novel summation inequalities and their applications to stability analysis for systems with time varying delay the inequality technique plays an important role in stability analysis for time delay systems this technical note presents a new sequence of novel summation inequalities by introducing some free matrices which includes the newly developed wirtinger based and free matrix based summation inequalities as special cases moreover the idea can be easily extended to the multiple summation inequality case based on the proposed inequalities relaxed stability conditions are obtained for systems with time varying delay numerical examples are given to demonstrate the effectiveness of the proposed approach

connective constant of the self avoiding walk on the triangular lattice the author uses a recently developed monte carlo method to study the connective constant of the self avoiding walk saw on the triangular lattice assuming gamma 43 32 one finds mu t 4 15096 or 0 00010 or 0 00026 95 confidence limits and mu t 4 15096 or 0 00020 or 0 00037 99 confidence limits where the first error bar represents systematic error due to corrections to scaling and the second error bar represents statistical error this rules out the conjecture mu t mu h 6 at the greater than 99 confidence level

stability analysis for discrete time delay systems based on new finite sum inequalities a new finite sum inequality is derived that includes the discrete jensen s inequality and the abel lemma based finite sum inequality as special cases another new inequality which needs fewer decision variables than the first one and provides a tighter lower bound than the abel lemma based finite sum inequality is also given applying these new inequalities yields new results on stability analysis for discrete time delay systems two numerical examples demonstrate the effectiveness and superiority of the proposed methods

correction to scaling exponents and amplitudes for the correlation length of linear polymers in two dimensions we consider the scaling behaviour of the radius of gyration for a system of dilute linear polymers we focus on pn the mean square end to end distance of an n step self avoiding walk for which an additional two terms were recently calculated for the close packed triangular lattice combining several extrapolation methods we find that a consistent description of the scaling behaviour exists if and only if the correction to scaling exponent a is roughly half as large as commonly believed we conclude that all data ale consistent with the equation pn an2 1 b na c n where y a a 1 j2 abso 21 and c lo

simulation of the three dimensional ising model on the creutz cellular automaton abstract the three dimensional ising model is simulated on the creutz cellular automaton the values computed for the critical temperature and the static critical exponents within the framework of the finite size scaling theory are in agreement with their values obtained by different methods

new results on stability analysis for systems with discrete distributed delay the integral inequality technique is widely used to derive delay dependent conditions and various integral inequalities have been developed to reduce the conservatism of the conditions derived in this study a new integral inequality was devised that is tighter than existing ones it was used to investigate the stability of linear systems with a discrete distributed delay and a new stability condition was established the results can be applied to systems with a delay belonging to an interval which may be unstable when the delay is small or nonexistent three numerical examples demonstrate the effectiveness and the smaller conservatism of the method

further improvement of jensen inequality and application to stability of time delayed systems this paper is concerned with the delay dependent stability for the linear systems with a time varying delay to get a result in the form of lmi from a lyapunov krasovskii functional an integral inequality is necessary and jensen inequality has been a most powerful inequality in the last few years recently based on wirtinger inequality an improved integral inequality encompassing jensen inequality was proposed and its application to the stability showed a quite improvement in this paper without using wirtinger inequality a further improved integral inequality in the form of infinite series is derived and based on this a delay dependent stability condition in the form of lmi is derived finally its contribution on the stability criterion is shown by well known two examples

a note on relationship between two classes of integral inequalities this technical note firstly introduces two classes of integral inequalities with and without free matrices respectively and points out that they although in different forms are actually equivalent in the sense of conservatism i e the two corresponding ones produce the same tight upper bounds secondly the relationship between the method of integral inequalities with free matrices and the free weighting matrix technique is intensively investigated it is shown that these two different methods are actually equivalent in assessing the stability of time delay systems

single multiple integral inequalities with applications to stability analysis of time delay systems this technical note is concerned with the problem of stability analysis for time delay systems a new series of integral inequalities to bound a single integral term is presented by introducing some free matrices which produces tighter bounds than some existing ones similarly based on orthogonal polynomials defined in integral inner spaces new series of multiple integral inequalities are presented as well which include the existing double ones to show the effectiveness of the proposed inequalities their applications to stability analysis of systems with discrete and distributed delays are provided with numerical examples

stability of linear systems with time varying delays using bessel legendre inequalities this paper addresses the stability problem of linear systems with a time varying delay hierarchical stability conditions based on linear matrix inequalities are obtained from an extensive use of the bessel inequality applied to legendre polynomials of arbitrary orders while this inequality has been only used for constant discrete and distributed delays this paper generalizes the same methodology to time varying delays we take advantages of the dependence of the stability criteria on both the delay and its derivative to propose a new definition of allowable delay sets a light and smart modification in their definition leads to relevant conclusions on the numerical results

critical slowing down in ising model for creutz algorithm abstract dynamical critical exponents for the two dimensional ising model are computed on a cellular automaton from the relaxations of the time displaced correlation and auto correlation functions for the order parameter and the internal energy at the critical temperatures and from the nonlinear relaxation time for the order parameter near the critical temperature the analysis of the data within the frame of the dynamical finite size scaling hypothesis gives z m z e 2 20 and z a m 1 91 and z a e 0 19 for the linear dynamical exponents corresponding respectively to the relaxations of the correlation and the autocorrelation functions for the order parameter and the internal energy and nl m 2 07 for the nonlinear dynamical exponent for the order parameter these values verify the scaling relations between the dynamical exponents

critical exponents of the two dimensional ising model with next nearest neighbor and four spin interactions on the creutz cellular automaton abstract the two dimensional ising model with nearest neighbor next nearest neighbor and four spin interactions is simulated on the creutz cellular automaton for j 0 the values of critical exponents in the antiferromagnetic interaction region are estimated within the framework of the finite size scaling theory the calculated exponents for interaction ratios r 1 20 and j 0 are in a good agreement with their theoretical values of 0 1 8 7 4 1 4 and v 1 for the ising model

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

failure of universal finite size scaling above the upper critical dimension abstract previous theories have predicted that o n symmetric systems in a finite cubic geometry with periodic boundary conditions have universal finite size scaling functions near criticality in d 4 dimensions on the basis of exact results for the o n symmetric 4 model in the large n limit we show that universal finite size scaling does not hold in the predicted form because of significant cut off and lattice effects for d 4 it is shown that finite size scaling is valid with two reference lengths which turn out to be identical with the amplitudes of the bulk correlation length for the 4 field theory the finite size scaling functions are shown to be non universal i e to depend explicitly on the cut off and on the bare four point coupling constant whereas for a 4 lattice model the finite size scaling functions have a different form that is independent of the lattice spacing and the four point coupling constant

free matrix based integral inequality for stability analysis of systems with time varying delay the free weighting matrix and integral inequality methods are widely used to derive delay dependent criteria for the stability analysis of time varying delay systems because they avoid both the use of a model transformation and the technique of bounding cross terms this technical note presents a new integral inequality called a free matrix based integral inequality that further reduces the conservativeness in those methods it includes well known integral inequalities as special cases using it to investigate the stability of systems with time varying delays yields less conservative delay dependent stability criteria which are given in terms of linear matrix inequalities two numerical examples demonstrate the effectiveness and superiority of the method

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

comparison of some bounding inequalities applied in stability analysis of time delay systems abstract this paper gives comparisons of several bounding inequalities proposed recently for the estimation of integrals and sums of quadratic functions which plays an important role in stability analysis of time delay systems via the lyapunov krasovskii approach the equivalences of several known variants of the free matrix based inequalities and their generalized and simplified forms are shown secondly the relationship between the simplified free matrix based inequality and the combination of the bessel based inequality with different bounding inequalities affine in the length of the intervals are investigated finally we shall demonstrate the theoretical results with the application of the various estimations for the stability analysis of a linear system with time varying delay using the same lyapunov krasovskii functional

self avoiding walks in two to five dimensions exact enumerations and series study the method of concatenation the addition of precomputed shorter chains to the ends of a centrally generated longer chain has permitted the extension of the exact series for cn the number of distinct configurations for self avoiding walks of length n the authors report on the leading exponent y and xc the reciprocal of the connectivity constant for the 2d honeycomb lattice 42 terms 1 3437 0 541 1968 the 2d square lattice 30 terms 1 3436 0 379 0520 the 3d simple cubic lattice 23 terms 1 161 932 0 213 4987 the 4d hypercubic 18 terms y approximately 1 0 147 60 and the 5d hypercubic lattice 13 terms y or 1 025 0 113 05 in addition they have also evaluated the leading correction terms honeycomb delta approximately 1 square delta approximately 0 85 simple cubic delta approximately 1 0 and the 4d hypercubic logarithmic correction with delta approximately 0 25

critical behavior of the two dimensional ising model with next nearest neighbor antiferromagnetic interaction on the creutz cellular automaton the two dimensional ising model with nearest and next nearest neighbor antiferromagnetic interactions of ratio r is simulated on the creutz cellular automaton the values of the critical temperature and the critical exponents for o r are estimated within the framework of the finite size scaling theory the dependence of tc on r agrees with the crossover scaling theory for 1 75 the results for 0 r are compatible with ising critical behavior

correction to scaling exponents and critical properties of the n vector model with dimensionality 4 a perturbation calculation is given which implies that the susceptibility of the five and six dimensional n vector models can be written x approximately at 1 1 bt12 and x approximately at 1 1 bt in t respectively independent of n for n 0 and 1 it is shown that series analysis techniques can extract the correction to scaling exponents and that estimates of the critical temperatures and critical amplitudes can also be obtained the correction to scaling exponents found are in agreement with those known to exist in the case of the spherical model

reciprocally convex approach to stability of systems with time varying delays whereas the upper bound lemma for matrix cross product introduced by park 1999 and modified by moon park kwon and lee 2001 plays a key role in guiding various delay dependent criteria for delayed systems the jensen inequality has become an alternative as a way of reducing the number of decision variables it directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities resulting in a linear combination of positive functions weighted by the inverses of convex parameters this paper suggests the lower bound lemma for such a combination which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables comparable to those based on the jensen inequality lemma

the two dimensional spin 1 ising system and related models although the two dimensional spin 1 2 ising model was solved in zero field in 1944 no exact results are yet available for the spin 1 model this model should have the same dominant critical exponents as the spin 1 2 model but appears to exhibit non analytic corrections to scaling as expected for a phi 4 model and unlike the spin 1 2 model in this paper new 45 term low temperature series for the magnetisation specific heat and susceptibility of the spin 1 model on the square lattice are presented and analysed a new 24 term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model which is also in the phi 4 universality class are also considered for both models a non analytic confluent correction with exponent 1 0 delta 1 1 3 is found the validity of this result is enhanced by a comparison with the s 1 2 case

multiple integral inequalities and stability analysis of time delay systems abstract this paper is devoted to stability analysis of continuous time delay systems based on a set of lyapunov krasovskii functionals new multiple integral inequalities are derived that involve the famous jensen s and wirtinger s inequalities as well as the recently presented bessel legendre inequalities of seuret and gouaisbaut 2015 and the wirtinger based multiple integral inequalities of park et al 2015 and lee et al 2015 the present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of lmis establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods numerical examples illustrate the efficiency of the method

stability analysis of systems with time varying delays via the second order bessel legendre inequality this paper is concerned with the delay dependent stability of systems with time varying delays the novelty relies on the use of the second order bessel legendre integral inequality which is less conservative than the jensen and wirtinger based inequalities unlike similar contributions the features of this inequality are fully integrated into the construction of augmented lyapunov krasovskii functionals leading to novel stability criteria expressed in terms of linear matrix inequalities the stability condition is tested on some classical numerical examples illustrating the efficiency of the proposed method

series and monte carlo study of high dimensional ising models ising models in high dimensions are used to compare high temperature series expansions with monte carlo simulations simulations of the magnetization on four six and seven dimensional hypercubic lattices give consistent values of the critical temperature from both equilibrium and nonequilibrium data ford 6 and 7 we tabulate 15 terms of series expansions for the susceptibility for generald and givej kbtc 0 092295 3 and 0 077706 2 ford 6 and 7 in contrast to five dimensions where earlier series found nonanalytic scaling corrections for d 6 and 7 the leading scaling correction may be analytic int tc in most cases these expansions gave more accurate results than these simulations

microcanonical simulations with a large number of demons we generalize the microcanonical algorithm developed by creutz et al 1986 and make a detailed comparison with the exact solution in the case of a two dimensional ising model at finite volume we present a new numerical method to compute the temperature in the microcanonical ensemble this allows us to define a thermalization criterion to estimate the point where the differences between canonical and microcanonical results are the smallest this criterion is shown to work well in the case of the two dimensional ising system

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

stability analysis of linear systems with interval time varying delays utilizing multiple integral inequalities this paper is devoted to stability analysis of continuous time delay systems with interval time varying delays having known bounds on the delay derivatives a parameterized family of lyapunov krasovskii functionals involving multiple integral terms is introduced and novel multiple integral inequalities are utilized to derive sufficient stability condition for systems with time varying delays the efficiency of the proposed method is illustrated by numerical examples

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

the test of the finite size scaling relations for the six dimensional ising model on the creutz cellular automaton the six dimensional ising model is simulated on the creutz cellular automaton using the finite size lattices with the linear dimension 4 l 10 the exponents in the finite size scaling relations for the magnetic susceptibility and the order parameter at the infinite lattice critical temperature are computed to be 3 00 12 and 1 47 10 using 6 l 10 respectively which are in very good agreement with the theoretical predictions of and the critical temperature for the infinite lattice is found to be 10 835 5 using 4 l 10 which is also in very good agreement with the precise results the finite size scaling relation for the magnetic susceptibility at the infinite lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite size lattices the finite size scaling plots of the magnetic susceptibility and the order parameter verify the finite size scaling relations about the infinite lattice critical temperature

summation inequalities to bounded real lemmas of discrete time systems with time varying delay summation inequality is an important technique for analysis of discrete time systems with a time varying delay it seems that from the literature a tighter inequality usually leads to a less conservative criterion based on h infty performance analysis problem this note presents different findings on the relationship between the conservatism of bounded real lemma brl and the tightness of summation inequality firstly the brl obtained by the wirtinger based inequality wbi is not always less conservative than the one by the jensen based inequality although the wbi is tighter secondly the wbi is tighter than a general free matrix based inequality gfmbi developed in this note while the brl obtained via the gfmbi is less conservative than the wbi based brl finally a numerical example is given to demonstrate those findings

fractals in monte carlo simulations of a short polyelectrolyte fractal dimensionalities for charged polymers are calculated over a continuous range of bjerrum length for chains with various numbers of beads n drastically decreasing values of characteristic of a phase transition are found as increases from zero in the range a the fractal dimensionality approaches unity as predicted by de gennes et al de gennes p g pincus p valesco r m and brochard f 1976 j physique 37 1461 close to the onset of order a newly developed smoothing algorithm yields a substantial improvement in the mc results and reveals an interesting fan shaped behaviour of exponents and prefactor components at slightly above zero it also enables discernment of a non monotonic behaviour of versus n at and close to zero differences found in bending between short and long chains may provide an additional explanation for the stability of benzene and the low stability of conjugated rings of n 6 based on fractal concepts monomer densities are derived at various and it is suggested that the drastic density changes at a are evidence of the first order nature of this phase transition

on three dimensional self avoiding walks long self avoiding walks of up to 2400 steps have been generated on the sc and bcc lattices using an improved monte carlo technique analysis of the asymptotic length dependence of the end to end distance and radius of gyration of the walks leads to values in the range 0 591 0 593 for the critical exponent nu the simple power law dependence on length provides an excellent fit to the data over walk lengths between 120 and 2400 with fluctuations around the asymptotic results averaging to a mere 0 13 systematic deviations from asymptotic behaviour in short self avoiding walks have also been examined using monte carlo generated walks ranging in length from 12 to 60 the results do not support the existence of the non analytic correction predicted by the renormalisation group in the light of this unexpected result available series expansions for walks on the fcc lattice have been re examined and previous claims to have observed non analytic behaviour questioned equal if not better convergence of the extrapolated series can be obtained without resorting to non analytic correction terms finally an analysis has been made of the radius of gyration series to which several new terms have been added

on the correction to scaling exponent of linear polymers in two dimensions the authors present new results which indicate that the leading correction to scaling exponent in the mean squared end to end distance in two dimensions is the analytic term a potential source of the various correction to scaling terms reported in the literature is pointed out

a new algorithm to enumerate the self avoiding random walk presents a new algorithm for enumerating the self avoiding walk and extend the existing series of 19 terms by one more term on a triangular lattice the leading correction to scaling exponent delta 1 for the mean square end to end distance of the self avoiding walk is found to be less than one

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

two dimensional polymers universality and correction to scaling improved monte carlo methods have been used to study the asymptotic behaviour of self avoiding walks on two dimensional lattices the mean square end to end distance rn2 and radius of gyration sn2 are both found to scale as n2v with exponent values very close to the expected v 3 4 and an amplitude ratio that is universal small n deviations of rn2 from the asymptotic results indicate a correction to scaling exponent delta 1 a value that differs from previous estimates

computation of correlation functions of the two dimensional ising model abstract the spin spin and energy energy correlation functions of the two dimensional ising model are computer on the creutz cellular automaton from the data for the spin spin correlation functions the values of the critical exponents and are obtained they are in agreement with their theoretical values

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

an exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases

a note on wirtinger type integral inequalities for time delay systems different integral inequalities play important role when problems of stability analysis and controller synthesis for time delay systems are considered the connection of jensen s inequality and extended jensen s inequality is well understood now to reduce the conservativeness introduced by the application of jensen s inequality several versions of the wirtinger integral inequality have been published recently this note presents a comparison between some of these inequalities