existence of solutions to an initial value problem for the differential equation f t x x 0 using barrier strips abstract under sign assumptions we investigate the global existence of solutions of the initial value problem f t x x x 0 a where the function f t x p may be singular at x a

applications of the parametric programming procedure abstract many classes of mathematical programming problems can be formulated as a linear program with a parametric objective function gass and saaty developed in the early 1950 s a parametric programming procedure for solving the latter problem this procedure is relevant to solution strategies for many problem types but is underutilized often because the relationship of the procedure to the relevant problem class is not recognized in this paper five recent applications of the parametric programming procedure are presented

on the solvability of a singular initial value problem x f t x x using barrier strips abstract in this paper conditions which guarantee the existence of solutions to x f t x x x 0 a are given

matrix rational h 2 approximation a gradient algorithm based on schur analysis this paper deals with the rational approximation of specified order n to transfer functions which are assumed to be matrix valued functions in the hardy space for the complement of the closed unit disk endowed with the l2 norm an approach is developed leading to a new algorithm the first one to our knowledge which concerns matrix transfer functions in l2 norm this approach generalizes the ideas presented in l baratchart m cardelli and m olivi automatica 27 1991 pp 413 418 in the scalar case but involves substantial new difficulties using the douglas shapiro shields factorization of transfer functions the criterion for the rational approximation problem above is expressed in terms of inner matrix functions of mcmillan degree n these functions which possess a manifold structure are represented by means of local coordinate maps obtained in d alpay l baratchart and a gombani oper theory adv appl 73 1994 pp 30 66 from a tangential schur algorithm and for which the coordinates range over n copies of the unit ball a gradient algorithm is then employed to solve the approximation problem using the coordinate maps to describe the manifold locally and changing from one coordinate map to another when required however while processing the gradient algorithm a boundary point can be reached it is proved that such a point can be considered as an initial point for searching for a local minimum of lower degree while a local minimum of mcmillan degree k n provides a starting point for searching for a local minimum at degree k 1 the minimization process then pursues through different degrees the convergence of this algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

a generalized upper and lower solution method for singular initial value problems abstract this paper presents new existence results for singular initial value problems in particular our nonlinearity may be singular in its dependent variable and is allowed to change sign

a mathematical programming approach to clusterwise regression model and its extensions abstract the clusterwise regression model is used to perform cluster analysis within a regression framework while the traditional regression model assumes the regression coefficient to be identical for all subjects in the sample the clusterwise regression model allows to vary with subjects of different clusters since the cluster membership is unknown the estimation of the clusterwise regression is a tough combinatorial optimization problem in this research we propose a generalized clusterwise regression model which is formulated as a mathematical programming mp problem a nonlinear programming procedure with linear constraints is proposed to solve the combinatorial problem and to estimate the cluster membership and simultaneously moreover by integrating the cluster analysis with the discriminant analysis a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis the cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model

a lagrangian treatment of certain nonlinear clustering problems abstract a clustering problem is described in which the objective function involving both within and between subset distances in order to induce separation between clusters and yield a realistic measure is highly nonlinear and difficult to analyze by fixing the number of within group distances however a linear objective function is achieved and the problem becomes of the more common set partitioning kind with the further assumption of collinear objects yielding an important subclass of problem the spp matrix is unimodular and the simplex method suffices to solve the integer programming problem a lagrange relaxation on the constraint for the number of within group distances allows then the nonlinear clustering problem to be solved exactly up to the occurrence of gaps as a sequence of lp s an implementation for small problems using the mpsx of ibm is presented as are the results from various examples for the non collinear case the mapping of data into a single dimension is discussed as are strategies for solving larger problems the behavior of the lagrangian generated function values is examined and found to be exploitable useful approximations in the event of gaps are found to be easily computed

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

singular and nonsingular first order initial value problems abstract under barrier strip type arguments we investigate the existence of global solutions to the initial value problem x f t x x x 0 a where the scalar function f t x p may be singular at x a

existence of closed solutions for a polynomial first order differential equation we find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the abel equation as a particular case such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields

the monotone method for first order problems with linear and nonlinear boundary conditions in this paper we develop the monotone method for the first order problem u t f t u t for a e t a b when f is a caratheodory function and u w 1 1 a b we consider the nonlinear boundary conditions l u a u b 0 with l c r 2 r nondecreasing in x or nonincreasing in y and the linear boundary conditions a 0u 0 b 0u t l 0 with a 0 b 0 and l 0 r we prove the existence of solutions and the validity of the monotone method if there exists a lower solution a and an upper solution b with either a b for the linear conditions we obtain eight new concepts of lower and upper solutions which generalize previous known cases

existence and approximation of solutions for discontinuous functional differential equations in this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly by applying the method of upper and lower solutions which may be discontinuous some existence results of extremal solutions are obtained furthermore we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences

existence of multiple solutions for some nonlinear boundary value problems abstract by imposing one sided conditions on the nonlinearity where neither regularity nor uniformity is required we prove the existence of either a nonnegative or a nonpositive solution for first and second order ordinary differential equations with periodic neumann or dirichlet boundary conditions positone and nonpositone problems are considered some nonexistence results are also obtained when using generalized ambrosetti prodi type conditions we get the existence of nonnegative and nonpositive solutions for first and second order periodic or neumann boundary value problems our method of proof makes use of topological degree arguments in cones

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

a multiplicity result for a nonlinear boundary value problem abstract in this paper we establish existence and multiplicity results for the problemx x f t x x s t a e in 0 x 0 x under the ambrosetti prodi type condition withfbeing a caratheodory function

existence of periodic solutions for fully nonlinear first order differential equations in this paper we establish some existence results for periodic solutions of fully nonlinear first order scalar differential equation f t x x 0 the approach here uses the viscosity solutions method developed in recent decades and the classical upper lower solutions method

a recursive lagrangian method for clustering problems abstract the problem of partitioning a finite set of objects into subsets or clusters so as to optimize an objective function is treated it is shown that for the particular case where all the objects lie upon a line the clustering problem may be solved exactly by an algorithm utilizing only lagrangian relaxation and dynamic programming by exploiting the approximately unimodal shape of the envelope of optimal objective values a search procedure in the space is devised and this allows the solution in reasonable computation time of the combinatorially large problem several examples are presented and the method is applied to one set of noncollinear data which is first transformed so as to become collinear further promising directions for computational enhancement are noted the important aspects of computer implementation are given

constructive algebra methods for the l 2 problem for stable linear systems abstract we investigate the feasibility of using computer algebra for solving l 2 system approximation problems the first order optimality conditions yield a set of polynomial equations which can in principle be solved using grobner basis methods a general solution along these lines would be tremendously useful although it does not appear to be feasible at present except for rather low mcmillan degrees we demonstrate that it can be feasible for specific examples in such cases global optima can be found reliably we show that the use of a schwarz like canonical form simplifies the structure of the problem

existence and multiplicity results for some three point boundary value problems abstract using the method of upper and lower solutions and some degree theory arguments we establish the existence of at least three solutions of some second order nonlinear differential equations of the type x f t x x t i 0 1 subject to nonlinear three point boundary conditions x 0 0 x 1 g x 0 1 the growth of f with respect to x is allowed to be quadratic

balanced realizations of discrete time stable all pass systems and the tangential schur algorithm in this paper the connections are investigated between two different approaches towards the parametrization of multivariable stable all pass systems in discrete time the first approach involves the tangential schur algorithm which employs linear fractional transformations it stems from the theory of reproducing kernel hilbert spaces and enables the direct construction of overlapping local parametrizations using schur parameters and interpolation points the second approach proceeds in terms of state space realizations in the scalar case a balanced canonical form exists that can also be parametrized by schur parameters this canonical form can be constructed recursively using unitary matrix operations here this procedure is generalized to the multivariable case by establishing the connections with the first approach it gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential schur algorithm

existence of at least three solutions of nonlinear three point boundary value problems with super quadratic growth abstract we establish the existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x 0 t 1 x 0 0 x 1 x 0 1 0 1 we employ a condition weaker than the well known nagumo condition which allows the nonlinearity f t x x to grow faster than quadratically with respect to x in some cases our method uses some degree theory arguments

the number of periodic solutions of some analytic equations of abel type new results are proved to estimate the number of periodic solutions of a differential equation of abel type by using a modification of a technique introduced by ilyashenko the main tool is an estimate on the number of zeros of a holomorphic function a concrete example is analyzed but the results are presented to make the method flexible and applicable to other equations

overlapping block balanced canonical forms for various classes of linear systems abstract through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension a possible problem of these parametrizations is that they are not overlapping this could be a drawback for the application of balanced parametrizations in such areas as system identification model reduction and optimization it is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations we first introduce input normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix these canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered then we introduce so called block balanced input normal forms which use the previously constructed input normal forms as building blocks the classes of systems for which such parametrizations are given are the stable minimal systems positive real minimal systems bounded real minimal systems and the class of all minimal systems of given mcmillan degree the results include both the single input single output and the multivariable case in the single input single output case however the issue of choosing nice selections of columns does not occur therefore in this case the derivation and presentation of the results is considerably simplified

existence of solutions to initial value problems for first order differential equations under sign assumptions we investigate the global existence of solutions of the initial value problem f t x x 0 x a a where the function f t x p may be singular at t a

exact multiplicity for periodic solutions of a first order differential equation abstract consider the differential equation 1 x f t x h t where h t is a 1 periodic continuous function and f t x c3 is concave convex in x and 1 periodic in t we obtain the complete structure of 1 periodic solutions by means of singularity theory more precisely we show that the image of singularities f consists of a codimension 1 manifold that divides the c r z into two open sets a1 a3 i equation 1 has a unique 1 periodic solution for h t a1 ii the equation has exactly three 1 periodic solutions for h t a3 furthermore if the image of cusp singularities f c is a codimension 1 manifold of f the differential equation has exactly two solutions for h t f f c and has a unique 1 periodic solution for h t f c

upper and lower solutions for second order boundary value problems with nonlinear boundary conditions on demontre des theoremes d existence en utilisant des solutions superieures et inferieures