separate joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two dimensional state space digital filters this paper is concerned with the minimization of roundoff noise subject to l sub 2 norm dynamic range scaling constraints in two dimensional 2 d state space digital filters two methods are proposed with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization in the first method several techniques for the determination of optimal full scale block diagonal diagonal and scalar error feedback matrices for a given 2 d state space digital filter are proposed in the second method an iterative approach for minimizing the roundoff noise under l sub 2 norm dynamic range scaling constraints is developed by jointly optimizing a scalar error feedback matrix and a coordinate transformation matrix which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2 d filter structure with minimum roundoff noise an analytical method for the joint optimization of a general error feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed a numerical example is presented to illustrate the utility of the proposed techniques

critical behavior of sandpile vortices under variable charge in the following we consider a sandpile cellular automaton model height version which also takes account of lattice cyclicity for variable charge of grains projected upon the board the size of the charge or additional grains of sand used to upset the equilibrium and to induce an avalanche is found to affect the distributional forms and in particular the convergence of the estimate for the dynamic exponent exponent estimates show slow variation with lattice dimensionl and some evidence of evolution with charge the scaling region is limited in all cases with noisy decay at all levels of charge results presented extend the work of duarte and goncalves 1990 on the triangular lattice

power law sensitivity to initial conditions for abelian directed self organized critical models abstract the power law sensitivity to initial conditions is investigated for self organized critical soc models within the damage spreading framework a class of two dimensional abelian directed models are analyzed results for the time evolution of the normalized squared euclidian distance indicate that the propagation of a small perturbation damage has the same behavior even assuming different parameters of these models the same technique applied to a non abelian complete toppling version of a directed model leads to a completely different behavior results also suggest that there might exist a connection between the multifractal spectra of a potential energy measure and the power law sensitivity to initial conditions for the abelian soc models as observed for low dimensional systems

a levinson type algorithm for modeling fast sampled data the standard discrete time autoregressive model is poorly suited for modeling series obtained by sampling continuous time processes at fairly rapid rates large computational errors can occur when the levinson algorithm is used to estimate the parameters of this model because the toeplitz covariance matrix is ill suited for inversion an alternative model is developed based on an incremental difference operator rather than the shift operator it is shown that as the sampling period goes to zero unlike the standard autoregressive parameters the coefficients of this model converge to certain parameters that depend directly on the statistics of the continuous time process a levinson type algorithm for efficiently estimating the parameters of this model is derived numerical examples are given to show that when the sampling interval is small this algorithm is considerably less sensitivity to arithmetic roundoff errors than the levinson algorithm

self organized criticality with and without conservation abstract the self organized sandpile models lose criticality if dissipation is introduced recently christensen et al have shown that dissipative automata based on the burridge knopoff earthquake model exhibit critical behavior criticality is qualitatively different for the cases with and without conservation a new characteristic length appears for the dissipative case which diverges slower than the system size for all dissipative models we have found a characteristic frequency in the power spectrum of the released energy which is absent for the conservative case the exponents describing criticality change continuously as a function of the strength of dissipation and crossover phenomena occur in the vicinity of conservation disorder is irrelevant if conservation is present while it destroys criticality in the dissipative case

statistics of epicenters in the olami feder christensen model in two and three dimensions abstract recently abe and suzuki pointed out that epicenters of earthquakes could be connected in order to generate a graph with properties of a scale free network we have shown that the olami feder christensen ofc model for the dynamics of earthquakes defined both in square and cubic lattices is able to reproduce this new behavior the distribution of distances between successive earthquakes is also presented our results indicate the robustness of the ofc model to describe earthquake dynamics surprisingly we found that only the non conservative version of the ofc model generates a network with scale free properties the conservative version instead behaves like a random graph the distribution of distances in 3 d and the distribution of connectivities in a 2 d lattice confirm the differences observed between the conservative and non conservative regime

directed abelian algebras and their application to stochastic models with each directed acyclic graph this includes some d dimensional lattices one can associate some abelian algebras that we call directed abelian algebras daas on each site of the graph one attaches a generator of the algebra these algebras depend on several parameters and are semisimple using any daa one can define a family of hamiltonians which give the continuous time evolution of a stochastic process the calculation of the spectra and ground state wave functions stationary state probability distributions is an easy algebraic exercise if one considers d dimensional lattices and chooses hamiltonians linear in the generators in finite size scaling the hamiltonian spectrum is gapless with a critical dynamic exponent z d one possible application of the daa is to sandpile models in the paper we present this application considering one and two dimensional lattices in the one dimensional case when the daa conserves the number of particles the avalanches belong to the random walker universality class critical exponent ensuremath sigma ensuremath tau 3 2 we study the local density of particles inside large avalanches showing a depletion of particles at the source of the avalanche and an enrichment at its end in two dimensions we did extensive monte carlo simulations and found ensuremath sigma ensuremath tau 1 780 ifmmode pm else textpm fi 0 005

a new scheme for designing iir filters with finite wordlength coefficients a scheme for designing iir digital filters with specified coefficient wordlengths based on successive digitization and reoptimization is presented and compared to related approaches via a number of design examples the ordered digitization of the coefficients based on their sensitivity with each digitization followed by reoptimization of the remaining coefficients results in a much reduced computation effort in comparison to related design approaches

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

continuous transforming the btw to the manna model abstract in this paper we define some stochastic perturbations of the btw model which make it into manna model these models have a continuous parameter p where p 0 and 1 correspond to the btw and manna models respectively we have investigated the properties of the statistical observables of the waves of avalanches for various values of p our data supports the expectation of a crossover in such systems at large scales manna model is dominant therefore we find strong evidence in favor of the universality classes being distinct also it is observed that the btw fixed point is unstable when manna type perturbation is added to the model

phase transition and critical behavior in a model of organized criticality we study a model of organized criticality where a single avalanche propagates through an a priori static i e organized sandpile configuration the latter is chosen according to an i i d distribution from a borel probability measure on 0 1 the avalanche dynamics is driven by a standard toppling rule however we simplify the geometry by placing the problem on a directed rooted tree as our main result we characterize which are critical in the sense that they do not admit an infinite avalanche but exhibit a power law decay of avalanche sizes our analysis reveals close connections to directed site percolation both in the characterization of criticality and in the values of the critical exponents

universality classes in directed sandpile models we perform large scale numerical simulations of a directed version of the two state stochastic sandpile model numerical results show that this stochastic model defines a new universality class with respect to the abelian directed sandpile the physical origin of the different critical behaviour has to be ascribed to the presence of multiple topplings in the stochastic model these results provide new insight into the long debated question of universality in abelian and stochastic sandpiles

reduced dimensionality geometric approach to fault identification in stochastic structural systems an information based reduced dimensionality formulation of the recently introduced geometric approach to fault identification in stochastic structural systems is presented the feature vector which conveys the system information used in fault identification and is a key element of the general geometric approach is transformed into a suitable coordinate system within which information compression may be best achieved dimensionality selection is subsequently based on bounding of the information loss expressed in terms of logarithmic entropy below a certain threshold the information based reduced dimensionality formulation thus presents a formal and systematic procedure for the proper selection of a minimal dimensionality feature vector and leads to substantial simplification and potentially improved fault identification its application to fault identification in a laboratory scale beam and a planar truss structure demonstrates high effectiveness and performance characteristics essentially equivalent to those of the full dimensionality formulation

three level description of the domino cellular automaton motivated by the approach of kinetic theory of gases a three level description microscopic mesoscopic and macroscopic of cellular automaton is presented to provide an analytical treatment a simple domino cellular automaton with avalanches is constructed formulas concerning exact relations for density clusters avalanches and other parameters in a stationary state are derived some relations approximately valid for deviations from the stationary state are found and the adequate ito equation is constructed it provides the time evolution description of the density on the macroscopic level the results support an idea of application of the ito equation to some natural time series

on the problem of stochastic experimental modal analysis based on multiple excitation multiple response data part ii the modal analysis approach in this part of the paper the stochastic multiple excitation multiple response modal analysis problem is considered the relationship between the actual structural and noise dynamics and their discrete special form armax type representation is studied for each one of the vibration displacement velocity and acceleration data cases and a novel and effective modal analysis approach is introduced that unlike previous schemes is capable of operating on any one of these types of data records by accounting for issues such as the required excitation signal type and stochastic model form algorithmic instability occurrence and other well known estimation difficulties model structure estimation and model validation as well as model reduction and analysis based on the dispersion analysis methodology introduced in the first part of the paper 1 the proposed approach not only overcomes the limitations and drawbacks of current schemes but also constitutes the first comprehensive procedure for stochastic multiple excitation multiple response experimental modal analysis the effectiveness of the approach is demonstrated through numerical experiments with structural systems characterized by well separated and closely spaced modes and data records of various lengths and signal to noise ratios comparisons with the classical frequency domain method and the deterministic eigensystem realization algorithm are also made and the approach is finally used for the experimental modal analysis of a three span beam from laboratory data

self organized critically in vector state automata we propose a class of cellular automata where on each lattice site the state is characterized by a vector these vector state models are observed to display self organized criticality and lie in the same universality class as that of the sandpile height model it is found that a conservative quantity is necessary for the systems to reach the self organized criticality

geometric approach to nondestructive identification of faults in stochastic structural systems a novel geometric approach to the nondestructive identification of faults in stochastic structural systems based on vibration test data is introduced the approach uses a partial and reduced order dynamic system model a properly selected stochastic feature space the notion of fault mode as the union of all faults of one particular cause and its approximate geometric representation as a subspace of the feature space fault identification is accomplished by determining the fault mode subspace within which the current fault lies the geometric approach offers important advantages over alternative structural fault identification schemes including its inherent accommodation of stochastic effects the use of a minimal number of measurement locations and its ability to operate on any type of vibration data and use only partial and reduced order models while relaxing the need for stiffness matrix or modal parameter estimates its effectiveness is demonstrated via fault identification in a laboratory scale beam and a finite element model of a planar truss structure where the importance of stochastic effect accommodation is addressed

algebraic aspects of abelian sandpile models the abelian sandpile models feature a finite abelian group g generated by the operators corresponding to particle addition at various sites we study the canonical decomposition of g as a product of cyclic groups g z d1 z d2 z d3 z dg where g is the least number of generators of g and di is a multiple of di 1 the structure of g is determined in terms of the toppling matrix delta we construct scalar functions linear in the height variables of the pile that are invariant under toppling at any site these invariants provide convenient coordinates to label the recurrent configurations of the sandpile for an l l square lattice we show that g l in this case we observe that the system has non trivial symmetries transcending the obvious symmetries of the square namely those coming from the action of the cyclotomic galois group gall of the 2 l 1 th roots of unity which operates on the set of eigenvalues of h these eigenvalues are algebraic integers the product of which is the order mod g mod with the help of gall we are able to group the eigenvalues into certain subsets the products of which are separately integers and thus obtain an explicit factorization of mod g mod we also use gall to define other simpler sets of toppling invariants

comparative study of finite wordlength effects in shift and delta operator parameterizations the sensitivity of the transfer function to finite wordlength effect errors in the implementation of the coefficients of both shift operator and delta operator parameterizations is analyzed the relationships between optimal realization sets in shift and delta operator are established it is shown that the optimal realization in delta operator parameterizations have better performance than those in their shift operator counterparts when the poles of the systems are clustered around z 1 a numerical example is given

steady state and relaxation spectrum of the oslo rice pile model we show that the one dimensional oslo rice pile model is a special case of the abelian distributed processors model the exact steady state of the model is determined we show that the time evolution operator w for the system satisfies the equation w n 1 w n where n l l 1 2 for a pile with l sites this is used to prove that w has only one eigenvalue 1 corresponding to the steady state and all other eigenvalues are exactly zero also all connected time dependent correlation functions in the steady state of the pile are exactly zero for time difference greater that n generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed

exact solution of the totally asymmetric oslo model recently it has been found pruessner g and jensen h j 2003 phys rev lett 91 244303 that a totally asymmetric variant of the oslo model christensen k et al 1996 phys rev lett 77 107 represents the entire universality class of the oslo model with anisotropy the totally asymmetric model can be solved without scaling assumptions by finding recursively the eigenvectors of the markov matrix which can be suitably modified to produce the moment generating function of the relevant observable this method should be applicable to many other stochastic processes

additivity property and emergence of power laws in nonequilibrium steady states we show that an equilibriumlike additivity property can remarkably lead to power law distributions observed frequently in a wide class of out of equilibrium systems the additivity property can determine the full scaling form of the distribution functions and the associated exponents the asymptotic behavior of these distributions is solely governed by branch cut singularity in the variance of subsystem mass to substantiate these claims we explicitly calculate using the additivity property subsystem mass distributions in a wide class of previously studied mass aggregation models as well as in their variants these results could help in the thermodynamic characterization of nonequilibrium critical phenomena

model for force fluctuations in bead packs we study theoretically the complex network of forces that is responsible for the static structure and properties of granular materials we present detailed calculations for a model in which the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead of the pile we compare our results for the force distribution function for this model including exact results for certain contact angle probability distributions with numerical simulations of force distributions in random sphere packings this model reproduces many aspects of the force distribution observed both in experiment and in numerical simulations of sphere packings our model is closely related to some that have been studied in the context of self organized criticality we present evidence that in the force distribution context critical power law force distributions occur only when a parameter hidden in other interpretations is tuned our numerical mean field and exact results all indicate that for almost all contact distributions the distribution of forces decays exponentially at large forces

jointly optimal high order error feedback and realization for roundoff noise minimization in 1 d and 2 d state space digital filters joint optimization of high order error feedback and state space realization for minimizing roundoff noise at filter output subject to l2 scaling constraints is investigated for one and two dimensional state space digital filters linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored and an efficient quasi newton algorithm is then applied to solve the unconstrained optimization problem iteratively in this connection closed form formulas are derived for fast and accurate gradient evaluation finally case studies are presented to demonstrate that the high order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state space coordinate transformation and high order error feedback matrices are optimized separately

height correlations in the abelian sandpile model the authors study the distribution of heights in the self organized critical state of the abelian sandpile model on a d dimensional hypercubic lattice they calculate analytically the concentration of sites having minimum allowed value in the critical state they also calculate in the critical state the probability that the heights at two sites separated by a distance r would both have minimum values and show that the lowest order r dependent term in it varies as r 2d for large r

alternative discrete time operators an algorithm for optimal selection of parameters we consider the issue of parameter sensitivity in models based on alternative discrete time operators adtos a generic first order adto is proposed that encompasses all the known first order adtos introduced so far in the literature new bounds on the operator parameters are derived and a new algorithm is given for optimally selecting the parameters to give minimum parameter sensitivity

area distribution for directed random walks we study the probability distribution for the area under a directed random walk in the plane the walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed this model captures some of the qualitative features of earthquakes avalanches and other self organized critical phenomena in one dimension by finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4 3

dynamical and spatial aspects of sandpile cellular automata the bak tang and wiesenfeld cellular automaton is simulated in 1 2 3 4 and 5 dimensions we define a new set of scaling exponents by introducing the concept of conditional expectation values scaling relations are derived and checked numerically and the critical dimension is discussed we address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2 the scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation primarily the results of our work show that the flow of sand down the slope does not have a 1 f power spectrum in any dimension although the model does show clear critical behavior with scaling exponents depending on the dimension the power spectrum behaves as 1 f 2 in all the dimensions considered

analytical and numerical studies on deterministic sandpiles abstract an analytical method of calculation of attractor periods and spilling numbers for higher dimensional deterministic sandpiles is used for models with periodic boundary conditions numerical results for lattices up to 9 9 are presented for the open boundary deterministic model the dependence of attractor periods and exponents of avalanche size distributions on input site position is shown

roundoff noise minimization using delta operator realizations digital filters the authors examine the advantages of using delta operator state space realizations rather than shift operator realizations of transfer functions in terms of minimizing roundoff noise gain they give several conditions under which the optimal roundoff noise gain for delta operator realizations is smaller than the optimal gain for shift operator realizations they then illustrate that even sparse and hence nonoptimal delta operator realizations can have smaller roundoff noise gain than the optimal shift operator realizations

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

landslides on sandpiles some moment relations in one dimension several relations between the structure of stable recurrent states and the statistics of avalanches in a one dimensional sandpile automaton are derived and numerically verified in particular it is shown that the average avalanche size is determined by the second rather than the first moment of the distribution of trough distances the two moments scale differently with system size which implies multiscaling for the distribution moreover the scaling of edge events avalanches which fall off the pile is shown to differ from that of bulk events avalanches which remain on the pile

statistical mechanics of the chinese restaurant process lack of self averaging anomalous finite size effects and condensation the pitman yor or chinese restaurant process is a stochastic process that generates distributions following a power law with exponents lower than 2 as found in numerous physical biological technological and social systems we discuss its rich behavior with the tools and viewpoint of statistical mechanics we show that this process invariably gives rise to a condensation i e a distribution dominated by a finite number of classes we also evaluate thoroughly the finite size effects finding that the lack of stationary state and self averaging of the process creates realization dependent cutoffs and behavior of the distributions with no equivalent in other statistical mechanical models

roundoff noise minimization and pole zero sensitivity in fixed point digital filters using residue feedback the problem of designing a finite wordlength fixed point realization of an nth order digital filter which uses residue feedback to minimize the output roundoff noise subject to l 2 scaling is considered the new structures require n extra additions but no more multiplications than the earlier low noise structures of mullis and roberts and have lower roundoff noise for sufficiently narrow bandwidth filters a new set of filter invariants called the residue modes are defined which characterize the new low noise structures and determine the output noise variance if the sum of the residue modes is less than the sum of the second order modes of mullis and roberts then lower roundoff noise is achieved every filter structure is shown to define a unique symmetrizing matrix q pole zero sensitivities and a new noise measure are defined in terms of q numerical results are included which compare the noise and pole zero sensitivity characteristics of different filter structures

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

exact results for the directed abelian sandpile models we define the directed abelian sandpile models by introducing a parameter c representing the degree of anisotropy in the avalanche processes where c 1 is for the isotropic case we calculate some quantities characterizing the self organized critical states on the one and two dimensional lattices in particular we obtain the expected number of topplings per added particle t which shows the dependence on the lattice size l as lx for large l we show that the critical exponent x does not depend on the dimensionality d at least for d 1 and 2 and that when any anisotropy is included in the system x 1 while x 2 in the isotropic system this result gives a rigorous proof of the conjecture by kadanoff et al 1989 a 39 6524 37 that the anisotropy will distinguish different universality classes we introduce a new critical exponent defined by x liml t l with as for both in d 1 and 2 we obtain 1

low roundoff noise augmented realizations of iir filters the scaling factor in a direct form ii infinite impulse response iir digital filter depends on only the system poles not its zeros adding pole zero cancellation pairs changes the scaling factor and the associated roundoff noise without affecting the transfer function it is shown that with l sub 2 scaling the upper bound of the noise reduction for direct form ii iir digital filters using k pole zero cancellation pairs is about 6k db the authors consider parallel form 3p to be the basic structure in which each subfilter is augmented with one or two pole zero cancellation pairs and realized in direct form ii they seek an iterative optimization process for each subfilter to find the optimal pole zero cancellation pairs that minimize the roundoff noise experimental results indicate the global optimal solution for these augmented subfilters can be obtained by using simple criteria to determine the starting points of the iterative processes alternative structures illustrative examples and overflow limit cycles are examined

identification of the structure parameters using short time non stationary stochastic excitation abstract in this paper we propose an approach to the flexural stiffness or eigenvalue frequency identification of a linear structure using a non stationary stochastic excitation process the idea of the proposed approach lies within time domain input output methods the proposed method is based on transforming the dynamical problem into a static one by integrating the input and the output signals the output signal is the structure reaction i e structure displacements due to the short time irregular load of random type the systems with single and multiple degrees of freedom as well as continuous systems are considered

exact solutions for a mean field abelian sandpile we introduce a model for a sandpile with n sites critical height n and each site connected to every other site it is thus a mean field model in the spin glass sense we find an exact solution for the steady state probability distribution of avalanche sizes and discuss its asymptotics for large n

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

some results and a conjecture for manna s stochastic sandpile model we present some analytical results for the stochastic sandpile model studied earlier by manna in this model the operators corresponding to particle addition at different sites commute the eigenvalues of operators satisfy a system of coupled polynomial equations for an l l square we construct a nontrivial toppling invariant and hence a ladder operator which acting on eigenvectors of the evolution operator gives new eigenvectors with different eigenvalues for periodic boundary conditions in one direction one more toppling invariant can be constructed we show that there are many forbidden subconfigurations and only an exponentially small fraction of all stable configurations are recurrent we obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration and conjecture a formula for its exact value for finite size rectangles

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

delay replacement in direct form structures a class of high order low sensitivity digital filter structures is generated by means of a direct transformation of a high order direct form df structure the proposed structures have low coefficient dynamic range and exhibit low coefficient frequency response sensitivity and roundoff noise characteristics yet require only 3n multiplications per output sample as compared to 2n in the direct form case

reunion and survival of interacting walkers the reunion and survival probabilities of p random walkers in d dimensions with a mutual repulsive interaction are formulated via appropriate partition functions of directed polymers the exponents that describe the decay of these probabilities with length are obtained through renormalization group theory to o mathrm ensuremath epsilon 2 where ensuremath epsilon 2 d the distribution function and the probability of n out of p walkers meeting are also discussed to first order the distribution function is a gaussian one modified by an anomalous exponent of the length of the polymer n the procedure is generalized to multicritical many body interactions for these multicritical cases the exponents are obtained to second order in the relevant ensuremath epsilon s at the upper critical dimension of the interaction there is a logarithmic correction other than the gaussian exponent an interesting consequence is a logarithmic correction for one dimensional walkers with a three body repulsive interaction

on the problem of stochastic experimental modal analysis based on multiple excitation multiple response data part i dispersion analysis of continuous time structural systems despite its importance and the undisputable significance of stochastic effects the problem of multiple excitation multiple response experimental modal analysis has thus far been almost exclusively considered within a deterministic framework in this paper a novel comprehensive and effective stochastic approach that unlike alternative schemes can operate on vibration displacement velocity or acceleration data is introduced the proposed approach is capable of effectively dealing with noise corrupted vibration data while also being characterized by unique features that enable it to overcome major drawbacks of current modal analysis methods and achieve high performance characteristics by employing a proper and mutually compatible force excitation signal type and stochastic model forms b an estimation scheme that circumvents problems such as algorithmic instability wrong convergence and high computational complexity while requiring no initial guess parameter values c effective model structure estimation and model validation procedures and d appropriate model transformation reduction and analysis procedures based on a novel dispersion analysis methodology this dispersion analysis methodology is a physically meaningful way of assessing the relative importance of the estimated vibrational modes based on their contributions dispersions to the vibration signal energy the effects of modal cross correlations are fully accounted for physical interpretations are provided in both the correlation and spectral domains and the phenomenon of negative dispersion modes is investigated and physically interpreted the effectiveness of the proposed approach is finally verified via numerical and laboratory experiments as well as comparisons with the classical frequency domain method and the deterministic eigensystem realization algorithm era the paper is divided into two parts the proposed dispersion analysis methodology is introduced in the first one whereas the complete stochastic experimental modal analysis approach is presented in the second 23

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

scale free network on a vertical plane a scale free network is grown in the euclidean space with a global directional bias on a vertical plane nodes are introduced at unit rate at randomly selected points and a node is allowed to be connected only to the subset of nodes which are below it using the attachment probability pi i t approximately k i t l alpha our numerical results indicate that the directed scale free network for alpha 0 belongs to a different universality class compared to the isotropic scale free network for alpha infinity the link length distribution is calculated analytically for all values of alpha

roundoff noise minimization for 2 d state space digital filters using joint optimization of error feedback and realization the joint optimization problem of error feedback and realization for two dimensional 2 d state space digital filters to minimize the effects of roundoff noise at the filter output subject to l2 norm dynamic range scaling constraints is investigated it is shown that the problem can be converted into an unconstrained optimization problem by using linear algebraic techniques the unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi newton algorithm with closed form formulas for key gradient evaluation analytical details are given as to how the proposed technique can be applied to the cases where the error feedback matrix is a general block diagonal diagonal or block scalar matrix a case study is presented to illustrate the utility of the proposed technique