a renormalisation group analysis of the semi finite potts model we discuss the ordinary transition of the semi infinite q state potts model within the migdal kadanoff renormalisation group scheme approximate surface free energies are calculated in dimension d 2 in addition we consider the effect of symmetry breaking surface fields our scheme predicts that only one such field is relevant

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

on symmetries conservation laws and variational problems for partial differential equations the problem of the correspondence between symmetries and conservation laws for partial differential equations is considered for lagrangian systems the set of noether variational symmetries can be shown to lead to the set of all local conservation laws for partial differential equations without well defined lagrangian functions there is no universal correspondence between symmetries and conservation laws in this article it is shown that for a large class of differential equations there is a natural way to associate conservation laws with symmetries the class consists of many interesting equations e g korteweg de vries equation kadomtsev petviashvili equation boussinesq equation nonlinear diffusion equation monge ampere equation regularized long wave equation and navier stokes equations characteristics of the corresponding conservation laws are calculated and examples are given for lagrangian systems the consistency of the approach with the standard noether results is discussed

effects of surface wedge corner and mixed boundary conditions on the local critical behaviour critical exponents for the o n spin model local field near a surface an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or comer is formed by mutually perpendicular d 1 dimensional walls each having von neumann special transition or dirichlet ordinary transitions boundary conditions the combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order e e 4 d the case of two or more von neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities

symmetry transformations isovectors and conservation laws a system of second order partial differential equations is considered it is shown that a conservation law may be associated with any pair consisting of i a symmetry transformation ii a symmetry transformation if the system is self adjoint a solution to the adjoint of the equations of variation otherwise such conservation laws continue to hold when symmetry transformations are replaced by isovectors it is also proved that isovectors identify additional conservation laws by deformation of a given one whence it follows that there exists a natural conserved current associated with every isovector also an application is made to find conserved currents for the navier stokes equations

zn n 1 generalization of the chiral potts model we show that ther matrix which intertwines twon by nn 1 state cyclicl operators related with a generalization ofuq sl n algebra can be considered as a boltzmann weight of four spin box for a lattice model with two spin interaction just as ther matrix of the checkerboard chiral potts model the rapidity variables lie on the algebraic curve of the genusg n2 n 1 n 1 n n 1 defined by 2n 3 independent moduli this curve is a natural generalization of the curve which appeared in the chiral potts model factorization properties of thel operator and its connection to the sos models are also discussed

generalized chiral potts models and minimal cyclic representations of u_q widehat mathfrak g mathfrak l n c finite dimensional cyclic representations of the quantized universal enveloping algebra u q g exist only when the deformation parameter q is a root of unity q n 1 in such representations the n th power of each chevalley generator acts as a non zero scalar these values of e i n f i n and t i n are part of a continuous family of parameters upon which the cyclic representations depend

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

noether type conservation laws for perfect fluid motions a general approach is developed for the derivation of conservation laws in continuum physics a noether type theorem is applied in connection with transformations which leave the action functional invariant to within the integral of a divergence specific results are derived in the case of fluid dynamics the pertinent equations are considered within the lagrangian material description and are associated with a genuine variation formulation the physical meaning of the conservation laws is emphasized and the greater generality of the approach is commented upon

percolation in an edge percolation in a three dimensional lattice bounded by two free plane surfaces meeting at an angle chi is studied within a real space renormalisation scheme the edge critical behaviour at various surface and bulk transitions is analysed the author does not find evidence for a first order edge transition at higher values of alpha as has recently been observed for an ising model at an edge dependence of the edge scaling power ye on the opening angle alpha has been confirmed

the connection between variational principles in eulerian and lagrangian descriptions the question of whether there exists a connection between variational principles in eulerian and lagrangian descriptions is investigated by having recourse to a proper view of the eulerian description it is shown that a variational principle in one description holds whenever a corresponding variational principle in the other description is given this theoretical conclusion is operative in that a precise rule for writing the new lagrangian is exhibited as an application a new lagrangian for fluid dynamics in the eulerian description is determined

symmetries and conservation laws of the euler equations in lagrangian coordinates abstract we consider the euler equations of incompressible inviscid fluid dynamics we discuss a variational formulation of the governing equations in lagrangian coordinates we compute variational symmetries of the action functional and generate corresponding conservation laws in lagrangian coordinates we clarify and demonstrate relationships between symmetries and the classical balance laws of energy linear momentum center of mass angular momentum and the statement of vorticity advection using a newly obtained scaling symmetry we obtain a new conservation law for the euler equations in lagrangian coordinates in n dimensional space the resulting integral balance relates the total kinetic energy to a new integral quantity defined in lagrangian coordinates this relationship implies an inequality which describes the radial deformation of the fluid and shows the non existence of time periodic solutions with nonzero finite energy

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

is the zamolodchikov model critical evidence is presented in favour of the hypothesis that the zamolodchikov model an exactly solvable three dimensional lattice model without any temperature like parameters is critical the evidence is obtained by generalising the zamolodchikov model to include a temperature like variable the magnetisation curve of this model is then studied using a modified form of a variational approximation formulated earlier also the authors show the two layer zamolodchikov model corresponds to a critical free fermion model

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

composite variational principles and the determination of conservation laws sufficiently regular systems of partial differential equations can always be derived from a composite variational principle via the introduction of a suitable set of adjoint variables by applying an appropriate formulation of noether s theorem to such composite variational principles an operative procedure leading to the determination of the conservation laws of the given system is determined as an application conservation laws for a viscous incompressible fluid and an inviscid barotropic compressible fluid are examined in detail in particular new conservation laws for euler flows are found

higher order symmetries of the compressible one dimensional isentropic fluid equations the equations of compressible one dimensional isentropic flow are shown to have infinitely many higher order symmetries by means of recursion operators furthermore a new first order conserved functional is found to be associated with a third order symmetry via noether s theorem

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

ising model at an edge a position space renormalisation group approach the critical behaviour of a three dimensional ising model bounded by two plane surfaces meeting at an angle alpha is studied using the migdal kadanoff and two terminal cluster renormalisation methods the authors obtain expressions for the edge fixed point and for the independent edge exponent yhe at the various transitions they find that yhe is not completely universal but depends on the angle alpha and that it shows some unexpected features

new solvable lattice models in three dimensions in this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models the first is the zamolodchikov model which is the only nontrivial model on a three dimen sional lattice so far solved the second is the chiral potts model on the square lattice and its generalization associated with theu q sl n algebra which is of current interest due to its connections with high genus algebraic curves and with representations of quantum groups at roots of unity we show that this last sl n generalized chiral potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square lattice layers with ann valued n 2 spin at each site further in then 2 case this three dimen sional model reduces after a modification of the boundary conditions to the zamolodchikov model we mentioned above