a derivative free bracketing scheme for univariate minimization abstract a derivative free scheme for univariate minimization is developed this scheme has a quadratic convergence rate and requires two function evaluations each iteration

initial value problems for lattice equations we describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils in finitely many directions we arrive at discrete goursat problems and in the remaining directions we find cauchy problems next we consider s1 s2 periodic initial value problems in the goursat directions the periodic solutions are generated by correspondences in the cauchy directions assuming the equation to be multi linear the periodic solution can be obtained uniquely by iteration of a particularly simple mapping whose dimension is a piecewise linear function of s1 s2

variable metric relaxation methods part ii the ellipsoid method the deepest or least shallow cut ellipsoid method is a polynomial time and space method which finds an ellipsoid representable by polynomial space integers such that the maximal ellipsoidal distance relaxation method using this fixed ellipsoid is polynomial this is equivalent to finding a linear transforming such that the maximal distance relaxation method of agmon motzkin and schoenberg in this transformed space is polynomial if perfect arithmetic is used then the sequence of ellipsoids generated by the method converges to a set of ellipsoids which share some of the properties of the classical hessian at an optimum point of a function and thus the ellipsoid method is quite analogous to a variable metric quasi newton method

an isochronous variant of the ruijsenaars toda model equilibrium configurations behavior in their neighborhood diophantine relations an isochronous variant of the ruijsenaars toda integrable many body problem is introduced an equilibrium configuration of this dynamical system is identified and by investigating the motions in its neighborhood diophantine relations are obtained

perturbations around the zeros of classical orthogonal polynomials starting from degree n solutions of a time dependent schrodinger like equation for classical orthogonal polynomials a linear matrix equation describing perturbations around the n zeros of the polynomial is derived the matrix has remarkable diophantine properties its eigenvalues are independent of the zeros the corresponding eigenvectors provide the representations of the lower degree 0 1 n 1 polynomials in terms of the zeros of the degree n polynomial the results are valid universally for all the classical orthogonal polynomials including the askey scheme of hypergeometric orthogonal polynomials and its q analogues

evolution equations associated with the discrete analog of the matrix schrodinger spectral problem solvable by the inverse spectral transform through the generalized wronskian technique we derive the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix schrodinger spectral problem for such equations we briefly discuss soliton solutions continuum limit and backlund transformations

novel solvable variants of the goldfish many body model a recent technique to identify solvable many body problems in two dimensional space yields via a new twist new many body problems of goldfish type some of these models are isochronous namely their generic solutions are completely periodic with a fixed period independent of the initial data the investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations

formulation and analysis of fully mixed methods for stress assisted diffusion problems abstract this paper is devoted to the mathematical and numerical analysis of a mixed mixed pde system describing the stress assisted diffusion of a solute into an elastic material the equations of elastostatics are written in mixed form using stress rotation and displacements whereas the diffusion equation is also set in a mixed three field form solving for the solute concentration for its gradient and for the diffusive flux this setting simplifies the treatment of the nonlinearity in the stress assisted diffusion term the analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of schauder and banach fixed point theorems together with the babuska brezzi and lax milgram theories concerning numerical discretization we propose two families of finite element methods based on either peers or arnold falk winther elements for elasticity and a raviart thomas and piecewise polynomial triplet approximating the mixed diffusion equation we prove the well posedness of the discrete problems and derive optimal error bounds using a strang inequality we further confirm the accuracy and performance of our methods through computational tests

on the solution of variational inequalities by the ellipsoid method variational inequalities provide a convenient mathematical approach for unifying results relating to extremum and equilibrium problems in this paper we demonstrate the applicability of the ellipsoid method to solve monotone variational inequalities a convergence proof is given which for example in case of convex optimization explains the linear convergence of the record points we compare this result with goffin s convergence proof by extending his proof technique to the constrained case in the appendix