stagnation analysis of dgmres we study problems for which the iterative method dgmres for solving drazin inverse solution of singular linear systems of equations which makes no progress in its initial iterations particularly our tool for analysis is nonlinear system of equations the stagnation system that characterize this behavior for problems of dimension two and index one we can solve this system explicitly we partially extend this result to higher dimensions for a class of eigenvector matrices called extreme we also study the real matrices

dgmres a gmres type algorithm for drazin inverse solution of singular non symmetric linear systems abstract in a recent work by the author linear algebra appl 298 1999 99 krylov subspace methods were derived for drazin inverse solution of consistent or inconsistent linear systems of the form ax b where a c n n is a singular and in general non hermitian matrix that has arbitrary index one of these methods modeled after the generalized conjugate residual method gcr and denoted dgcr is considered in the present work again it is shown that all of the approximations produced by dgcr exist and a gmres like algorithm denoted dgmres for its implementation is derived like gmres dgmres too is economical computationally and storagewise

a unified approach to krylov subspace methods for the drazin inverse solution of singular nonsymmetric linear systems abstract consider the linear system ax b where a c n n is a singular matrix in the present work we propose a general framework within which krylov subspace methods for drazin inverse solution of this system can be derived in a convenient way the krylov subspace methods known to us to date treat only the cases in which a is hermitian and its index ind a is unity necessarily in the present work a is not required to be hermitian it can have any type of spectrum and ind a is arbitrary we show that as is the case with nonsingular systems the krylov subspace methods developed here terminate in a finite number of steps that is at most n ind a for one of the methods derived here we also provide an analysis by which we are able to bound the errors the relevant bounds decreasing with increasing dimension of the krylov subspaces involved the results of this paper are applicable to consistent systems as well as to inconsistent ones an interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard krylov subspace approach to nonsingular systems indeed our approach here reduces to that relevant for nonsingular systems upon setting ind a 0 everywhere

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

some block matrices with signed drazin inverses abstract the sign pattern of a real matrix m is the 0 1 1 matrix obtained from m by replacing each entry by its sign let q m be the set of real matrices with the same sign pattern as m for any m q m if the drazin inverses of m and m have the same sign pattern then m is said to have signed drazin inverse in this paper we give a complete characterization for a class of anti triangular matrices with signed drazin inverse and present a complete characterization for sign symmetric bipartite matrices with signed drazin inverse

on the convergence rate of dgmres abstract the dgmres algorithm was designed to compute the drazin inverse solution of consistent or inconsistent linear systems a x b where a is a square singular matrix with arbitrary index in this paper we compare the convergence rate of dgmres with that of gmres applied to the nonsingular part of a

the boolean hierarchy ii applications the boolean hierarchy i structural properties j cai et al siam j comput 17 1988 pp 1232 252 explores the structure of the boolean hierarchy the closure of np with respect to boolean

transition from anticipatory to lag synchronization via complete synchronization in time delay systems the existence of anticipatory complete and lag synchronization in a single system having two different time delays that is feedback delay tau1 and coupling delay tau2 is identified the transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay tau2 with a suitable stability condition is discussed in particular it is shown that the stability condition is independent of the delay times tau1 and tau2 consequently for a fixed set of parameters all the three types of synchronizations can be realized further the emergence of exact anticipatory complete or lag synchronization from the desynchronized state via approximate synchronization when one of the system parameters b2 is varied is characterized by a minimum of the similarity function and the transition from on off intermittency via periodic structure in the laminar phase distribution

stability of synchronization in coupled time delay systems using krasovskii lyapunov theory stability of synchronization in unidirectionally coupled time delay systems is studied using the krasovskii lyapunov theory we have shown that the same general stability condition is valid for different cases even for the general situation but with a constraint where all the coefficients of the error equation corresponding to the synchronization manifold are time dependent these analytical results are also confirmed by the numerical simulation of paradigmatic examples

complexity measures for public key cryptosystems a general theory of public key cryptography is developed that is based on the mathematical framework of complexity theory two related approaches are taken to the development of this theory and th

the boolean hierarchy i structural properties in this paper we study the complexity of sets formed by boolean operations union intersection and complement on np sets these are the sets accepted by trees of hardware with np predicates as leaves and together these form the boolean hierarchy we present many results about the structure of the boolean hierarchy separation and immunity results natural complete languages and structural asymmetries between complementary classes we show that in some relativized worlds the boolean hierarchy is infinite and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels we prove natural languages variations of vertex cover complete for the various levels of the boolean hierarchy we show the following structural asymmetry though no set in the boolean hierarchy is text d text p immune there is a relativized world in which the boolean hierarchy contains text cod text p immune sets thus this paper explores the structural properties of the

index sets and presentations of complexity classes this paper draws close connections between the ease of presenting a given complexity class b and the position of the index sets ib lcub i l mi b rcub and jb lcub i miis total l mi b rcub in the arithmetical hierarchy for virtually all classes b studied in the literature the lowest levels attainable are ib 03 and jb 02 the first holds iff b is 02 presentable and the second iff b is recursively presentable a general kind of priority argument is formulated and it is shown that every property enforcible by it is not recursively presentable it follows that the classes of p immune and p biimmune languages in exponential time are not recursively presentable it is shown that for all b with ib 03 many members of b do not provably in true 2 arithmetic belong to b a class h is exhibited such that ih 03 is open and ih 03 implies that the polynomial hierarchy is infinite

global phase synchronization in an array of time delay systems we report the identification of global phase synchronization gps in a linear array of unidirectionally coupled mackey glass time delay systems exhibiting highly non phase coherent chaotic attractors with complex topological structure in particular we show that the dynamical organization of all the coupled time delay systems in the array to form gps is achieved by sequential synchronization as a function of the coupling strength further the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster we have confirmed these results by estimating instantaneous phases including phase difference average phase average frequency frequency ratio and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non phase coherent attractors the results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non phase coherent attractors directly without explicitly introducing the measure of phase

development of iterative techniques and extrapolation methods for drazin inverse solution of consistent or inconsistent singular linear systems abstract consider the linear system of equations bx where b is an nxn singular matrix in an earlier work by the author it was shown that iterative techniques coupled with standard vector extrapolation methods can be used to obtain or approximate a solution of this system when it is consistent in the present work we expand on that approach to treat the case in which this system is in general inconsistent starting with richardson s iterative method we develop a family of new iterative techniques and vector extrapolation methods that enable us to obtain or approximate the drazin inverse solution of this system whether the index of b is 1 or greater than 1 we show that the drazin inverse solution can be constructed from a finite number of iterations this number being at most n 2 we also provide detailed convergence analyses of the new iterative techniques and vector extrapolation methods and give their precise rates of convergence

a two step algorithm for solving singular linear systems with index one abstract in this paper a two step algorithm for solving singular linear systems is presented we compare this algorithm with a dgcr type algorithm a sidi a unified approach to krylov subspace methods for the drazin inverse solution of singular non symmetric linear systems linear algebra appl 298 1999 99 113 by numerical experiments an error analysis is given

iterative methods for the drazin inverse of a matrix with a complex spectrum iterative methods for the drazin inverse of a square matrix with a real spectrum have been developed recently these methods are generalized in the case of matrices with complex spectra semiiterative method for the drazin inverse is also discussed numerical examples are given to illustrate the theoretic results

intermittency transition to generalized synchronization in coupled time delay systems we report the nature of the transition to generalized synchronization gs in a system of two coupled scalar piecewise linear time delay systems using the auxiliary system approach we demonstrate that the transition to gs occurs via an on off intermittency route and that it also exhibits characteristically distinct behaviors for different coupling configurations in particular the intermittency transition occurs in a rather broad range of coupling strength for the error feedback coupling configuration and in a narrow range of coupling strength for the direct feedback coupling configuration it is also shown that the intermittent dynamics displays periodic bursts of periods equal to the delay time of the response system in the former case while they occur in random time intervals of finite duration in the latter case the robustness of these transitions with system parameters and delay times has also been studied for both linear and nonlinear coupling configurations the results are corroborated analytically by suitable stability conditions for asymptotically stable synchronized states and numerically by the probability of synchronization and by the transition of sub lyapunov exponents of the coupled time delay systems we have also indicated the reason behind these distinct transitions by referring to the unstable periodic orbit theory of intermittency synchronization in low dimensional systems

two remarks on the power of counting the relationship between the polynomial hierarchy and valiant s class p is at present unknown we show that some low portions of the polynomial hierarchy namely deterministic polynomial algorithms using an np oracle at most a logarithmic number of times can be simulated by one p computation we also show that the class of problems solvable by polynomial time nondeterministic turing machines which accept whenever there is an odd number of accepting computations is idempotent that is closed under usage of oracles from the same class

gradient methods for computing the drazin inverse solution the problem of computing the vector a db in the cases when it provides a solution of a singular linear system is reduced to the problem of finding a least squares solution of an appropriate linear system later the presented ideas are extended to arbitrary linear systems as a consequence we present iterative methods for computing the vector a db in general the functionality of the exposed algorithms is based on a specific representation of the drazin inverse solution as well as the properties that we introduce convergence results of the proposed methods are also considered illustrative numerical examples are presented

transition from intermittency to periodicity in lag synchronization in coupled rossler oscillators the dynamical and statistical behavior of lag synchronization in two coupled self sustained chaotic rossler oscillators is reexamined the lack of uniqueness in the conventional characterization of lag synchronization based on the similarity function has caused much skepticism about the existence of lag synchronization we provide an evidence that the emergence of lag synchronization is associated with the transition from on off intermittency to a periodic structure in the laminar phase distribution

phase synchronization in time delay systems though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay it has not been realized yet in chaotic time delay systems exhibiting non phase coherent hyperchaotic attractors in this paper we report identification of phase synchronization in coupled time delay systems exhibiting hyperchaotic attractor we show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength these transitions are characterized by recurrence quantification analysis by phase differences based on a transformation of the attractors and also by the changes in the lyapunov exponents we have found these transitions in coupled piecewise linear and in mackey glass time delay systems

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

dfom algorithm and error analysis for projection methods for solving singular linear system in this paper dfom algorithm is presented we compare dgmres algorithm with dfom algorithm by numerical experiment error analysis for projection method is also given

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

near testable sets in this paper a new property of sets near testability is introduced a set s is near testable s in nt if the membership relation for all immediate neighbors is polynomially computable i e if the function t x chi s x chi s x 1 bmod 2 is polynomially computable the near testable sets form a subclass of the class oplus p parity polynomial time introduced by papadimitriou and zachos and goldschlager and parberry oplus p has a complete set oplus sat that has recently been shown by valiant and vazirani to be hard for np under randomized polynomial time reductions it is proved that there is a uniform polynomial one one reduction that takes every set in oplus p to a near testable set and it is shown that the image of oplus sat under this reduction which we call ntsat is polynomially isomorphic to oplus sat as corollaries it is shown that ntsat is complete for both nt and for oplus sat that ntsat is hard for np under randomized polynom

perturbation analysis of singular linear systems with arbitrary index a new perturbation result for singular but consistent linear systems with coefficient matrix of arbitrary index however if the system is inconsistent we also derive a perturbation bound of the generalized linear least squares solution our treatment extends a classical perturbation result in numerical linear algebra

rice s theorem for the limit sets of cellular automata abstract rice s theorem is a well known result in the theory of recursive functions a corresponding theorem for cellular automata limit sets is proved all nontrivial properties of limit sets of cellular automata cas are shown undecidable the theorem remains valid even if only one dimensional cas are considered

the analysis of restart dgmres for solving singular linear systems abstract a surprising phenomenon concerning with restart dgmres a sidi dgmres a gmres type algorithm for drazin inverse solution of singular nonsymmetric linear systems linear algebra appl 335 2001 189 204 is presented that small values of the restart parameter may converge faster than larger values we take three examples where dgmres 2 converge while dgmres 3 stagnates to interpret the phenomenon two of these examples reveals that dgmres convergence can be extremely sensitive to small changes in the initial residual

a second step towards complexity theoretic analogs of rice s theorem abstract rice s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable borchert and stephan 1997 initiated the search for complexity theoretic analogs of rice s theorem in particular they proved that every nontrivial counting property of circuits is up hard and that a number of closely related problems are spp hard the present paper studies whether their up hardness result itself can be improved to spp hardness we show that their up hardness result cannot be strengthened to spp hardness unless unlikely complexity class containments hold nonetheless we prove that every p constructibly bi infinite counting property of circuits is spp hard we also raise their general lower bound from unambiguous nondeterminism to constant ambiguity nondeterminism

synchronization of time delayed systems in this paper we study synchronization in linearly coupled time delayed systems we first consider coupled nonidentical ikeda systems with a square wave coupling rate using the theory of the time delayed equation we derive less restrictive synchronization conditions than those resulting from the krasovskii lyapunov theory yang kuang delay differential equations academic press new york 1993 then we consider a wide class of nonlinear nonidentical time delayed systems we also propose less restrictive synchronization conditions in an approximative sense even if the coefficients in the linear time delayed equation on the synchronization error are time dependent theoretical analysis and numerical simulations fully verify our main results

new methods for computing the drazin inverse solution of singular linear systems the dgmres method is an iterative method for computing the drazin inverse solution of consistent or inconsistent linear systems of the form a x b where a c n n is a singular and in general non hermitian matrix that has an arbitrary index this method is generally used with restarting but the restarting often slows down the convergence and dgmres often stagnates based on the lgmres and gmres e methods we present two new techniques for accelerating the convergence of restarted dgmres by adding some approximate error vectors or approximate eigenvectors corresponding to a few of the smallest eigenvalues to the krylov subspace we derive the implementation of these methods and present some numerical examples to show the advantages of these methods

orthogonal polynomials and semi iterative methods for the drazin inverse solution of singular linear systems in this work we present a novel class of semi iterative methods for the drazin inverse solution of singular linear systems whether consistent or inconsistent the matrices of these systems are allowed to have arbitrary index and arbitrary spectra in the complex plane the methods we develop are based on orthogonal polynomials and can all be implemented by 4 term recursion relations independently of the index we give all the computational details of the associated algorithms we also give a complete convergence analysis for all methods

p printable sets p printable sets arise naturally in the studies of generalized kolmogorov complexity and data compression as well as in other areas we present new characterizations of the p printable sets and present necessary and sufficient conditions for the existence of sparse sets in p that are not p printable as a corollary to one of our results we show that the class of sets of small generalized kolmogorov complexity is exactly the class of sets which are p isomorphic to a tally language

a semi iterative method for real spectrum singular linear systems with an arbitrary index in this paper we develop a semi iterative method for computing the drazin inverse solution of a singular linear system ax b where the spectrum of a is real but its index i e the size of its largest jordan block corresponding to the eigenvalue zero is arbitrary the method employs a set of polynomials that satisfy certain normalization conditions and minimize some well defined least squares norm we develop an efficient recursive algorithm for implementing this method that has a fixed length independent of the index of a following that we give a complete theory of convergence in which we provide rates of convergence as well we conclude with a numerical application to determine eigenprojections onto generalized eigenspaces our treatment extends the work of hanke and hochbruck 1993 that considers the case in which the index of a is 1 keyword singular systems iterative methods polynomial acceleration

lower bounds and the hardness of counting properties rice s theorem states that all nontrivial language properties of recursively enumerable sets are undecidable borchert and stephan math logic quart 46 4 2000 489 504 started the search for complexity theoretic analogs of rice s theorem and proved that every nontrivial counting property of boolean circuits is up hard hemaspaandra and rothe theoret comput sci 244 1 2 2000 205 217 improved the up hardness lower bound to upo 1 hardness the present paper raises the lower bound for nontrivial counting properties from upo 1 hardness to fewp hardness i e from constant ambiguity nondeterminism to polynomial ambiguity nondeterminism furthermore we prove that no relativizable technique can raise this lower bound to fewp 1 ttp hardness we also prove a rice style theorem for np namely that every nontrivial language property of np sets is np hard and for a broad class of leaf language classes we prove a sufficient condition for the natural analog of rice s theorem to hold

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

a note on the drazin inverse of an anti triangular matrix abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m under some conditions moreover some particular cases of these results are also considered