a combined phase i phase ii scaled potential algorithm for linear programming we develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program without the addition of any m terms the method and its lower bounding procedure are particularly simple compared with previous interior algorithms not requiring feasibility

eigenvalues of elliptic boundary value problems with an indefinite weight function on etudie les problemes aux valeurs propres elliptiques autoadjoints au r x u dans un ensemble ouvert r k a est un operateur positif d ordre 2m et le poids r est une fonction qui change de signe dans et peut etre discontinue on determine le comportement asymptotique des valeurs propres

on anstreicher s combined phase i phase ii projective algorithm for linear programming anstreicher has proposed a variant of karmarkar s projective algorithm that handles standard form linear programming problems nicely we suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm much of the analysis depends on a two constraint linear programming problem that is a relaxation of the scaled original problem

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

projective transformations for interior point algorithms and a superlinearly convergent algorithm for the w center problem the purpose of this study is to broaden the scope of projective transformation methods in mathematical programming both in terms of theory and algorithms we start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w center of a polyhedral system which stands for weighted center where there is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronx we prove basic results regarding contained and containing ellipsoids centered at the w center of the systemx we next shift attention to projective transformations and we exhibit an elementary projective transformation that transforms the polyhedronx to another polyhedronz and that transforms the current interior point to the w center of the transformed polyhedronz we work throughout with a polyhedral system of the most general form namely both inequality and equality costraints

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

a two dimensional haar wavelets method for solving systems of pdes in this paper we modify the idea of haar wavelets method to obtain semi analytical solutions for the systems of three dimensional nonlinear partial differential equations theoretical considerations are discussed to demonstrate the efficiency of the method a test problem is presented the approximate solutions of the system of three dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature the numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions

asymptotics of eigencurves for second order ordinary differential equations i abstract the regular two parameter sturm liouville equation py qy f r y is studied for l1 coefficients with p r 0 for each fixed number n of internal zeros of the eigenfunctions y n is analytic in necessary and sufficient conditions which are in fact independent of n are given for lim n gl to exist as or asymptotic expansions for n are derived in cases of existence and non existence of lim n

a potential function reduction algorithm for solving a linear program directly from an infeasible warm start this paper develops an algorithm for solving a standard form linear program directly from an infeasible warm start i e directly from a given infeasible solution hat x that satisfies a hat x b but hat x ngeqslant 0 the algorithm is a potential function reduction algorithm but the potential function is somewhat different than other interior point method potential functions and is given by f x b q ln c t x b sum limits j 1 n ln x j h j c t x b where q n sqrt n is a given constant h is a given strictly positive shift vector used to shift the nonnegativity constaints andb is a lower bound on the optimal value of the linear program the duality gapc t x b is used both in the leading term as well as in the barrier term to help shift the nonnegativity constraints the algorithm is shown under suitable conditions to achieve a constant decrease in the potential function and so achieves a constant decrease in the duality gap and hence also in the infeasibility in o n iterations under more restrictive assumptions regarding the dual feasible region this algorithm is modified by the addition of a dual barrier term and will achieve a constant decrease in the duality gap and in the infeasibility in o sqrt n iterations

modified barrier functions theory and methods the nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and or the objective function into an equivalent problem the classical lagrangian which has important properties on the primal and the dual spaces the application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions mbf s and mbf methods mbfm s being classical lagrangians cl s for an equivalent problem the mbf s combine the best properties of the cl s and classical barrier functions cbf s but at the same time are free of their most essential deficiencies due to the excellent mbf properties new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed the mbfm have up to a superlinear rate of convergence and are to the classical barrier functions cbf s method as the multipliers method for augmented lagrangians is to the classical penalty function method based on the dual theory associated with mbf the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed the application of the mbf to linear lp and quadratic qp programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the cbf methods the numerical realization of the mbfm leads to the newton modified barrier method nmbm the excellent mbf properties allow us to discover that for any nondegenerate constrained optimization problem there exists a hot start from which the nmbm has a better rate of convergence a better complexity bound and is more stable than the interior point methods which are based on the classical barrier functions

a method for the parametric center problem with a strictly monotone polynomial time algorithm for linear programming given a system of linear inequalities ax b dt and equalities mx g ht where the right hand sides are parametrically deformed over the scalar t the parametric center problem is to trace the parametric family of approximate solutions x t to the center problems p t where p t is the problem maximize i 1m ln bi dit aix subject to ax b dt and mx g ht we present an algorithm for tracing the parametric family of solutions x t over some given range at the kth iterate of the algorithm the point x tk is an approximate solution to the center problem p tk in an appropriate measure of approximation the value of tk is increased to tk 1 and a newton step is used to generate x tk 1 which is an approximate solution to the center problem p tk 1 the sequence of values of t exhibit at least one of the following geometric rates of change tmax tk 1 tmax tk 1 1 128m tk 1 tmin tk tmin 1 1 128m where tmin tmax is the maximum minimum number t such t

computation of eigenvalues and solutions of regular sturm liouville problems using haar wavelets the paper presents a novel method for the computation of eigenvalues and solutions of sturm liouville eigenvalue problems sleps using truncated haar wavelet series this is an extension of the technique proposed by hsiao to solve discretized version of variational problems via haar wavelets the proposed method aims to cover a wider class of problems by applying it to historically important and a very useful class of boundary value problems thereby enhancing its applicability to demonstrate the effectiveness and efficiency of the method various celebrated sturm liouville problems are analyzed for their eigenvalues and solutions also eigensystems are investigated for their asymptotic and oscillatory behavior the proposed scheme unlike the conventional numerical schemes such as rayleigh quotient and rayleigh ritz approximation gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue and it is found to have quadratic convergence with increase in resolution

quadratic growth of convergence radii for eigenvalues of two parameter sturm liouville equations abstract the n th eigenvalue of the equation y r x y 0 a x b subject to self adjoint boundary conditions admits a power series expansion into powers of for sufficiently small it is proved that the convergence radii of these series grow like n 2 as n tends to infinity provided r x is analytic on a b applications to the airy and mathieu equation are given

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

eigencurves of non definite sturm liouville problems for the p laplacian abstract the eigencurves are studied for a weighted quasi linear sturm liouville type problem of the form p y p 1 r q s sgn y y p 1 on 0 1 with sturmian type boundary conditions p being the p laplacian topics include holder classes c k and asymptotics including existence of oblique asymptotes an example illustrates precision of our holder exponents when p 2

following a balanced trajectory from an infeasible point to an optimal linear programming solution with a polynomial time algorithm this paper is concerned with the problem of following a trajectory from an infeasible starting point directly to an optimal solution of the linear programming problem a class of trajectories for the problem is defined based on the notion of a balanced solution to the problem given a prespecified positive balancing constant an infeasible solution x is said to be balanced if the optimal value gap is less than or equal to times the infeasibility gap mathematically this can be written as ctx z tx where the linear form tx is the phase i objective function and z is the optimal objective value of the linear program the concept of a balanced solution is used to define a class of trajectories from an infeasible point to an optimal solution of a given linear program each trajectory has the property that all points on or near the trajectory in a suitable metric are balanced the main thrust of the paper is the development of an algorithm that traces a given balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial time more specifically the algorithm allows for fixed improvement in the bound on the phase i and the phase ii objectives in on newton steps

spectral inversion of an indefinite sturm liouville problem due to richardson we study an indefinite sturm liouville problem due to richardson whose complicated eigenvalue dependence on a parameter has been a puzzle for decades in atomic physics a process exists that inverts the usual schr dinger situation of an energy eigenvalue depending on a coupling parameter into the so called sturmian problem where the coupling parameter becomes the eigenvalue which then depends on the energy we observe that the richardson equation is of the sturmian type this means that the richardson and its related schr dinger eigenvalue functions are inverses of each other and that the richardson spectrum is therefore no longer a puzzle

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

an infeasible interior point predictor corrector algorithm for linear programming a predictor corrector method is proposed for solving standard form linear programming problems starting from initial points that strictly satisfy the positivity constraints but do not necessarily satisfy the equality constraints the algorithm is globally convergent under the assumption that the linear program has an optimal solution under some additional assumptions on the starting point we prove that epsilon feasibility and epsilon complementarily can be obtained in o n ln frac 1 epsilon iterations

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

a combined phase i phase ii projective algorithm for linear programming we devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution inclusion of such a constraint allows the algorithm to be applied to a possibly infeasible standard form linear program without the addition of any bigm terms or conversion to a primal dual problem

a potential reduction algorithm with user specified phase i phase ii balance for solving a linear program from an infeasible warm start this paper develops a potential reduction algorithm for solving a linear programming problem directly from a warm start initial point that is neither feasible nor optimal the algorithm is an interior point variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement phase i and objective value improvement phase ii the key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense given a prespecified balancing parameter beta 0 the algorithm maintains the following phase i phase ii beta balancing constraint throughout left c t x z right beta xi t x where c t x is the objective function z is the unknown optimal objective value of the linear program and xi t x measures the infeasibility of the current iterate x this balancing constraint can be used to either emphasize rapid attainment of feasibility set beta large

polynomial time algorithms for linear programming based only on primal scaling and projected gradients of a potential function this paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function the paper contains extensions and analysis of two polynomial time algorithms for linear programming we first present an extension of gonzaga s o nl iteration algorithm that computes dual variables and does not assume a known optimal objective function value this algorithm uses only affine scaling and is based on computing the projected gradient of the potential function q ln x t s sum limits j 1 n ln x j wherex is the vector of primal variables ands is the vector of dual slack variables and q n sqrt n the algorithm takes either a primal step or recomputes dual variables at each iteration we next present an alternate form of ye s o sqrt n l iteration algorithm that is an extension of the first algorithm of the paper but uses the potential function q ln x t s sum limits j 1 n ln x j sum limits j 1 n ln s j where q n sqrt n we use this alternate form of ye s algorithm to show that ye s algorithm is optimal with respect to the choice of the parameterq in the following sense suppose thatq n n t wheret 0 then the algorithm will solve the linear program in o n r l iterations wherer max t 1 t thus the value oft that minimizes the complexity bound ist 1 2 yielding ye s o sqrt n l iteration bound

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

spectral properties of two parameter eigenvalue problems ii we study the self adjoint eigenvalue problem w x 0 in hilbert space for one equation in two parameters here is bounded below with compact resolvent for each 1 2 we give necessary and sufficient conditions for the existence of so that holds with w 0 and we investigate the geometry of the set z 0 of such we also discuss higher order solution sets z i where the ith eigenvalue of w vanishes deriving various asymptotic results in a unified fashion

theoretical efficiency of a shifted barrier function algorithm for linear programming abstract this paper examines the theoretical efficiency of solving a standard form linear program by solving a sequence of shifted barrier problems of the form minimize x c t x e j 1 n ln x j eh j s t a x b x eh 0 for a given and fixed shift vector h 0 and for a sequence of values of 0 that converges to zero the resulting sequence of solutions to the shifted barrier problems will converge to a solution to the standard form linear program the advantage of using the shifted barrier approach is that a starting feasible solution is unnecessary and there is no need for a phase i phase ii approach to solving the linear program either directly or through the addition of an artificial variable furthermore the algorithm can be initiated with a warm start i e an initial guess of a primal solution x that need not be feasible the number of iterations needed to solve the linear program to a desired level of accuracy will depend on a measure of how close the initial solution x is to being feasible and optimal the number of iterations will also depend on the judicious choice of the shift vector h if an approximate center of the dual feasible region is known in advance then h can be chosen so that the guaranteed fractional decrease in at each iteration is 1 1 6 n which contributes afactor of 6 n to the number of iterations needed to solve the problem

on coupled multiparameter nonlinear elliptic systems on considere le systeme de problemes non lineaires aux valeurs limites de dirichlet lu x f u x v x lv x g u x v x x domaine borne de r n l est un operateur aux derivees partielles lineaire fortement et uniformement elliptique et sont des parametres reels f g r 2 r des applications continues telles que f 0 0 0 g 0 0 on presente une analyse detaillee de l ensemble des solutions du point de vue de la theorie de la bifurcation a plusieurs parametres

non real eigenvalues of indefinite sturm liouville problems abstract the present paper deals with non real eigenvalues of regular indefinite sturm liouville problems a priori bounds and sufficient conditions of the existence for non real eigenvalues are obtained under mild integrable conditions of coefficients

theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane emden type equations abstract in this article a new method is generated to solve nonlinear lane emden type equations using legendre hermite and laguerre wavelets we are interested to note that these wavelets will give same solutions with good accuracy theorems on convergence analysis are stated and proved on the spaces which are created by legendre hermite and laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis the main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients a very high level of accuracy reflects the reliability of this scheme for such problems

polynomial affine algorithms for linear programming the method of steepest descent with scaling affine scaling applied to the potential functionq logc x i 1n logxi solves the linear programming problem in polynomial time forq n ifq n then the algorithm terminates in no more than o n2l iterations if q n sqrt n withq o n then it takes no more than o nl iterations a modified algorithm using rank 1 updates for matrix inversions achieves respectively o n4l and o n3 5l arithmetic computions