analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

impedance boundary conditions for finite planar and curved frequency selective surfaces we consider the time harmonic electromagnetic scattering problem from a finite planar or curved infinitesimally thin frequency selective surface fss the periodic unit cells of which are constituted exclusively by electric conductors and free space in order to avoid the meshing of these cells the problem is solved by employing an integral equation formulation in conjunction with approximate impedance boundary conditions ibc prescribed on the sheet that models the fss the impedance in the ibc is derived from the exact reflection coefficient calculated for the fundamental floquet mode on the infinite planar fss illuminated by a plane wave at a given incidence when the fss is curved and or the direction of the incident wave is unknown higher order ibcs are proposed that are valid in a large angular range and can be implemented in a standard method of moments formulation also a simple technique is presented that allows to reproduce the radiating floquet modes in the scattered field even though those are not accounted for in these ibcs their numerical efficiencies are evaluated for a curved strip grating translationally invariant along one direction finally we present an alternative approach where the impedance is approximated by its truncated fourier series that considerably enhances the accuracy of the results at the cost however of a denser mesh of the sheet

directed percolation shape of the percolation cone conductivity exponents and high dimensionality behaviour presents several results for the directed percolation problem the shape of the percolation cone is discussed showing how the operating angle behaves near threshold and explaining the orientation dependence of the correlation exponents found by domany and kinzel 1981 the cayley tree model is used to discuss the high dimensionality limit the conductivity problem on a directed network is introduced and the high dimensionality values for the conductivity exponents s and tau are found

examples of phase equilibrium in lattice models the general z4 and z5 models on the simple quadratic lattice and the pair triplet ising model on the triangular lattice are employed to illustrate an extension of the finite size scaling technique which obtains the qualitative structure of the phase equilibrium surface in terms of multiple phase coexistence

finite range scaling study of the 1d long range ising model the critical behaviour of the one dimensional ising model with long range ferromagnetic interactions decaying with distance r as 1 r1 sigma has been studied by scaling the range of interactions exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0 sigma or 1 applying the range scaling the critical temperature critical exponent nu and the anomalous dimension of the order parameter have been calculated additional analysis of the convergence of the method has been performed by applying the vanden broeck and schwartz extrapolation procedure in addition to the simple least squares approximation and by evaluating the convergence exponents

scaling of overhangs in 1 1 dimensional directed processes in a gradient when a gradient is imposed in the control parameter of a directed process in a 1 1 dimensional directed process a front may occur the position of this front evolves in steps of size one in the direction opposite to the gradient and in steps of size equal to or larger than one in the other direction the larger steps are due to overhangs there is a critical value for the control parameter only if the first moment of the step size distribution in the direction of the gradient is not singular i e it takes on a finite value in the thermodynamic limit if there is a critical point this step size distribution is a power law at this point with an exponent larger than two and which the authors conjecture to be equal to 3 beta nu perpendicular to beta is the order parameter exponent and nu perpendicular to is the spatial correlation length exponent in a finite gradient there is a second exponent governing the upper cut off in the step size distribution this exponent is equal to nu perpendicular to 1 nu perpendicular to these two exponents govern the convergence of the position of the front towards the critical value of the control parameter the authors exemplify their discussion with directed site percolation and with two versions of a model originally designed to study catalytic reactions

estimates of the site percolation probability exponents for some directed lattices series expansion analysis has been used to obtain estimates of the percolation probability exponent beta for directed site percolation on the square triangular simple cubic and body centred cubic lattices these estimates are consistent with those obtained by please 1977 for the corresponding bond problems and the results for site and bond problems on all lattices considered are summarised by beta 0 28 or 0 01 d 2 and beta 0 59 or 0 02 d 3

application of logical transport to determine the directed and isotropic percolation thresholds the authors present a monte carlo algorithm based on logical transport which determines the effective isotropic percolation and directed percolation thresholds of finite size lattices they apply the algorithm to bond and site directed percolation in two dimensions and determine the asymptotic percolation thresholds by finite size scaling

monte carlo and series study of corrections to scaling in two dimensional percolation corrections to scaling for percolation cluster numbers in two dimensions are studied by monte carlo simulations of very large systems up to 17 109 lattice sites and by series analysis both series and monte carlo work suggests that the value of the correction to scaling exponent is slightly lower at the percolation threshold than away from it moreover the corrections to scaling observed at pc omega equivalent to 0 64 might be due to the mixing of scaling fields rather than to the irrelevant scaling fields the monte carlo results are compatible with finite size scaling and finite size scaling corrections are estimated technical problems associated with monte carlo simulation of very large systems are discussed in an appendix

directed percolation mean field theory and series expansions for some two dimensional lattices the shape of percolation clusters in the directed percolation problem is studied within mean field theory low density series expansions are obtained for bond and site problems on the square and triangular lattices these are used to test a scaling formula for the pair connectedness which involves two lengths xi sub rho and xi perpendicular to rho determination of the exponents nu sub and nu perpendicular to for these lengths shows that all four problems are in the same universality class and the values support the hyperscaling relation beta 1 2 d nu perpendicular to nu sub gamma an independent argument is given for this relation

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

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

topology and geometry of single hidden layer network least squares weight solutions in this paper the topological and geometric properties of the weight solutions for multilayer perceptron mlp networks under the mse error criterion are characterized the characterization is obtained by analyzing a homotopy from linear to nonlinear networks in which the hidden node function is slowly transformed from a linear to the final sigmoidal nonlinearity two different geometric perspectives for this optimization process are developed the generic topology of the nonlinear mlp weight solutions is described and related to the geometric interpretations error surfaces and homotopy paths both analytically and using carefully constructed examples these results illustrate that although the natural homotopy provides a practically valuable heuristic for training it suffers from a number of theoretical and practical difficulties the linear system is a bifurcation point of the homotopy equations and solution paths are therefore generically discontinuous bifurcations and infinite solutions further occur for data sets that are not of measure zero these results weaken the guarantees on global convergence and exhaustive behavior normally associated with homotopy methods however the analyses presented provide a clear understanding of the relationship between linear and nonlinear perceptron networks and thus a firm foundation for development of more powerful training methods the geometric perspectives and generic topological results describing the nature of the solutions are further generally applicable to network analysis and algorithm evaluation

impedance boundary conditions for finite planar or curved frequency selective surfaces embedded in dielectric layers we consider the time harmonic electromagnetic scattering problem from a finite planar or curved structure made up of infinitesimally thin frequency selective surfaces fsss embedded in dielectric layers with possibly nearby located objects in order to avoid the meshing of the unit cells that constitute the fsss this problem is solved by employing an integral equation ie or finite element fe formulation in conjunction with approximate impedance boundary conditions ibcs prescribed on the sheets that model the fsss the impedances in the ibcs are derived from the exact reflection and transmission coefficients calculated for the fundamental floquet mode on the infinite planar structure illuminated by a planewave at a given incidence when the structure is curved and or the direction of the incident wave is unknown higher order ibcs are proposed that are valid in a large angular range and can be implemented in a standard ie or fe formulation their numerical efficiencies are evaluated for finite planar or curved two dimensional structures or radomes where the fsss are strip gratings as an example for a curved radome surrounding a conducting plate it is shown that when the floquet modes of the gratings are evanescent these ibcs allow an accurate calculation of the radar cross section of the whole structure with far smaller computing resources than would have been required by a full wave formulation

basis vector importance sampling for hamiltonian lattice spectrum calculations different numerical methods for accurate calculation of low lying eigenvalues of lattice hamiltonians are proposed and critically compared a dynamical procedure called basis vector importance sampling is shown to select the relevant subspace of the hilbert space very effectively this method is used to compute the mass gap of o 2 symmetric quantum chains up to a length of nine sites kosterlitz thouless type freezing transitions of z p symmetric chains are also studied via the spectrum of quantum kinks

mean field renormalisation group calculations for directed percolation on the square lattice a simple extrapolation technique for mean field renormalisation group calculations is described application of this method to large cell calculations for percolation on the directed square lattice yields pc 0 7067 or 0 0001 site problem and pc 0 6454 or 0 0002 bond problem

directed percolation series expansion for some three dimensional lattices low density series expansions have been obtained for the directed cubic lattice bond problem and the directed cubic and body centred cubic lattice site problems estimates of the critical probabilities and critical exponents gamma gamma 0 nu nu perpendicular to for these problems are obtained the exponent values are found to be consistent with the universality of directed percolation and reggeon field theory and the hyperscaling relation beta d nu perpendicular to nu gamma 2

finite range scaling for a one dimensional system with long range interactions a procedure similar to the finite size scaling is proposed for systems with long range interactions it is applied to the one dimensional ising model with interactions decaying as 1 mod i j mod 1 sigma with distance mod i j mod the critical temperature and exponent nu are calculated for sigma or 1

series expansion study of the distribution of currents in the elements of a random diode insulator network the critical exponent zeta k of the kth moment of the current distribution for random diode insulator networks on the square and simple cubic lattices is calculated for a range of k using low density series expansion techniques it is also shown that zeta 1 nu sub where nu sub is the critical exponent of the parallel connectedness length for directed percolation the values of zeta k for the simple cubic lattice are well fitted by a simple exponential formula with zeta k 1 zeta k 1 1 1 2 the skal shklovskii scaling relation for the conductivity is generalised to the kth moment and it follows that the exponent k describing the divergence of flicker noise is given by kappa approximately d 1 nu perpendicular to nu sub zeta 4 2 zeta 2 the authors estimate this for both lattices

high order impedance boundary conditions for multilayer coated 3 d objects the scattering problem by a multilayer coated three dimensional 3 d object where the coating is modeled by an impedance boundary condition ibc is considered first the exact boundary condition is obtained for an infinite planar coating with an arbitrary number of layers then various approximations for the pseudodifferential operators involved in this exact condition are proposed in the expressions of the resulting ibcs all tangential derivatives of the fields of order higher than two are suppressed these ibcs are compared in terms of numerical efficiency by computing either the reflection coefficients on an infinite planar metal backed coating or the radar cross section rcs of a perfectly conducting coated sphere using the tangent plane approximation in both cases it is found that the highest order ibc models the coating with a good accuracy finally some guidance is given on how this ibc may be numerically implemented in an integral equation or a finite element formulation for an arbitrarily shaped object

series expansion analysis of corrections to scaling in the three state potts model several extant series for the d 2 q 3 potts model are analysed with a method that explicitly accounts for the effect of confluent corrections to scaling excellent agreement with the conjectured exponents for this model is found as well as confirmation of renormalisation group estimates of the sub leading eigenvalue

low density series expansions for directed percolation i a new efficient algorithm with applications to the square lattice a new algorithm for the derivation of low density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems numerical evidence shows that the computational complexity grows exponentially but with a growth factor which is much smaller than the growth factor of the previous best algorithm for bond site percolation on the directed square lattice the series has been extended to order 171 158 analysis of the series yields sharper estimates of the critical points and exponents

a monte carlo study of scaling and universality in 2d directed percolation monte carlo data for the pair correlation function is obtained on five different lattices for the directed percolation problem in two dimensions scaling is tested and found to hold beyond 20 lattice spacings for all models evidence is found to support the universality of the scaling function the values of the exponents gamma v sub and vperpendicular to v sub are extracted from the data the results for all models are summarised by gamma v sub 1 305 or 0 007 and vperpendicular to v sub 0 629 or 0 010 comparison is made with estimates derived by other means

semi directed percolation in two dimensions abstract we studied a model of semi directed percolation on finite strips of the square and triangular lattices using the transfer matrix method combined with phenomenological renormalization group approach we obtain good numerical estimates for critical probabilities and correlation lengths critical exponents our results confirm the conjecture that semi directed percolation belongs to the universality class of the usual fully directed percolation model

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

backpropagation with homotopy when training a feedforward neural network with backpropagation rumelhart et al 1986 local minima are always a problem because of the nonlinearity of the system there have been several ways to attack this problem for example to restart the training by selecting a new initial point to perform the preprocessing of the input data or the neural network here we propose a method which is efficient in computation to avoid some local minima

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

series expansion studies of directed percolation estimates of the correlation length exponents series expansion methods have been used to estimate the longitudinal and transverse correlation length exponents for directed bond percolation on the square lattice the central estimate for the ratio of these exponents phi 1 579 or 0 006 is in good agreement with the value obtained by the phenomenological scaling renormalisation group method

percolation properties of a three dimensional random resistor diode network studies percolation in a random resistor diode network on the simple cubic lattice here the occupied bonds joining nearest neighbour sites may be either resistor like transmitting information or connectivity in either direction along a bond or diode like transmitting in only one direction they consider a model in which there exist both positive diodes which transmit in the x y or z directions and negative diodes which transmit oppositely they apply positive space renormalisation group methods to map out the phase diagram and calculate the exponents associated with the phase transitions of this system they find three novel types of transitions due to the presence of the diodes with only one species of diodes and vacancies unoccupied bonds present a directed threshold occurs at a critical concentration of diodes here an infinite cluster forms which transmits only in the direction of the average diode polarisation a reverse threshold occurs when resistors and only one species of diodes exist at this point the resistors mediate information flow in the infinite cluster opposite to the diode polarisation finally with all bond elements present a mixed threshold occurs

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

yang lee edge singularity in the one dimensional long range ising model the finite range scaling method has been used to study the yang lee edge singularity problem for the one dimensional ferromagnetic ising model with long range interactions decaying with distance r as r d sigma the phase diagram and the critical exponent nu have been calculated for different values of sigma by using finite ranges l or 9 the critical value sigma c dividing the long range from the short range critical regime has been discussed

sufficient uniqueness conditions for the solution of the time harmonic maxwell s equations associated with surface impedance boundary conditions we consider the time harmonic maxwell s equations for the scattering or radiating problem from a 3 d object that is either a metallic surface coated with material layers mcs or a dichroic structure ds made up of multiple frequency selective surfaces fss embedded in materials low or high order impedance boundary conditions ibc are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation an ibc links the tangential components of the electric field to those of the magnetic field on the outer surface of the mcs or on the fsss and avoids the solution of maxwell s equations inside the inhomogeneous domain for a mcs or for a ds the meshing of the numerous unit cells in a fss sufficient uniqueness conditions suc are established for the solutions of maxwell s equations associated with these ibcs the performances of which when constrained by the corresponding sucs are numerically evaluated for an infinite or finite planar structure

on a natural homotopy between linear and nonlinear single layer networks in this paper we formulate a homotopy approach for solving for the weights of a network by smoothly transforming a linear single layer network into a nonlinear perceptron network while other researchers have reported potentially useful numerical results based on heuristics related to this approach the work presented here provides the first rigorous exposition of the deformation process results include a complete description of how the weights relate to the data space a proof of the global convergence and validity of the method and a rigorous formulation of the generalized orthogonality theorem to provide a geometric perspective of the solution process this geometric interpretation clarifies conditions resulting in the appearance of local minima and infinite weights in network optimization procedures and the similarities of and differences between optimizing the weights in a nonlinear network and optimizing the weights in a linear network the results provide a strong theoretical foundation for quantifying performance bounds on finite neural networks and for constructing globally convergent optimization approaches on finite data sets

implementation of high order impedance boundary conditions in some integral equation formulations the scattering problem from an object coated with materials is solved in the frequency domain the coating is modeled by an impedance boundary condition ibc implemented in an integral equation ie prescribed on its outer surface the leontovich ibc ibc0 is local and constitutes a poor approximation for low index coatings a possible remedy is to employ higher order ibcs these as well as the ibc0 are implemented in two ie formulations 1 cfie with the electric current as sole unknown and 2 efie mfie where the unknowns are both the electric and magnetic currents the problems raised by the discretization of the surface div and surface curl operators involved in the ie and ibc formulations are solved by employing a simple and computationally cheap technique the accuracy of which is numerically assessed the performances of the various ibc and ie formulations are evaluated by calculating the radar cross section rcs of some axisymmetric objects

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

one way domain decomposition method with exact radiation condition and fast gmres solver for the solution of maxwell s equations for the solution of the time harmonic electromagnetic scattering problem by inhomogeneous 3 d objects a one way domain decomposition method ddm is considered the computational domain is partitioned into concentric subdomains on the interfaces of which robin type transmission conditions tcs are prescribed an integral representation of the electromagnetic fields on the outer boundary constitutes an exact radiation condition the global system obtained after discretization of the finite element fe formulations is solved via a krylov subspace iterative method gmres it is preconditioned in such a way that essentially only the solution of the fe subsystems in each subdomain is required this is made possible by a computationally cheap h curl h div transformation performed on the interfaces that separate the two outermost subdomains the eigenvalues of the preconditioned matrix of the system are bounded by two and optimized values of the coefficients involved in the local tcs on the interfaces are determined so as to maximize the minimum eigenvalue numerical experiments are presented that illustrate the numerical accuracy of this technique its fast convergence and legitimate the choices made for the optimized coefficients

a homotopy approach to improving pem identification of armax models homotopy methods are applied to the non convex optimization problem that arises in prediction error estimation using armax model structures a concrete algorithm is proposed and its performance is studied by monte carlo simulation

the resistive susceptibility of random diode insulator networks the authors consider regular crystal lattices in which the bonds are either diodes probability p or insulators the mean number of backbone bonds lbb p when averaged over lattice points is found to be related to the mean size of clusters s p by lbb p zps p where z is the number of bonds directed away from any lattice site thus lbb diverges at pc with the mean size exponent gamma the resistive susceptibility chi r p of harris and fisch 1977 1978 is expanded graphically as a power series in p and it is found that term by term chi r p psi r p s p 2 where psi r p is obtained from chi r p by ignoring contributions from nodal graphs the above results are valid for bond and site dilution in any dimension for bond dilution on the square lattice the authors have determined psi r p to p17 and pade analysis of the resulting series for chi r p shows that it diverges with exponent gamma r 3 654 or 0 017 using a scaling relation the exponent t for the conductivity of the infinite cluster is estimated to be t 0 75 or 0 02

estimations for asymptotic series using a modified romberg algorithm i finite size scaling calculations a new numerical method for estimating asymptotic properties is proposed a generalisation of the romberg algorithm leads to a systematic elimination of higher order analytical terms in finite size calculations and determines both the asymptotic value and the leading convergence exponent in a self consistent way

mean field renormalisation group transformations for geometric phase transitions it is shown that the mean field renormalisation group technique of indekeu et al 1982 may be applied directly to the self avoiding walk saw and percolation problems without reference to the n to 0 limit of interacting spin systems numerical results for the saw and bond and site percolation both directed and undirected problems on the square lattice obtained by using a variety of small cells are reported an isotropic transformation for the percolation problems on the d dimensional hypercubic lattice is also discussed

directed compact percolation cluster size and hyperscaling exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice the corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained the work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice a duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve the moments of both distributions have critical exponents which satisfy a constant gap hypothesis with gap exponents nu 2 and delta 3 corresponding to the scaling length and scaling size respectively the usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by delta nu d nu perpendicular to where nu perpendicular to is the exponent corresponding to the cluster width and d is the number of transverse dimensions 1 for the square lattice

conductivity exponents from the analysis of series expansions for random resistor networks there has been considerable controversy in recent years over the value of the conductivity exponent t this exponent can be deduced from series expansions via the scaling relations t zeta d 2 nu where zeta is deduced from differences between the exponents of the resistive gamma r percolative gamma p and conductive gamma c susceptibility the author finds that allowance for non analytic confluent corrections to scaling and the use of recent pc estimates leads to estimates for gamma r gamma p and gamma c that are somewhat different to those of fisch and harris 1978 however the differences between these exponents do not change significantly moreover the change in accepted estimates of nu in the last five years cancels some of this remaining discrepancy and she concludes using the relation zeta gamma r gamma p that t 1 31 d 2 t 2 04 d 3 t 2 39 d 4 t 2 72 d 5 with an error of about or 0 10 in each case the d 2 estimate is in significantly better agreement with those of other methods than that of fisch and harris

finite size effects on the critical structure factor of the two dimensional ising model explicit calculations of the critical structure factor of the planar ising model on square lattices with a variety of boundaries are compared with the results of conformal invariance monte carlo methods are used for square circular and 2 1 rectangular geometries with free boundary conditions transfer matrices are used for an infinite strip with periodic boundary conditions good agreement is found for wavevectors less than a quarter of a reciprocal lattice spacing

partition function zeros and leading order scaling correction of the 3d ising model from multicanonical simulations the density of states for the three dimensional ising model is calculated with high precision by means of multicanonical simulations this allows us to estimate the leading partition function zeros for lattice sizes up to l 32 we have evaluated the critical exponent and the correction to scaling through an analysis of a multi parameter fit and of the bulirsch stoer bst extrapolation algorithm the performance of the bst algorithm is also explored in case of the 2d ising model where the exact partition function zeros are known

diffraction at a curved grating tm and te cases homogenization a hierarchy of effective boundary conditions is derived for a helmholtz equation with dirichlet or neumann boundary conditions these generalized impedance boundary conditions approximate the effect of thin dielectric layers with perfectly conducting inclusions covering an object well posedness of the problems with these new boundary conditions is proved and optimal error estimates are given

magnetization at corners in two dimensional ising models we investigate the corner spin magnetization of two dimensional ferromagnetic ising models in various wedge geometries results are obtained for triangular and square lattices by numerical studies on finite wedges using the star triangle transformation as well as analytic calculations using corner transfer matrices and a fermionic representation of the row to row transfer matrix the corner magnetizations vanish at the bulk critical temperature tc with an exponent c differing from the bulk exponent for isotropic systems with free edges we find that c is given simply by c 2 where is the angle at the corner for apex magnetizations of conical lattices we obtain the strikingly similar result a 4 these formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle eff determined by the relative strengths of the interactions

approximate impedance for time harmonic maxwell s equations in a non planar domain with contrasted multi thin layers abstract the aim of this paper is to give asymptotic models for the impedance of contrasted multi thin layers for the harmonic maxwell s equations we start from a transmission problem which describes the scattering of electromagnetic waves by an obstacle covered with a thin coating superposition of different thin layers of dielectric materials we show how to model the effect of the thin coating by an impedance boundary condition on the boundary of the propagation domain to this end we use a technique of abstract differential equations

a transfer matrix study of directed lattice animals and directed percolation on a square lattice we studied the large scale properties of directed lattice animals and directed percolation on a square lattice using a transfer matrix approach on strips of finite widths we generated relatively long sequences of estimates for effective values of critical fugacity percolation threshold and correlation length critical exponents we applied two different extrapolation methods to obtain estimates for infinite systems the precision of our final estimates is comparable to or better than the precision of the best currently available results

approximations of mean spherical type for lattice percolation models the authors develop a general class of approximations of mean spherical msa type as a method for studying lattice percolation problems they review the msa and test certain extensions of it on lattice spin models the relations between thermal spin models and percolation models are then reviewed in order to identify natural extensions of the msa to percolation models these extensions are used to treat both site and bond percolation models in one low density formulation of msa the threshold for bond percolation on a lattice is found to equal the value at the origin of the corresponding lattice green s function this formula gives accurate results for all lattices studied and in all space dimensions d or 3 an accurate treatment is also given of the general site bond problem the entire percolation locus for this problem agrees closely with the results of simulation they also introduce a new method for studying percolation transitions which is a hybrid of the kikuchi cluster approximation scheme and the msa the method is shown to give good values for percolation thresholds while preserving the valuable features of the standard msa in particular it provides a convenient means of computing the pair connectedness function they also explore extensions of their approximations to treat directed site and bond percolation

series and monte carlo studies of 2 and 3 dimensions for axial hyperscaling in directed percolation abstract the relationship between the exponent 0 characterizing the divergence at criticality of the cluster mass measured on the anisotropic axis of directed percolation and the susceptibility exponent while well established in two dimensions both by series and simulation work has been doubtful for a decade for three and higher dimensions in the present paper we furnish extensive simulations for the body centered cubic site problem that focus on that quantity and study further the ascending moments of the number of sites on the axis these numerical results are highly compatible with the scaling relation 0 d where d 2 is the transverse dimensionality they also indicate the existence of a fixed gap exponent for the size moments of this quantity extensive series expansions in two dimensions are in acceptable agreement with the same picture with d 1 above while direct determination of 0 is still not satisfactory for some of the highly connected three dimensional lattices we have adopted the existence of a fixed gap exponent is confirmed in fairly reasonable agreement with the concurrent evidence of the bcc monte carlo studies

a long range domany kinzel model of directed percolation a long range domany kinzel 1981 model proposed by wu and stanley 1982 is solved using random walk formulations in this model for every site i j in a two dimensional lattice there is a directed bond present from site i j to i 1 j with probability one there are also m 1 directed bonds present from i j to i k j 1 k 1 0 1 2 3 m 1 with respective probabilities pk 1 where m is any positive integer an exact expression is obtained to determine the critical percolation angle theta c for any distribution of pn the system percolates in the region theta c theta or 0 with probability one and zero outside this region where theta is the angle measured from the x axis if the authors let m go to infinity and pn varies as p1n s they find that when s or approximately 2 when m is large but finite theta c is also obtained for the following two distributions a pn a a n with a 0 b pn beta m and beta 0

conformal invariance and correction to finite size scaling applications to the three state potts model corrections to finite size scaling are determined numerically for several levels of the three state potts quantum chain with various boundary conditions it is found that the leading correction term behaves like wo in the case of periodic and twisted boundary conditions the coefficients of the n o term are determined by the three point correlation functions of the conformal theory

on the uniqueness of weights in single layer perceptrons in this paper the geometric formulation of the single layer perceptron weight optimization problem previously described by coetzee et al 1993 1996 is combined with results from other researchers on nonconvex set projections to describe sufficient conditions for uniqueness of weight solutions it is shown that the perceptron data surface is pseudoconvex and has infinite folding allowing for the specification of a region of desired vectors having unique projections purely in terms of the local curvature of the data surface no information is therefore required regarding the global curvature or size of the data surface these results in principle allow for a posteriori evaluation of whether a weight solution is unique or globally optimal and for a priori scaling of desired vector values to ensure uniqueness through analysis of the input data the practical applicability of these results from a numerical perspective is evaluated on some carefully chosen examples

two dimensional lattice tree exponents and amplitudes simulation algorithms versus series we use a local monte carlo algorithm to simulate lattice trees in two dimensions for the site and bond problem we investigate the properties of radius of gyration perimeter to site ratio and vertex degree in a tree adding some new results in the site problem compare our results on their noncritical properties with those obtained from earlier reversible and slightly nonreversible algorithms and combine our determinations with new exact series expansion data on the controversy surrounding the possible lack of universality for the first confluent singularity for the gyration radius we feel that conclusions must be guarded

telescoping recursive representations and estimation of gauss markov random fields we present telescoping recursive representations for both continuous and discrete indexed noncausal gauss markov random fields our recursions start at the boundary a hypersurface in and telescope inwards for example for images the telescoping representation reduce recursions from to i e to recursions on a single dimension under appropriate conditions the recursions for the random field are linear stochastic differential difference equations driven by white noise for which we derive recursive estimation algorithms that extend standard algorithms like the kalman bucy filter and the rauch tung striebel smoother to noncausal markov random fields

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

critical exponents and corrections to scaling for bond trees in two dimensions the author has analysed the newly obtained series of the radius of gyration rn and the number of clusters nn for n bond trees i e branch polymers without loops on the square n or 14 and triangular n or 11 lattices to estimate the critical parameters respective estimates of the exponents nu and theta for rn and nn are consistent with the corresponding values for lattice animals while the correction to scaling exponent delta 1 is inconsistent with the animal value in addition delta 1 has different values for rn delta 1 0 635 and nn delta 1 1 3 for bond trees he has also estimated an exponent delta characterising the density distribution delta 2 69 for the triangular lattice

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

mean field theory of percolation with application to surface effects a probabilistic formulation of mean field theory for percolation in a general medium is given the theory covers the whole density range and above the critical density provides the necessary generalisation of the random walk approximation application to surface percolation enables the calculation of scaling functions and critical exponents for the ordinary surface special and extraordinary transitions considered by previous authors to the context of ferromagnetism

real space renormalisation theory of self avoiding walks on a manhattan lattice the real space renormalisation group theory is applied to the self avoiding walk saw problem on a two dimensional oriented lattice known as the manhattan lattice a finite lattice renormalisation transformation is used to calculate the connectivity constant mu and the size exponent nu a more general saw model is defined that contains the oriented and nonoriented lattice problems as special cases a renormalisation group analysis of this model provides evidence that the saw on the manhattan lattice is the same universality class as the ordinary nonoriented saw problem

finite size corrections and numerical calculations for long spin 1 2 heisenberg chains in the critical region leading and next to leading order finite size corrections to the ground and first excited states are calculated for the spin 1 2 anisotropic heisenberg model in the critical region the analytic results are compared to numerical data obtained for chains up to a length of n 1024 it is found that near the isotropic point the asymptotic region where the results obtained for n to infinity are applicable sets in at very large n values and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms even at large n 100 chain lengths

finite size scaling for directed bond percolation with and without cycles on a triangular lattice the authors apply finite size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic directed cyclic and undirected bond percolation on a triangular lattice the results are in good agreement with known estimates and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases as well as the distinction between locally directed and fully isotropic problems in addition the authors draw tentative comments on the extension to directed problems of the relation a pi eta where a is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point known to hold for various isotropic systems

asymmetric fluid criticality ii finite size scaling for simulations the vapor liquid critical behavior of intrinsically asymmetric fluids is studied in finite systems of linear dimensions l focusing on periodic boundary conditions as appropriate for simulations the recently propounded complete thermodynamic stackrel ensuremath rightarrow l ensuremath infty scaling theory incorporating pressure mixing in the scaling fields as well as corrections to scaling phys rev e 67 061506 2003 is extended to finite l initially in a grand canonical representation the theory allows for a yang yang anomaly in which when stackrel ensuremath rightarrow l ensuremath infty the second temperature derivative d 2 ensuremath mu ensuremath sigma dt 2 of the chemical potential along the phase boundary ensuremath mu ensuremath sigma t diverges when stackrel ensuremath rightarrow t t mathrm c ensuremath the finite size behavior of various special critical loci in the temperature density or t ensuremath rho plane in particular the k inflection susceptibility loci and the q maximal loci derived from q l t ensuremath rho l ensuremath equiv m 2 l 2 m 4 l where m ensuremath equiv ensuremath rho ensuremath ensuremath rho l is carefully elucidated and shown to be of value in estimating t mathrm c and ensuremath rho mathrm c concrete illustrations are presented for the hard core square well fluid and for the restricted primitive model electrolyte including an estimate of the correlation exponent ensuremath nu that confirms ising type character the treatment is extended to the canonical representation where further complications appear

directed percolation a monte carlo approach describes a monte carlo study of two dimensional directed site percolation the authors show that the anisotropic behaviour resulting from the imposition of a directional bias must be accounted for in the finite size scaling analysis of the monte carlo data if reliable results are to be obtained using suitably modified finite size scaling they obtain estimates for the percolation threshold pc and the exponents nu p nu r and beta which are in good agreement with results obtained by other methods

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

on the number of lattice animals embeddable in the square lattice enumeration of lattice animals embeddable in a square lattice has been extended to 24 cell animals by redelmeier 1981 it is shown that the number of animals per site an approximately 0 317 4 0626 nn 1 exp 0 465 n 0 87 to a high degree of accuracy

magnetisations from finite size scaling a method of obtaining the spontaneous magnetisation from finite lattice matrix elements of the magnetic field operator due to yang 1952 and uzelac 1980 is discussed the method is demonstrated for the case of the ising model in 1 1 dimensions and is shown to provide smooth and rapidly convergent finite lattice sequences applied to the case of the three state potts z3 model in 1 1 dimensions the method yields an estimate beta 0 111 09 or 0 000 05 for the critical exponent this confirms alexander s conjecture of universality with the hard hexagonal model

finite size corrections for ground states of the xxz heisenberg chain the method of de vega and woynarovitch 1985 is used to calculate finite size corrections to the ground state energy in different sectors for the xxz heisenberg chain finite size scaling amplitudes and correction to scaling exponents in the critical region are derived using conformal invariance a scaling dimension x pi gamma 2 pi is extracted corresponding to the electric field operator in the 8 vertex model this confirms a conjecture of baxter and kelland 1974 finite size scaling properties near the kosterlitz thouless critical point delta 1 are discussed

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

numerical study of conformal invariance in rsos models the author studies conformal invariance in the rsos models of andrews et al 1985 at both the ferromagnetic and antiferromagnetic transitions using finite size scaling of the transfer matrix the author confirms the conjecture of huse 1984 concerning the conformal classification of the ferromagnetic transitions and gives evidence that the antiferromagnetic transitions lie in the universality class of zn models in addition he shows that conformal results can be applied to anisotropic systems if logarithmic corrections are taken into account

series expansion analysis of directed site bond percolation on the square and simple cubic lattices analysis of low density series for site bond percolation on the directed square sq and simple cubic sc lattice and related series for bond percolation on the honeycomb h and diamond di lattices is found to be consistent with pc sq site bond pc h bond 0 8228 or 0 0002 pc sc site bond pc di bond 0 637 or 0 002 and previous estimates of gamma nu sub and nu perpendicular to analysis of the square lattice series supports the validity of the scaling relation gamma 0 gamma d 1 nu perpendicular to for the two dimensional lattices site percolation on the honeycomb and diamond lattices is also considered

partially directed site percolation on the square and triangular lattices using the phenomenological renormalisation pr method the authors obtain the critical probability for partially directed site percolation on the square and triangular lattices pc 0 6317 or 0 006 and pc 0 5468 or 0 005 respectively they discuss the corrections to scaling in the calculation of the anisotropy exponent theta nu sub nu perpendicular to and the critical probability and show that it is necessary to work with at least two correction terms this is in contrast to the usual analysis where only one correction exponent is used in pr calculations

phenomenological renormalisation of monte carlo data for percolation the accuracy of a phenomenological renormalisation which is based on monte carlo data is tested by investigating site percolation in a simple cubic lattice the method appears to be very accurate and can yield precise estimates of the quantities of interest with small to moderate lattice sizes the site percolation threshold of the lattice is predicted to be 0 3115 or 0 0005 if nu is the exponent of correlation length the author finds beta nu approximately 0 48 or 0 01 for the exponent of percolation probability and beta b nu approximately 1 1 or 0 03 for the backbone exponent in agreement with most accurate data currently available an analysis of the data at the percolation threshold based on the standard finite size scaling method also supports the results

one way domain decomposition method with adaptive absorbing boundary condition for the solution of maxwell s equations for the solution of the time harmonic electromagnetic scattering problem by inhomogeneous 3 d objects a previously published one way domain decomposition method ddm is considered the computational domain is partitioned into concentric subdomains on the interfaces of which robin type transmission conditions tcs are prescribed with an integral equation ie on the outer boundary of the computational domain ddm ie on account of the large computing time required by the solution of the isolated ie system in this paper the ie is replaced by the integral representations irs of the fields that requires only a few matrix vector products adaptive absorbing boundary condition aabc the irs necessitate the calculation of the electric and magnetic currents on some inner surface s that is chosen to be the interface between the last two subdomains taking advantage of the tcs the unknown current on s here the magnetic current is obtained via a change of bases h rot to h div that allows the accurate computation of the ir integrals involving the surface divergence terms and permits the separate solution of the fe systems in the last two subdomains the matrix vector products in the aabc are performed only once per ddm iteration numerical results are presented that illustrate the accuracy of the ddm aabc and its superiority in terms of computing time over the ddm ie also some indications are given on how to estimate numerically the convergence rate of the ddm aabc

a new convex objective function for the supervised learning of single layer neural networks this paper proposes a novel supervised learning method for single layer feedforward neural networks this approach uses an alternative objective function to that based on the mse which measures the errors before the neuron s nonlinear activation functions instead of after them in this case the solution can be easily obtained solving systems of linear equations i e requiring much less computational power than the one associated with the regular methods a theoretical study is included to proof the approximated equivalence between the global optimum of the objective function based on the regular mse criterion and the one of the proposed alternative mse function furthermore it is shown that the presented method has the capability of allowing incremental and distributed learning an exhaustive experimental study is also presented to verify the soundness and efficiency of the method this study contains 10 classification and 16 regression problems in addition a comparison with other high performance learning algorithms shows that the proposed method exhibits in average the highest performance and low demanding computational requirements

finite lattice calculation of the quantum mechanical n vector model the truncated quantum mechanical n vector model in one dimension is studied by means of the phenomenological renormalisation group prg and the block renormalisation group brg for general values of n particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis accurate estimates of the critical properties can be made for 0 or approximately n or approximately 2 the two methods used are critically compared

the yang lee edge singularity by the phenomenological renormalisation group the phenomenological renormalisation group is applied to the 1d transverse ising model with longitudinal imaginary field in order to investigate the yang lee edge singularity 1952 of the equivalent 2d ising model the method is extended to calculate directly the exponent sigma of the magnetisation