stability of holomorphic foliations with split tangent sheaf we show that the set of singular holomorphic foliations on projective spaces with split tangent sheaf and good singular set is open in the space of holomorphic foliations we also give a cohomological criterion for the rigidity of holomorphic foliations induced by group actions and prove the existence of rigid codimension one foliations of degree n 1 on p n for every n 3

complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

irreducible components of the space of holomorphic foliations a holomorphic foliation with singularities in a complex manifold m consists of a pair 2 y where the singular set z is an analytic subset of m and 7 is a holomorphic foliation on m x if the codimension of the foliation 7 is one and the codimension of the singular set 7 is at least 2 then y may be represented by an equivalence class of holomorphic sections v of t m l where l is a holomorphic line bundle and v satisfies the frobenius integrability condition w a dw 0 if m is compact the set fo l m l of those foliations represented by sections of t m l is an algebraic variety the purpose of this paper is to describe the deformations of logarithmic foliations which are represented by sections of the type

numerical validation of solutions of complementarity the nonlinear case this paper proposes a validation method for solutions of nonlinear complementarity problems the validation procedure performs a computational test if the result of the test is positive then it is guaranteed that a given multi dimensional interval either includes a solution or excludes all solutions of the nonlinear complementarity problem

interval algorithms for finding the minimal root in a set of multiextremal one dimensional nondifferentiable functions two problems arising very often in applications are considered the first problem consists of finding the minimal root of an analytic one dimensional function over a given interval it is supposed that the objective function can be multiextremal and nondifferentiable three new algorithms based on interval analysis and branch and bound global optimization approaches are proposed for solving this problem the novelty of the new algorithms is in improving the elimination criteria and the order in which interval and point evaluations are realized the techniques introduced accelerate the search in comparison with interval analysis methods traditionally used for finding roots of equations the second problem considered in the paper is a generalization of the first one and deals with the search for the minimal root in a set of multiextremal and nondifferentiable functions the methods proposed for solving the first problem are generalized for solving the second the main idea is to use the information obtained from any of the functions to reduce the search domain associated with all the functions numerical experiments carried out on a wide set of test functions demonstrate a quite satisfactory performance of the new algorithms in both cases

rigorous sensitivity analysis for parameter dependent systems of equations abstract in this paper we construct rigorous and realistic enclosures for the solutions of parameter dependent systems of equations f x t 0 the enclosure of the solution x x t is given in the form x 0 s t t 0 with interval vectors x0 t0 and an interval matrix s this allows a detailed analysis of the sensitivity of the solution x t to small changes of t and the influence of specific changes in selected parameters on the solution the linear case is treated separately

exclusion regions for systems of equations branch and bound methods for finding all zeros of a nonlinear system of equations in a box frequently have the difficulty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box this has the effect that near each zero many small boxes are created by repeated splitting whose processing may dominate the total work spent on the global search this paper discusses the reasons for the occurrence of this so called cluster effect and how to reduce the cluster effect by defining exclusion regions around each zero found that are guaranteed to contain no other zero and hence can safely be discarded such exclusion regions are traditionally constructed using uniqueness tests based on the krawczyk operator or the kantorovich theorem these results are reviewed moreover refinements are proved that significantly enlarge the size of the exclusion region existence and uniqueness tests are also given

empirical evaluation of innovations in interval branch and bound algorithms for nonlinear systems interval branch and bound algorithms for finding all roots use a combination of a computational existence uniqueness procedure and a tessellation process generalized bisection such algorithms identify with mathematical rigor a set of boxes that contains unique roots and a second set within which all remaining roots must lie though each root is contained in a box in one of the sets the second set may have several boxes in clusters near a single root thus the output is of higher quality if there are relatively more boxes in the first set in contrast to previously implemented similar techniques a box expansion technique in this paper based on using an approximate root finder epsilon inflation and exact set complementation decreases the size of the second set increases the size of the first set and never loses roots in addition to the expansion technique use of second order extensions to eliminate small boxes that do not contain roots and interval slopes versus interval derivative matrices are studied these items are evaluated empirically on a significant test problem set within a fortran 90 environment designed for such purposes the results are compared with previous results and show that careful incorporation of the techniques yields both quantitatively and qualitatively superior computer codes

nonlinear optimization exclusion tests for finding all solutions of nonlinear equations abstract exclusion tests are a well known tool in the area of interval analysis for finding the zeros of a function over a compact domain recently k georg developed linear programming exclusion tests based on taylor expansions in this paper we modify his approach by choosing another objective function and using nonlinear constraints to make the new algorithm converges faster than the algorithm in k georg a new exclusion test j comput appl math 152 2003 147 160 in this way we reduce the number of subinterval in each level the computational complexity for the new tests are investigated also numerical results and comparisons will be presented

analyzing ultimate positivity for solvable systems the positivity problem is a foundational decision problem it asks whether a dynamical system would keep the observing expression over its states positive it has a derivative the ultimate positivity problem which allows that the observing expression is non positive within a bounded time interval for the two problems most existing results are established on discrete time dynamical systems specifically on linear recurrence sequences in this paper however we study the ultimate positivity problem for a class of continuous time dynamical systems called solvable systems they subsume linear systems for the general solvable system we present a sufficient condition for inferring ultimate positivity the validity of the condition can be algorithmically checked once it is valid we can further find the time threshold after which the observing expression would be always positive on the other hand we show that the ultimate positivity problem is decidable for some special classes of solvable systems such as linear systems of dimension up to five we present sufficient conditions for the ultimate positivity of solvable systems we algorithmically check the validity of those conditions we decide the ultimate positivity of linear systems of dimension up to five

the presence of a zero in an integer linear recurrent sequence is np hard to decide we show that the problem of determining if a given integer linear recurrent sequence has a zero a problem that is known as pisot s problem is np hard with a similar argument we show that the problem of finding the minimal realization dimension of a one letter max plus rational series is np hard this last result answers a folklore question raised in the control literature on the max plus approach to discrete event systems our results are simple consequences of a construction due to stockmeyer and meyer c 2002 elsevier science inc all rights reserved

feasible method for semi infinite programs a new numerical method is presented for semi infinite optimization problems which guarantees that each iterate is feasible for the original problem the basic idea is to construct concave relaxations of the lower level problem to compute the optimal values of the relaxation problems explicitly and to solve the resulting approximate problems with finitely many constraints the concave relaxations are constructed by replacing the objective function of the lower level problem by its concave upper bound functions under mild conditions we prove that every accumulation point of the solutions of the approximate problems is an optimal solution of the original problem an adaptive subdivision algorithm is proposed to solve semi infinite optimization problems it is shown that the karush kuhn tucker points of the approximate problems converge to a karush kuhn tucker point of the original problem within arbitrarily given tolerances numerical experiments show that our algorithm is much faster than the existi

on proving existence of feasible points in equality constrained optimization problems various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems in exhaustive search algorithms for global optimizers and other contexts it is of interest to construct bounds around such approximate feasible points then to verify computationally but rigorously that an actual feasible point exists within these bounds hansen and others have proposed techniques for proving the existence of feasible points within given bounds but practical implementations have not to our knowledge previously been described various alternatives are possible in such an implementation and details must be carefully considered also in addition to hansen s technique for handling the underdetermined case it is important to handle the overdetermined case when the approximate feasible point corresponds to a point with many active bound constraints the basic ideas along with experimental results from an actual implementation are summarized here

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

a new exclusion test exclusion tests are a well known tool for obtaining all solutions of a nonlinear system of equations see e g the work of krawczyk or moore recently yamamura and collaborators have developed linear programming exclusion tests that turned out to be highly successful for nonlinear problems in electrical networking the author has developed higher order exclusion tests based on taylor expansions in the present paper it is shown that the ideas behind both approaches can be combined we present a new class of higher order linear programming exclusion tests investigate their computational complexity and illustrate their performance on several examples

new techniques for the analysis of linear interval equations abstract the basic properties of interval matrix multiplication and several well known solution algorithms for linear interval equations are abstracted by the concept of a sublinear map the new concept coupled with a systematic use of ostrowski s comparison operator in a form generalized to interval matrices is used to derive quantitative information about the result of interval gauss elimination andthe limit of various iterative schemes for the solution of linear interval equations moreover optimality results are prove for 1 the use of the midpoint inverse as a preconditioning matrix and 2 gauss seidel iteration with componentwise intersection this extends and improves results by scheu krawczyk and alefeld and herzberger

interval slopes for rational functions and associated centered forms for an arithmetic expression f x involving n rational operations an o n algorithm is given which computes an interval enclosure for the set of slopes f x z where x ranges over an interval x applications to real and complex centered forms are given resulting in improvements over previous results by ratschek siam j numer anal 17 1980 pp 656 662 and petkovic freiburger intervall berichte 83 2 1983 pp 33 50

the complexity of some reachability problems for a system on a finite group abstract we consider a system defined as the product of a finite set of periodic systems on cyclic groups it is of interest to determine if certain subgroups and unions of subgroups of the state set are reachable from a specified initial state and in particular to determine the computational complexity of verifying such reachability these questions are motivated by certain problems that arise in the modelling and control of discrete event systems and certain forms of periodic scheduling our main result is that deciding whether or not the union of a certain set of subgroups is reachable or not is np complete

on the complexity of the orbit problem we consider higher dimensional versions of kannan and lipton s orbit problem determining whether a target vector space may be reached from a starting point x under repeated applications of a linear transformation a answering two questions posed by kannan and lipton in the 1980s we show that when has dimension one this problem is solvable in polynomial time and when has dimension two or three the problem is in nprp

balanced truncation for a class of stochastic jump linear systems and model reduction for hidden markov models this paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete time stochastic jump linear systems the approximation error which is captured by means of the stochastic l 2 gain is bounded from above by twice the sum of singular numbers associated to the truncated states similar to the case of linear time invariant systems a two step model reduction algorithm for hidden markov models is also developed the first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden markov models and a subclass of stochastic jump linear systems in a second step the positivity constraints which reflect the hidden markov model structure are enforced by solving a low dimensional optimization problem

nonlinear equations and optimization abstract interval analysis has led to improved numerical algorithms both for the solution of nonlinear equations and for nonlinear optimization in fact interval analysis has made possible algorithms for solving a problem often predicted to be numerically insoluble the nonlinear global optimization problem in this paper we survey interval methods for solving nonlinear equations and optimization

abstract generalized bisection and a cost bound etude des proprietes des algorithmes de recherche binaires pour la separation de toutes les solutions des systemes d equations non lineaires dans un domaine compact donne

a new exclusion test for finding the global minimum exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain these algorithms are based on a minimization condition that can be applied to each cell in the domain in this paper we consider lipschitz functions of order a and give a new minimization condition for the exclusion algorithm furthermore convergence and complexity results are presented for such algorithm

on the boolean minimal realization problem in the max plus algebra one of the open problems in the max plus algebraic system theory for discrete event systems is the minimal realization problem in this paper we present some results in connection with the minimal realization problem in the max plus algebra first we characterize the minimal system order of a max linear discrete event system we also introduce a canonical representation of the impulse response of a max linear discrete event system next we consider a simplified version of the general minimal realization problem the boolean minimal realization problem i e we consider models in which the entries of the system matrices are either equal to the max plus algebraic zero element or to the max plus algebraic identity element we give a lower bound for the minimal system order of a max plus algebraic boolean discrete event system we show that the decision problem that corresponds to the boolean realization problem i e deciding whether or not a boolean realization of a given order exists is decidable and that the boolean minimal realization problem can be solved in a number of elementary operations that is bounded from above by an exponential of the square of any upper bound of the minimal system order we also point out some open problems the most important of which is whether or not the boolean minimal realization problem can be solved in polynomial time

quasiconvex relaxations based on interval arithmetic abstract interval analysis provides a tool for i forward error analysis ii estimating and controlling rounding and approximation errors automatically and iii proving existence and uniqueness of solutions in this context the terms self validating methods inclusion methods orverification methods are in use in this paper we present a new self validating method for solving global constrained optimization problems this method is based on the construction of quasiconvex lower bound and quasiconcave upper bound functions of a given function the latter defined by an arithmetical expression no further assumptions about the nonlinearities of the given function are necessary these lower and upper bound functions are rigorous by using the tools of interval arithmetic in its easiest form they are constructed by taking appropriate linear and or quadratical estimators which yield quasiconvex quasiconcave bound functions we show how these bound functions can be used to define rigorous quasiconvex relaxations for constrained global optimization problems and nonlinear systems these relaxations can be incorporated in a branch and bound framework yielding a self validating method

minimal max realization of convex sequences we show that the minimal dimension of a linear realization over the max semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps the minimal dimensional realization of any convex realizable sequence can thus be found in linear time the result is based on a bound in terms of minors of the hankel matrix

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable

minimal realization in the max algebra is an extended linear complementarity problem we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an extended linear complementarity problem elcp we use an algorithm that finds all solutions of an elcp to find all equivalent minimal state space realizations of a single input single output siso discrete event system we also give a geometrical description of the set of all minimal realizations of a siso max linear discrete event system

the continuous skolem pisot problem we study decidability and complexity questions related to a continuous analogue of the skolem pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence in particular we show that the continuous version of the nonnegativity problem is np hard in general and we show that the presence of a zero is decidable for several subcases including instances of depth two or less although the decidability in general is left open the problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations which are interesting problems in their own right