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

effects of random and deterministic discrete scale invariance on the critical behavior of the potts model the effects of disorder on the critical behavior of the q state potts model in noninteger dimensions are studied by comparison of deterministic and random fractals sharing the same dimensions in the framework of a discrete scale invariance we carried out intensive monte carlo simulations in the case of a fractal dimension slightly smaller than two d f 1 974636 we give evidence that the disorder structured by discrete scale invariance does not change the first order transition associated with the deterministic case when q 7 furthermore the study of the high value q 14 shows that the transition is a second order one both for deterministic and random scale invariance but that their behavior belongs to different university classes

rigidity of inversive distance circle packings revisited abstract inversive distance circle packing metric was introduced by p bowers and k stephenson 7 as a generalization of thurston s circle packing metric 34 they conjectured that the inversive distance circle packings are rigid for nonnegative inversive distance guo 22 proved the infinitesimal rigidity and then luo 27 proved the global rigidity in this paper based on an observation of zhou 37 we prove this conjecture for inversive distance in 1 by variational principles we also study the global rigidity of a combinatorial curvature introduced in 14 16 19 with respect to the inversive distance circle packing metrics where the inversive distance is in 1

combination of improved multibondic method and the wang landau method we propose a method for monte carlo simulation of statistical physical models with discretized energy the method is based on several ideas including the cluster algorithm the multicanonical monte carlo method and its acceleration proposed recently by wang and landau as in the multibondic ensemble method proposed by janke and kappler the present algorithm performs a random walk in the space of the bond population to yield the state density as a function of the bond number a test on the ising model shows that the number of monte carlo sweeps required of the present method for obtaining the density of state with a given accuracy is proportional to the system size whereas it is proportional to the system size squared for other conventional methods in addition the method shows a better performance than the original wang landau method in measurement of physical quantities

first order phase transitions of the potts model in fractal dimensions abstract the phase diagram of the q state potts model in fractal dimensions is studied with the help of wang landau monte carlo simulations on sierpinski and menger fractal structures a particular attention is paid to first order transitions just above the border separating the second order phase transition regime from the first order one although the translation invariance is strongly broken in deterministic fractals evidence is given that such a deviation from the translational symmetry is not able to induce second order transitions for large values of q when the dimension lies between 1 9746 and 3 moreover the occurrence of second order transitions for very large values of q in the case of hierarchically weakly connected systems that is when the fractal dimension is significantly smaller than 2 is pointed out at last the evolution of first order physical averages such as the latent heats and the interfacial free energies with the space dimensionality and the number of spin states is discussed

novel robust stability criteria for uncertain systems with time varying delay this paper is concerned with the delay dependent stability and robust stability criteria for linear systems with time varying delay and norm bounded uncertainties through constructing a general form of lyapunov krasovskii functional and using integral inequalities some slack matrices and newly established convex combination condition in the calculation the delay dependent stability criteria are derived in terms of linear matrix inequalities numerical examples are given to illustrate the improvement on the conservatism of the delay bound over some reported results in the literature

a discrete uniformization theorem for polyhedral surfaces a notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper this discrete conformality is shown to be computable it is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying gauss bonnet formula furthermore the hyperbolic polyhedral metric with given curvature can be obtained using a discrete yamabe flow with surgery in particular each hyperbolic polyhedral metric on a closed surface with negative euler characteristic is discrete conformal to a unique hyperbolic metric

multicanonical monte carlo simulations canonical monte carlo simulations of disordered systems like spin glasses and systems undergoing first order phase transitions are severely hampered by rare event states which lead to exponentially diverging autocorrelation times with increasing system size and hence to exponentially large statistical errors one possibility to overcome this problem is the multicanonical reweighting method using standard local update algorithms it could be demonstrated that the dependence of autocorrelation times on the system size v is well described by a less divergent power law v with 1 3 depending on the system after a brief review of the basic ideas combinations of multicanonical reweighting with non local update algorithms will be discussed with the multibondic algorithm which combines multicanonical reweighting with cluster updates the dynamical exponent can be reduced to unity the optimal value one would expect from a random walk argument asymptotically for large system sizes the multibondic algorithm therefore always performs better than the standard multicanonical method finally it is shown that a combination with multigrid update techniques improves the performance of multicanonical simulations by roughly one order of magnitude uniformly for all system sizes

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram we describe an efficient monte carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models the density of states is modified at each step when the energy level is visited to produce a flat histogram by carefully controlling the modification factor we allow the density of states to converge to the true value very quickly even for large systems from the density of states at the end of the random walk we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature using this method we not only can avoid repeating simulations at multiple temperatures but we can also estimate the free energy and entropy quantities that are not directly accessible by conventional monte carlo simulations this algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability as with the multicanonical monte carlo technique our method overcomes the tunneling barrier between coexisting phases at first order phase transitions in this paper we apply our algorithm to both first and second order phase transitions to demonstrate its efficiency and accuracy we obtained direct simulational estimates for the density of states for two dimensional ten state potts models on lattices up to 200 x 200 and ising models on lattices up to 256 x 256 our simulational results are compared to both exact solutions and existing numerical data obtained using other methods applying this approach to a three dimensional j spin glass model we estimate the internal energy and entropy at zero temperature and using a two dimensional random walk in energy and order parameter space we obtain the rough canonical distribution and energy landscape in order parameter space preliminary data suggest that the glass transition temperature is about 1 2 and that better estimates can be obtained with more extensive application of the method this simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space

monte carlo study of the random field ising model using a cluster flipping monte carlo algorithm combined with a generalization of the histogram reweighting scheme of ferrenberg and swendsen we have studied the equilibrium properties of the thermal random field ising model on a cubic lattice in three dimensions we have equilibrated systems of l x l x l spins with values of l up to 32 and for these systems the cluster flipping method appears to a large extent to overcome the slow equilibration seen in single spin flip methods from the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling for the exponents we find v 1 02 0 06 b 0 06 0 07 y 1 9 0 2 and mean y 2 9 0 2

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

understanding and improving the wang landau algorithm we present a mathematical analysis of the wang landau algorithm prove its convergence and identify sources of errors and strategies for optimization in particular we found the histogram increases uniformly with small fluctuations after a stage of initial accumulation and the statistical error is found to scale as sqrt mathrm ln phantom rule 0 2em 0ex f with the modification factor f this has implications for strategies for obtaining fast convergence

improved approach to robust stability and h performance analysis for systems with an interval time varying delay abstract an improved delay and range dependent robust stability and h performance analysis criterion for systems with an interval time varying delay is investigated a novel lyapunov krasovskii functional containing triple integral terms is established and an extended free weighting matrix method double integral inequality method is subsequently proposed numerical examples validate the effectiveness of the proposed double integral inequality method and show that our approach is less conservative than existing methods

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

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

robust exponential stability and stabilizability of linear parameter dependent systems with delays abstract the robust exponential stability and stabilizability problems are addressed in this paper for a class of linear parameter dependent systems with interval time varying and constant delays in this paper restrictions on the derivative of the time varying delay is not required which allows the time delay to be a fast time varying function based on the lyapunov krasovskii theory we derive delay dependent exponential stability and stabilizability conditions in terms of linear matrix inequalities lmis which can be solved by various available algorithms numerical examples are given to illustrate the effectiveness of our theoretical results

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

first order transition in the one dimensional three state potts model with long range interactions the first order phase transition in the three state potts model with long range interactions decaying as 1 r 1 ensuremath sigma has been examined by numerical simulations using the recently proposed luijten bl ote algorithm by applying scaling arguments to the interface free energy the binder s fourth order cumulant and the specific heat maximum the change in the character of the transition through variation of parameter ensuremath sigma was studied

new stability criteria of neural networks with interval time varying delay a piecewise delay method abstract this paper provides improved conditions for the global asymptotic stability of a class of neural networks with interval time varying delays a piecewise delay method is firstly proposed in this method the variation interval of the time delay is divided into two subintervals by introducing its central point then by constructing a new lyapunov krasovskii functional and checking its variation in the two subintervals respectively some new delay dependent stability criteria for the addressed neural networks are derived numerical examples are provided to show that the achieved conditions are less conservative than some existing ones in the literature

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

the critical behaviour of the long range potts chain from the largest cluster probability distribution abstract we present the numerical study of the one dimensional potts model with power law decaying ferromagnetic interactions the largest cluster probability distribution is obtained by monte carlo simulations using the swendsen wang cluster algorithm with cumulative probabilities the finite size scaling analysis of the largest cluster is used to derive the critical behaviour in the non classical regime of this model for various values of q

one dimensional ising models with long range interactions we consider ising models with long range ferromagnetic pair interactions decaying as for we first find approximate values for the critical temperature we use a cluster mean field approach combined with finite size scaling and vanden broeck and schwartz transformations for we find which can be compared with recent results of luijten and blote who found and which is two orders of magnitude more accurate than any previous results since we use a mean field cluster approximation as part of our approach the accuracy for larger values of decreases significantly in addition to we obtain approximate values for the critical exponents and using the coherent anomaly method for we obtain and all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of

monte carlo investigations of phase transitions status and perspectives abstract using the concept of finite size scaling monte carlo calculations of various models have become a very useful tool for the study of critical phenomena with the system linear dimension as a variable as an example several recent studies of ising models are discussed as well as the extension to models of polymer mixtures and solutions it is shown that using appropriate cluster algorithms even the scaling functions describing the crossover from the ising universality class to the mean field behavior with increasing interaction range can be described additionally the issue of finite size scaling in ising models above the marginal dimension d 4 is discussed

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

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

test of renormalization predictions for universal finite size scaling functions we calculate universal finite size scaling functions for systems with an n component order parameter and algebraically decaying interactions just as previously found for short range interactions this leads to a singular ensuremath varepsilon expansion where ensuremath varepsilon is the distance to the upper critical dimension subsequently we check the results by numerical simulations of spin models in the same universality class our systems offer the essential advantage that ensuremath varepsilon can be varied continuously allowing an accurate examination of the region where ensuremath varepsilon is small the numerical calculations turn out to be in striking disagreement with the predicted singularity

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

critical behavior of an ising system on the sierpinski carpet a short time dynamics study the short time dynamic evolution of an ising model embedded in an infinitely ramified fractal structure with noninteger hausdorff dimension was studied using monte carlo simulations completely ordered and disordered spin configurations were used as initial states for the dynamic simulations in both cases the evolution of the physical observables follows a power law behavior based on this fact the complete set of critical exponents characteristic of a second order phase transition was evaluated also the dynamic exponent ensuremath theta of the critical initial increase in magnetization as well as the critical temperature were computed the exponent ensuremath theta exhibits a weak dependence on the initial small magnetization on the other hand the dynamic exponent z shows a systematic decrease when the segmentation step is increased i e when the system size becomes larger our results suggest that the effective noninteger dimension for the second order phase transition is noticeably smaller than the hausdorff dimension even when the behavior of the magnetization in the case of the ordered initial state and the autocorrelation in the case of the disordered initial state with time are very well fitted by power laws the precision of our simulations allows us to detect the presence of a soft oscillation of the same type in both magnitudes that we attribute to the topological details of the generating cell at any scale

the one dimensional potts model with long range interactions a renormalization group approach the one dimensional q state potts model with ferromagnetic pair interactions which decay with the distance r as is considered we calculate through a real space renormalization group technique using kadanoff blocks of length b the critical temperature and the correlation length critical exponent as a function of for different values of q some of the very few known rigorous results for general q are reproduced by our approach several asymptotic behaviours are derived analytically for q 2 3 in the limit we also obtain extrapolated critical temperatures for arbitrary values of and for q 2 3 4 which we believe approximate the exact ones well except in the region near furthermore the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q we also verify that which is consistent with a recent conjecture of tsallis

delay dependent stabilization of linear systems with time varying state and input delays the integral inequality method is a new way of tackling the delay dependent stabilization problem for a linear system with time varying state and input delays email protected t ax t a 1x t h 1 t b 1u t b 2u t h 2 t in this paper a new integral inequality for quadratic terms is first established then it is used to obtain a new state and input delay dependent criterion that ensures the stability of the closed loop system with a memoryless state feedback controller finally some numerical examples are presented to demonstrate that control systems designed based on the criterion are effective even though neither a b 1 nor a a 1 b 1 is stabilizable

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

fast flat histogram method for generalized spin models we present a monte carlo method that efficiently computes the density of states for spin models having any number of interaction per spin by combining a random walk in the energy space with collective updates controlled by the microcanonical temperature our method yields dynamic exponents close to their ideal random walk values reduced equilibrium times and very low statistical error in the density of states the method can host any density of states estimation scheme including the wang landau algorithm and the transition matrix method our approach proves remarkably powerful in the numerical study of models governed by long range interactions where it is shown to reduce the algorithm complexity to that of a short range model with the same number of spins we apply the method to the q state potts chains 3 ensuremath leqslant q ensuremath leqslant 12 with power law decaying interactions in their first order regime we find that conventional local update algorithms are outperformed already for sizes above a few hundred spins by considering chains containing up to 2 16 spins which we simulated in fairly reasonable time we obtain estimates of transition temperatures correct to five figure accuracy finally we propose several efficient schemes aimed at estimating the microcanonical temperature

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

critical behaviour of the 1d q state potts model with long range interactions the critical behaviour of the one dimensional q state potts model with long range interactions decaying with distance r as r 1 sigma has been studied in the wide range of parameters 0 sigma or 1 and 1 16 or q or 64 a transfer matrix has been constructed for a truncated range of interactions for integer and continuous q and finite range scaling has been applied results for the phase diagram and the correlation length critical exponent are presented

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

robust state estimation for a class of uncertain time delay systems this paper considers a robust state estimation problem for a class of uncertain time delay systems in this problem the noise and uncertainty are modelled deterministically via an integral quadratic constraint the robust state estimation problem involves constructing the set of all possible states at the current time consistent with given output measurements and the integral quadratic constraint this set is found to be an ellipsoid which is constructed via a linear state estimator

synchronization criteria of fuzzy complex dynamical networks with interval time varying delays this paper presents new delay dependent synchronization criteria for fuzzy complex dynamical networks with interval time varying delays by construction of a suitable lyapunov krasovskii s l k functional and utilization of finsler s lemma new synchronization criteria for the networks are established in terms of linear matrix inequalities lmis which can be easily solved by various effective optimization algorithms one numerical example is given to illustrate the effectiveness of the proposed methods

reexamination of the long range potts model a multicanonical approach we investigate the critical behavior of the one dimensional q state potts model with long range lr interaction 1 r d sigma using a multicanonical algorithm the recursion scheme initially proposed by berg is improved so as to make it suitable for a large class of lr models with unequally spaced energy levels the choice of an efficient predictor and a reliable convergence criterion is discussed we obtain transition temperatures in the first order regime which are in far better agreement with mean field predictions than in previous monte carlo studies by relying on the location of spinodal points and resorting to scaling arguments we determine the threshold value sigma c q separating the first and second order regimes to two digit precision within the range 3 leq q leq 9 we offer convincing numerical evidence supporting sigma c q 1 0 for all q by virtue of an unusual finite size effect namely finite size scaling predicts a continuous transition in the thermodynamic limit despite the first order nature of the transition at finite size a qualitative account in terms of correlation lengths is provided finally we find the crossover between the lr and short range regimes to occur inside a narrow window 1 0 sigma 1 2 thus lending strong support to sak s scenario

new monte carlo algorithm entropic sampling we present a new monte carlo sampling algorithm with which one can obtain any desired distribution of the sampling in one monte carlo simulation the free energy and the entropy of a system can thus be obtained from a simple exercise of this algorithm the main idea is to sample directly the entropy of a system at infinite temperature importance sampling is shown to be a particular case of the new algorithm the algorithm is tested against the exact partition function of the ital l 4 simple cubic ising model a comparison with the multicanonical ensemble for the ital l 12 ital q 10 potts model shows that the new algorithm is more general and more efficient

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

letter to the editor on the exponential stability of a class of nonlinear systems including delayed perturbations in this letter the problem of robust stabilization of a class of nonlinear dynamical systems with delayed perturbations is considered based on the stability of the nominal systems a new stabilizing control law for exponential stability of the system is designed using lyapunov stability theory

dynamical behavior of the multibondic and multicanonic algorithm in the 3d q state potts model we investigate the dynamical behavior of the recently proposed multibondic cluster monte carlo algorithm in applications to the three dimensional q state potts models with q 3 4 and 5 in the vicinity of their first order phase transition points for comparison we also report simulations with the standard multicanonical algorithm similar to the findings in two dimensions we show that for the multibondic cluster algorithm the dependence of the autocorrelation time on the system size vis well described by the power law v and that the dynamical exponent is consistent with the optimal random walk estimate 1 for the multicanonical simulations we obtain as expected a larger value of 1 2

rigidity of polyhedral surfaces ii we study the rigidity of polyhedral surfaces using variational principles the action functionals are derived from the cosine laws the main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics several of these cosine laws were first discovered and used by fenchel and nielsen by studying the derivative of the cosine laws we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary as a consequence the work of penner bobenko and springborn and thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

overcoming the slowing down of flat histogram monte carlo simulations cluster updates and optimized broad histogram ensembles we study the performance of monte carlo simulations that sample a broad histogram in energy by determining the mean first passage time to span the entire energy space of d dimensional ferromagnetic ising potts models we first show that flat histogram monte carlo methods with single spin flip updates such as the wang landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased markovian random walk in energy space for the d 1 2 3 ising model the mean first passage time ensuremath tau scales with the number of spins n l d as ensuremath tau ensuremath propto n 2 l z the exponent z is found to decrease as the dimensionality d is increased in the mean field limit of infinite dimensions we find that z vanishes up to logarithmic corrections we then demonstrate how the slowdown characterized by zg0 for finite d can be overcome by two complementary approaches char22 cluster dynamics in connection with wang landau sampling and the recently developed ensemble optimization technique both approaches are found to improve the random walk in energy space so that ensuremath tau ensuremath propto n 2 up to logarithmic corrections for the d 1 2 ising model

further refinements in stability conditions for time varying delay systems abstract this paper addresses the problem of assessing the stability of linear time invariant lti systems with time varying delay the first contribution is a new stability criterion specified as a negativity condition for a quadratic function parameterized by the delay this result mainly follows from an augmented affine parameter dependent lyapunov krasovskii functional which in turn takes advantages of convexity properties then as a second contribution we invoke a result from robust control literature to show how the proposed stability condition can be checked exactly in terms of linear matrix inequality lmi conditions the improvements obtained by the proposed refinements are illustrated via numerical examples drawn from the literature

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

direct evidence for weak universality on fractal structures abstract we provide a direct quantitative evidence that the critical behavior of the ising model on self similar lattices depends upon the geometrical features of the underlying structure intensive monte carlo simulations enable to show that the values of the ratio of exponents depend upon the lacunarity of the structure which measures the deviation from the translational symmetry finally the topological character of the scaling corrections when the fractal dimension is smaller than 2 is discussed

delay dependent robust stability and stabilization of uncertain linear delay systems a linear matrix inequality approach this paper considers the problems of robust stability analysis and robust control design for a class of uncertain linear systems with a constant time delay the uncertainty is assumed to be norm bounded and appears in all the matrices of the state space model we develop methods for robust stability analysis and robust stabilization the proposed methods are dependent on the size of the delay and are given in terms of linear matrix inequalities

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

fractal structure of spin clusters and domain walls in the two dimensional ising model the fractal structure of spin clusters and their boundaries in the critical two dimensional ising model is investigated numerically the fractal dimensions of these geometrical objects are estimated by means of monte carlo simulations on relatively small lattices through standard finite size scaling the obtained results are in excellent agreement with theoretical predictions and partly provide significant improvements in precision over existing numerical estimates

delay dependent exponential stabilization for uncertain linear systems with interval non differentiable time varying delays abstract in this paper the problem of exponential stabilization for a class of linear systems with time varying delay is studied the time delay is a continuous function belonging to a given interval which means that the lower and upper bounds for the time varying delay are available but the delay function is not necessary to be differentiable based on the construction of improved lyapunov krasovskii functionals combined with leibniz newton s formula new delay dependent sufficient conditions for the exponential stabilization of the systems are first established in terms of lmis numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

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

exponential stability of time delay systems via new weighted integral inequalities in this paper new weighted integral inequalities wiis are first derived based on jensen s integral inequalities in single and double forms it is theoretically shown that the newly derived inequalities in this paper encompass both the jensen inequality and its most recent improvement based on wirtinger s integral inequality the potential capability of wiis is demonstrated through applications to exponential stability analysis of some classes of time delay systems in the framework of linear matrix inequalities lmis the effectiveness and least conservativeness of the derived stability conditions using wiis are shown by various numerical examples

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

delay range dependent stabilization of uncertain dynamic systems with interval time varying delays in this paper the problem of delay range dependent stabilization criterion for uncertain dynamic systems with time varying delays is considered the time varying delays considered is assumed to be belong to a given interval in which lower bound of delay is not restricted to zero by constructing a suitable augmented lyapunov s functional and utilizing free weight matrices the criterion for stabilization is established in terms of linear matrix inequalities three numerical examples are given to show the effectiveness of proposed method

the correlation length of the potts model at the first order transition point the authors consider the q state two dimensional potts model for q g i e in the first order phase transition regime following a scheme given by p martin 1991 the authors prove an identity between the spectra of the transfer matrices of the potts model and the transfer matrices connecting the diagonals of a 6 vertex model by using a bethe ansatz for the latter they obtain an exact expression for the correlation length of the potts model at the transition point

brief paper new delay dependent stability criteria for systems with interval delay this paper provides a new delay dependent stability criterion for systems with a delay varying in an interval with a different lyapunov functional defined a tight upper bound of its derivative is given the resulting criterion has advantages over some previous ones in that it involves fewer matrix variables but has less conservatism which is established theoretically examples are provided to demonstrate the advantage of the stability result

stability criteria of delay impulsive systems on time scales abstract in this paper we study stability of delay impulsive systems on time scales by using lyapunov functional and analysis method some stability and instability criteria for the delay system with impulses at fixed times on time scales are obtained two examples are presented to illustrate the efficiency of results proposed in this paper

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

q state potts models in d dimensions migdal kadanoff approximation the first and second order phase transitions of the q state potts models are obtained in arbitrary dimension d critical and tricritical behaviours merge and annihilate at qc d clearing the way to first order transitions at q qc d by the condensation of effective vacancies the value of qc d decreases with increasing d from diverging as exp 2 d 1 at d to 1 to qc 2 3 81 cf exact value of 4 to lower values at d 2 for given d a changeover in critical behaviour occurs at q1 d as the critical fixed points merge from the potts lattice gas region to the undiluted potts limit it is suggested that the power law singularities of the percolation problem q to 1 have logarithmic corrections

optimized energy calculation in lattice systems with long range interactions we discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long range interactions although since the introduction of the luijten blote algorithm monte carlo simulations of these systems no longer pose a fundamental problem the energy calculation is still an o n 2 problem for systems of size n we show how this can be reduced to an o n log n problem with a break even point that is already reached for very small systems this allows the study of a variety of until now hardly accessible physical aspects of these systems in particular we combine the optimized energy calcula tion with histogram interpolation methods to investigate the specific heat of the ising model and the first order regime of the three state potts model with long range interactions

stability by linear approximation for time scale dynamical systems abstract we study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods in this paper we consider linear systems and their small nonlinear perturbations in terms of time scales and of eigenvalues of matrices we formulate conditions sufficient for stability by linear approximation for non periodic time scales we use techniques of central upper lyapunov exponents a common tool of the theory of linear odes to study stability of solutions also time scale versions of the famous chetaev s theorem on conditional instability are proved in a nutshell we have developed a completely new technique in order to demonstrate that methods of non autonomous linear ode theory may work for time scale dynamics