a computational method for finding all the roots of a vector function based on dynamical systems theory a computational method is proposed to locate all the roots of a nonlinear vector function the computational approach utilizes the cell mapping method this method relies on discretization of the state space and is a convenient and powerful numerical tool for analyzing the global behavior of nonlinear systems our study shows that it is efficient and effective for determining roots because it minimizes and simplifies computations of system trajectories since the roots are asymptotically stable equilibrium points of the autonomous dynamical system it also provides the domains of attraction associated with each root other numerical techniques based on iterative and homotopic methods can make use of these domains to choose appropriate initial guesses singular manifolds play an important role in limiting the extent of these domains of attraction both a theoretical basis and a computational algorithm for locating the singular manifolds are also provided they make use of similar state space discretization frameworks examples are given to illustrate the computational approaches it is demonstrated that for one of the examples a mechanical system the method yields many more solutions than those previously reported

on the preservation of log convexity and log concavity under some classical bernstein type operators this paper analyzes the preservation of both the log convexity and the log concavity under certain bernstein type operators some results are provided for the bernstein szasz baskakov the gamma type and the weierstrass operators probabilistic methods support the proofs of these results

on uniform approximation by some classical bernstein type operators abstract we investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval the discussed examples include the szasz operators the szasz durrmeyer operators the gamma operators the baskakov operators and the meyer konig and zeller operators we show that some results of totik remain valid for unbounded functions at the same time that we give simple rates of convergence in terms of the usual modulus of continuity we also show by a counterexample that the result for meyer konig and zeller operators does not extend to cheney and sharma operators

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

logspace hierarchies polynomial time and the complexity of fairness problems concerning ω machines in this paper we define a restricted logspace oracle hierarchy which turns out to be equivalent to the logspace alternation hierarchy of chandra kozen and stockmeyer and thus is contained within the second level of the logspace oracle heirarchy of ruzzo simon and tompa we then examine problems concerning various types of fair computations with respect to omega finite state machines omega fsm s and omega one counter machines omega 1cm s for example we consider the nonemptiness problem for omega fsm s and omega 1cm s where acceptance is defined in the usual fashion but with a fairness constraint imposed on accepting computations our results yield problems that are complete not only for logspace and ptime but the second and third levels of the restricted logspace oracle hierarchy as well as far as we know these are the first natural problems shown to be complete for various levels of the logspace alternation hierarchy the problems are also of independent interest

petri nets and large finite sets an upper bound is given for the complexity of the karp miller decision procedure for the finite containment problem for pairs of petri nets the procedure is shown to be primitive recursive in the ackermann function bounds for the lengths of the searches involved are obtained in terms of large finite sets in the sense of paris harrington and of ketonen solovay

on the finite containment problem for petri nets abstract we prove that the finite containment problem fcp for petri nets is dtime ackermann complete for the reducibility p t thus sharpening previous results due to mcaloon 1984 and to mayr and meyer 1981 our principal technique is to replace an application of the infinite ramsey theorem by a certain finite ramsey theorem previously studied by paris 1980 and by ketonen and solovay 1981 such techniques may have further applications in obtaining upper bounds for combinatorial problems

a numerical method for computing domains of attraction for dynamical systems the backward mapping approach for computation of global domains of attraction of asymptotically stable non critical equilibrium points of dynamical systems is presented a basis for the proposed approach is an extension of lyapunov s direct method due to lasalle and lefschetz an iterative process that converges to the global domain of attraction of an asymptotically stable equilibrium point is formulated the method applies to both continuous time and discrete time multidimensional systems it is shown that the backward mapping approach proposed by c s hsu for spiral equilibrium points of second order discrete time systems is a particular case of the algorithm presented here the proposed method can be used for autonomous systems as well as for systems with periodic coefficients when applied to discrete time formulation of dynamical systems the method can be used to determine the regions of stability of periodic solutions the paper concludes with a number of illustrative examples that demonstrate the usefulness of the proposed approach

on an application of dynamical systems theory to determine all the zeros of a vector function abstract a continuous time dynamical system is constructed and analyzed in this paper to help locate all the zeros of nonlinear vector functions lyapunov stability technique is used to show that the zeros of the vector function become asymptotically stable equilibria for this dynamical system a strict lyapunov function is constructed to indicate local stability and to estimate the domains of attraction of all equilibria manifolds on which the jacobian of the vector field is singular play a significant role in characterizing the global behavior of the system examples are provided to illustrate the extent of the theory new computational techniques to determine all the zeros of a vector function can be developed based on the dynamical aspects of the theory given in this paper

the complexity of the finite containment problem for petri nets if the reachability set of a petri net or equivalently vector addition system is finite it can be effectively constructed furthermore the finiteness is decidable thus the containment and equality problem for finite reachability sets become solvable we investigate the complexity of decision procedures for these two problems and show by reducing a bounded version of hilbert s tenth problem to the finite containment problem that these two problems are extremely hard that in fact the complexity of each decision procedure exceeds any primitive recursive function infinitely often the finite containment and quality problem are thus the first uncontrived decidable problems with provably non primitive recursive complexity

nonprimitive recursive complexity and undecidability for petri net equivalence the aim of this note is twofold firstly it shows that the undecidability result for bisimilarity in theor comput sci 148 1995 281 301 can be immediately extended for the whole range of equivalences and preorders on labelled petri nets secondly it shows that restricting our attention to nets with finite reachable space the respective decidable problems are nonprimitive recursive this approach also applies to mayr and meyer s result j acm 28 1981 561 576 for the reachability set equality yielding a more direct proof

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

log concavity for bernstein type operators using stochastic orders abstract this paper aims to study the preservation of log concavity for bernstein type operators in particular attention is focused on positive linear operators defined on the positive semi axis admitting a probabilistic representation in terms of a process with independent increments this class includes the classical gamma szasz and szasz durrmeyer operators with respect to the first and second operators the results of this paper correct two erroneous counterexamples in 10 as a main tool in our results we use stochastic order techniques our results include as a particular case the log concavity of certain functions related to the incomplete gamma function

on the role of singularities in branin s method from dynamic and continuation perspectives a global analysis of branin s method originally due to davidenko for finding all the real zeros of a vector function is carried out the analysis is based on a global study of this method perceived of as a dynamical system since branin s algorithm is closely related to homotopy methods this paper sheds some light on the global performance of these methods when employed for locating all the zeros of a vector function following the dynamical system approach the performance of branin s algorithm is related to the existence of extraneous singularities as well as to the relative spatial distribution of the zeros of the vector function and singular manifolds branin s conjectures regarding the types and the role of extraneous singularities are examined and counterexamples are provided to disprove them we conclude that the performance of branin s method for locating all the zeros of a vector function is questionable even in the absence of extraneous singularities

a general cdc formulation for specializing the cell exclusion algorithms of finding all zeros of vector functions let j be the family of analytic compositions of analytic mappings and their absolute values a novel inequality on the range of the mappings in j is proved which is then used to yield a universal cdc cell discarded condition formulation for the system of equations f x 0 with f in j with this cdc formulation the cell exclusion algorithm developed in 1 is generalized it is shown that the algorithm generalized can not only be applied assuredly to locate all the zeros of any functions f in j but it can do so much more efficiently and conveniently than the original algorithms as well

bernstein durrmeyer operators abstract in this paper we consider the durrmeyer type modifications of the classical bernstein szasz and baskakov operators and deal with two different kind of problems firstly we obtain new results concerning preservation of shape properties lipschitz constants and global smoothness as well as monotonic convergence under convexity to do this we use a probabilistic approach based on representations of these operators in terms of stochastic processes having a s nondecreasing paths and satisfying a suitable martingale type condition secondly we show that the szasz durrmeyer operator is the limit in an appropriate sense of both the bernstein durrmeyer and the baskakov durrmeyer operators we provide rates of convergence which are derived from the bounds for the total variation distance between the probability measures involved

problems concerning fairness and temporal logic for conflict free petri nets abstract in this paper we examine the complexity of the fair nontermination problem for conflict free petri nets under several definitions of fairness for each definition of fairness we are able to show the problem to be complete for either np ptime or nlogspace we then address the question of whether these results extend to the more general model checking problem with respect to the temporal logic for petri nets introduced by suzuki since many of the model checking problems concerning finite state systems can be reduced to a version of the fair nontermination problem it would seem plausible that the model checking problem for conflict free petri nets would be decidable however it turns out that unless the logic is severely restricted model checking is undecidable for conflict free petri nets in particular the problem is undecidable even when formulas are of the form g invariantly where contains no temporal logic operators on the other hand we show that model checking for conflict free petri nets is np complete for l f x the logic restricted to the operators f eventually x next time and with negations allowed only on the predicates

weak singularities and the continuous newton method abstract some issues related to the determination of the singular roots of a nonlinear vector function f r n r n are addressed in this paper it is usually assumed that newton like fields are not defined at singularities thus a particular treatment for these points is necessary nevertheless in dimension 1 and in several higher dimensional instances it is possible to make a smooth extension of the field to singular points when this is the case for a singular root it can be treated in a way similar to that of regular ones necessary and sufficient conditions for this extension to be possible are given under some structural assumptions through the concept of weak singularity the actual setting for this result is a general class of quotient functions which includes in particular the newton field for the specific case of the continuous time newton method we enlarge some previous results concerning the relation between singular roots and equilibrium points of the extended field as well as their asymptotic stability finally a computational tool obtained from the extended continuous newton method by means of the cell mapping technique is shown to be well behaved for the location of these singular roots

a cell exclusion algorithm for determining all the solutions of a nonlinear system of equations abstract a new group of methods named cell exclusion algorithms ceas is developed for finding all the solutions of a nonlinear system of equations these types of algorithms different in principle from those of homotopy interval and cell mapping dynamical analysis approaches are based on cellular discretization and the use of a certain simple necessity test of the solutions the main advantages of the algorithms are their simplicity reliability and general applicability having all features of interval techniques but without using interval arithmetic and with complexity o log 1 e the algorithms improve significantly on both the interval algorithms and the cell mapping techniques theoretical analysis and numerical simulations both demonstrate that ceas are very efficient

best constants for tensor products of bernstein type operators abstract for the tensor product of k copies of the same one dimensional bernstein type operator l we consider the problem of finding the best constant in preservation of the usual modulus of continuity for the l p norm on r k two main results are obtained the first one gives both necessary and sufficient conditions in order that 1 k 1 1 p is the best uniform constant for a single operator the second one gives sufficient conditions in order that 1 k 1 1 p is the best uniform constant for a family of operators the general results are applied to several classical families of operators usually considered in approximation theory throughout the paper probabilistic concepts and methods play an important role

some complexity bounds for problems concerning finite and 2 dimensional vector addition systems with states abstract in this paper we analyse the complexity of the reachability containment and equivalence problems for two classes of vector addition systems with states vasss finite vasss and 2 dimensional vasss both of these classes are known to have effectively computable semilinear reachibility sets slss by giving upper bounds on the sizes of the sls representations we achieve upper bounds on each of the aforementioned problems in the case of the finite vasss the sls representation is simply a listing of the reachability set therefore we derive a bound on the norm of any reachable vector based on the dimension number of states and amount of increment caused by any move in the vass the bound we derive shows an improvement of two levels in the primitive recursive hierarchy over results previously obtained by mcaloon 1984 thus answering a question posed by clote 1986 we then show this bound to be optimal we feel that the techniques we use in deriving our upper bounds represent an original approach to the problem and since they yield improvements over previous results we feel these techniques may have applications to other problems in the case of 2 dimensional vasss we analyse an algorithm given by hopcroft and pansiot 1979 that generates an sls representation of the reachability set specifically we show that the algorithm operates in 2 2 cln nondeterministic time where l is the length of the binary representation of the largest integer in the vass n is the number of transitions and c is some fixed constant we also give examples for which this algorithm will take 2 2 dln nondeterministic time for some positive constant d finally we give a method of determinizing the algorithm in such a way that it requires no more than 2 2 cln deterministic time from this upper bound and special properties of the generated slss we derive upper bounds of dtime 2 2 cln for the three problems mentioned above

on the approximation of convex functions by bernstein type operators abstract as a consequence of jensen s inequality centered operators of probabilistic type also called bernstein type operators approximate convex functions from above starting from this fact we consider several pairs of classical operators and determine in each case which one is better to approximate convex functions in almost all the discussed examples the conclusion follows from a simple argument concerning composition of operators however when comparing szasz mirakyan operators with bernstein operators over the positive semi axis the result is derived from the convex ordering of the involved probability distributions analogous results for non centered operators are also considered

adaptive cellular integration of linearly implicit differential equations the use of cell to cell mapping methods for the global analysis of linearly implicit odes is addressed in this paper limitations of simple and adjoining cell mapping when applied to linearly implicit equations motivate the introduction of a finite time convergence model for cellular methods which unifies time and spatial discretization with numerical aspects in a comprehensive framework the resulting formal characterization of some properties of simple and adjoining cell mapping leads to a hybridization of both techniques which is shown to be well behaved for the analysis of linearly implicit odes

a taxonomy of fairness and temporal logic problems for petri nets in this paper we define a temporal logic for reasoning about petri nets we show the model checking problem for this logic to be ptime equivalent to the petri net reachability problem using this logic and two refinements we show the fair nontermination problem to be ptime equivalent to reachability for several definitions of fairness for other versions of fairness this problem is shown to be either ptime equivalent to the boundedness problem or highly undecidable in all 24 versions of fairness are examined

analysis of newton s method at irregular singularities recent work on newton s method at singularities of the jacobian has established linear convergence under certain regularity assumptions here newton s method is analyzed in the neighborhood of irregular singularities which include all minimizers at which the hessian has a one dimensional null space depending on a parameter specific to the underlying optimization problem or system of simultaneous equations newton s method is found either to converge with a limiting ratio of about frac 2 3 or to diverge from arbitrarily close starting points or to behave in a certain sense chaotically

an algorithm for the general petri net reachability problem an algorithm is presented for the general petri net reachability problem it is based on a generalization of the basic reachability tree construction which is made symmetric with respect to the ini

completeness results for single path petri nets we define a new subclass of persistent petri nets called single path petri nets our intention is to provide a class of petri nets whose study might yield some insight into the mathematical properties of persistent petri nets or even general petri nets we conjecture that the karp miller coverability tree for a persistent net is small enough to be searched in polynomial space although we are unable to prove this conjecture we do show that single path petri nets have this property we then use this fact to show that the canonical analysis problems i e boundedness reachability containment and equivalence for single path petri nets are pspace complete in the strong sense furthermore we show that the problem of recognizing a single path petri net is also pspace complete

optimal algorithms for the coverability the subword the containment and the equivalence problems for commutative semigroups abstract in this paper we present decision procedures for the coverability the subword the containment and the equivalence problems for commutative semigroups these procedures require at most space 2c n where n is the size of the problem instance and c is some problem independent constant furthermore we show that the exponential space hardness of the above problems follows from the work of mayr and meyer thus the presented algorithms are space optimal our results close the gap between the 2c n log n space upper bound shown by rackoff for the coverability problem and shown by huynh for the containment and the equivalence problems and the exponential space lower bound resulting from the corresponding bound for the uniform word problem established by mayr and meyer

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem