free energy and correlations of the number theoretical spin chain we analyze the number theoretical spin chain with partition function z 1 using the polymer model technique the finite grand canonical chains give bounds for the limit free energy and internal energy the correlation functions for inverse temperature 1 are products of two point functions a combinatorial result for general interval graphs is derived

markov branching processes with instantaneous immigration markov branching processes with instantaneous immigration possess the property that immigration occurs immediately the number of particles reaches zero i e the conditional expectation of sojourn time at zero is zero in this paper we consider the existence and uniqueness of such a structure we prove that if the sum of the immigration rates is finite then no such structure can exist and we provide a necessary and sufficient condition for existence for the case in which this sum is infinite study of the uniqueness problem shows that for honest processes the solution is unique

phases of the number theoretic spin chain we present numerical and analytical evidence for a first order phase transition of the ferromagnetic spin chain with partition functionz 1 at the inverse temperature cr 2

discontinuous displacement approximation for capturing plastic localization it is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries interfaces of the finite element subdivision the regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept as a by product of the regularization an intrinsic band width is introduced as a constitutive property rather than a geometric feature of the finite element mesh in this way the spurious mesh sensitivity which is obtained when the displacement approximation is continuous can be avoided another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material an interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface up to within a scalar factor an analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a shear band for a general plasticity model the significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress it is shown that a unique structural post peak response in accordance with non linear fracture mechanics can be achieved when the plastic softening modulus is properly related to the bandwidth the paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope

on a ferromagnetic spin chain the quotient s 1 s of riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperatures

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

on asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics let let n be a positive integer and define n as the number of matrices c which are products of a and b where both a and b must occur such that the trace tr c n it has been conjectured that see 10 in this note we consider the summatory function and show that

a farey fraction spin chain we introduce a new number theoretic spin chain and explore its thermodynamics and connections with number theory the energy of each spin configuration is defined in a translation invariant manner in terms of the farey fractions and is also expressed using pauli matrices we prove that the free energy exists and a phase transition occurs for positive inverse temperature 2 the free energy is the same as that of related non translation invariant number theoretic spin chain using a number theoretic argument the low temperature 3 state is shown to be completely magnetized for long chains the number of states of energy e log n summed over chain length is expressed in terms of a restricted divisor problem we conjecture that its asymptotic form is n log n consistent with the phase transition at 2 and suggesting a possible connection with the riemann function the spin interaction coefficients include all even many body terms and are translation invariant computer results indicate that all the interaction coefficients except the constant term are ferromagnetic

generalized number theoretic spin chain connections to dynamical systems and expectation values we generalize the number theoretic spin chain a one dimensional statistical model based on the farey fractions by introducing a parameter x geqslant 0 this allows us to write recursion relations in the length of the chain these relations are closely related to the lewis three term equation which is useful in the study of the selberg function we then make use of these relations and spin orientation transformations we find a simple connection with the transfer operator of a model of intermittency in dynamical systems in addition we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length some of these expectation values appear to be directly connected with the mechanism of the phase transition

generalized farey trees transfer operators and phase transitions we consider a family of markov maps on the unit interval interpolating between the tent map and the farey map the latter is not uniformly expanding each map being composed of two fractional linear transformations the family generalizes many particular properties which for the case of the farey map have been successfully exploited in number theory we analyze the dynamics through the spectral properties of the generalized transfer operator application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function

dimension spectra of hyperbolic flows for flows with a conformal hyperbolic set we establish a conditional variational principle for the dimension spectra of holder continuous functions we consider simultaneously birkhoff averages into the future and into the past we emphasize that the description of the spectra is not a consequence of the existing results for birkhoff averages into the future or into the past the main difficulty is that even though the local product structure is bi lipschitz the level sets of the birkhoff averages are never compact our proof is based on the use of markov systems and is inspired in earlier arguments in the case of discrete time

application of material forces to hyperelastostatic fracture mechanics ii computational setting the concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics to this end we consider the continuum mechanics of material forces as outlined in part i of this work p steinmann int j solid struct 37 7371 7391 which act contrary to the common physical forces on the material manifold or rather in the material space in the sequel it is proposed to discretize the corresponding quasi static balance of pseudo momentum by a standard galerkin finite element procedure as a result we obtain global discrete node point quantities the material node point forces which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical j integral

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

theory and numerics for finite deformation fracture modelling using strong discontinuities a general finite element approach for the modelling of fracture is presented for the geometrically non linear case the kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements thus the deformation map is defined in terms of one continuous and one discontinuous portion considered as mutually independent giving rise to a weak formulation of the equilibrium consisting of two coupled equations in addition two different fracture criteria are considered firstly a principle stress criterion in terms of the material mandel stress in conjunction with a material cohesive zone law relating the cohesive mandel traction to a material displacement jump associated with the direct discontinuity secondly a criterion of griffith type is formulated in terms of the material crack driving force mcdf with the crack propagation direction determined by the direction of the force corresponding to the direction of maximum energy release apart from the material modelling the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria copyright 2005 john wiley sons ltd

a transfer operator approach to random fibonacci sequences we consider two stochastic variants of a fibonacci sequence generated by xn 1 xn xn 1 and xn 1 xn xn 1 respectively both the sequences begin with x0 x1 1 and the signs in each iteration are independently chosen with probability p for we focus on the generalized lyapunov exponent q of xn which gives the growth when q 0 or decay when q 0 rate of the ensemble average of xn q when n by using a technique that reduces the non increasing segments in the sequences we map the random sequences to a product of random 2 2 nonnegative matrices which is closely related to the farey fraction spin chain model we show that there exists a critical value q below which the ensemble average of xn q is dominated by the non increasing or linearly increasing samples and hence q is a constant both q and q and its derivatives can be precisely calculated with the transfer operator method moreover when q is a positive integer q is exactly determined by a system of polynomial equations the analysis can be readily generalized to the random sequence generated by xn 1 2cos k xn xn 1 or xn 1 2cos k xn xn 1 for any integer k 3

a framework for fracture modelling based on the material forces concept with xfem kinematics a theoretical and computational framework which covers both linear and non linear fracture behaviour is presented as a basis for the formulation we use the material forces concept due to the close relation between on one hand the eshelby energy momentum tensor and on the other hand material defects like cracks and material inhomogeneities by separating the discontinuous displacement from the continuous counterpart in line with the extended finite element method xfem we are able to formulate the weak equilibrium in two coupled problems representing the total deformation however in contrast to standard xfem where the direct motion discontinuity is used to model the crack we rather formulate an inverse motion discontinuity to model crack development the resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion which may be used to simulate linear as well as non linear fracture in one and the same formulation in fact the linear fracture formulation can be retrieved from the non linear cohesive zone formulation simply by confining the cohesive zone to the crack tip these features are clarified in the two numerical examples which conclude the paper copyright 2005 john wiley sons ltd

on a ferromagnetic spin chain ii thermodynamic limit the existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the riemann zeta function is proven

cluster approximation for the farey fraction spin chain we consider the farey fraction spin chain in an external field h utilising ideas from dynamical systems the free energy of the model is derived by means of an effective cluster energy approximation this approximation is valid for divergent cluster sizes and hence appropriate for the discussion of the magnetizing transition we calculate the phase boundaries and the scaling of the free energy at h 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy for h 0 our results are largely consistent with those found previously using mean field theory and renormalization group arguments