fractals in surface growth with power law noise the authors present a microscopic description of interface growth with power law noise distribution in the form p eta eta 1 approximately 1 mu which exhibits non universal roughening for the mu d 1 case in d 1 dimensions the existence of a fractal pattern in the bulk of the aggregate is explained leading trivially to the proof of the identity alpha z 2 for the roughening and the dynamical scaling exponents alpha and z respectively investigations on the distribution of step sizes of the discretized interface and the saturated growth speed further support the arguments

coupling eigenvalues and eigenvectors a tool for investigating the vibroacoustic behaviour of coupled vibrating systems the transmission of vibrations between coupled subsystems is treated by using coupling eigenvalues and eigenvectors it is demonstrated through a basic equation that coupling eigenvalues and eigenvectors characterize the energy exchanges between subsystems due to the coupling the coupling eigenvalue relates to the coupling strength and coupling eigenvectors to the coupling transmission path in addition in the case of several couplings a simplified method is presented in which only the prevailing modal path between subsystems is used the results obtained by using this method compare well with the reference calculation the transmission of vibrations by coupling can be calculated with a small number of variables and offers a new perspective in the range of medium frequency

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

exact results for the critical behaviour of a nienhuis o n model on the square lattice exact results are obtained for a square lattice fifteen vertex model and related o n model recently proposed by nienhuis 1990 the partition function per site and finite size corrections are obtained along with the critical exponents via the temperley lieb equivalent six vertex model the exact exponents nu 4 7 and gamma 6 7 follow in the limit n 0 for even system sizes odd system sizes are also discussed the fifteen vertex model is also the vertex formulation of the anisotropic or q deformed spin 1 biquadratic model

a class of interaction round a face models and its equivalence with an ice type model a new model called the temperley lieb interactions model is introduced in two dimensional lattice statistics on a square lattice the temperley lieb equivalence of this model to the six vertex self dual potts critical hard hexagons and critical nonintersecting string models is established a graphical equivalence of this model to the six vertex model generalizes this equivalence to noncritical cases of the above models the order parameters of a specialization of this model are studied

new results for the xx model with boundaries in this paper we study the xx model with particular non diagonal boundaries using a fermionization technique for an extended version of this chain we are able to present a new operator which commutes with the extended and also the original hamiltonian it is a highly non trivial conserved quantity in the spin chain formalism which gives rise to an interesting combinatorial partition function

response probability distribution of built up vibro acoustic systems the vibro acoustic response of built up structures consisting of stiff components with low modal density and flexible components with high modal density is sensitive to small imperfections in the flexible components in this paper the uncertainty of the response is considered by modeling the low modal density master system as deterministic and the high modal density subsystems in a nonparametric stochastic way i e carrying a diffuse wave field and by subsequently computing the response probability density function the master system s mean squared response amplitude follows a singular noncentral complex wishart distribution conditional on the subsystem energies for a single degree of freedom this is equivalent to a chi square or an exponential distribution depending on the loading conditions the subsystem energies follow approximately a chi square distribution when their relative variance is smaller than unity the results are validated by application to plate structures and good agreement with monte carlo simulations is found

on the bogomolny schmit conjecture bogomolny and schmit proposed that the critical edge percolation on the square lattice is a good model for the nodal domains of a random plane wave based on this they made a conjecture about the number of nodal domains recent computer experiments showed that the mean number of clusters per vertex and the mean number of nodal domains per unit area are very close but different since the original argument was mostly supported by numerics it was believed that the percolation model is wrong in this paper we give some numerical evidence in favour of the percolation model

iterative solvers for the stochastic finite element method this paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations pdes a stochastic galerkin finite element discretization is applied to transform the pde into a coupled set of deterministic pdes specialized solvers are required to solve the very high dimensional systems that result after a finite element discretization of the resulting set this paper discusses one level iterative methods based on matrix splitting techniques multigrid methods which apply a coarsening in the spatial dimension and multilevel methods which make use of the hierarchical structure of the stochastic discretization also krylov solvers with suitable preconditioning are addressed a local fourier analysis provides quantitative convergence properties the efficiency and robustness of the methods are illustrated on two nontrivial numerical problems the multigrid solver with block smoother yields the most robust convergence properties though a cheaper point smoother performs as well in most cases multilevel methods based on coarsening the stochastic dimension perform in general poorly due to a large computational cost per iteration moderate size problems can be solved very quickly by a krylov method with a mean based preconditioner for larger spatial and stochastic discretizations however this approach suffers from its nonoptimal convergence properties

logarithmic operators and logarithmic conformal field theories logarithmic operators and logarithmic conformal field theories are reviewed prominent examples considered here include c 2 and c 0 logarithmic conformal field theories c 0 logarithmic conformal field theories are especially interesting since they describe some of the critical points of a variety of longstanding problems involving a two dimensional quantum particle moving in a spatially random potential as well as critical two dimensional self avoiding random walks and percolation lack of classification of logarithmic conformal field theories remains a major impediment to progress towards finding complete solutions to these problems

modification of the maximin and ϕp phi criteria to achieve statistically uniform distribution of sampling points this article proposes a sampling technique that delivers robust designs that is point sets selected from a design domain in the shape of a unit hypercube the designs are guaranteed to provide a

on the applicability of the lognormal distribution in random dynamical systems this paper is concerned with the probability density function of the energy of a random dynamical system subjected to harmonic excitation it is shown that if the natural frequencies and mode shapes of the system conform to the gaussian orthogonal ensemble then under common types of loading the distribution of the energy of the response is approximately lognormal providing the modal overlap factor is high typically greater than two in contrast it is shown that the response of a system with poisson natural frequencies is not approximately lognormal numerical simulations are conducted on a plate system to validate the theoretical findings and good agreement is obtained simulations are also conducted on a system made from two plates connected with rotational springs to demonstrate that the theoretical findings can be extended to a built up system the work provides a theoretical justification of the commonly used empirical practice of assuming that the energy response of a random system is lognormal

ensemble averaged vibration energy flows in a three element structure abstract closed form expressions are presented for ensemble averaged vibration energies and energy flows in a three element structure the ensemble consist of one dimensional elements having random lengths with an uncertainty that is plus or minus a quarter of the wavelength the ensemble averages are functions of only the ensemble averaged input power the modal densities the dissipation loss factors and the transmission factors for the couplings found from travelling wave analysis the ensemble averages are not related to each other according to the sea hypothesis that energy flow is proportional to the difference of coupled elements modal energies however energy balance is governed by terms directly proportional to differences of modal energies while also relating non directly coupled elements numerical investigations of this indirect energy flow indicate a measure of coupling strength i e the ratio of the transmission factor to the product of the two modal overlaps in both connected elements based on the proposed measure series expansions are made showing that when coupling is weak and when the response in each element is reverberant the sea hypothesis is true also the ensemble averaged energies are then determined by a standard sea model by using travelling wave estimates of the coupling loss factors

boundary correlation functions of the six vertex model we consider the six vertex model on an n n square lattice with the domain wall boundary conditions boundary one point correlation functions of the model are expressed as determinants of n n matrices generalizing the known result for the partition function in the free fermion case explicit answers are obtained the introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings

transformations and hardy krause variation using a multivariable faa di bruno formula we give conditions on transformations tau 0 1 m to cal x where cal x is a closed and bounded subset of mathbb r d such that f circ tau is of bounded variation in the sense of hardy and krause for all f in c m cal x we give similar conditions for f circ tau to be smooth enough for scrambled net sampling to attain o n 3 2 epsilon accuracy some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition some other transformations due to fang and wang monogr statist appl probab 51 crc press boca raton fl 1993 satisfy the first but not the second condition we provide transformations for the simplex that makes f circ tau smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials we also find sufficient conditions for conditional inversion in mathbb r 2 and for importance sampling to be of bounded variation in the sense of ha

holomorphic symplectic fermions let v be the even part of the vertex operator super algebra of r pairs of symplectic fermions up to two conjectures we show that v admits a unique holomorphic extension if r is a multiple of 8 and no holomorphic extension otherwise this is implied by two results obtained in this paper 1 if r is a multiple of 8 one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice d r with shifted stress tensor 2 we classify lagrangian algebras in mathcal s mathcal f mathfrak h a ribbon category associated to symplectic fermions the classification of holomorphic extensions of v follows from 1 and 2 if one assumes that mathcal s mathcal f mathfrak h is ribbon equivalent to mathrm rep v and that simple modules of extensions of v are in one to one relation with simple local modules of the corresponding commutative algebra in mathcal s mathcal f mathfrak h

the logarithmic triplet theory with boundary the boundary theory for the c 2 triplet model is investigated in detail in particular we show that there are four different boundary conditions that preserve the triplet algebra and check the consistency of the corresponding boundary operators by constructing their ope coefficients explicitly we also compute the correlation functions of two bulk fields in the presence of a boundary and verify that they are consistent with factorization

manhattan lattice theta point exponents from kinetic growth walks and exact results from the nienhuis o n model kinetic growth walks kgw on the manhattan lattice have previously been shown to be equivalent to the static problem of interacting self avoiding walks on that lattice at the theta temperature here we illustrate how a complete set of exponents for the static problem including the crossover exponent phi and surface exponents at the ordinary and special transition points may be obtained from simulations of kinetic walks in the process we find that phi m 0 430 or 0 006 which encompasses the conjectured value 3 7 approximately 0 42857 for the theta point on an isotropic lattice our numerics confirm a predicted set of exponents for both the bulk and surface transitions in addition to results such as the exact internal energy at the bulk transition furthermore we point out that a recently examined variant of the nienhuis o n model can be mapped onto theta point walks on the manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures

proof of the macdonald conjecture a plane partition of n is an array a i not necessarily square of positive integers which has non increasing rows and columns such that the sum of its elements is n with such an array we associate the set d which is the set of all integer points i j k in the first octant such that aij is defined and 1 k aij a plane partition is cyclically symmetric if d is invariant under the mapping i j k o k i let m m n be the number of such cyclically symmetric partitions of n such that none of i j aij exceed m macdonald has conjectured that the generating function

nonmeromorphic operator product expansion and c2 cofiniteness for a family of w algebras we prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras the triplet algebras using results from p z tensor product theory while doing this we also show that all these vertex operator algebras are c2 cofinite

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

on the triplet vertex algebra w p abstract we study the triplet vertex operator algebra w p of central charge 1 6 p 1 2 p p 2 we show that w p is c 2 cofinite but irrational since it admits indecomposable and logarithmic modules furthermore we prove that w p is of finite representation type and we provide an explicit construction and classification of all irreducible w p modules and describe block decomposition of the category of ordinary w p modules all this is done through an extensive use of zhu s associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms moreover we obtain an upper bound for dim a w p finally for p prime we completely describe the structure of a w p the methods of this paper are easily extendable to other w algebras and superalgebras

nonequilibrium stationary states and equilibrium models with long range interactions it was recently suggested by blythe and evans that a properly defined steady state normalization factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities in analogy with the lee yang description of phase transition of equilibrium systems they studied the zeros in the complex plane of the normalization factor in order to find phase transitions in nonequilibrium steady states we show that like for equilibrium systems the densities associated with the rates are nondecreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the densities we illustrate this phenomenon for the asymmetric exclusion process we actually show that its normalization factor coincides with an equilibrium partition function of a walk model in which the densities have a simple physical interpretation

energy flow models from finite element analysis computationally efficient methods are described by which the results of a finite element analysis of a system may be post processed to form energy flow models yielding time and perhaps frequency average subsystem energies as well as input and dissipated powers the methods are particularly efficient for excitation which is spatially distributed or broadband e g rain on the roof or if the frequency average response is required first a method based on a global finite element analysis is presented this involves a global modal decomposition and a reordering of the subsequent numerical calculations the properties of the distribution of the excitation and the system s mass and stiffness lead to subsystem force distribution mass distribution and stiffness distribution matrices the response is given by a sum of terms involving the interaction of a pair of global modes the contribution of each pair depending on the appropriate elements of the distribution matrices frequency averaging is performed by separating the resulting frequency dependent terms and integrating in most practical cases this integration can be done analytically next an alternative method involving component mode synthesis is described in this individual finite element analyses are performed for each subsystem using here the fixed interface method these are then assembled to perform a global modal analysis with the order of the model being much reduced the consequent results are then post processed in the same way finally a system comprising three coupled plates is presented as a numerical example

twist operator correlation functions in o n loop models using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the o n model at the critical point we focus on correlation functions containing twist operators combining these with anchored loops boundaries with sle processes and with double sle processes we focus further upon n 0 representing self avoiding loops which corresponds to a logarithmic conformal field theory lcft with c 0 in this limit the twist operator plays the role of a 0 weight indicator operator which we verify by comparison with known examples using the additional conditions imposed by the twist operator null states we derive a new explicit result for the probabilities that an sle8 3 winds in various ways about two points in the upper half plane e g that the sle passes to the left of both points the collection of c 0 logarithmic cft operators that we use deriving the winding probabilities is novel highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory we argue that both partners do appear in the theory one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk boundary fusion rules

a general derivation of the statistical energy analysis equations for coupled dynamic systems abstract the response of a multi coupled dynamic system to random excitation is considered with the aim of providing a theoretical derivation of the statistical energy analysis sea approach to vibration prediction it is shown that the general form of the principal sea equation is widely applicable provided that the coupling between the subsystems is conservative and a suitable definition of the subsystem energy is used under the assumption of weak coupling the standard sea equations are recovered expressions for the coupling loss factors are obtained in terms of the frequency and space averaged green functions of the coupled system

an efficient probabilistic approach to vibro acoustic analysis based on the gaussian orthogonal ensemble vibro acoustic analysis of complex systems at higher frequencies faces two challenges how to compute the response without using an excessive number of degrees of freedom dofs and how to quantify the uncertainty of the response due to small spatial variations in geometry material properties and boundary conditions which have a wave scattering effect in this study a general method of analysis is presented that provides an answer to both questions while overcoming most limitations of statistical energy analysis the fundamental idea is to numerically compute an artificial ensemble of realizations for the components of the built up system that are highly sensitive to small random wave scatterers this can be efficiently performed because their eigenvalue spacings and mode shapes conform to gaussian orthogonal ensemble spacings and gaussian random fields respectively the dofs of the overall system are therefore limited to those of the deterministic components and the interface dofs of the random components the method is extensively validated by application to plate structures good agreement between the predicted response probability distributions and the results of detailed parametric probabilistic models is obtained also for cases of low modal overlap single point loading and strong subsystem coupling

conformal algebras of two dimensional disordered systems we discuss the structure of two dimensional conformal field theories at a central charge c 0 describing critical disordered systems polymers and percolation we construct a novel extension of the c 0 virasoro algebra characterized by a number b measuring the effective number of massless degrees of freedom and by a logarithmic partner of the stress tensor it is argued to be present at a generic random critical point lacking super kac moody or other higher symmetries and is a tool to describe and classify such theories interestingly this algebra is not only consistent with but indeed naturally accommodates in general an underlying global supersymmetry polymers and percolation realize this algebra unexpectedly we find that the c 0 kac table of the degenerate fields contains two distinct theories with b 5 6 and b 5 8 which we conjecture to correspond to percolation and polymers respectively a given kac table field can be degenerate only in one of them remarkably we also find this algebra and thereby an ensuing hidden supersymmetry realized at general replica averaged critical points for which we derive an explicit formula for b

open string vertex algebras tensor categories and operads we introduce notions of open string vertex algebra conformal open string vertex algebra and variants of these notions these are open string theoretic noncommutative generalizations of the notions of vertex algebra and of conformal vertex algebra given an open string vertex algebra we show that there exists a vertex algebra which we call the meromorphic center inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center this result gives us a general method for constructing open string vertex algebras besides obvious examples obtained from associative algebras and vertex super algebras we give a nontrivial example constructed from the minimal model of central charge open image in new window we establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable vertex operator algebra and the grading restricted conformal open string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra we also give a geometric and operadic formulation of the notion of grading restricted conformal open string vertex algebra we prove two isomorphism theorems and in particular we show that such an algebra gives a projective algebra over what we call the swiss cheese partial operad

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

an approach to the theoretical background of statistical energy analysis applied to structural vibration rayleigh s classical approach to the study of vibration of systems having a finite number of degrees of freedom is applied to the problem of coupling of subsystems in a complicated structure in order to probe the regions of applicability of the approach to vibration analysis usually known as statistical energy analysis sea the classical method has advantages of simplicity and rigor over previous approaches to the background of sea in certain cases and provides extensions and simplifications in several areas of the theory it also suggests modifications to sea modeling strategy depending on the type of coupling involved even when that coupling is weak so that earlier analyses might be thought to apply

polynomial fusion rings of w extended logarithmic minimal models the countably infinite number of virasoro representations of the logarithmic minimal model lm p p can be reorganized into a finite number of w representations with respect to the extended virasoro algebra symmetry w using a lattice implementation of fusion we recently determined the fusion algebra of these representations and found that it closes albeit without an identity for p 1 here we provide a fusion matrix realization of this fusion algebra and identify a fusion ring isomorphic to it we also consider various extensions of it and quotients thereof and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings one particular extension is reminiscent of the fundamental fusion algebra of lm p p and offers a natural way of introducing the missing identity for p 1 working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual moore penrose inverses intertwining bet

boundary polarization in the six vertex model vertical arrow fluctuations near the boundaries in the six vertex model on the two dimensional n n square lattice with the domain wall boundary conditions are considered the one point correlation function boundary polarization is expressed via the partition function of the model on a sublattice the partition function is represented in terms of standard objects in the theory of orthogonal polynomials this representation is used to study the large n limit the presence of the boundary affects the macroscopic quantities of the model even in this limit the logarithmic terms obtained are compared with predictions from conformal field theory

virasoro representations and fusion for general augmented minimal models in this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models the augmented c2 3 0 model as well as the augmented yang lee model at c2 5 22 5 we analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra we also describe the first few higher rank representations in detail in particular we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented cp q models

symmetry classes of alternating sign matrices under one roof in a previous article math co 9712207 we derived the alternating sign matrix asm theorem from the izergin korepin determinant for a partition function for square ice with domain wall boundary here we show that the same argument enumerates three other symmetry classes of alternating sign matrices vsasms vertically symmetric asms even htsasms half turn symmetric asms and even qtsasms quarter turn symmetric asms the vsasm enumeration was conjectured by mills the others by robbins math co 0008045 we introduce several new types of asms uasms asms with a u turn side uuasms two u turn sides osasms off diagonally symmetric asms oosasms off diagonally off antidiagonally symmetric and uosasms off diagonally symmetric with u turn sides uasms generalize vsasms while uuasms generalize vhsasms vertically and horizontally symmetric asms and another new class vhpasms vertically and horizontally perverse osasms oosasms and uosasms are related to the remaining symmetry classes of asms namely dsasms diagonally symmetric dasasms diagonally anti diagonally symmetric and tsasms totally symmetric asms we enumerate several of these new classes and we provide several 2 enumerations and 3 enumerations our main technical tool is a set of multi parameter determinant and pfaffian formulas generalizing the izergin korepin determinant for asms and the tsuchiya determinant for uasms solv int 9804010 we evaluate specializations of the determinants and pfaffians using the factor exhaustion method

w extended fusion algebra of critical percolation two dimensional critical percolation is the member lm 2 3 of the infinite series of yang baxter integrable logarithmic minimal models lm p p we consider the continuum scaling limit of this lattice model as a rational logarithmic conformal field theory with extended w w2 3 symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture we find that the representation content of the ensuing closed fusion algebra contains 26 w indecomposable representations with eight rank1 representations 14 rank 2 representations and four rank 3 representations weidentifytheserepresentationswithsuitablelimitsofyang baxterintegrable boundary conditions on the lattice and obtain their associated w extended characters the latter decompose as finite non negative sums of w irreducible characters of which 13 are required implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit cayley table the closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry

integrable boundary conditions and w extended fusion in the logarithmic minimal models lm 1 p we consider the logarithmic minimal models lm 1 p as rational logarithmic conformal field theories with extended w symmetry to make contact with the extended picture starting from the lattice we identify 4p 2 boundary conditions as specific limits of integrable boundary conditions of the underlying yang baxter integrable lattice models specifically we identify 2p integrable boundary conditions to match the 2p known irreducible w representations these 2p extended representations naturally decompose into infinite sums of the irreducible virasoro representations r s a further 2p 2 reducible yet indecomposable w representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank 2 virasoro representations the fusion rules in the extended picture are deduced from the known fusion rules for the virasoro representations of lm 1 p and are found to be in agreement with previous works the closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach comment 15 pages

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

a braided monoidal category for free super bosons the chiral conformal field theory of free super bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian lie super algebra h with non degenerate super symmetric pairing the mode algebras of a single free boson and of a single pair of symplectic fermions arise for even odd dimension 1 0 and 0 2 of h respectively in this paper the representations of the untwisted mode algebra of free super bosons are equipped with a tensor product a braiding and an associator in the symplectic fermion case i e if h is purely odd the braided monoidal structure is extended to representations of the z 2z twisted mode algebra the tensor product is obtained by computing spaces of vertex operators the braiding and associator are determined by explicit calculations from three and four point conformal blocks

the statistics of power flow between two continuous one dimensional subsystems abstract the power flow between two continuous subsystems is examined using a wave solution 1 one of the subsystems is excited by time harmonic forces and both subsystems can support only one power transmitting wave mode uncertainty exists in the subsystems which are assumed to be drawn at random from ensembles those ensembles members that have a natural frequency at the excitation frequency are determined the particular members that are resonant are seen to depend on the strength of coupling the statistics of power flow are considered in particular the maximum minimum ensemble arithmetic and geometric means the variance and the cumulative probability distribution of the power flow the maximum power flow undergoes a qualitative change at a particular coupling strength for weak coupling the peak power flows occur at the uncoupled natural frequencies while for stronger coupling they occur at the coupled frequencies the ensemble average power flow may be significantly less than that expected from a normal sea approach due to power re radiation especially for a lightly damped receiving subsystem and strong coupling conversely it may be much greater due to power re injection which occurs particularly for a lightly damped excited subsystem and weak coupling the power flow variance is largest for small damping and weak coupling under which circumstances the cumulative probability distribution has an infrequent but dominant tail of very large power flows

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

a derivation of the coupling loss factors used in statistical energy analysis abstract the conditions under which an exact theory concerning the energy power flow relationships for a complex dynamic system may be reduced to the standard statistical energy analysis sea equations are considered a definition of weak coupling is introduced which encompasses a number of previous definitions and it is shown that the presence of weak coupling does not guarantee the validity of the standard sea assumption that the coupling loss factor clf between two subsystems is zero unless the subsystems are directly connected the standard assumption is likely to be reasonable for systems which are both weakly coupled and reverberant attention is then turned to the clf between two connected subsystems and a general result is obtained which involves a space and frequency averaged green function it is shown that under the appropriate assumptions a number of existing results may be derived from this expression including those produced by both the wave and the modal approaches to sea the exact theory may be used as a basis for developing improved models of the system dynamics

superspace conformal field theory conformal sigma models and wess zumino witten wzw models on coset superspaces provide important examples of logarithmic conformal field theories they possess many applications to problems in string and condensed matter theory we review recent results and developments including the general construction of wzw models on type i supergroups the classification of conformal sigma models and their embedding into string theory

a model of subdiffusive interface dynamics with a local conservation of minimum height we define a new model of interface roughening in one dimension which has the property that the minimum of interface height is conserved locally during the evolution this model corresponds to the limit q of the q color dimer deposition evaporation model introduced by us earlier hari menon and dhar j phys a math gen 28 6517 1995 we present numerical evidence from monte carlo simulations and the exact diagonalization of the evolution operator on finite rings that growth of correlations in this model is subdiffusive with dynamical exponent z 2 5 for periodic boundary conditions the variation of the gap in the relaxation spectrum with system size appears to involve a logarithmic correction term some generalizations of the model are briefly discussed

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

a crossing probability for critical percolation in two dimensions langlands et al considered two crossing probabilities and in their extensive numerical investigations of critical percolation in two dimensions cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the q state potts model we extend his results to find an analogous formula for which compares very well with the numerical results

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

explicit constructions of quasi monte carlo rules for the numerical integration of high dimensional periodic functions in this paper we give explicit constructions of point sets in the s dimensional unit cube yielding quasi monte carlo algorithms which achieve the optimal rate of convergence of the worst case error for numerically integrating high dimensional periodic functions in the classical measure p alpha of the worst case error introduced by korobov the convergence for every even integer alpha ge 1 is of cal o n min alpha d log n s alpha 2 where d is a parameter of the construction which can be chosen arbitrarily large and n is the number of quadrature points this convergence rate is known to be the best possible up to some log n factors we prove the result for the deterministic and also a randomized setting the construction is based on a suitable extension of digital t m s nets over the finite field mathbb z b

critical behaviour in a non local interface model the raise and peel model is a recently proposed one dimensional statistical model describing a fluctuating interface the evolution of the model follows from the competition between adsorption and desorption processes the model is non local due to the possible occurrence of avalanches at a special ratio of the adsorption desorption rates the model is integrable and many rigorous results are known off the critical point the phase diagram and scaling properties are not known in this paper we search for indications of phase transition studying the gap in the spectrum of the non hermitian generator of the stochastic interface evolution we present results for the gap obtained from exact diagonalization and from monte carlo estimates derived from temporal correlations of various observables

representation theory of sl 2 1 abstract in this note we present a complete analysis of finite dimensional representations of the lie superalgebra click to view the mathml source this includes in particular the decomposition of all tensor products into their indecomposable building blocks our derivation makes use of a close relation with the representation theory of click to view the mathml source for which analogous results are described and derived keywords lie superalgebras representation theory

walsh spaces containing smooth functions and quasi monte carlo rules of arbitrary high order we define a walsh space which contains all functions whose partial mixed derivatives up to order delta ge 1 exist and have finite variation in particular for a suitable choice of parameters this implies that certain sobolev spaces are contained in these walsh spaces for this walsh space we then show that quasi monte carlo rules based on digital t alpha s sequences achieve the optimal rate of convergence of the worst case error for numerical integration this rate of convergence is also optimal for the subspace of smooth functions explicit constructions of digital t alpha s sequences are given hence providing explicit quasi monte carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions

on the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies a recent paper has shown that under certain conditions the cross spectral matrix of the forces exerted by a vibrational or acoustic wave field on its surrounding boundaries can be expressed in terms of i the energy of the wave field and ii the direct field dynamic stiffness matrix of the boundary this diffuse field reciprocity relation was derived using wave mechanics and it is not immediately clear how the required wave field properties translate to conditions on the vibrational modes of the system or the applied forcing this issue is addressed here by deriving an extended version of the reciprocity relation using modal methods and the conditions required for the extended version to reduce to the existing relation are delineated it is shown that the existing diffuse field reciprocity relation leads to an anomalous result when used to predict the energy variance of a subsystem and that this anomaly is resolved by using the present extended version of the relation a supplementary result arising from the analysis is that for systems with a sufficient degree of randomness the ensemble average of the dynamic stiffness matrix of a random subsystem is equal to the inverse of the ensemble average of the receptance matrix

response variance prediction for uncertain vibro acoustic systems using a hybrid deterministic statistical method imperfections during the manufacturing process can cause significant variations in the noise and vibration levels exhibited by nominally identical structures any response calculations employed during the design process should ideally take account of these uncertainties and predict the expected range in performance recently a hybrid method has been developed to predict the ensemble average response of a built up system by combining a deterministic model of parts of the system with a statistical model of other components shorter p j and langley r s 2005 j sound vib 288 669 700 in this paper the method is extended to predict the ensemble variance of the response expressions are derived for the variance of the vibrational energies in the statistical components and for the variance of the cross spectrum of the response of the deterministic components which augment the mean values of these quantities predicted by the original theory the method employs a nonparametric model of uncertainty in the sense that the statistical components are taken to carry diffuse wave fields and this obviates the requirement for a detailed description of the system uncertainties the method is validated by application to a range of coupled plate structures and good agreement with detailed monte carlo simulations is found

nonsemisimple fusion algebras and the verlinde formula we find a nonsemisimple fusion algebra open image in new window associated with each 1 p virasoro model we present a nonsemisimple generalization of the verlinde formula which allows us to derive open image in new window from modular transformations of characters

the sea of two coupled plates an investigation into the effects of subsystem irregularity abstract a system comprising two edge coupled plates is considered theoretical predictions of the coupling power and coupling loss factor are made using traditional asymptotic sea wave theory and an analytical wave solution application to rectangular plates these are compared to numerical frequency averages found by fea of the complete system results are presented for variously shaped plates and different levels of damping if the damping is large enough i e for weak coupling the response is independent of the shape of the plates for lighter damping i e strong coupling the response depends significantly on the specific geometry of each plate the coupling power is often substantially less than that predicted by traditional sea theory which also overpredicts the coupling loss factor both the coupling power and coupling loss factor are least for rectangular plates for which the irregularity of the subsystems is least the reasons for this behaviour are attributed to wave coherence or in modal terms to localisation of the global modes of the structure within one or other subsystem a parameter 0is given which provides an estimate of the strength of coupling this parameter arises in the analysis of coupled rectangular plates and gives a conservative estimate for irregularily shaped plates

on the reciprocity relationship between direct field radiation and diffuse reverberant loading this analysis is concerned with the derivation of a diffuse field reciprocity relationship between the diffuse field excitation of a connection to a structural or acoustic subsystem and the radiation impedance of the connection such a relationship has been derived previously for connections described by a single degree of freedom in the present work it is shown that the diffuse field reciprocity relationship also arises when describing the ensemble average response of connections to structural or acoustic subsystems with uncertain boundaries furthermore it is shown that the existing diffuse field reciprocity relationship can be extended to encompass connections that possess an arbitrary number of degrees of freedom the present work has application to i the calculation of the diffuse field response of structural acoustic systems modeled by finite elements boundary elements and infinite elements ii the general calculation of the coupling loss factors employed in statistical energy analysis sea and iii the derivation of an alternative analysis method for describing the dynamic interactions of coupled subsystems with uncertain boundaries a generalized boundary approach to sea

shared information in stationary states of stochastic processes we present four estimators of the shared information or interdepency in ground states given that the coefficients appearing in the wave function are all real non negative numbers and therefore can be interpreted as probabilities of configurations such ground states of hermitian and non hermitian hamiltonians can be given for example by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models we consider in detail the last case the system being a classical not a quantum one using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive conformal invariant and nonconformal invariant massless phases we show that like in the case of the quantum problem the estimators verify the area law with logarithmic corrections when phase transitions take place

logarithmic torus amplitudes abstractfor the example of the logarithmic triplet theory at c 2 the chiral vacuumtorus amplitudes are analysed it is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations as well as a function thatcan be associated to the logarithmic extension of the vacuum representation a fewimplications and generalisations of this result are discussed 1 introduction during the last twenty years much has been understood about the struc ture of rational conformal eld theories rational conformal eld theories are characterisedby the property that they have only nitely many irreducible highest weight representationsof the chiral algebra or vertex operator algebra and that every highest weight represen tation is completely decomposable into irreducible representations the structure of thesetheories is well understood in particular the characters of the irreducible representationstransform into one another under modular transformations 1 see also 2 and the modulars matrix determines the fusion rules via the verlinde formula 3 a general proof for thishas only recently been given in 4 on the other hand it is clear that rational conformal eld theories are rather special andit is therefore important to understand the structure of more general classes of conformal eld theories one such class are the rational logarithmic theories that possess only nitelymany indecomposable representations but for which not all highest weight representationsare completely decomposable the name logarithmic comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts the rst example of a non rational logarithmic conformal eld theory was found in 5 see also 6 and the rstrational example that shall also concern us in this paper was constructed in 7 for somerecent reviews see 8 9 10 from a physics point of view logarithmic conformal eldtheories appear naturally in various models of statistical physics for example in the theory

a modular invariant bulk theory for the c 0 triplet model a proposal for the bulk space of the logarithmic cal w 2 3 triplet model at central charge zero is made the construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory the resulting bulk space is a quotient of the direct sum of projective representations which is isomorphic as a vector space to the direct sum of tensor products of the irreducible representations with their projective covers as a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant and that the boundary state analysis is compatible with the proposed annulus partition functions of this model

dispersion and pollution of the fem solution for the helmholtz equation in one two and three dimensions for high wave numbers the helmholtz equation suffers the so called pollution effect this effect is directly related to the dispersion a method to measure the dispersion on any numerical method related to the classical galerkin fem is presented this method does not require to compute the numerical solution of the problem and is extremely fast numerical results on the classical galerkin fem p method is compared to modified methods presented in the literature a study of the influence of the topology triangles is also carried out the efficiency of the different methods is compared the numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well the most effective modified method is the qsfem 1 2 but it is also very complicated in the general setting the residual free bubble 3 4 is effective in one dimension but not in higher dimensions the least square method 1 5 approach lowers the dispersion but relatively little the results for triangular meshes show that the best topology is the criss cross pattern copyright 1999 john wiley sons ltd

scaling fields in the two dimensional abelian sandpile model we consider the unoriented two dimensional abelian sandpile model from a perspective based on two dimensional conformal field theory we compute lattice correlation functions for various cluster variables at and off criticality from which we infer the field theoretic description in the scaling limit we find perfect agreement with the predictions of a c 2 conformal field theory and its massive perturbation thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory the question of the height 2 variable is also addressed with however no definite conclusion yet

stochastic processes and conformal invariance we discuss a one dimensional model of a fluctuating interface with a dynamic exponent z 1 the events that occur are adsorption which is local and desorption which is nonlocal and may take place over regions of the order of the system size in the thermodynamic limit the time dependence of the system is given by characters of the c 0 logarithmic conformal field theory of percolation this implies in a rigorous way a connection between logarithmic conformal field theory and stochastic processes the finite size scaling behavior of the average height interface width and other observables are obtained the avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite size scaling with new critical exponents

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

validity diagrams of statistical energy analysis abstract this paper is concerned with the validity domain of statistical energy analysis sea which is defined in terms of four criteria the mode count n and the modal overlap m must be high the normalized attenuation factor m and the coupling strength must be small the application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of sea must be delimited this domain is discussed on the basis of geometry of the surfaces delimiting it the diagrams of validity of sea are introduced and discussed a numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach

boundary effects on the vibration statistics of a random plate abstract this paper considers the estimation of statistics of displacement of a vibrating rectangular plate with random wave scatterers the influence of uncertainty is investigated using point impedance theory coherent boundary effects are seen which decrease when the number of scatterers increases the boundary effect is investigated using images and the first side and corner reflections are found to be a minimum requirement to estimate the spatial correlation statistics for point driven response are investigated under the assumption that the statistics of the natural frequencies follow those of the gaussian orthogonal ensemble goe the estimates are compared with monte carlo simulation results and they show good agreement

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

exact results for an asymmetric annihilation process with open boundaries we consider a nonequilibrium reaction diffusion model on a finite one dimensional lattice with bulk and boundary dynamics inspired by the glauber dynamics of the ising model we show that the model has a rich algebraic structure that we use to calculate its properties in particular we show that the markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes this remark leads us to devise a technique which we call the transfer matrix ansatz that allows us to determine the steady state distribution and correlation functions furthermore we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of markov matrices finally we compare the transfer matrix ansatz used here with the matrix product representation of the steady state of one dimensional stochastic models

smooth phases roughening transitions and novel exponents in one dimensional growth models a class of solid on solid growth models with short range interactions and sequential updates is studied the models exhibit both smooth and rough phases in dimension d 1 some of the features of the roughening transition which takes place in these models are related to contact processes or directed percolation type problems the models are analyzed using a mean field approximation scaling arguments and numerical simulations in the smooth phase the symmetry of the underlying dynamics is spontaneously broken a family of order parameters which are not conserved by the dynamics is defined as well as conjugate fields which couple to these order parameters the corresponding critical behavior is studied and novel exponents identified and measured we also show how continuous symmetries can be broken in one dimension a field theory appropriate for studying the roughening transition is introduced and discussed

a quantitative criterion validating coupling power proportionality in statistical energy analysis the response of two general spring coupled elements is investigated to develop a unifying approach to the weak coupling criterion in statistical energy analysis sea first the coupled determinis

the raise and peel model of a fluctuating interface we propose a one dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter the adsorption rate at a special value of this parameter the model has some interesting features for example the spectrum is given by conformal field theory and the stationary non equilibrium probability distribution is given by the two dimensional equilibrium distribution of the ice model with domain wall type boundary conditions this connection is used to find exact analytic expressions for several quantities of the stochastic model vice versa some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model at the special point we study several properties of the model such as the height fluctuations as well as cluster and avalanche distributions the latter has a long tail which shows that the model exhibits self organized criticality we also find in the stationary state a special surface phase transition without enhancement and with a crossover exponent 2 3 furthermore we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken