guiding functions for generalized periodic problems and applications abstract in this paper developing the method of guiding functions for differential inclusions with a generalized periodic condition we obtain the existence theorem for such problems it is shown how the abstract result can be applied to the study of differential games and anti periodic problems

on a class of second order anti periodic boundary value problems of concern is the existence and uniqueness of anti periodic solutions to a class of abstract nonlinear second order differential equations the investigation relies on the theory of m accretive operators in banach or hilbert spaces examples of boundary value problems for ordinary and partial differential equations are also discussed

fusion of dilute al lattice models the fusion procedure is implemented for the dilute al lattice models and a fusion hierarchy of functional equations with an su 3 structure is derived for the fused transfer matrices we also present the bethe ansatz equations for the dilute al lattice models and discuss their connection with the fusion hierarchy the solution of the fusion hierarchy for the eigenvalue spectra of the dilute al lattice models will be presented in a subsequent paper

recent progress in solving a d e lattice models abstract there are many families of solvable a d e lattice models exhibiting order disorder transitions these represent many different universality classes of critical behaviour some a d e models can be solved off criticality but most can only be solved at criticality here we review the methods being developed to solve these models to gain a detailed understanding of their critical behaviour

new construction of solvable lattice models including an ising model in a field in this letter we report a new construction to obtain restricted solid on solid rsos models out of loop models the method is a generalization of ideas developed by owczarek and baxter and by pasquier in particular we consider a solvable o n model and point out that some of the rsos models thus obtained admit an off critical extension among these models we find a spin 1 ising model which is solvable not only at the critical point but also in a fieldlike deviation away from it we calculate the critical exponent ensuremath delta 15 directly from the relation between the free energy and the field this is the first determination of this exponent without the use of scaling relations

analysis and control of differential inclusions with anti periodic conditions we deal with the optimal control of systems driven by differential inclusions with anti periodic conditions in banach spaces the operators in the differential inclusions describing the control system all depend on control variables by a solution of the control system we mean a trajectory control pair we first prove that for fixed control variables the set of solutions is non empty for the corresponding differential inclusion with anti periodic conditions we then show the functional topological properties of the set of solutions finally we investigate the existence of solutions to the optimal control problem these abstract results can be used to study optimal control systems of evolutionary hemi variational inequalities

weighted pseudo antiperiodic solutions for fractional integro differential equations in banach spaces in this paper we prove the existence of weighted pseudo antiperiodic mild solutions for fractional integro differential equations in the form d u t au t t a t s au s ds f t u t t r where f u is stepanov like weighted pseudo antiperiodic and a generates a resolvent family of bounded and linear operators on a banach space x a l loc 1 r and 0 here the fractional derivative is considered in the sense of weyl also we give a short proof to show that the vector valued space of stepanov like weighted pseudo antiperiodic functions is a banach space

a class of interaction round a face models and its equivalence with an ice type model a new model called the temperley lieb interactions model is introduced in two dimensional lattice statistics on a square lattice the temperley lieb equivalence of this model to the six vertex self dual potts critical hard hexagons and critical nonintersecting string models is established a graphical equivalence of this model to the six vertex model generalizes this equivalence to noncritical cases of the above models the order parameters of a specialization of this model are studied

the odd primary h structure of low rank lie groups and its application to exponents a compact connected simple lie group g localized at an odd prime p is shown to be homotopy equivalent to a product of homotopy associative homotopy commutative spaces provided the rank of g is low this holds for su n for example if n p 1 p 3 the homotopy equivalence is usually just as spaces not multiplicative spaces nevertheless the strong multiplicative features of the factors can be used to prove useful properties which after looping can be transferred multiplicatively to g this is applied to prove useful information about the torsion in the homotopy groups of g including an upper bound on its exponent

modular representations and the homotopy of low rank p local cw complexes fix an odd prime p and let x be the p localization of a finite suspended cw complex given certain conditions on the reduced mod p homology widetilde h x mathbb z p of x we use a decomposition of x due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that i x is a homotopy retract of x this implies the stable homotopy groups of x are in a certain sense retracts of the unstable homotopy groups and by a result of stanley one can confirm the moore conjecture for x under additional assumptions on widetilde h x mathbb z p we generalize a result of cohen and neisendorfer to produce a homotopy decomposition of x that has infinitely many finite h spaces as factors

hard squares with diagonal attractions the authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next nearest neighbour diagonal interactions on the special surface corresponding to regimes iii and iv of the generalised hard hexagon model the interfacial tension correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row to row and corner transfer matrices the associated critical exponents are found to be mu v 5 4 beta 3 32 in regime iii and mu v 5 2 beta 1 4 in regime iv in particular their results confirm the recent proposal by huse that regime iii is the first order coexistence surface separating the disordered fluid phase from the square ordered solid phases and that the regime iii regime iv boundary is a line of tricritical points

the dual polyhedral product cocategory and nilpotence abstract the notion of a dual polyhedral product is introduced as a generalization of hovey s definition of lusternik schnirelmann cocategory properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if x is a simply connected space then the weak cocategory of x equals the homotopy nilpotency class of x this answers a fifty year old problem posed by ganea the methods are applied to determine the homotopy nilpotency class of quasi p regular exceptional lie groups and sporadic p compact groups

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

critical and multicritical o n models soluble o n models on the square lattice are found via the yang baxter equations in a parameter space of three coupling constants a number of distinct soluble cases are found for each value of n some of these constitute critical or multicritical points in the phase diagram for all solutions the free energy per spin can be determined using the bethe ansatz by studying the finite size corrections at the critical points the critical exponents and the central charge can be found in some cases analytically and failing that numerically by extrapolation of exact results for finite size systems

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

critical points in coupled potts models and critical phases in coupled loop models we show how to couple two critical q state potts models to yield a new self dual critical point we also present strong evidence of a dense critical phase near this critical point when the potts models are defined in their completely packed loop representations in the continuum limit the new critical point is described by an su 2 coset conformal field theory while in this limit of the critical phase the two loop models decouple using a combination of exact results and numerics we also obtain the phase diagram in the presence of vacancies we generalize these results to coupling two potts models at different q

lattice realizations of unitary minimal modular invariant partition functions the conformal spectra of the critical dilute a d e lattice models are studied numerically the results strongly indicate that in branches one and two these models provide realizations of the complete a d e classification of unitary minimal modular invariant partition functions given by cappelli itzykson and zuber 1987 in branches three and four the results indicate that the modular invariant partition functions factorize similar factorization results are also obtained for two colour lattice models

existence of weighted pseudo anti periodic solutions to some non autonomous differential equations abstract in this paper we first introduce two new concepts called respectively weighted pseudo periodicity and weighted pseudo anti periodicity which generalize respectively the well known notions of periodicity and that of anti periodicity basic properties of these new functions are discussed furthermore the existence of weighted pseudo anti periodic solutions to some damped non autonomous second order differential equations will be investigated to illustrate our abstract results the existence of weighted pseudo anti periodic solutions to a plate like equation is discussed

existence of anti periodic mild solutions for semilinear evolution equations in this paper we study the existence of anti periodic mild solutions for a class of semilinear evolution equations in banach spaces and extend some related results in this direction an example is given to illustrate our results

a universal amplitude ratio for the q 4 potts model from a solvable lattice model the universal amplitude ratio r for the q 4 state potts model in two dimensions is determined by using results for the dilute a model in regime 1 the nature of the relationship between the potts model and the dilute a model is discussed

order parameters of the dilute a models the free energy and local height probabilities of the dilute a models with borken 2 symmetry are calculated analytically using inversion and corner transfer matrix methods these models possess four critical branches the first two branches provide new realizations of the unitary minimal series and the other two branches give a direct product of this series with an ising model we identify the integrable perturbations which move the dilute a models away from the critical limit generalized order parameters are defined and their critical exponents extracted the associated conformal weights are found to occur on the diagonal of the relevant kac table in an appropriate regime the dilute a3 model lies in the universality class of the ising model in a magnetic field in this case we obtain the magnetic exponent 15 directly without the use of scaling relations

susceptibility amplitudes for the three and four state potts models abstract the finite lattice method of series expansions is used to calculate a number of extended series for susceptibilities of the three and four state potts models on the square lattice we analyse all these series and estimate their critical amplitudes we resolve an uncertainty in amplitude ratios for high to low temperature susceptibilities and thus provide confirmation of field theoretic predictions

critical properties of the izergin korepin and solvable o n models and their related quantum spin chains the central charge of the izergin korepin model the corresponding quantum spin chain and the o n model is calculated analytically via the bethe ansatz the calculation extends a technique recently developed for the zamolodchikov fateev model in addition critical exponents and the central charge for these models are obtained from numerical solutions of bethe ansatz equations for finite systems as a physical application the authors find the exponents v 12 23 and gamma 53 46 for the theta transition of polymers in two dimensions

anti periodic solutions to nonlinear evolution equations abstract we deal with anti periodic problems for nonlinear evolution equations with nonmonotone perturbations the main tools in our study are the maximal monotone property of the derivative operator with anti periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings

exceptional structure of the dilute a 3 model e 8 and e 7 rogers ramanujan identities the dilute a3 lattice model in regime 2 is in the universality class of the ising model in a magnetic field here we directly establish the existence of an e8 structure in the dilute a3 model in this regime by expressing the one dimensional configuration sums in terms of fermionic sums which involve the e8 root system explicitly in the thermodynamic limit these polynomial identities yield a proof of the e8 rogers ramanujan identity recently conjectured by kedem et al 1993 the polynomial identities also apply to regime 3 which is obtained by transforming the modular parameter by q to 1 q in this case we find an a1xe7 structure and prove a rogers ramanujan identity of a1xe7 type finally in the critical q to 1 limit we give some intriguing expressions for the number of l step paths on the a3 dynkin diagram with tadpoles in terms of the e8 cartan matrix all our findings confirm the e8 and e7 structure of the dilute a3 model found recently by means of the thermodynamic bethe ansatz

locus of the tricritical transition in a two dimensional q state potts model a simple two dimensional potts model with vacancies is introduced the phase transition between paramagnetic and ferromagnetic is located by simple symmetry arguments it is shown that the transition has a continuous and a first order segment separated by a tricritical transition the locus of the tricritical point is determined by a transformation of the partition sum into that of an o n model at the tricritical point and at some of the critical points this potts model is soluble

the o n model on the triangular lattice to find critical points of o n models on the triangular lattice we apply two methods first we investigate the yang baxter equations on the triangular lattice we find only solvable points directly related to those for the square lattice second we construct intersections with the potts model this yields eight branches of critical points parametrized by n we establish the equivalence of these branches with the known critical points of the o n model on the square lattice transfer matrix calculations are performed to obtain the conformal anomaly and the thermal exponent of these branches these results include a numerical analysis of a q 3 potts tricritical point we find analytic relations between potts and o n models as well as among o n models with different values of n and among potts models with different values of q

the dilute al models and the integrable perturbations of unitary minimal cfts recently a set of thermodynamic bethe ansatz tba equations has been proposed by dorey pocklington and tateo for unitary minimal models perturbed by the 1 2 or 2 1 operator we examine their results in view of the lattice analogues dilute al models in regimes 1 and 2 taking m5 6 1 2 and m3 4 2 1 as the simplest examples we will explicitly show that the conjectured tba equations can be recovered from the lattice model in a scaling limit

quenched bond dilution in two dimensional potts models we report a numerical study of the bond diluted two dimensional potts model using transfer matrix calculations for different numbers of states per spin we show that the critical exponents at the random fixed point are the same as in self dual random bond cases in addition we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin spin correlation function in the cylinder geometry we show that the behaviour is fully compatible with the one observed in the random bond case confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder dilution self dual binary random bond self dual continuous random bond

solvable lattice models labelled by dynkin diagrams an equivalence between generalized restricted solid on solid models associated with sets of graphs and multi colour loop models is established as an application the authors consider solvable loop models and in this way obtain new solvable families of critical rsos models these families can all be classified by the dynkin diagrams of the simply laced lie algebras for one of the rsos models labelled by the lie algebra pair al al and related to the c2 1 vertex model they give an off critical extension which breaks the z2 symmetry of the dynkin diagrams

existence of solutions for anti periodic boundary value problems abstract by using schauder s fixed point theorem and the lower and upper solutions method we discuss the anti periodic boundary value problem for a class of second order differential equations some sufficient conditions for the existence of solutions are obtained

scaling limit of the ising model in a field the dilute mathrm a 3 model is a solvable interaction round a face model with three local states and adjacency conditions encoded by the dynkin diagram of the lie algebra mathrm a 3 it can be regarded as a solvable spin 1 ising model at the critical temperature in a magnetic field one therefore expects the scaling limit to be governed by zamolodchikov s integrable perturbation of the c 1 2 conformal field theory indeed a recent thermodynamic bethe ansatz approach succeeded in unveiling the corresponding mathrm e 8 structure under certain assumptions on the nature of the bethe ansatz solutions in order to check these conjectures we perform a detailed numerical investigation of the solutions of the bethe ansatz equations for the critical and off critical models scaling functions for the ground state corrections and for the lowest spectral gaps are obtained which give very precise numerical results for the lowest mass ratios in the massive scaling limit while these agree perfectly with the mathrm e 8 mass ratios we observe one state that seems to violate the assumptions underlying the thermodynamic bethe ansatz calculation we also analyze the critical spectrum of the dilute mathrm a 3 model which exhibits excitations with a finite gap on top of the massless spectrum of the ising conformal field theory

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

existence of anti periodic mild solutions to semilinear nonautonomous evolution equations abstract in this paper applying the theory of evolution family and schauder s fixed point theorem we prove the anti periodic mild solutions for semilinear nonautonomous evolution equations in banach space under conditions furthermore an example is given to illustrate our results

free field approach to the dilute al models we construct a free field realization of vertex operators of the dilute al models along with the felder complex for l 3 we also study an e8 structure in terms of the deformed virasoro currents

antiperiodic solutions of semilinear integrodifferential equations in banach spaces abstract we are concerned in this paper with the existence of antiperiodic mild solutions of some semilinear integrodifferential equations in a banach space we study further properties of antiperiodic functions including a new composition theorem and introduce the new concept of asymptotically antiperiodic functions

an exact universal amplitude ratio for percolation the universal amplitude ratio for percolation in two dimensions is determined exactly using results for the dilute a model in regime 1 by way of a relationship with the q state potts model with q 4

on the existence of anti periodic solutions to a nonlinear evolution equation associated with odd subdifferential operators the existence of anti periodic solutions hence also that of periodic solutions to the nonlinear parabolic type evolution equation ddt u t u t f t u t 0 t r is shown under the assumptions that the subdifferential operator is odd and that f t t r is a family of monotone operators satisfying the anti periodicity condition f t t v f t v v d f t hence f t t v f t v v d d f t t r for a fixed t r here coerciveness of is not assumed an application to a heat equation is given without assuming boundedness of the domain of rxn

anti periodic solutions for semilinear evolution equations in this paper we use the homotopy method to establish the existence and uniqueness of anti periodic solutions for the nonlinear anti periodic problem a t x bx f t a e t r x t t x t where a t x is a nonlinear map and b is a bounded linear operator from r n to r n sufficient conditions for the existence of the solution set are presented also we consider the nonlinear evolution problems with a perturbation term which is multivalued we show that for this problem the solution set is nonempty and weakly compact in w 1 2 i r n for the case of convex valued perturbation and prove the existence theorems of anti periodic solutions for the nonconvex case all illustrative examples are provided

dilute birman wenzl murakami algebra and dn 1 2 models a dilute generalization of the birman wenzl murakami algebra is considered it can be baxterized to a solution of the yang baxter algebra the dn 1 2 vertex models constitute a series of solvable lattice models which realize this algebraic structure they can be regarded as a dilute version of the bn 1 vertex models

q trinomial coefficients and the dilute a model we express the configuration sums of the dilute model in regimes and in terms of q trinomial coefficients we then extend previous results to give expressions for all phases and regimes for odd l for finite system size

second order differential variational inequalities involving anti periodic boundary value conditions abstract this paper deals with a new class of second order nonlinear differential variational inequalities firstly we study the properties of a solution set for a variational inequality by using the kkm technique and a monotonicity property then we prove the existence of solutions to a second order nonlinear differential variational inequality with anti periodic boundary value conditions not assuming the lipschitz continuity of the nonlinear term our approaches which are essentially different from previous research methods in differential variational inequalities are based on the topological degree theory and the filippov implicit function lemma