coherent anomaly method in self avoiding walk problems abstract self avoiding walk saw being a nonequilibrium cooperative phenomenon is investigated with a finite order restricted walk finite orw or forw coherent anomaly method cam the coefficient 1 r in the asymptotic form c nr 1 r n 1 r for the total number c nr of r orw s with respect to the step number n is investigated for the first time an asymptotic form for saw s is thus obtained form the series of forw approximants c nr br g n 1 a r n as the envelope curve c n b ae g g n n g numerical results are given by c n 1 424 n 0 2788 4 1507 n and c n 1 179 n 0 1587 10 005 n for the plane triangular lattice and f c c lattice respectively a good coincidence of the total numbers estimated from the above simple formulae with exact enumerations for finite step saw s implies that the essential nature of saw non markov process can be understood from forw markov process in the cam framework

positive solutions for three point boundary value problems with dependence on the first order derivative abstract a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second order three point boundary value problem x f t x x 0 0 x 0 0 x 1 x where f is a nonnegative continuous function 0 0 1

certainty equivalence in nonlinear output regulation with unmeasured error in this article we consider a nonlinear output regulation problem in which the controlled output and the measured output are not the same it is assumed that the controlled plant has a single control input and that it can be transformed into gauthier kupka s observability canonical form then it is shown that a design based on certainty equivalence is effective for determining a controller that solves the given problem

critical temperature in the two layered ising model abstract using the transfer matrix mean field approximation we precisely calculate the critical temperature in the two layered ising model when the intra layer interactions are the same our estimations of the shift exponent are in agreement with some scaling arguments which predict where is the susceptibility exponent however for unequal intra layer interactions our result 0 5 show that scaling arguments prediction 2 is incorrect in this case an approximate decimation scheme is proposed which gives consistent with numerical calculations

approximations and rapid convergence of solutions of nonlinear three point boundary value problems abstract the method of quasilinearization for second order differential equation x f t x x t 0 1 with nonlinear nonlocal three point boundary conditions x 0 0 x 1 g x 0 1 is developed a monotone sequence of solutions converging uniformly and rapidly to a unique solution of the problem is obtained

application of implicit function theorems to existence of solutions to ordinary differential equations with nonlocal boundary conditions abstract we consider the problem y a x y y b y 0 0 y 1 g y y 0 fixed in 0 1 as a model of steady state heat conduction in a rod when the heat flux at the end x 1 is determined by observation of the temperature and heat flux at some interior point we establish conditions sufficient for existence uniqueness and positivity of solutions existence and uniqueness for a semi infinite problem with a boundary condition of this form are also examined

independent parameterization of two degree of freedom compensators in general robust tracking systems consideration is given to a general robust tracking problem in which the controlled object is a linear multivariable system having measured variables that are not necessary coincident with the output variables to be controlled a necessary and sufficient condition for the two degree of freedom compensator to achieve a robust tracking system with internal stability is given an existence condition is derived for such a compensator and the class of all such compensators is parameterized so that two specifications with respect to responses for reference commands and feedback properties can be satisfied independently by the two free parameters

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

robust tracking relation between the numerators mode readability and inverse internal model in the robust multivariable asymptotic tracking problem with two output plants it is shown that if the numerator related to the controlled output is fixed while the other numerator and the denominator are perturbed then there is no solution to the problem if however the whole part of the plant related to the controlled output is stable and fixed while the other part is arbitrarily perturbed the problem has a solution and the compensator that solves the problem incorporates an inverse internal model of the signal to be tracked

the effect of the number of simulations on the exponents obtained by finite size scaling relations for the seven dimensional ising model on the creutz cellular automaton abstract the seven dimensional ising model is simulated on the creutz cellular automaton using the finite size lattice with the linear dimension 4 l 8 the exponents in the finite size scaling relations for the magnetic susceptibility the order parameter and the specific heat at the infinite lattice critical temperature are computed to be 3 544 24 3 505 55 3 501 52 1 78 64 1 77 38 1 76 46 and 0 02 11 0 01 17 0 01 16 using 4 l 8 for 7 14 and 21 independent simulations respectively as the number of independent simulations increase exponents are in very good agreement with the theoretical predictions of 7 2 7 4 and 0 respectively the values for the critical temperature of infinite lattice 12 870 12 12 870 10 12 869 3 and 12 871 15 12 871 5 12 871 5 are obtained from the straight line fit of the magnetic susceptibility maxima and the specific heat using 4 l 8 for 7 14 and 21 independent simulations respectively as the number of independent simulations increases the obtained results are in very good agreement with the more precise value of 12 86902 33

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

positive solutions of four point singular boundary value problems abstract existence of c 1 positive solutions for a class of second order nonlinear singular equations of the type x t x t f t x t t 0 1 subjectto four point boundary conditions of the type x 0 ax x 1 bx 0 1 is established existence of c 1 positive solution is proved with the upper and lower solutions method examples are included to show the validity of our results finally the method of quasilinearization is developed to approximate the solution we show that under suitable conditions on f there exists a sequence of solutions of linear problems that converges monotonically and quadratically to the solution of the original nonlinear problem

computation of critical exponents for two dimensional ising model on a cellular automaton static critical exponents for the two dimensional ising model are computed on a cellular automaton the analysis of the data within the framework of the finite size scaling theory reproduces their well established values

shift exponent and breakdown of universality for the two layer ising model on a square lattice abstract by using a function for a spin 3 2 ising model we investigate the critical phenomena of a general two layer ising model of square lattice we find that the value of the shift exponent is 1 73 0 04 for the system with the same intra layer interaction and 0 5 for the system with different intra layer interactions we also find an evidence that the value of critical exponent exceeds 1 in some region of interaction parameter space where the degeneracy of the ground state is fourfold

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

an approach to critical phenomena from one subsystem based on the cam analysis two systematic series of multi body effective field approximations are constructed and the cam analyses are performed for the ising mode on the two dimensional square lattice in the first series a two body effective field is introduced in addition to the usual one body one and a set of strips are used the critical temperature and the critical index of the susceptibility are estimated as t c approximately 2 2697 j kb and gamma approximately 1 749 in the second series only a three line strip is used and applied with increasing numbers of multi body effective fields the expected cam scaling behaviour has also been observed

general solution of robust tracking problem in two degree of freedom control systems in this note we study linear time invariant multivariable systems with two degree of freedom compensators first we give a new parametrization of all stabilizing compensators from the viewpoint of independent design of transfer function and robustness property second utilizing this parametrization we solve a general robust tracking problem including the case of nonsquare plants and show that tracking and robustness can be independently specified in two degree of freedom control systems

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

multiplicity results for positive solutions of some semi positone three point boundary value problems in this paper some multiplicity results for positive solutions of some singular semi positone three point boundary value problem be obtained by using the fixed point index method

robust regulation with h 2 performance abstract we consider the problem of minimizing the h 2 norm of a feedback system subject to the constraint that the controller robustly regulate against constants and sinusoids we show that for any 0 there exists a controller that both achieves robust regulation and renders the closed loop h 2 norm within of the optimal norm the norm achievable without the robust regulation constraint we also provide conditions that are both necessary and sufficient for there to exist a robustly regulating controller that achieves the optimal h 2 norm all proofs are constructive

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

generalized approximations and rapid convergence of solutions of m point boundary value problems abstract the upper and lower solutions method and the generalized quasilinearization technique for second order nonlinear m point boundary value problem of the type x f t x x t 0 1 x 0 x 0 0 x 1 i 1 m 2 i x i i 0 1 is developed a monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained in the c 1 norm

the free energies of six vertex models and the n equivalence relation the free energies of six vertex models on a general domain d with various boundary conditions are investigated with the use of the n equivalence relation which help classify the thermodynamic limit properties it is derived that the free energy of the six vertex model on the rectangle is unique in the limit height width it is derived that the free energies of the model on the domain d are classified through the densities of left down arrows on the boundary specifically the free energy is identical to that obtained by lieb phys rev lett 18 1046 1967 19 108 1967 phys rev 162 162 1967 and sutherland phys rev lett 19 103 1967 with the cyclic boundary condition when the densities are both equal to 12 this fact explains several results already obtained through the transfer matrix calculation the relation to the domino tiling or dimer or matching problems is also noted

existence of multiple solutions for second order boundary value problems we give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y f x y y 0 x epsilon 0 1 y 0 0 y 1 in the special case f x y y f y greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and of lakshmikantham et al

a quasilinearization method for second order four point boundary value problems abstract in this paper we obtain the existence and uniqueness results for general second order four point boundary value problems by applying the method of upper and lower solutions together with leray schauder degree also we obtain monotone iteration schemes which converge quadratically to the unique solution of some specific second order four point boundary value problems by using the method of quasilinearization

spin 1 2 ising heisenberg model with the pair xyz heisenberg interaction and quartic ising interactions as the exactly soluble zero field eight vertex model the spin 1 2 ising heisenberg model with the pair xyz heisenberg interaction and quartic ising interactions is exactly solved by establishing a precise mapping relationship with the corresponding zero field symmetric eight vertex model it is shown that the ising heisenberg model with the ferromagnetic heisenberg interaction exhibits a striking critical behavior which manifests itself through re entrant phase transitions as well as continuously varying critical exponents the changes in critical exponents are in accordance with the weak universality hypothesis in spite of a peculiar singular behavior that emerges at a quantum critical point of the infinite order which occurs at the isotropic limit of the heisenberg interaction on the other hand the ising heisenberg model with the antiferromagnetic heisenberg interaction surprisingly exhibits less significant changes in both critical temperatures and critical exponents upon varying the strength of the exchange anisotropy in the heisenberg interaction

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

three positive solutions of a nonlinear three point boundary value problem abstract in this paper existence criteria for three positive solutions of the nonlinear three point boundary value problem u t a t u t b t u t h t f u 0 t 0 1 u 0 0 u 1 u are established by using the leggett williams fixed point theorem an example is also included to illustrate the importance of the result obtained

brief paper algebraic and topological aspects of the regulator problem for lumped linear systems the regulator problem with robustness is solved for systems modelled by rational transfer matrices a topology for possibly unstable plants is presented a necessary and sufficient condition is derived for the existence of a proper controller which provides internal stability and output regulation throughout an open neighbourhood of the plant a characterization of all such controllers is determined

multi effective field theory applications to the cam analysis of the two dimensional ising model abstract a multi effective field theory is formulated and applied to the two dimensional ising model from the viewpoint of the coherent anomaly method cam two necessary conditions to construct the cam canonical series are shown two series of approximations are derived on 3 3 and 4 4 clusters and with the use of the cam we estimate the critical exponent of the susceptibility within an error of 0 37 and 0 45 percent respectively for the exact value 1 75 where the exact t c is assumed this accuracy of the estimation shows the effectiveness of this theory it is generally proved that certain kinds of approximations with different combinations of effective fields yield exactly the same approximate critical temperature and mean field coefficient

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

a multiplicity result for a nonlinear boundary value problem abstract in this paper we establish existence and multiplicity results for the problemx x f t x x s t a e in 0 x 0 x under the ambrosetti prodi type condition withfbeing a caratheodory function

re entrant phase transitions of the blume emery griffiths model for a simple cubic lattice on the cellular automaton the spin 1 ising beg model with the nearest neighbor bilinear and biquadratic interactions and single ion anisotropy is simulated on a cellular automaton which is improved from the creutz cellular automaton cca for a simple cubic lattice the simulations have been made for several sets of parameters k jk j and d jd j in the 3 d j 0 3 d j 0 and 1 k j 0 1 k j 0 parameter regions the re entrant and double re entrant phase transitions of the beg model are determined from the temperature variations of the thermodynamic quantities m q and the phase diagrams characterizing phase transitions are compared with those obtained from other methods

robust regulation in the presence of norm bounded uncertainty we consider robust regulation against steps and sinusoids in the presence of unstructured uncertainty the unstructured uncertainty is norm bounded by a constant that is given a priori this problem is equivalent to a certain multiobjective problem where one objective is robust regulation and the other is the standard objective of h sub spl infin suboptimal control it is shown that a solution to this problem exists if and only if the standard h sub spl infin problem admits a solution and certain matrix inequalities are satisfied these solvability conditions are readily computable controller synthesis is also addressed

the two layer ising film correlation length and critical curve by cle and cvm methods abstract on the basis of an improved elaboration of the method already proposed by the present authors the correlation length along a vertical section of a two layer ferromagnetic ising film is calculated for all t tc as a by product the critical curve is determined and a very interesting factorization property related to scaling limits is pointed out a comparison is made with previous results from other methods

the simple quadratic ising model with crossing bonds abstract we investigate the ferromagnetic critical line of the simple quadratic ising model with nearest neighbour and next nearest neighbour interactions by means of monte carlo renormalisation group and analytic methods a comparison of the monte carlo results with the renormalisation group trajectories yields the asymptotic behaviour of the critical line in the strong coupling limit a number of figures of spin configurations is included to illustrate the physical behaviour of the model

on robust asymptotic tracking perturbations on coprime factors and parameterization of all solutions the robust asymptotic tracking problem is analyzed in this paper relative to unstructured perturbations on each coprime factor of the transfer functions from the control input to the measured and controlled outputs in each case necessary and sufficient conditions for the existence of solutions are presented which are explicitly given in terms of problem data under such conditions explicit parameterizations are given of all controllers which achieve robust asymptotic tracking in terms of free rational proper and stable matrices

coherent anomaly method applied to the eight vertex model abstract the coherent anomaly method cam in terms of the multi effective field theory is applied to the eight vertex model in which critical exponents depend on interaction energies of the hamiltonian and vary continuously with them series of approximations are constructed by the rule which we have already proposed and the critical exponent is estimated using the exact critical temperature t c the results that are obtained lie within an error of 1 2 percent for the exact value in a region of interaction energies and this shows that the cam is applicable to a model which does not satisfy the ordinary universality hypothesis

critical exponents of the two layer ising model the symmetric two layer ising model tlim is studied by the corner transfer matrix renormalization group method the critical points and critical exponents are calculated it is found that the tlim belongs to the same universality class as the ising model the shift exponent is calculated to be 1 773 which is consistent with the theoretical prediction of 1 75 with 1 3 deviation

a two dimensional ising model with non universal critical behaviour it is shown that the two dimensional square lattice ising model with the nearest neighbour ferromagnetic interaction j the next nearest neighbour antiferromagnetic interaction j 0 and the four body interaction jint shows non universal critical behaviour when j is small the estimations of critical temperature tc and the critical exponent of susceptibility are performed within errors of 0 003 and 1 2 respectively

the layered ising model mean field and interfacial approximations the critical temperature for layered ising models on the s c lattice is calculated using the recently proposed transfer matrix version of a mean field approximation for two layered case the present results tc j 3 20755 5 and 3 62279 2 for the open and periodic boundary conditions respectively improve considerably on the previous estimation obtained by the high temperature series expansion this model is also studied using the interfacial approximation i e the muller hartmann zittartz method we show that in this method the behaviour of interface in the weak interlayer coupling limit is similar to that of interacting quantum particles for the strong interlayer interaction limits some results for the two layered case are also obtained

existence and multiplicity results for some three point boundary value problems abstract using the method of upper and lower solutions and some degree theory arguments we establish the existence of at least three solutions of some second order nonlinear differential equations of the type x f t x x t i 0 1 subject to nonlinear three point boundary conditions x 0 0 x 1 g x 0 1 the growth of f with respect to x is allowed to be quadratic

constrained variational approach to second order phase transitions application to a two layer ising film within the framework of the transfer matrix formalism a variational method is elaborated with a constraint related to the fundamental role of the correlation length near a critical point the method is applied to a two layer ferromagnetic film made of two horizontal ising planes interacting through a vertical coupling the critical curve of the model as well as the correlation length above and near tc are calculated the critical exponent nu obtained agrees with the expected two dimensional exact value for the two particular values of the vertical coupling for which there are numerical estimates of tc through series analysis or the monte carlo method an essential agreement with these estimates is found analytical expressions are given in the weak and strong vertical coupling regimes

the effect of the increase of linear dimensions on exponents obtained by 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

cam approach to the ground state phase transition in the two dimensional transverse ising model the double cluster approximation dca and the multi effective field approximation mefa are formulated for the two dimensional s 1 2 transverse ising model on a square lattice in the ground state these two cluster effective field approximations expand the sizes of clusters effectively combining some suitable series of these approximations with the coherent anomaly method cam the authors have analysed the critical phenomenon of this model they have obtained g c 1 528 or 0 016 and gamma 1 25 or 0 08 using the dca and gamma 1 26 or 0 10 using the mefa these estimates are consistent with the other results g c approximately 1 52 and gamma approximately 1 24 their scheme is expected to be useful in the study of frustrated systems because various shapes of clusters can be used in their calculations

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

generalized transformation for decorated spin models abstract the paper discusses the transformation of decorated ising models into an effective undecorated spin model using the most general hamiltonian for interacting ising models including a long range and high order interactions the inverse of a vandermonde matrix with equidistant nodes s s is used to obtain an analytical expression of the transformation this kind of transformation is very useful to obtain the partition function of decorated systems the method presented by fisher is also extended in order to obtain the correlation functions of the decorated ising models transforming into an effective undecorated ising model we apply this transformation to a particular mixed spin 1 2 1 and 1 2 2 square lattice with only nearest site interaction this model could be transformed into an effective uniform spin s square lattice with nearest and next nearest interaction furthermore the effective hamiltonian also includes combinations of three body and four body interactions in particular we considered spin 1 and 2

critical behavior of the blume emery griffiths model for a simple cubic lattice on the cellular automaton the spin 1 ising model with the nearest neighbour bilinear and biquadratic interactions and single ion anisotropy is simulated on a cellular automaton which improved from the creutz cellular automaton cca for a simple cubic lattice the simulations have been made for several k k j and d d j in the 0 d 3 and 2 k 0 parameter regions we confirm the existence of the re entrant and the successive re entrant phase transitions near the phase boundary the phase diagrams characterizing phase transitions are presented for comparison with those obtained from other calculations the static critical exponents are estimated within the framework of the finite size scaling theory at d 0 1 and 2 in the interval 2 k 0 the results are compatible with the universal ising critical behavior

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

further results on the robust tracking problem in two degree of freedom control systems the robust tracking problem in two degree of freedom control systems is formulated in a general setting which includes the case where the controlled output is not available directly a complete solution of the problem which is an extension of previous results is derived in a considerably simplified and transparent manner furthermore we show what kind of information about the output is needed to achieve robust tracking

completeness of the bethe ansatz for the six and eight vertex models we discuss some of the difficulties that have been mentioned in the literature in connection with the bethe ansatz for the six vertex model and xxz chain and for the eight vertex model in particular we discuss the beyond the equator infinite momenta and exact complete string problems we show how they can be overcome and conclude that the coordinate bethe ansatz does indeed give a complete set of states as expected

six vertex model with domain wall boundary conditions and one matrix model the partition function of the six vertex model on a square lattice with domain wall boundary conditions dwbc is rewritten as a hermitian one matrix model or a discretized version of it similar to sums over young diagrams depending on the phase the expression is exact for finite lattice size which is equal to the size of the corresponding matrix in the thermodynamic limit the matrix integral is computed using traditional matrix model techniques thus providing a complete treatment of the bulk free energy of the six vertex model with dwbc in the different phases in particular in the antiferroelectric phase the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions

obtaining critical point and shift exponent for the anisotropic two layer ising and potts models cellular automata approach abstract using probabilistic cellular automata with the glauber algorithm we have precisely calculated the critical points for the anisotropic two layer ising and potts models k x k y k z of the square lattice where k x and k y are the nearest neighbor interactions within each layer in the x and y directions respectively and k z is the inter layer coupling a general equation is obtained as a function of the inter and intra layer interactions for both the two layer ising and potts models separately where k z k x and k y k x furthermore the shift exponent for the two layer ising and potts models is calculated it was demonstrated that in the case of 1 for the two layer ising model the value of 1 756 0 0078 supports the scaling theories prediction that however for the unequal intra layer couplings for the two layer ising model and also in the case of both equal and unequal intra layer interactions for the two layer potts model our results are different from those obtained from the scaling theories finally an equation is obtained for the shift exponent as a function of intra layer couplings for the two layer ising and potts models

calculation of critical properties for the anisotropic two layer ising model on the kagome lattice cellular automata approach abstract the critical point k c of the two layer ising model on the kagome lattice has been calculated with a high precision using the probabilistic cellular automata with the glauber algorithm the critical point is calculated for different values of the inter and intra layer couplings k 1 k 2 k 3 k z where k 1 k 2 and k 3 are the nearest neighbor interactions within each layer in the 1 2 and 3 directions respectively and k z is the intralayer coupling a general ansatz equation for the critical point is given as a function of the inter and intra layer interactions k 3 k 1 k 2 k 1 and k z k 1 for the one and two layer ising models on the kagome lattice

quadratic approximation of solutions for boundary value problems with nonlocal boundary conditions in this paper using the quasilinearization method coupled with the method of upper and lower solutions we study a class of second order nonlinear boundary value problems with nonlocal boundary conditions we establish some sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem an example is also included to illustrate the main result

non universal critical behaviour of the two dimensional ising model with crossing bonds abstract the two dimensional square lattice ising model with the nearest neighbour ferromagnetic interaction j and the next nearest neighbour antiferromagnetic interaction j is investigated in the interaction range j j 0 8 this model is shown to violate the ordinary universality hypothesis estimations of critical temperature as a function of j j by means of an extrapolation method results in the exact critical temperature t c when j j 0 and in an approximate t c accurate up to the fourth digit when j j is small the critical exponent is estimated as a function of j j with the use of the multi effective field theory and the coherent anomaly method it is shown consequently that varies continuously as j j increases

interface properties of the two dimensional blume emery griffiths model the authors study the interface behaviour of the two dimensional blume emery griffiths model 1971 which is described by the hamiltonian h j sigma ij sisj k sigma ij si2sj 2 d sigma isi2 si 1 0 1 an interface is introduced into the system by fixing the spins on opposite boundaries in two different states 1 and 1 they pay particular attention to the appearance of an excess of the third state 0 in the vicinity of the interface the net adsorption of the non boundary state is studied near critical first order and tricritical transitions two methods have been used to attack this problem the monte carlo technique and a modified version of the interface free energy approximation of muller hartmann and zittartz 1977 which is seen to give a surprisingly good description of interface properties in three state systems

positive solutions of nonlinear three point boundary value problems abstract let a c 0 1 b c 0 1 0 let 1 t be the unique solution of the linear boundary value problem u t a t u t b t u t 0 t 0 1 u 0 0 u 1 1 we study the existence of positive solutions to the nonlinear boundary value problem u t a t u t b t u t h t f u 0 t 0 1 u 0 0 u u 1 where 0 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones

generalized approximations for nonlinear three point boundary value problems abstract the generalization of the method of quasilinearization for a class of second order nonlinear nonlocal three point boundary value problems of the types x f t x x t 0 1 x 0 0 x 1 g x 0 1 is developed a monotone sequence of solutions of linear problems converging uniformly and rapidly to a solution of the problem is obtained in the c 1 norm

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

critical indices for the two dimensional ising model with nearest neighbor and next nearest neighbor interactions ii strip cluster approximation the spin pair correlation functions are calculated for the ising model with nearest neighbor interaction j 0 and next nearest neighbor interaction j on the square lattice in terms of three approximations of the simplified cluster variation method with strip clusters as the basic clusters the critical indices and are estimated from the obtained spin pair correlation functions with the aid of the coherent anomaly method for j j 0 5 we confirm that the system belongs to the onsager universality class in the sense of the weak universality hypothesis