stability of linear time varying delay systems and applications to control problems this paper deals with the stability of a class of linear time varying systems with multiple delays using the lyapunov function method we give sufficient delay dependent conditions for the exponential stability with a given convergence rate which are described in terms of linear matrix inequalities lmi and the solution of riccati differential equations rde the results are applied to the problem of stabilization of linear time varying control systems with multiple delays numerical examples are given to illustrate the results

generating functions and duality for non crossing walks on a plane graph the generating function for the number of non crossing walk configurations of n walks between the roots of a two rooted directed plane graph is introduced this is shown to be a rational function and the structure and symmetry property of its numerator are discussed the walk configurations correspond to flows and the equivalent dual generating function for potentials is investigated independently also equivalences are drawn with the partially order sets that can be constructed from the walk configurations finally the general results developed here are applied to the directed square and honeycomb lattices this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

low density series expansions for directed percolation i a new efficient algorithm with applications to the square lattice a new algorithm for the derivation of low density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems numerical evidence shows that the computational complexity grows exponentially but with a growth factor which is much smaller than the growth factor of the previous best algorithm for bond site percolation on the directed square lattice the series has been extended to order 171 158 analysis of the series yields sharper estimates of the critical points and exponents

series expansion of the directed percolation probability using a transfer matrix technique the authors obtain extended series expansion of the percolation probability for the directed site percolation problem on the square lattice their approach reveals a previously unsuspected connection between his problem and the enumeration of the number of ways of dissecting a ball they show that the method can also be used to determine a series expansion for the mean cluster size an analysis based on pade approximants gives estimates of the critical threshold and also of the critical exponent beta

h stability for linear dynamic equations on time scales we study the h stability for linear dynamic equations on time scales and their perturbations by using the bihari type inequality on time scales and the unified time scale quadratic lyapunov functions

stability for time varying linear dynamic systems on time scales we study conditions under which the solutions of a time varying linear dynamic system of the form x t a t x t are stable on certain time scales we give sufficient conditions for various types of stability including lyapunov type stability criteria and eigenvalue conditions on slowly varying systems that ensure exponential stability finally perturbations of the unforced system are investigated and an instability criterion is also developed

adaptive sampling control of high gain stabilizable systems it is well known that proportional output feedback control can stabilize any relative degree one minimum phase system if the sign of the feedback is correct and the proportional gain is high enough moreover there exist simple adaptation laws for tuning the proportional gain so called high gain adaptive controllers which do not need to know the system and do not attempt to identify system parameters the authors consider sampled versions of the high gain adaptive controller the motivation for sampling arises from the possibility that the output of a system may not be available continuously but only at sampled times the main point of interest is the need to develop techniques for adapting the sampling rate since the stiffness of the system increases as the proportional gain is increased our main result shows that adaptive sampling stabilization is possible if the product hk of the decreasing sampling interval h and the increasing proportional gain k decreases at a rate proportional to 1 log k

a unified floquet theory for discrete continuous and hybrid periodic linear systems abstract in this paper we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts e g hybrid dynamical systems we develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function and use these results to study homogeneous as well as nonhomogeneous periodic problems furthermore we explore the connection between floquet multipliers and floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales finally we show this unified floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated but time varying system s poles in the complex plane we include several examples to show the utility of this theory

hearing the music of the primes auditory complementarity and the siren song of zeta a counting function for the primes can be rendered as a sound signal whose harmonies spanning the gamut of musical notes are the riemann zeros but the individual primes cannot be discriminated as singularities in this music because the intervals between them are too short conversely if the prime singularities are detected as a series of clicks the riemann zeros correspond to frequencies too low to be heard the sound generated by the riemann zeta function itself is very different a rising siren howl which can be understood in detail from the riemann siegel formula

thermal transmissivity in discrete spin systems formulation and applications thermal transmissivity a notion which arises empirically and leads to considerable simplification of analyses of discrete spin systems is of fundamental interest in graphical and real space renormalization group approaches the authors present a formulation of the theory of thermal transmissivity they show that the situation is different depending on whether the interaction matrices commute in the commuting case the thermal transmissivity is given by the eigenvalues of the interaction matrix while in the non commuting case it is given by the eigenvalues as well as matrices which block diagonalize the interaction matrix the meaning of the thermal transmissivity in diagrammatic analyses of spin models is also elucidated and their results are illustrated by examples finally they present a general formulation of disorder solutions for spin models in terms of thermal transmissivities

the discrete n vector ferromagnet connection to a percolation with frustration features we extend the kasteleyn fortuin formalism to the discrete n vector ferromagnet we show that the free energy and the correlation functions of this model are related when the number of states tends to 1 to the mean number of clusters and to the pair connectedness of a polychromatic bond percolation type problem which combines frustration and connectivity features

stability criteria for set dynamic equations on time scales very recently a new theory known as set dynamic equations on time scales has been built in this paper notions of stability for the solutions of set dynamic equations on time scales using lyapunov like functions are considered criteria for the equistability equiasymptotic stability uniform and uniform asymptotic stability are developed

asymptotical properties of social network dynamics on time scales in this paper we develop conditions for various types of stability in social networks governed by imitation of success principle considering so called prisoner s dilemma as the base of node to node game in the network we obtain well known hopfield neural network model asymptotic behavior of the original model and dynamic hopfield model has a certain correspondence to obtain more general results we consider hopfield model dynamic system on time scales developed stability conditions combine main parameters of network structure such as network size and maximum relative nodes degree with the main characteristics of time scale nodes inertia and resistance rate of input output response

extension of the arrowsmith essam formula to the domany kinzel model arrowsmith and essam gave an expansion formula for point to point connectedness functions of the mixed site bond percolation model on oriented lattices in which each term is characterized by a graph we extend this formula to general k point correlation functions which are point to set with k points connectivities in the context of percolation of the two neighbor discrete time markov process stochastic cellular automata with two parameters in one dimension called the domany kinzel model which includes the mixed site bond oriented percolation model on a square lattice as a special case our proof of the formula is elementary and based on induction with respect to time step which is different from the original graph theoretical one given by arrowsmith and essam we introduce a system of m interacting random walkers called m friendly walkers m fw with two parameters following the argument of cardy and colaiori it is shown that our formula is useful to derive a theorem that the correlation functions of the domany kinzel model are obtained as an m 0 limit of the generating functions of the m fw

on stabilisability of nonlinear systems on time scales in this article stabilisability of nonlinear finite dimensional control systems on arbitrary time scales is studied the classical results on stabilisation of nonlinear continuous time and discrete time systems are extended to systems on arbitrary time scales with bounded graininess function it is shown that uniform exponential stability of the linear approximation of a nonlinear system implies uniform exponential stability of the nonlinear system then this result is used to show a similar implication for uniform exponential stabilisability

stability criteria for impulsive systems on time scales in this paper we study stability of impulsive system on time scales by using comparison method lyapunov function and analysis method the asymptotic stability criteria for system with impulses at fixed times and impulses at variable times on time scales are obtained respectively an example is presented to illustrate the efficiency of proposed result

variationally stable difference systems we characterize the h stability in variation for nonlinear difference systems via n similarity and lyapunov functions furthermore using lyapunov s method and comparison principle we obtain some results related to stability for the perturbations of nonlinear difference systems

improved stability criteria for linear time varying systems on time scales in this paper asymptotic stability problems of linear time varying ltv systems on time scales are considered based on a less conservative lyapunov inequality whose right side is not required to

on gain adaptation in adaptive control the adaptive high gain output feedback strategy u t k t y t d dt k t spl par y t spl par sup 2 is well established in the context of linear minimum phase m input m output systems a b c with the property that spec cb spl sub spl copf sub the strategy applied to any such linear system achieves the performance objectives of 1 global attractivity of the zero state and 2 convergence of the adapting gain to a finite limit here these results are generalized in three aspects first the class of systems is enlarged to a class n sub h spl mu encompassing nonlinear systems modeled by functional differential equations where the parameter h spl ges 0 quantifies system memory and the continuous function spl mu 0 spl infin spl rarr 0 spl infin with spl mu 0 0 relates to the allowable system nonlinearities next the linear control law is replaced by u t k t y t spl mu spl par y t spl par spl par y t spl par y t wherein the additional nonlinear term counteracts the system nonlinearities then the quadratic adaptation law is replaced by the law d dt k t spl psi spl par y t spl par where the continuous function spl psi satisfies certain growth conditions determined by spl mu in particular cases e g linear systems a bounded function spl psi is admissible the above performance objectives 1 and 2 are shown to persist in the generalized framework

self affine fractal clusters conceptual questions and numerical results for directed percolation in this paper we address the question of the existence of a well defined non trivial fractal dimensiond of self affine clusters in spite of the obvious relevance of such clusters to a wide range of phenomena this problem is still open since thedifferent published predictions ford have not been tested yet an interesting aspect of the problem is that a nontrivial global dimension for clusters is in contrast with the trivial global dimension of self affine functions as a much studied example of self affine structures we investigate the infinite directed percolation cluster at the threshold we measuredd ind 2 dimensions by the box counting method using a correction to scaling analysis we obtainedd 1 765 10 this result does not agree with any of the proposed relations but it favorsd 1 1 where and are the correlation length exponents and is a fisher exponent in the cluster scaling

integral inequalities of gronwall bellman bihari type and asymptotic behavior of certain second order nonlinear differential equations abstract several integral inequalities similar to gronwall bellmann bihari inequalities are obtained these inequalities are used to discuss the asymptotic behavior of certain second order nonlinear differential equations

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

existence uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales abstract in this paper we study the existence and uniqueness of solution and stability results of mixed integral dynamic system with instantaneous and noninstantaneous impulses on time scales by using the fixed point method the main tools to establish our results are the gronwall s inequality on time scales picard operator and abstract gronwall lemma some assumptions are made to overcome the hurdles in achieving our results at the end an example is given that shows the validity of our main results

instability results for slowly time varying linear dynamic systems on time scales abstract we develop eigenvalue criteria under which the solutions of a slowly time varying linear dynamic system of the form x t a t x t are unstable

extension of the kasteleyn fortuin formulas to directed percolation it is shown that the pair connectedness for percolation on an arbitrary directed graph may be obtained by setting 1 in the transmissivity function for a state chiral potts model the odd flow model and that the pair connectedness of the dual percolation model is expressible in terms of the correlation function of a second chiral potts model the odd potential difference model there is an important distinction between odd and even and only those models with odd exhibit chirality the duality relation between the transmissivity and correlation functions leads to a duality relation for pair connectedness of directed percolation theory which extends the result of dhar barna and phani for percolation probabilities

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

on the exponential stability of dynamic equations on time scales in this paper we deal with some theorems on the exponential stability of trivial solution of time varying non regressive dynamic equation on time scales with bounded graininess in particular well known perron s theorem is generalized on time scales under rather restrictive condition that is integral boundedness of coefficient operators we obtain a characterization of the uniformly exponential stability

perturbations of nonlinear systems of difference equations in this paper we study a general variational stability introduced mainly for non autonomous difference systems under summability conditions these systems remain stable for general perturbations

high gain robust adaptive controllers for multivariable systems abstract this paper extends the switching free high gain stabilizing adaptive control rules of byrnes and willems to a wide class of adaptive schemes capable of tolerating nonlinear state feedback perturbations

extensions of razumikhin type stability theorems for nonlinear time delay systems on time scales this paper focuses on the stability analysis of nonlinear time delay systems on time scales which are of generality as they can include not only the traditional continuous and discrete ones but a

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

novel α stability criterion of linear systems with multiple time delays abstract in this paper the stability is discussed for a class of linear systems with multiple time delays based on the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi technique new sufficient condition is derived for the exponential stability with a given convergence rate the obtained result is less conservative than those in the literature two numerical examples are given to illustrate the superiority of our one to the previous

the riemann siegel expansion for the zeta function high orders and remainders on the critical line s i t t real riemann s zeta function can be calculated with high accuracy by the riemann siegel expansion this is derived here by elementary formal manipulations of the dirichlet series it is shown that the expansion is divergent with the high orders r having the familiar 9factorial9 divided by power9 dependence decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly terms of the series decrease until r r 2 t and then increase the form of the remainder when the expansion is truncated near r is determined it is of order exp t indicating that the critical line is a stokes line for the riemann siegel expansion these conclusions are supported by computations of the first 50 coefficients in the expansion and of the remainders as a function of truncation for several values of t