the potts model and flows iii standard and subgraph break collapse methods for pt ii see ibid vol 19 p 1655 1986 an algorithm is developed for the exact calculation of the many spin correlation functions of potts model clusters which is more efficient than the standard break collapse method successfully used in real space renormalisation group calculations the improved performance is based on a relationship which at any stage of the calculation allows the replacement of certain subgraphs by single effective edges thereby decreasing the number of iterations needed the authors method avoids as in the standard one the time consuming summation over spin states and may be used to extend series expansion and real space renormalisation group calculations on crystal lattices both methods are based on a number of combinatorial formulae the proofs of which are given in this paper

distribution of the eigenvalues of random block matrices the asymptotic behaviour of the eigenvalues of random block matrices is investigated with block sizes tending to infinity in the same order in the proofs some extended version of wigner s semi circle law as well as perturbation results for symmetric matrices are used the paper also deals with the asymptotic distribution of the protruding eigenvalues and some applications to random graphs

a semicoarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients the purpose of this paper is to present a semicoarsening multigrid algorithm for solving the finite difference discretization of symmetric and nonsymmetric two and three dimensional elliptic partial differential equations with highly discontinuous and anisotropic coefficients the discrete equations are assumed to be defined on a logically rectangular grid obtained possibly through grid generation for a problem defined on an irregular domain the basic algorithm is described along with some modifications which are designed to improve its efficiency and robustness for certain types of problem cases fortran codes that implement the two and three dimensional semicoarsening multigrid algorithms are described briefly and numerical results are presented

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

refined factorizations of solvable potentials a generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in quantum mechanics the method is applied in the study of radial oscillator morse and coulomb potentials to obtain a wide set of raising and lowering operators and to show clearly the connection that links these systems

properties of dense partially random graphs we study the properties of random graphs where for each vertex a neighborhood has been previously defined the probability of an edge joining two vertices depends on whether the vertices are neighbors or not as happens in small world graphs swg s but we consider the case where the average degree of each node is of order of the size of the graph unlike swg s which are sparse this allows us to calculate the mean distance and clustering which are qualitatively similar although not in such a dramatic scale range to the case of swg s we also obtain analytically the distribution of eigenvalues of the corresponding adjacency matrices this distribution is discrete for large eigenvalues and continuous for small eigenvalues the continuous part of the distribution follows a semicircle law whose width is proportional to the disorder of the graph whereas the discrete part is simply a rescaling of the spectrum of the substrate we apply our results to the calculation of the mixing rate and the synchronizability threshold

scattering potentials with ls terms from first order casimir operators using a first order casimir operator calculated in a non standard realization for the so 3 1 algebra we obtain a one dimensional scattering problem with ls type interaction terms it is shown that for this realization the square of this operator can be expressed in terms of the usual quadratic casimir due to this constraint the scattering states are completely specified by restricting the possible set of eigenvalues accordingly the results show that the use of extra casimir operators can provide additional insight into the group theoretical structure of the scattering problem a generalization for the so 2n 1 1 n 2 case is also given the underlying supersymmetry of the resulting schrodinger equations is pointed out the supersymmetric charge operators are related to our first order casimir operators

random incidence matrices moments of the spectral density we study numerically and analytically the spectrum of incidence matrices of random labeled graphs on n vertices any pair of vertices is connected by an edge with probability p we give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in n and p for large n and fixed p the spectrum contains a large eigenvalue at np and a semicircle of small eigenvalues for large n and fixed average connectivity pn dilute or sparse random matrices limit we show that the spectrum always contains a discrete component an anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed we develop recursion relations to compute the moments as explicit polynomials in pn their growth is slow enough so that they determine the spectrum the extension of our methods to the laplacian matrix is given in appendix

a multiphase field concept numerical simulations of moving phase boundaries and multiple junctions we present numerical simulations which support the formal asymptotic analysis relating a multiorder parameter allen cahn system to a multiphase interface problem with curvature dependent evolution of the interfaces and angle conditions at triple junctions within the gradient energy of the allen cahn system the normal to an interface between phases i and j is modeled by the irreducible representations u i nabla u j u j nabla u i u i nabla u j u j nabla u i where ui and uj are the ith and jth components of the vectorial order parameter bu in rz n in the vectorial case the dependence of the limiting surface tensions and mobilities on the bulk potentials of the allen cahn system is not given explicitly but in terms of all the n components of the planar stationary wave solutions one of the issues of this paper is to find bulk potentials which allow a rather easy access to the resulting surface tensions and mobilities we compare numerical computations for planar and circular phase bound

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

a nonstiff adaptive mesh refinement based method for the cahn hilliard equation we present a nonstiff fully adaptive mesh refinement based method for the cahn hilliard equation the method is based on a semi implicit splitting in which linear leading order terms are extracted and discretized implicitly combined with a robust adaptive spatial discretization the fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size we demonstrate the efficacy of the method with numerical examples both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy employing this approach we also identify several stationary solutions of that decomposition on the 2d torus

computer simulations of phase field drops on super hydrophobic surfaces abstract we present a novel quasi newton continuation procedure that efficiently solves the system of nonlinear equations arising from the discretization of a phase field model for wetting phenomena we perform a comparative numerical analysis that shows the improved speed of convergence gained with respect to other numerical schemes moreover we discuss the conditions that on a theoretical level guarantee the convergence of this method at each iterative step a suitable continuation procedure develops and passes to the nonlinear solver an accurate initial guess discretization performs through cell centered finite differences the resulting system of equations is solved on a composite grid that uses dynamic mesh refinement and multi grid techniques the final code achieves three dimensional realistic computer experiments comparable to those produced in laboratory settings this code offers not only new insights into the phenomenology of super hydrophobicity but also serves as a reliable predictive tool for the study of hydrophobic surfaces

factorization of spin dependent hamiltonians we generalize the formalism of the factorization method by introducing spin degrees of freedom in addition to local potential terms we compare this construction with conventional factorization techniques of non relativistic hamiltonians and the formalism of supersymmetric quantum mechanics the characteristic isospectrality of the hamiltonians h1 qr and h2 rq is maintained in our more general framework as illustrative examples we consider two exactly solvable models related to the harmonic oscillator and the coulomb problems and some quasi exactly solvable problems in all three cases the zero energy ground state of h1 is found to be infinitely degenerate further possible generalizations of our approach are also outlined

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

wetting of rough surfaces a homogenization approach the contact angle of a drop in equilibrium on a solid is strongly affected by the roughness of the surface on which it rests we study the roughness induced enhancement of the hydrophobic or hydrophilic properties of a solid surface through homogenization theory by relying on a variational formulation of the problem we show that the macroscopic contact angle is associated with the solution of two cell problems giving the minimal energy per unit macroscopic area for a transition layer between the rough solid surface and a liquid or vapor phase our results are valid for both chemically heterogeneous and homogeneous surfaces in the latter case a very transparent structure emerges from the variational approach the classical laws of wenzel and cassie baxter give bounds for the optimal energy and configurations of minimal energy are those leading to the smallest macroscopic contact angle in the hydrophobic case to the largest one in the hydrophilic case

the potts model and flows i the pair correlation function it is shown that the partition function for the a state potts model with pair interactions is related to the expected number of integer mod a flows in a percolation model the relation is generalised to the pair correlation function the resulting high temperature expansion coefficients are shown to be the flow polynomials of graph theory we also prove an observation of tsallis and levy concerning the equivalent transmissivity of a cluster

potts model and graph theory elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the potts model topics discussed include graphical expansions of the potts partition function and correlation functions and their relationships with the chromatic dichromatic and flow polynomials occurring in graph theory it is also shown that the potts model realization of these classic graph theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory

pauli equation and the method of supersymmetric factorization we consider different variants of factorization of a 2 2 matrix schrodinger pauli operator in two spatial dimensions they allow us to relate its spectrum to the sum of spectra of two scalar schrodinger operators in a manner similar to one dimensional darboux transformations we consider both the case when such factorization is reduced to the ordinary two dimensional supersymmetric quantum mechanics quasifactorization and a more general case which involves covariant derivatives the admissible classes of electromagnetic fields are described and some illustrative examples are given

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph