two dimensional percolation logarithmic corrections to the critical behaviour from series expansions analyses several extant series for the mean cluster size the zeroth moment of the pair connectedness and the percolation probability for bond and site percolation on two dimensional lattices with a view to detecting possible logarithmic corrections the logarithmic correction exponent z is found to be in the range 0 or z or 0 15 and our analysis also provides some new information about the critical exponents gamma and beta

analyzing head related transfer function measurements using surface spherical harmonics a continuous functional representation of a large set of head related transfer function measurements hrtfs is developed the hrtfs are represented as a weighted sum of surface spherical harmonics sshs up to degree 17 a gaussian quadrature method is used to pick out a set of experimentally efficient measurement directions anechoic impulse responses are measured for these directions between a source loudspeaker and the entrance to the ear canal of a head and torso simulator hats three separate ssh analyses are carried out the first forms a ssh representation from the time responses with the variable onset delay caused by interaural differences intact by applying the analysis to each time sample in turn the second ssh model is formed in exactly the same way except using impulse responses in which the variable onset delays have been equalized the final ssh analysis is carried out in the frequency domain by applying the technique on a frequency bin by frequency bin basis to the magnitude and unwrapped phase responses of the hrtfs the accuracy and interpolation performance of each of the computed ssh models is investigated and the usefulness of the ssh technique in analyzing directional hearing and particularly in spatializing sounds is discussed

discrete vakonomic mechanics discretizations of vakonomic mechanics are developed by using the methodology of variational integrators the resulting algorithms are symplectic in some sense that we will explain and preserve the momentum map associated with a lie group of symmetries a specific example illustrating the theory is tested numerically

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

invariant conservation law preserving discretizations of linear and nonlinear wave equations symmetry and conservation law preserving finite difference discretizations are obtained for linear and nonlinear one dimensional wave equations on five and nine point stencils using the theory of lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction in particular for the linear wave equation an explicit five point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws for a class of nonlinear wave equations arising in hyperelasticity a nine point implicit scheme is constructed preserving four point symmetries and three local conservation laws other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed

triviality problem and high temperature expansions of higher susceptibilities for the ising and scalar field models in four five and six dimensional lattices high temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar field models on high dimensional lattices the critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state we have obtained a substantial extension through order 24 of the high temperature expansions of the free energy in presence of a magnetic field for the ising models with spin s 1 2 and for the lattice scalar field theory with quartic self interaction on the simple cubic and the body centered cubic lattices in four five and six spatial dimensions a numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas based on the renormalization group and the constructive approach to euclidean quantum field theory concerning the no interaction triviality property of the continuum scaling limit of spin s ising and lattice scalar field models at and above the upper critical dimensionality

discrete time lagrangian mechanics on lie groups with an application to the lagrange top we develop the theory of discrete time lagrangian mechanics on lie groups originated in the work of veselov and moser and the theory of lagrangian reduction in the discrete time setting the results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics the lagrange top we recall the derivation of the euler poinsot equations of motion both in the frame moving with the body and in the rest frame the latter ones being less widely known we find a discrete time lagrange function turning into the known continuous time lagrangian in the continuous limit and elaborate both descriptions of the resulting discrete time system namely in the body frame and in the rest frame this system naturally inherits poisson properties of the continuous time system the integrals of motion being deformed the discrete time lax representations are also found kirchhoff s kinetic analogy between elastic curves and motions of the lagrange top is also generalised to the discrete context

scaling and density of lee yang zeros in the four dimensional ising model the scaling behaviour of the edge of the lee yang zeroes in the four dimensional ising model is analyzed this model is believed to belong to the same universality class as the phi 4 4 model which plays a central role in relativistic quantum field theory while in the thermodynamic limit the scaling of the yang lee edge is not modified by multiplicative logarithmic corrections such corrections are manifest in the corresponding finite size formulae the asymptotic form for the density of zeroes which recovers the scaling behaviour of the susceptibility and the specific heat in the thermodynamic limit is found to exhibit logarithmic corrections too the density of zeroes for a finite size system is examined both analytically and numerically

self avoiding walks in four dimensions logarithmic corrections we present simulation results for long n or 4000 self avoiding walks in four dimensions we find definite indications of logarithmic corrections but the data are poorly described by the asymptotically leading terms detailed comparisons are presented with renormalization group flow equations derived by direct renormalization and with the results of a field theoretic calculation

mechanical systems subject to holonomic constraints differential algebraic formulations and conservative integration abstract the numerical integration in time of the equations of motion for mechanical systems subject to holonomic constraints is considered schemes are introduced for the direct treatment of a differential algebraic form of the equations that preserve the constraints the total energy and other integrals such as linear and angular momentum arising from affine symmetries moreover the schemes can be shown to preserve the property of time reversibility in an appropriate sense an example is given to illustrate various aspects of the proposed methods

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

d alembert lagrange analytical dynamics for nonholonomic systems the d alembert lagrange principle dlp is designed primarily for dynamical systems under ideal geometric constraints although it can also cover linear velocity constraints its application to nonlinear kinematic constraints has so far remained elusive mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint on recognition that the commutation rule traditionally accepted for velocity displacements in lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements which can be adjoined via lagrange s multipliers to the d alembert lagrange equation to yield the equations of state and also new transpositional relations for nonholonomic systems the equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived in appendix a from the quite different gauss principle the present advance therefore solves a long outstanding problem on the application of dlp to ideal nonholonomic systems and as an aside provides validity to axioms as the chetaev rule previously left theoretically unjustified a more general transpositional form of the boltzmann hamel equation is also obtained

high temperature series for scalar field lattice models generation and analysis an implementation of the free embedding scheme for high temperature series generation on the body centered cubic family of lattices in arbitrary dimensiond is described series to order 21 in inverse temperature are tabulated for several scalar field models both for the magnetic susceptibility and for the second moment of the spin correlation function the critical behavior of a family of 3 dimensional double gaussian models which interpolate continuously between the spin 1 2 ising model and the gaussian model is analyzed in detail away from the gaussian model limit using confluent inhomogeneous secondorder differential approximants with our best estimate of the correction to scaling exponent 0 52 0 03 the leading exponents for the susceptibility and correlation length for this family are consistent with universality and are given by 1 237 0 002 and 0 630 0 0015 respectively and 2 0 0359 0 0007

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

geometric integrators and nonholonomic mechanics a geometric derivation of nonholonomic integrators is developed it is based in the classical technique of generating functions adapted to the special features of nonholonomic systems the theoretical methodology and the integrators obtained are different from those obtained in cortes and martinez nonholonomic integrators nonlinearity 14 1365 1392 2001 in the case of mechanical systems with linear constraints a family of geometric integrators preserving the nonholonomic constraints is given

dynamic critical behavior of the worm algorithm for the ising model we study the dynamic critical behavior of the worm algorithm for the two and three dimensional ising models by monte carlo simulation the autocorrelation functions exhibit an unusual three time scale behavior as a practical matter the worm algorithm is slightly more efficient than the swendsen wang algorithm for simulating the two point function of the three dimensional ising model

correction to scaling exponents and critical properties of the n vector model with dimensionality 4 a perturbation calculation is given which implies that the susceptibility of the five and six dimensional n vector models can be written x approximately at 1 1 bt12 and x approximately at 1 1 bt in t respectively independent of n for n 0 and 1 it is shown that series analysis techniques can extract the correction to scaling exponents and that estimates of the critical temperatures and critical amplitudes can also be obtained the correction to scaling exponents found are in agreement with those known to exist in the case of the spherical model

series and monte carlo study of high dimensional ising models ising models in high dimensions are used to compare high temperature series expansions with monte carlo simulations simulations of the magnetization on four six and seven dimensional hypercubic lattices give consistent values of the critical temperature from both equilibrium and nonequilibrium data ford 6 and 7 we tabulate 15 terms of series expansions for the susceptibility for generald and givej kbtc 0 092295 3 and 0 077706 2 ford 6 and 7 in contrast to five dimensions where earlier series found nonanalytic scaling corrections for d 6 and 7 the leading scaling correction may be analytic int tc in most cases these expansions gave more accurate results than these simulations

towards a hamilton jacobi theory for nonholonomic mechanical systems in this paper we obtain a hamilton jacobi theory for nonholonomic mechanical systems the results are applied to a large class of nonholonomic mechanical systems the so called caplygin systems

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

diversity of critical behavior within a universality class we study spatial anisotropy effects on the bulk and finite size critical behavior of the mathrm o n symmetric anisotropic ensuremath varphi 4 lattice model with periodic boundary conditions in a d dimensional hypercubic geometry above at and below t c the absence of two scale factor universality is discussed for the bulk order parameter correlation function the bulk scattering intensity and for several universal bulk amplitude relations the anisotropy parameters are observable by scattering experiments at t c for the confined system renormalization group theory within the minimal subtraction scheme at fixed dimension d for 2ldl4 is employed in contrast to the ensuremath epsilon 4 ensuremath d expansion the fixed d finite size approach keeps the exponential form of the order parameter distribution function unexpanded for the case of cubic symmetry and for n 1 our perturbation approach yields excellent agreement with the monte carlo mc data for the finite size amplitude of the free energy of the three dimensional ising model at t c by mon phys rev lett 54 2671 1985 the ensuremath epsilon expansion result is in less good agreement below t c a minimum of the scaling function of the excess free energy is found we predict a measurable dependence of this minimum on the anisotropy parameters the relative anisotropy effect on the free energy is predicted to be significantly larger than that on the binder cumulant our theory agrees quantitatively with the nonmonotonic dependence of the binder cumulant on the ferromagnetic next nearest neighbor nnn coupling of the two dimensional ising model found by mc simulations of selke and shchur j phys a 38 l739 2005 our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic nnn coupling and the existence of a lifshitz point at a larger value of this coupling the nonuniversal anisotropy effects in the finite size scaling regime are predicted to satisfy a kind of restricted universality the tails of the large l behavior at t ensuremath ne t c violate both finite size scaling and universality even for isotropic systems as they depend on the bare four point coupling of the ensuremath varphi 4 theory on the cutoff procedure and on subleading long range interactions

an on line library of extended high temperature expansions of basic observables for the spin s ising models on two and three dimensional lattices we present an on line library of unprecedented extension for high temperature expansions of basic observables in the ising models of general spin s with nearest neighbor interactions we have tabulated through order 25 the series for the nearest neighbor correlation function the susceptibility and the second correlation moment in two dimensions on the square lattice and in three dimensions on the simple cubic and the body centered cubic lattices the expansion of the second field derivative of the susceptibility is also tabulated through 23 for the same lattices we have thus added several terms from four up to thirteen to the series already published for spin s 1 2 1 3 2 2 5 2 3 7 2 4 5

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators