the 2d transverse ising model at t 0 a finite size rescaling transformation approach the critical behaviour of the spin 4 ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite size rescaling transformation this technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size in a generalisation of the so called phenomenological renormalisation group the critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy but calculations on larger clusters are limited by computer time and storage

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

non reciprocity in nonlinear elastodynamics abstract reciprocity is a fundamental property of linear time invariant lti acoustic waveguides governed by self adjoint operators with symmetric green s functions the break of reciprocity in lti elastodynamics is only possible through the break of time reversal symmetry on the micro level and this can be achieved by imposing external biases adding nonlinearities or allowing for time varying system properties we present a volterra series based asymptotic analysis for studying spatial non reciprocity in a class of one dimensional 1d time invariant elastic systems with weak stiffness nonlinearities we show that nonlinearity is neither necessary nor sufficient for breaking reciprocity in this class of systems rather it depends on the boundary conditions the symmetries of the governing linear and nonlinear operators and the choice of the spatial points where the non reciprocity criterion is tested extension of the analysis to higher dimensions and time varying systems is straightforward from a mathematical point of view but not in terms of new non reciprocal physical phenomena whereas the connection of non reciprocity and time irreversibility can be studied as well finally we show that suitably defined non reciprocity measures enable optimization and can provide physical understanding of the nonlinear effects in the dynamics enabling one to establish regimes of maximum nonlinearity we highlight the theoretical developments by means of a numerical example

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

some analytical consequences of the inverse relation for the potts model the authors present some analytical results for the potts model using the symmetry group acting on its parameters generated by the inverse relation and other symmetries of this model in particular they find the critical manifolds and study the relationship between this group and the lee yang singularities in the complex plane for those cases where the symmetry group is finite they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials

finite size scaling analysis of the dilute baxter wu model the finite size scaling method is used to study the critical properties of the spin 1 baxter wu model as function of the fugacity z of the vacant spin zero sites for z 0 the thermal exponent converges very quickly to the exact value yt 3 2 for z 0 yt monotonically increases beyond the value 2 this increase is interpreted as indicating a first order transition out of several possible renormalisation group flows the results seem to favour the one in which the critical baxter wu hamiltonian flows to the fixed point of the 4 state potts model with the amplitude of the marginal operator equal to zero

renormalisation group approach to delaunay percolation networks with topological disorder a renormalisation group approach is developed for delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space considering various renormalisation transformations for two and three dimensional delaunay lattices the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent the fixed points for the two dimensional bond lattice and the three dimensional site lattice are 0 3229 and 0 1443 respectively which are in excellent agreement with results of monte carlo simulations the fixed point for the two dimensional site lattice gives the value 1 2 for the critical percolation probability which is equal to the known result for a fully triangulated lattice

on separable schrodinger equations we classify 1 3 dimensional schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables as a result we get 11 classes of the vector potentials of the electromagnetic field a t x a0 t x a t x providing separability of the corresponding schrodinger equations it is established in particular that the necessary condition for the schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables next we prove that any schrodinger equation admitting variable separation into second order ordinary differential equations can be reduced to one of the 11 separable schrodinger equations mentioned above and carry out variable separation in the latter furthermore we apply the results obtained for separating variables in the hamilton jacobi equation

control of error in the homotopy analysis of nonlinear klein gordon initial value problems in the present paper we discuss the application of homotopy analysis to general nonlinear klein gordon type equations we first outline the method for general forms of the nonlinearity as well as for general functional forms of the initial conditions in particular we discuss a method of controlling the residual error in approximate solutions which may be found via homotopy analysis through adequate selection of the convergence control parameter with the general problem outlined we apply the method to various equations including the quasilinear cubic klein gordon equation the modified liouville equation the sinh gordon equation and the tanh gordon equation for each of these equations and related initial data we obtain residual error minimizing solutions which demonstrate the qualitative behavior of the true solutions in each case

correlated percolation and a sublattice dilute potts model at t 0 abstract it is shown that a sublattice dilute q state potts model at t 0 is equivalent to a correlated percolation which favours subgraphs with larger numbers of disconnected finite clusters the syozi model at t is equivalent to the above model with q 2 in the same limit and thus corresponds to the same correlated percolation a table is listed which reflects the influence of correlation on critical probabilities for such correlated percolation processes on some lattices

position space renormalisation group for isolated polymer chains we develop a cell position space renormalisation group psrg with which we study the scaling properties of isolated polymer chains we model a chain by a self avoiding walk constrained to a lattice for rescaling factors b s 6 we calculate recursion relations analytically on the square lattice with several different choices for the psrg weight function we also calculate implicit cell to cell transformations in which a cell of size b is rescaled to a cell of size b the results of these psrgs improve both as b increases and as b b 1 in addition we construct a true infinitesimal psrg transformation which appears to become exact as the dimensionality d approaches 1 we obtain the closed form expression for the correlation length exponent u d l d in d the flory formula deviates from this already at first order in d 1 we also develop a constant fugacity monte carlo method which enables us to simu late in an unbiased way within the grand canonical ensemble chains of up to lo3 bonds with this method we extend the psrg to larger cells b s 150 on the square lattice our numerical method provides high statistical accuracy for all cell sizes however in the range of b we study the asymptotic behaviour of our results appears to depend on the choice of weight function one weight function provides smooth behaviour as a function of b and with it we extrapolate to find u 0 756 0 004 further work is required to resolve the apparent anomalies in the results based on the other weight functions

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

linear defects in two dimensional systems a finite size investigation finite size scaling theory is employed to study non universal and crossover critical behaviour in two dimensional systems with linear defects the finite size method is shown to reproduce very accurately the non universality as known to the present in the case of the ising model the calculations show that the non universal behaviour of all the models investigated in the ising universality class can be described approximately by a single universal curve

stability of density based clustering high density clusters can be characterized by the connected components of a level set l x p x of the underlying probability density function p generating the data at some appropriate level 0 the complete hierarchical clustering can be characterized by a cluster tree t l in this paper we study the behavior of a density level set estimate l and cluster tree estimate t based on a kernel density estimator with kernel bandwidth h we define two notions of instability to measure the variability of l and t as a function of h and investigate the theoretical properties of these instability measures

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

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

threshold and scaling in percolation with restricted valence monte carlo calculations have been carried out to study the problem of percolation with restricted valence on the square and simple cubic lattices the results are in full agreement with the expectations there is no transition when the restriction nu is equal to two while even for nu 3 the transition occurs and the correlation length exponent determined via finite size scaling is within the numerical accuracy the same as that in the unrestricted random percolation the maximum random occupancy as a function of the restriction is also determined

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

a characterization of linkage based hierarchical clustering the class of linkage based algorithms is perhaps the most popular class of hierarchical algorithms we identify two properties of hierarchical algorithms and prove that linkage based algorithms are the only ones that satisfy both of these properties our characterization clearly delineates the difference between linkage based algorithms and other hierarchical methods we formulate an intuitive notion of locality of a hierarchical algorithm that distinguishes between linkage based and global hierarchical algorithms like bisecting k means and prove that popular divisive hierarchical algorithms produce clusterings that cannot be produced by any linkage based algorithm

gromov hausdorff stability of linkage based hierarchical clustering methods abstract a hierarchical clustering method is stable if small perturbations on the data set produce small perturbations in the result these perturbations are measured using the gromov hausdorff metric we study the problem of stability on linkage based hierarchical clustering methods we obtain that under some basic conditions standard linkage based methods are semi stable we also prove that avoiding exotic examples introducing any unchaining condition in the algorithm always produces unstable methods

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

the packing of two species of polygons on the square lattice we decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons we end up with a 24 vertex model which is known in the literature as the fully packed double loop model fpl2 in the particular case in which the fugacities of the polygons are the same the model admits an exact solution the solution is obtained using coordinate bethe ansatz and provides a closed expression for the free energy in particular we find the free energy of the four colouring model and the double hamiltonian walk and recover the known entropy of the ice model when both fugacities are set equal to 2 the model undergoes an infinite order phase transition

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

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

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

analytic properties of the partition function for statistical mechanical models the author explains with examples the relationship between zeros of the partition function analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models it is shown how to write down a polynomial representation of the partition function for any model with a two sheet largest eigenvalue the staggered ice model representation of the q state potts model on a sequence of semi infinite strips is solved and it is shown that for 0 q 4 the potts model result is only projected out by special boundary conditions the non potts part of the result is obtained for a 4 infinity strip and gives baxter s antiferromagnetic critical curve 1982

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

functionally separable solutions to nonlinear wave equations by group foliation method abstract we apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force u t t a x d u u x x b x q u a x 0 a classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

finite lattice methods in quantum hamiltonian field theory i o 2 and o 3 heisenberg models for pt i see ibid vol 14 p 241 57 1981 two efficient methods for finding the low lying states of hamiltonians on finite lattices are described the first involves constructing a finite representation of the hamiltonian using strong coupling eigenstates while the second is based on the lanczos recursion method the methods are used to determine the mass gap of the o 2 and o 3 heisenberg hamiltonians in 1 1 dimensions for a sequence of finite chains the critical behaviour of the infinite chain is then analysed by extrapolating the finite lattice estimates using finite size scaling a remarkably sensitive test is developed for the presence of a phase transition for the o 2 model data this test yields strong evidence for a phase transition with the weak coupling phase massless while in the o 3 case the test supports although more weakly the absence of any transition

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

phase equilibria and the scaling transformation it is argued that the scaling transformation which is currently being used to obtain numerical estimates of critical exponents cannot safely be employed without prior knowledge that the points in question are second order transition points the transformation when defined on finite or semi infinite systems will not distinguish between order disorder critical points and any other point on the coexistence surface this in turn means that the full phase diagram of the equilibrium surface can be found using the scaling transformation the three state potts model is used to illustrate this

separation of variables for scalar evolution equations in one space dimension the flow of an autonomous kth order evolution equation in one space dimension is generated by a kth order ordinary differential operator on the space of fields the characteristic fields are the solutions of the characteristic equation of the operator a solution of the evolution equation is separable if and only if it is a curve in the k c 1 parameter space of characteristic fields the non stationary characteristic fields with separable evolution are those which remain characteristic under small dilation every characteristic field has local separable evolution if and only if the characteristic equation is infinitesimally dilation invariant this is the case when the evolutionary generator is an infinitesimal symmetry of the characteristic equation so that its flow stabilizes the space of characteristic fields the evolution equation is then locally homogeneous if and only if the characteristic value is an invariant of the restricted flow which is a non generic property the separable kth order evolution equations are parametrized by the kth order invariants of the locally homogeneous k c 1 th order ordinary differential equations

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

finite size scaling in the 2 1 d ising model hamiltonian eigenvalues are calculated for the 2 1 d ising model on lattices up to 5 5 sites finite size scaling techniques are used to estimate the critical coupling xc 0 329 or 0 001 and index v 0 635 or 0 005 the prospects are discussed for further applications of these methods in three and four dimensions

asymptotic negative type properties of finite ultrametric spaces abstract negative type inequalities arise in the study of embedding properties of metric spaces but they often reduce to intractable combinatorial problems in this paper we study more quantitative versions of these inequalities involving the so called p negative type gap in particular we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining let x d be a given finite ultrametric space with minimum non zero distance then the p negative type gap x p of x d is positive for all p 0 in this paper we compute the value of the limit x lim p x p p it turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula this formula allows us to characterize when the ratio x p p is a constant independent of p the determination of x also leads to new asymptotically sharp families of enhanced p negative type inequalities for x d indeed suppose that g 0 x then for all sufficiently large p the inequality g p 2 k 1 n k 2 j i 1 n d z j z i p j i 0 holds for each finite subset z 1 z n x and each scalar n tuple 1 n r n that satisfies 1 n 0 notably these results do not extend to general finite metric spaces

nonlinear evolution equations admitting multilinear variable separable solutions based on the hirota s bilinear transformation method it is shown that a family of nonlinear partial differential equations admit multilinear variable separation solutions this extends the class of nonlinear partial differential equations which admit the multilinear variable separation solutions

ising like critical phenomena in two dimensional percolation numerical evidence a recent analytic theory of two dimensional isotropic percolation indicates that the singular behaviour near pc is identical to that appropriate for the associated dilute ising critical point a specific prediction supported by recent series expansion studies is that the mean number of clusters k p presents a singularity of the form k p approximately mod p pc mod 2ln mod ln mod p pc rather than the accepted form k p approximately mod p pc mod 2 alpha with alpha 2 3 a novel numerical and renormalisation group finite size scaling analysis of the nature of the singularity in k p is presented in support of the new theory which implies the absence of a separate universality class for two dimensional percolation processes this study is consistent with the effective values of the exponents alpha 0 and nu 1

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

polymer models and generalized potts kasteleyn models the well established relation between potts models withv spin values and random cluster models with intracluster bonding favored over intercluster bonding by a factorv is explored but with the random cluster model replaced by a much generalized polymer model implying a corresponding generalization of the potts model the analysis is carried out in terms a given defined functionr an entropy free energy density for the polymer model in the casev 1 expressed as a function of the density of units the aim of the analysis is to determine the analogrv ofr for general nonnegativev in terms ofr and thence to determine the critical value of density vg at which gelation occurs this critical value is independent ofv up to a valuevp the potts critical value what is principally required ofr is that it should show a certain given concave convex behavior although differentiability and another regularizing condition are required for complete conclusions under these conditions the unique evaluation ofrv in terms ofr is given in a form known to hold for integralv but not previously extended the analysis is carried out in terms of the legendre transforms of these functions in terms of which the phenomena of criticality gelation and potts criticality appear very transparently andvp is easily determined the value ofvp is 2 under mild conditions onr special interest attaches to the functionr0 which is shown to be the greatest concave minorant ofr the naturalness of the approach is demonstrated by explicit treatment of the first shell model

finite size scaling and crossover phenomena the xy chain in a transverse field at zero temperature the scaling theory of finite size effects in the limiting bulk behaviour is extended to treat crossover phenomena the method is used to study quantum critical phenomena in the xy chain in a transverse field at zero temperature through the scaling of the longitudinal susceptibilities and of the energy gaps between the ground and two first excited states as expected no abrupt change in critical exponents is observed for small anisotropy because of the finiteness of the system but the limiting isotropic and anisotropic regions display quite distinct critical behaviour in good agreement with known results another interesting result obtained is that the two energy gaps examined vanish at the critical line with the same critical exponent

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

persistent brain network homology from the perspective of dendrogram the brain network is usually constructed by estimating the connectivity matrix and thresholding it at an arbitrary level the problem with this standard method is that we do not have any generally accepted criteria for determining a proper threshold thus we propose a novel multiscale framework that models all brain networks generated over every possible threshold our approach is based on persistent homology and its various representations such as the rips filtration barcodes and dendrograms this new persistent homological framework enables us to quantify various persistent topological features at different scales in a coherent manner the barcode is used to quantify and visualize the evolutionary changes of topological features such as the betti numbers over different scales by incorporating additional geometric information to the barcode we obtain a single linkage dendrogram that shows the overall evolution of the network the difference between the two networks is then measured by the gromov hausdorff distance over the dendrograms as an illustration we modeled and differentiated the fdg pet based functional brain networks of 24 attention deficit hyperactivity disorder children 26 autism spectrum disorder children and 11 pediatric control subjects

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

approximation schemes for finite lattices two algorithms are formulated for systematically truncating the set of basis states required in lattice hamiltonian calculations thus allowing larger lattice abort sizes to be treated they are tested for the case of the 1 1 dimensional ising model it is shown that smoothly convergent finite lattice sequences are obtained which can successfully the extrapolated to the bulk limit the authors point out the relevance of the technique for lattice calculations in higher dimensions

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

statistical tests for large tree structured data we develop a general statistical framework for the analysis and inference of large tree structured data with a focus on developing asymptotic goodness of fit tests we first propose a consistent statistical model for binary trees from which we develop a class of invariant tests using the model for binary trees we then construct tests for general trees by using the distributional properties of the continuum random tree which arises as the invariant limit for a broad class of models for tree structured data based on conditioned galton watson processes the test statistics for the goodness of fit tests are simple to compute and are asymptotically distributed as 2 and f random variables we illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer we use a novel approach with tree based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

on the new approach to variable separation in the time dependent schrodinger equation with two space dimensions we suggest an effective approach to separation of variables in the schrodinger equation with two space variables using it we classify inequivalent potentials v x1 x2 such that the corresponding schrodinger equations admit separation of variables besides that we carry out separation of variables in the schrodinger equation with the anisotropic harmonic oscillator potential v k1x21 k2x22 and obtain a complete list of coordinate systems providing its separability most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free schrodinger equation v 0

multiple phase coexistence and the scaling transformation it is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible the method is illustrated by an application to the three four and five state potts models and even in its lowest order of approximation is able to distinguish the difference in the zero field critical behaviour between the five state and the three and four state models

consistency of single linkage for high density clusters abstract high density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form x f x c single linkage clustering is evaluated for consistency in detecting such high density clusters other standard hierarchical techniques such as average and complete linkage are hopelessly inconsistent for these clusters the asymptotic consistency of single linkage closely depends on the percolation problem of broadbent and hammersley if small spheres are removed at random from a solid at which density of spheres will water begin to flow through the solid if there is a single critical density such that no flow takes place below a certain density and flow occurs through a single connected set above that density then single linkage is consistent in separating high density clusters by disjoint single linkage clusters that include a positive fraction of sample points in the respective clusters and pass arbitrarily close to all points in the respective clusters

separation of variables in 1 2 dimensional schrodinger equations using our classification of separable schrodinger equations with two space dimensions published in j math phys 36 5506 1995 we give an exhaustive description of the coordinate systems providing their separability furthermore we apply these results to separate variables in the heat hamilton jacobi and fokker planck equations

hierarchical clustering given confidence intervals of metric distances this paper considers metric the exact dissimilarities between pairs of points are not unknown but known to belong to some interval the goal is to study methods for the determination of hierarchical clusters i e a family of nested partitions indexed by a resolution parameter induced from the given distance intervals of the dissimilarities our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that satisfy the axioms of value nodes in a metric space with two nodes are clustered together at the convex combination of the upper and lower bounds determined by a parameter and transformation when both distance bounds are reduced the output may become more clustered but not less two admissible methods are constructed and are shown to provide universal bounds in the space of admissible methods practical implications are explored by clustering moving points via snapshots and by clustering coauthorship networks representing collaboration between researchers from different communities the proposed clustering methods succeed in identifying underlying hierarchical clustering structures via the maximum and minimum distances in all snapshots as well as in differentiating collaboration patterns in journal publications between different research communities based on bounds of network distances

monte carlo method for critical systems in infinite volume the planar ising model in this paper we propose a monte carlo method for generating finite domain marginals of critical distributions of statistical models in infinite volume the algorithm corrects the problem of the long range effects of boundaries associated to generating critical distributions on finite lattices it uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of holographic boundary condition that encodes the presence of an infinite volume beyond it we check the quality of the distribution obtained in the case of the planar ising model by comparing various observables with their infinite plane prediction we accurately reproduce planar two three and four point of spin and energy operators we also define a lattice stress energy tensor and numerically obtain the associated conformal ward identities and the ising central charge

percolation in two dimensional quasicrystals by monte carlo simulations the authors examine whether or not two dimensional quasi periodic lattices belong to the same universality class as periodic lattices and they study bond and site percolation in penrose tiling and its dual lattice by making use of monte carlo simulations for the sake of comparison they also investigate percolation in periodic systems such as a square kagome and dice lattice for all these lattices they evaluate several critical exponents such as alpha related to the total number of clusters beta related to the percolation strength gamma related to the mean cluster size nu related to the correlation length tau related to cluster sizes and the fractal dimension d their results indicate that universality holds in two dimensional lattices with or without periodicity where the coordination number could be either single valued or multi valued

anisotropic cubic lattice potts ferromagnet renormalisation group treatment the choice of a convenient self dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q state potts ferromagnet criticality the exact critical frontier and dimensionality crossover exponent hf as well as the expected universality behaviour renormalisation flow sense are recovered for any linear scaling factor b and all values of q q or 4 the b 2 and b 3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den nijs conjecture the same calculation is performed for all values of b for the exponent nu d 1 associated with the one dimensional limit and the exact result nu d 1 1 is recovered in the limit b to infinity