geometric properties of positive definite matrices cone with respect to the thompson metric we shall discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to thompson metric

deterministic input driven queue automata we introduce and study the model of deterministic input driven queue automata on such devices the input letters uniquely determine the operations on the memory store which is organized as a queue in particular we consider the case where only a finite number of turns on the queue is allowed the resulting language families share with regular languages many desirable properties we show that emptiness and several other problems are decidable furthermore we investigate closure under boolean operations the existence of an infinite and tight hierarchy depending on the number of turns is also proved

ekeland s variational principle for vectorial multivalued mappings in a uniform space abstract in this paper we establish ekeland s variational principle and an equilibrium version of ekeland s variational principle for vectorial multivalued mappings in the setting of separated sequentially complete uniform spaces our approaches and results are different from those in chen et al 2008 hamel 2005 and lin and chuang 2010 13 14 15 as applications of our results we study vectorial caristi s fixed point theorems and takahashi s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated sequentially complete uniform space we also apply our results to study maximal element theorems which are unified methods of several variational inclusion problems our results contain many known results in the literature fang 1996 21 and will have many applications in nonlinear analysis

a remark on discontinuity at fixed point abstract in this note we give one more solution to the open question of the existence of contractive definitions which are strong enough to generate a fixed point but which do not force the mapping to be continuous at the fixed point r kannan 1969 9 b e rhoades 1988 14

the size cost of boolean operations on constant height deterministic pushdown automata we study the size cost of boolean operations on constant height deterministic pushdown automata i e on the traditional pushdown automata with a built in constant limit on the height of the pushdown we design a simulation showing that a complement can be obtained with a polynomial tradeoff for intersection and union we show an exponential simulation and prove that the exponential blow up cannot be avoided

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

transformations between discrete time and continuous time algebraic riccati equations we introduce a transformation between the discrete time and continuous time algebraic riccati equations we show that under mild conditions the two algebraic riccati equations can be transformed from one to another and both algebraic riccati equations share common hermitian solutions the transformation also sets up the relations about the properties commonly in system and control setting that are imposed in parallel to the coefficient matrices and hermitian solutions of two algebraic riccati equations the transformation is simple and all the relations can be easily derived we also introduce a generalized transformation that requires weaker conditions the proposed transformations may provide a unified tool to develop the theories and numerical methods for the algebraic riccati equations and the associated system and control problems

nonexpansiveness of the resolvent average abstract we show that the resolvent average for positive definite matrices which interpolates between the harmonic and the arithmetic means contracts the thompson metric on the convex cone of positive definite matrices classical results depending on the nonexpansive property of the arithmetic average are considered in the context of the resolvent average in particular resolvent power mean approximation to the karcher mean and ferrante and levy like matrix equations are investigated

a generalization of the banach contraction principle with high order of convergence of successive approximations in this paper we present the following generalization of the banach contraction principle let t d x xt d x x be a continuous operator on a complete metric space x d x d satisfying d tx t2x d x tx for all x d tx d with d x tx j where j r j r is an interval containing 0 j j j j is a gauge function of order r 1r 1 on jj in the sense that 0 0 t tr 0 0 t tr is nondecreasing on j 0 j 0 and t t t t on j 0 j 0 suppose that x0 dx0 d is an initial point such that d x0 tx0 jd x0 tx0 j and all of the iterates xn 1 txn n 0 1 belong to dd then the iterative sequence xn 1 txnxn 1 txn converges to a fixed point of tt with error estimates d xn sn r 1 rnd x0 tx0 andd xn d xn xn 1 1 d xn xn 1 where sn r 1 r rn 1 d x0 tx0 sn r 1 r rn 1 d x0 tx0 and is a nonnegative and nondecreasing function on jj such that t t t this fixed point theorem includes as special cases some results of mysovskih i p mysovskih on the convergence of l v kantorovic s method of solution of functional equations and its applications dokl acad nauk sssr 70 1950 565 568 in russian rheinboldt w c rheinboldt a unified convergence theory for a class of iterative processes siam j numer anal 5 1968 42 63 gel man a gel man a certain fixed point principle dokl akad nauk sssr 198 1971 506 508 in russian soviet math dokl 12 1971 813 816 engl transl huang z d huang on newton s method under holder continuous derivative j math anal appl 270 2002 332 339 and others

the drop theorem the petal theorem and ekeland s variational principle on montre que le principe variationnel d ekeland ou plus precisement une forme legerement modifiee est une consequence d un resultat geometrique connu sous le terme de theoreme de la goutte

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

vectorial form of ekeland type variational principle with applications to vector equilibrium problems and fixed point theory abstract this paper introduces a vectorial form of equilibrium version of ekeland type variational principle some equivalent results to our variational principle are given as applications we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi metric spaces with a w distance caristi kirk fixed point theorem for multivalued maps is also established in a more general setting

quasi gradient based inversion free iterative algorithm for solving a class of the nonlinear matrix equations abstract inspired by the gradient based and inversion free iterations a new quasi gradient based inversion free iterative algorithm is proposed for solving the nonlinear matrix equation x a t x n a i the convergence proof of the suggested algorithm is given several matrix norm inequalities are established to depict the convergence properties of this algorithm three numerical examples are given to illustrate the effectiveness of the suggested algorithms

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

some generalizations of ekeland type variational principle with applications to equilibrium problems and fixed point theory abstract in this paper we introduce the concept of a q function defined on a quasi metric space which generalizes the notion of a function and a w distance we establish ekeland type variational principles in the setting of quasi metric spaces with a q function we also present an equilibrium version of the ekeland type variational principle in the setting of quasi metric spaces with a q function we prove some equivalences of our variational principles with caristi kirk type fixed point theorems for multivalued maps the takahashi minimization theorem and some other related results as applications of our results we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps we also extend the nadler s fixed point theorem for multivalued maps to a q function and in the setting of complete quasi metric spaces as a consequence we prove the banach contraction theorem for a q function and in the setting of complete quasi metric spaces the results of this paper extend and generalize many results appearing recently in the literature

solution of a class of nonlinear matrix equations abstract in this paper we solve nonlinear matrix equations of the form x q i 1 p a i f i x a i r i and x q i 1 p a i f i x a i r i j 1 q b j g j x b j q j where 1 1 r i q j 1 1 q p n the collection of all n n hermitian positive definite matrices and a i b j s are n n matrices also f i g j s are monotone mappings from p n into p n examples are given to illustrate that the equations can not be solved by previously known theorems

remarks on contractive conditions of integral type abstract we show that most contractive conditions of integral type given recently by many authors coincide with classical ones this is done with the help of a geometric lemma on subsets of the quadrant 0 2 in particular we extend a recent result of suzuki who observed that an integral version of the banach contraction principle given by branciari could be derived from a classical fixed point theorem of meir and keeler nevertheless we also give a new contractive condition of integral type which is independent of classical ones since it does not force the nonexpansiveness of a mapping

on park s open questions and some fixed point theorems for general contractive type mappings in this paper we answer two fixed point questions of park by constructing ten nontrivial examples and prove some fixed point theorems for general contractive type mappings which in turn generalize improve and unify some results due to fisher hegedus hegedus and szilagyi hikida kasahara park park and rhoades and others

an extension of coupled fixed point s concept in higher dimension and applications we introduce the concept of fixed point of n order for mappings f x n x where n 2 and x is an ordered set endowed with a metric d we establish fixed point results for such mappings satisfying a given contractive condition presented theorems extend and generalize the coupled fixed point results of bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 7 2006 1379 1393 and the tripled fixed point results of berinde and borcut v berinde m borcut tripled fixed point theorems for contractive type mappings in partially ordered metric spaces nonlinear anal 74 2011 4889 4897 some applications to integral equations and to matrix equations are also presented in this work

ekeland s variational principle in locally convex spaces and the density of extremal points in this paper we prove a general version of ekeland s variational principle in locally convex spaces where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function from this we obtain a number of special versions of ekeland s principle which include all the known extensions of the principle in locally convex spaces moreover we give a general criterion for judging the density of extremal points in the general ekeland s principle which extends and improves the related known results

structure preserving algorithms for periodic discrete time algebraic riccati equations in this paper we investigate structure preserving algorithms for computing the symmetric positive semi definite solutions to the periodic discrete time algebraic riccati equations p dares using a structure preserving swap and collapse procedure a single symplectic matrix pair in standard symplectic form is obtained the p dares can then be solved via a single dare using a structure preserving doubling algorithm we develop the structure preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability with several numerical results the algorithm is shown to be efficient out performing other algorithms on a large set of benchmark problems

two double exponential gaps for automata with a limited pushdown abstract we shall consider nondeterministic and deterministic automata equipped with a limited pushdown constant height npda s and dpda s as well as their two way versions constant height 2 npda s and 2 dpda s we show two double exponential gaps for these devices namely i for complementing constant height one way npda s and ii for converting constant height 2 npda s or 2 dpda s into one way devices

thompson metric method for solving a class of nonlinear matrix equation based on the elegant properties of the thompson metric we prove that the general nonlinear matrix equation x q a f x a q q 1 always has a unique positive definite solution an iterative method is proposed to compute the unique positive definite solution we show that the iterative method is more effective as q increases a perturbation bound for the unique positive definite solution is derived in the end

on hermitian positive definite solution of the matrix equation x σ m i 1 a i x r a i q a conjecture that the nonlinear matrix equation email protected i 1 ma i x ra i q email protected r 0 or 0

equivalent contractive conditions in symmetric spaces in this paper we introduce 22 contractive conditions in symmetric spaces we prove that all these contractive conditions are equivalent then we use these conditions to get some common fixed point theorems for noncompatible maps

two kinds of condition numbers for the quadratic matrix equation in the situation that the quadratic matrix equation has the dominant or minimal solvents the mixed and componentwise condition numbers of the quadratic matrix equation are considered the explicit approximate expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed the theoretical results are illustrated by using some simple numerical examples

the uniqueness of endpoints for set valued dynamical systems of contractions of meir keeler type in uniform spaces abstract in this paper the concept of the set valued dynamical systems of contractions of meir keeler type in uniform spaces is introduced and conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions are established the definition and the result presented here are new for set valued dynamical systems in uniform locally convex and metric spaces and even for single valued maps examples show a fundamental difference between our result and the well known ones

characterization of glushkov automata abstract glushkov s algorithm computes a nondeterministic finite automaton without e transitions and with n 1 states from a regular expression having n occurrences of letters the aim of this paper is to give a set of necessary and sufficient conditions characterizing this automaton our characterization theorem is formulated in terms of directed graphs moreover these conditions allow us to produce an algorithm of conversion of a glushkov automaton into a regular expression of small size

perturbation analysis of the hermitian positive definite solution of the matrix equation x a x 2a i abstract consider the nonlinear matrix equation x a x 2a i where a is an n n complex matrix i the identity matrix and a the conjugate transpose of a matrix a in this paper it is proved that this matrix equation has a unique hermitian positive definite solution provided a 2

positive operator based iterative algorithms for solving lyapunov equations for ito stochastic systems with markovian jumps this paper studies the iterative solutions of lyapunov matrix equations associated with ito stochastic systems having markovian jump parameters for the discrete time case when the associated stochastic system is mean square stable two iterative algorithms with one in direct form and the other one in implicit form are established the convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system for the continuous time case a transformation is first performed so that it is transformed into an equivalent discrete time lyapunov equation then the iterative solution can be obtained by applying the iterative algorithm developed for discrete time lyapunov equation similar to the discrete time case an implicit iteration is also proposed for the continuous case for both discrete time and continuous time lyapunov equations the convergence rates of the established algorithms are analyzed and compared numerical examples are worked out to validate the effectiveness of the proposed algorithms

applications of hilbert s projective metric to a class of positive nonlinear operators the purpose of this paper is to study the eigenvalue problems for a class of positive nonlinear operators using projective metric techniques and the contraction mapping principle we establish existence uniqueness and continuity results for positive eigensolutions of a particular type of positive nonlinear operator in addition we prove the existence of a unique fixed point of the operator with explicit norm estimates applications to nonlinear systems of equations and to matrix equations are considered

weakly homogeneous variational inequalities and solvability of nonlinear equations over cones given a closed convex cone c in a finite dimensional real hilbert space h a weakly homogeneous map f c rightarrow h is a sum of two continuous maps h and g where h is positively homogeneous of degree gamma ge 0 on c and g x o x gamma as x rightarrow infty in c given such a map f a nonempty closed convex subset k of c and a q in h we consider the variational inequality problem text vi f k q of finding an x in k such that langle f x q x x rangle ge 0 for all x in k in this paper we establish some results connecting the variational inequality problem text vi f k q and the cone complementarity problem text cp f infty k infty 0 where f infty h is the homogeneous part of f and k infty is the recession cone of k we show for example that text vi f k q has a nonempty compact solution set for every q when zero is the only solution of text cp f infty k infty 0 and the topological index of the map x mapsto x pi k infty x g x at the origin is nonzero where g is a continuous extension of f infty to h as a consequence we generalize a complementarity result of karamardian j optim theory appl 19 227 232 1976 formulated for homogeneous maps on proper cones to variational inequalities the results above extend some similar results proved for affine variational inequalities and for polynomial complementarity problems over the nonnegative orthant in mathcal r n as an application we discuss the solvability of nonlinear equations corresponding to weakly homogeneous maps over closed convex cones in particular we extend a result of hillar and johnson proc am math soc 132 945 953 2004 on the solvability of symmetric word equations to euclidean jordan algebras

perturbation analysis of the maximal solution of the matrix equation x a x 1a p consider the nonlinear matrix equation x a x 1 a p where a p are n x n complex matrices with p hermitian positive definite and a denotes the conjugate transpose of a matrix a in this pape

generalized contractions in partially ordered metric spaces and applications to ordinary differential equations the purpose of this paper is to present some fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions we also present some applications to first and second order ordinary differential equations

the density of extremal points in ekeland s variational principle in this paper we investigate the density of extremal points appeared in ekeland s variational principle by introducing radial intersections of sets we give a very general result on the density of extremal points in the framework of locally convex spaces this solves a problem proposed by g isac in 1997 from the general result we deduce several convenient criterions for judging the density of extremal points which extend and improve a result of f cammaroto and a chinni using the equivalence between ekeland s variational principle and caristi s fixed point theorem we obtain some density results on caristi s fixed points

some φ coupled fixed point results via modified f control function s concept in metric spaces and its applications abstract the aim of this paper is to propose a new coupled fixed point theorem for a mixed monotone mapping in a metric space which is taken no account of the continuity of the mapping to this end we introduce the notions of coupled fixed point and modified f control function our obtained results are applied to derive some coupled fixed point theorems in partial metric spaces and quasi metric spaces respectively some applications to matrix equations and integral equations are also presented so as to illustrate our results

fixed point theorems in partially ordered metric spaces and applications abstract we prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order using a weak contractivity type of assumption besides including several recent developments our theorem can be used to investigate a large class of problems as an application we discuss the existence and uniqueness of solution for a periodic boundary value problem

on hermitian positive definite solutions of matrix equation x a x 2a i abstract the hermitian positive definite solutions of the matrix equation x a x 2 a i are studied a necessary and sufficient condition for existence of solutions is given in case a is normal the basic fixed point iterations for the equation in case a is nonnormal with a 2 3 3 are discussed in some detail some of ivanov s hasanov s and minchev s results in linear algebra appl 326 2001 27 are improved

removing nondeterminism in constant height pushdown automata we study the descriptional cost of removing nondeterminism in constant height pushdown automata i e pushdown automata with a built in constant limit on the height of the pushdown we show a double exponential size increase when converting a constant height nondeterministic pushdown automaton into an equivalent deterministic device moreover we prove that such a double exponential blow up cannot be avoided by certifying its optimality as a direct consequence we get that eliminating nondeterminism in classical finite state automata is single exponential even with the help of a constant height pushdown store

on the nonlinear matrix equation xp a mt x b m abstract the nonlinear matrix equation x p a m t x b m where p 1 is a positive integer m is an n n nonsingular matrix a is a positive semidefinite matrix and b is a positive definite matrix is considered we denote by c d the geometric mean of positive definite matrices c and d based on the properties of the thompson metric we prove that this nonlinear matrix equation always has a unique positive definite solution and that the fixed point iteration method can be efficiently employed to compute it in addition estimates of the positive definite solution and perturbation analysis are investigated numerical experiments are given to confirm the theoretical analysis

on matrix equations x a x 2a i the two matrix equations x a x 2 a i and x a x 2 a i are studied we construct iterative methods for obtaining positive definite solutions of these equations sufficient conditions for the existence of two different solutions of the equation x a x 2 a i are derived sufficient conditions for the existence of positive definite solutions of the equation x a x 2 a i are given numerical experiments are discussed 2001 elsevier science

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

on some nonlinear problems induced by an abstract maximal element principle abstract in this paper by applying an abstract maximal element principle on quasi ordered sets established by lin and du we obtain a generalized brezis browder principle system vectorial versions of ekeland s variational principle and maximal element principle and a vectorial version of takahashi s nonconvex minimization theorem moreover we investigate the equivalence between scalar versions and vectorial versions of these results

on the positive definite solutions of the matrix equations xs atx ta in abstract the two matrix equations x s a t x t a i n and x s a t x t a i n are studied based on the fixed point theory the existence of the symmetric positive definite solutions are proved sensitivity analysis of the maximal solution is presented some elegant estimates of the positive definite solutions are obtained three iterative methods for computing the positive solutions are proposed

best proximity points for cyclic meir keeler contractions abstract we introduce a notion of cyclic meir keeler contractions and prove a theorem which assures the existence and uniqueness of a best proximity point for cyclic meir keeler contractions this theorem is a generalization of a recent result due to eldred and veeramani

descriptional complexity of two way pushdown automata with restricted head reversals two way nondeterministic pushdown automata 2pda are classical nondeterministic pushdown automata pda enhanced with two way motion of the input head in this paper the subclass of 2pda accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects in particular the effect of reducing the number of pushdown reversals to a constant number is of interest it turns out that this reduction leads to an exponential blow up in case of nondeterministic devices and to a doubly exponential blow up in case of deterministic devices if the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped the resulting trade offs are no longer bounded by recursive functions and so called non recursive trade offs are shown

newton s method for the quadratic matrix equation abstract under suitable condition a quadratic matrix equation is equivalent to a nonlinear matrix equation we apply newton s method to the nonlinear matrix equation for computing the numerical solution of a class of quadratic matrix equations convergence theorems are derived and numerical results are also given

a variant of the meir keeler type theorem in ordered banach spaces abstract this paper discusses a projective variant of the fixed point theorem for a class of mappings satisfying the meir keeler type contraction condition in ordered banach spaces through projective metric we also prove a result on the chainability of the thompson s metric using an elementary argument

contractions of banach tarafdar meir keeler ciric jachymski matkowski and suzuki types and fixed points in uniform spaces with generalized pseudodistances in the uniform spaces sequentially complete and not sequentially complete we introduce the concept of the j families of generalized pseudodistances we apply it to construct j generalized contractions and we provide the conditions guaranteeing the existence and uniqueness of fixed points of these contractions and the convergence to these fixed points of each iterative sequence of these contractions our j generalized contractions essentially extend banach tarafdar meir keeler ciric jachymski matkowski and suzuki contractions to uniform spaces with j families of generalized pseudodistances examples showing a difference between our results and the well known ones are given the definitions the results and the method to investigate uniqueness and iterative approximation of fixed points of the maps presented here are new for maps in uniform and locally convex spaces and even in metric spaces

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

on maximal element theorems variants of ekeland s variational principle and their applications abstract in this paper we establish several different versions of generalized ekeland s variational principle and maximal element theorem for functions in complete metric spaces the equivalence relations between maximal element theorems generalized ekeland s variational principle generalized caristi s common fixed point theorems and nonconvex maximal element theorems for maps are also proved moreover we obtain some applications to a nonconvex minimax theorem nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces

new fixed point theorems for asymptotically regular multi valued maps abstract recently extending ciric s quasi contraction to a very general setting proinov petko d proinov fixed point theorems in metric spaces nonlinear anal 64 2006 546 557 obtained a remarkable fixed point theorem we utilize proinov s contractive type hypotheses for a self map of a metric space to obtain very general fixed point theorems for multi valued maps in metric spaces

improved methods and starting values to solve the matrix equations x a x 1a i iteratively the two matrix iterations x k 1 i a x 1 k a are known to converge linearly to a positive definite solution of the matrix equations x a x 1 a i respectively for known choices of x 0 and under certain restrictions on a the convergence for previously suggested starting matrices x 0 is generally very slow this paper explores different initial choices of x 0 in both iterations that depend on the extreme singular values of a and lead to much more rapid convergence further the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use newton s method in part

more concise representation of regular languages by automata and regular expressions we consider two formalisms for representing regular languages constant height pushdown automata and straight line programs for regular expressions we constructively prove that their sizes are polynomially related comparing them with the sizes of finite state automata and regular expressions we obtain optimal exponential and double exponential gaps i e a more concise representation of regular languages

fixed point theorems in metric spaces abstract in this paper we establish two general theorems for equivalence between the meir keeler type contractive conditions and the contractive definitions involving gauge functions one of these theorems is an extension of a recent result of lim on characterization of meir keeler contractive maps nonlinear anal 46 2001 113 120 also we establish the following new fixed point theorem suppose r r is a contractive gauge function in the sense that for any e 0 there exists e such that e t implies t e and suppose t is a continuous and asymptotically regular selfmapping on a complete metric space x d satisfying the following i d tx ty d x y for all x y x and ii d tx ty d x y for all x y x with x y where d x y d x y d x tx d y ty with 0 then t has a unique fixed point and all of the picard iterates of t converge to this fixed point this result includes those of jachimski equivalent conditions and the meir keeler type theorems j math anal appl 194 1995 293 303 matkowski fixed point theorems for contractive mappings in metric spaces cas pest mat 105 1980 341 344 and others

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

solving the nonlinear matrix equation x q i 1mmixδimi via a contraction principle abstract in this paper we consider the nonlinear matrix equation x q i 1 m m i x i m i where q is positive resp semidefinite definite and m i s are arbitrary resp nonsingular matrices we prove that if max i 1 i m 1 then the equation has a unique positive definite solution which is realized as the unique fixed point of a strict contraction with the lipschitz constant less than or equal to furthermore we show that the solution map varying over the determining coefficient matrices is continuous

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

generalized meir keeler type contractions on g metric spaces in this manuscript we introduce generalized meir keeler type contractions over g metric spaces moreover we show that every orbitally continuous generalized meir keeler type contraction has a unique fixed point on complete g metric spaces we illustrate our results by some given examples

on the existence of hermitian positive definite solutions of the matrix equation xs a x ta q abstract in this paper the existence of hermitian positive definite solutions of the general nonlinear matrix equation x s a x t a q is studied systematically and deeply a new estimate of hermitian positive definite solutions is derived based on a fixed point theorem some new sufficient conditions and new necessary conditions for the existence of hermitian positive definite solutions are obtained in the end a necessary and sufficient condition for the existence of a hermitian positive definite solution is proved

endpoints of set valued dynamical systems of asymptotic contractions of meir keeler type and strict contractions in uniform spaces abstract in this paper we introduce the concepts of the set valued dynamical systems of asymptotic contractions of meir keeler type and set valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions the result concerning the investigations of problems of the set valued asymptotic fixed point theory include some well known results of meir and keeler kirk and suzuki concerning the asymptotic fixed point theory of single valued maps in metric spaces the result concerning set valued strict contractions in which the contractive coefficient is not constant is different from the result of yuan concerning the existence of endpoints of tarafdar vyborny generalized contractions in which the contractive coefficient is constant in bounded metric spaces and provides some examples of tarafdar yuan topological contractions in compact uniform spaces definitions and results presented here are new for set valued dynamical systems in uniform locally convex and metric spaces and even for single valued maps examples show a fundamental difference between our results and the well known ones

uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

positive definite solutions of the matrix equations x a x qa q abstract in this paper we investigate nonlinear matrix equations x a x q a q and x a x q a q where q 0 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x q a q to have two different positive definite solutions and a sufficient condition for the equation x a x q a q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions for these equations

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

ekeland s variational principle minimax theorems and existence of nonconvex equilibria in complete metric spaces abstract in this paper we introduce the concept of function which generalizes the concept of w distance studied in the literature we establish a generalized ekeland s variational principle in the setting of lower semicontinuous from above and functions as applications of our ekeland s variational principle we derive generalized caristi s common fixed point theorems a generalized takahashi s nonconvex minimization theorem a nonconvex minimax theorem a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces we also prove that these theorems also imply our ekeland s variational principle

on the descriptional complexity of stateless deterministic ordered restarting automata abstract stateless deterministic ordered restarting automata accept exactly the regular languages here we study the descriptional complexity of these automata taking the size of the tape alphabet as the complexity measure we present a construction that turns a stateless deterministic ordered restarting automaton that works on an alphabet of size n into an equivalent nondeterministic finite state acceptor of size at most 2 o n and we prove that this bound is sharp with respect to its order of magnitude in addition we investigate the descriptional complexity of some operations for regular languages that are given through stateless deterministic ordered restarting automata based on these results we then show that many decision problems like emptiness finiteness and inclusion are pspace complete for these automata finally we propose three ways of associating relations to deterministic ordered restarting automata in the most general setting we obtain succinct representations for all rational relations in the most restricted setting we obtain exactly those rational functions that map the empty word to itself and by extending the stateless deterministic ordered restarting automaton into a transducer we obtain exactly all those transductions on words that are mso definable

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

on the nonlinear matrix equation x i 1mai xδiai q abstract based on fixed point theorems for monotone and mixed monotone operators in a normal cone we prove that the nonlinear matrix equation x i 1 m a i x i a i q 0 i 1 always has a unique positive definite solution a conjecture which is proposed in x g liu h gao on the positive definite solutions of the matrix equation x s a t x t a i n linear algebra appl 368 2003 83 97 is solved multi step stationary iterative method is proposed to compute the unique positive definite solution numerical examples show that this iterative method is feasible and effective

a note on φ contractions in probabilistic and fuzzy metric spaces abstract in a recent paper fang 2015 1 j x fang generalized a crucial fixed point theorem for probabilistic contractions on complete menger spaces due to jachymski 2010 3 in this note we show that actually fang s theorem is an easy consequence of jachymski s theorem we also observe that the proof of a fixed point theorem for complete metric spaces deduced by fang from his main result is not correct and present a new proof of it

some inequalities for the nonlinear matrix equation x s a x t a q trace determinant and eigenvalue the nonlinear matrix equation x s a x ta q is investigated where q is a hermitian positive definite matrix we present the upper bound of det aa 1 n when the nonlinear matrix equation x s a x ta q has hermitian positive definite solution the bound of det aa 1 n improves the corresponding result in duan and liao 2008 6 theorem 3 3 and yin et al 2009 17 theorem 2 1 we also get the bounds of detx 1 n and trx for the existence of a hermitian positive definite solution which generalize and improve the corresponding conclusions of zhao 2013 22 we obtain some bounds for the eigenvalues of the hermitian positive definite solution as compared with earlier works on these topics the results we present here are more general and the analysis here is much simpler finally we derive tight bounds about partial sum and partial product about the eigenvalues of the solution x for the nonlinear matrix equation x s a x ta q these present improvements and completions for the existing bounds

fixed point theorems for multivalued mappings on complete metric spaces on obtient une version multivaluee du theoreme de point fixe de caristi qui est equivalent au principe e variationnel du a ekeland on utilise ce resultat pour obtenir un theoreme qui deduit le theoreme de nadler et un theoreme de point fixe commun pour une application a 1 valeur et une application multivaluee

correlations in connected random graphs we study the properties of the giant connected component in random graphs with arbitrary degree distribution we concentrate on the degree degree correlations we show that the adjoining nodes in the giant connected component are correlated and derive analytic formulas for the joint nearest neighbor degree probability distribution using those results we describe correlations in maximal entropy connected random graphs we show that connected graphs are disassortative and that correlations are strongly related to the presence of one degree nodes leaves we propose an efficient algorithm for generating connected random graphs we illustrate our results with several examples

the power mean and the least squares mean of probability measures on the space of positive definite matrices abstract in this paper we derive properties of the least squares or karcher mean of probability measures on the open cone of positive definite matrices of some fixed dimension endowed with the trace metric that generalize known properties of the weighted least squares mean of finitely many positive definite matrices our approach is based on first defining the t power mean of a probability measure as the unique fixed point of the contractive map x x t z d z with respect to the thompson metric establishing its properties analogous to those of the power mean for a finite number of positive definite matrices and showing the t power means converge to the karcher mean as t 0 we carry out this program first of all for the compactly supported probability measures and show that theory including the monotonicity extends to the general case

equivalent conditions for generalized contractions on ordered metric spaces abstract we establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane as its application we get that various contractive conditions using the so called altering distance functions coincide with classical ones we consider several classes of mappings both on metric spaces and ordered metric spaces in particular we show that unexpectedly some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by harjani and sadarangani j harjani k sadarangani generalized contractions in partially ordered metric spaces and applications to ordinary differential equations nonlinear anal 72 2010 1188 1197 and amini harandi and emami a amini harandi h emami a fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations nonlinear anal 72 2010 2238 2242 do follow from an earlier result of o regan and petrusel d o regan and a petrusel fixed point theorems for generalized contractions in ordered metric spaces j math anal appl 341 2008 1241 1252

tripled fixed point theorems for contractive type mappings in partially ordered metric spaces abstract in this paper we introduce the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtain existence and existence and uniqueness theorems for contractive type mappings our results generalize and extend recent coupled fixed point theorems established by gnana bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 7 2006 1379 1393 examples to support our new results are given

core organization of directed complex networks the recursive removal of leaves dead end vertices and their neighbors from an undirected network results when this pruning algorithm stops in a so called core of the network this specific subgraph should be distinguished from k cores which are principally different subgraphs in networks if the vertex mean degree of a network is sufficiently large the core is a giant cluster containing a finite fraction of vertices we find that generalization of this pruning algorithm to directed networks provides a significantly more complex picture of cores by implementing a rate equation approach to this pruning procedure for directed uncorrelated networks we identify a set of cores progressively embedded into each other in a network and describe their birth points and structure

a generalized distance and enhanced ekeland s variational principle for vector functions we propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation this result is shown to contain probably a large part of existing versions of ekeland s variational principle evp we introduce the notion of a weak function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced evp for various settings under relaxed lower semicontinuity assumptions these principles conclude the existence not only of p strict minimizers of p perturbations of the considered vector function but also p sharp and p strong minimizers our results are proved to be stronger than the classical evp and many generalizations in the literature even in the usual one dimensional case by numerous corollaries and examples we include equivalent formulations of our enhanced evp as well

fixed points for generalized contractions and applications to control theory the concepts of weak strong topological contraction and a generalization of banach contraction mappings called p contraction are introduced and used to prove fixed point theorems for self mappings from a topological metric space into itself satisfying topological contraction metric p contraction respectively certain non linear integral equations defined on c a b satisfying generalized lipschitzian conditions can easily be solved by applying these theorems in the sequel we shall study the possibility of optimally controlling the solution of the ordinary differential equation via dynamic programming