the binary weight distribution of the extended 2m 2m 4 code of the reed solomon code over gf 2m with generator polynomial x α x α2 x α3 abstract consider an n k linear code with symbols from gf 2 m if each code symbol is represented by a binary m tuple using a certain basis for gf 2 m we obtain a binary nm km linear code called a binary image of the original code in this paper we present a lower bound on the minimum weight of a binary image of a cyclic code over gf 2 m and the weight enumerator for a binary image of the extended 2 m 2 m 4 code of the reed solomon code over gf 2 m with generator polynomial x x 2 x 3 and its dual code where is a primitive element in gf 2 m

geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

surface diffusion coefficients for room acoustics free field measures a surface diffusion coefficient is needed in room acoustics to enable the quality of diffusing surfaces to be evaluated it may also facilitate more accurate geometric room acoustic models this paper concentrates on diffusion coefficients derived from free field polar responses an extensive set of two and three dimensional measurements and predictions was used to test the worth of different diffusion coefficient definitions the merits and problems associated with these types of coefficients are discussed and past parameters reviewed two new coefficients are described the new measure based on the autocorrelation function is forwarded as the best free field coefficient the strengths and weaknesses of the coefficient are defined

critical exponents of the ising model on low dimensional fractal media abstract the critical behavior of the ising model on fractal substrates with noninteger hausdorff dimension d h 2 and infinite ramification order is studied by means of the short time critical dynamic scaling approach our determinations of the critical temperatures and critical exponents and are compared to the predictions of the wilson fisher expansion the wallace zia expansion the transfer matrix method and more recent monte carlo simulations using finite size scaling analysis we also determined the effective dimension d e f which plays the role of the euclidean dimension in the formulation of the dynamic scaling and in the hyperscaling relationship d e f 2 furthermore we obtained the dynamic exponent z of the nonequilibrium correlation length and the exponent that governs the initial increase of the magnetization our results are consistent with the convergence of the lower critical dimension towards d 1 for fractal substrates and suggest that the hausdorff dimension may be different from the effective dimension

dimer monomer model on the sierpinski gasket abstract we present the numbers of dimer monomers m d n on the sierpinski gasket s g d n at stage n with dimension d equal to two three and four the upper and lower bounds for the asymptotic growth constant defined as z s g d lim v ln m d n v where v is the number of vertices on s g d n are derived in terms of the results at a certain stage as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of z s g d can be evaluated with more than a hundred significant figures accurate from the results for d 2 3 4 we conjecture the upper and lower bounds of z s g d for general dimension the corresponding results on the generalized sierpinski gasket s g d b n with d 2 and b 3 4 are also obtained

on the yang lee edge singularity for ising model on nonhomogeneous structures abstract we studied distribution of zeros of the partition function of ferromagnetic ising model near the yang lee edge on two self similar structures we have shown that the nature of associated critical behavior crucially depends on the local lattice structure if the sites of higher coordination number make an infinite connected cluster then we find an usual power law behavior while a logarithmic divergence of the correlation length develops near the edge in the case that these sites make only finite islands we reveal also a close connection between yang lee edge critical behavior and critical behavior of a simple zero field gaussian model on the same structures

ultradiffusion on the fractal branching koch curves ultradiffusion is defined on a family of fractal branching koch curves by an exact renormalization decimation transformation the anomalous long time behaviour of the autocorrelation function is obtained for the particular hierarchy the authors find a non universal crossover c1 m 1 2m and a universal one c2 1 2 as the effective temperature c is increased where m is the maximum ramification number of the fractal they also find that for large m the dynamic behaviour of ultradiffusion on the fractal is independent of the effective temperature c

percolation thresholds on finitely ramified fractals exact renormalisation group recursion relations are used to estimate the effective percolation thresholds for site and bond percolation on finite generation sierpinski gaskets and for bond percolation on branching koch curves

counter based compaction delay and stuck open faults in this paper we study the properties of all major counter based compaction schemes when the circuit under test is affected by delay or stuck open faults we present an error model that accurately describes the behavior of such circuits the error model inherits from the asymmetric error model using this model we compute exact aliasing probability for any test session length we also determine the asymptotic aliasing probability examples that compare aliasing in counter based compaction with aliasing in lfsrs indicate that the latter is more predictable

test pattern generation based on reed muller coefficients reed muller coefficients are used to generate a test pattern selection procedure for detecting single stuck at faults this procedure is based on the heuristics deduced from the way in which the spectral coefficients are affected by such faults the changes that the spectral coefficients undergo are also compared with the fault models used most frequently to model defects in combinational circuits the proposed pattern generating procedure does not need a simulation of the circuit for each of the possible stuck at faults and its complexity is proportional to the number of gates in the circuit the proportionality constant increases exponentially as the number of inputs in the circuit increases to evaluate the performance of the proposed method its application to some benchmark circuits including the alu 74181 is presented

yang lee edge singularity on a class of tree like lattices the density of zeros of the partition function of the ising model on a class of tree like lattices is studied an exact closed form expression for the pertinent critical exponents is derived by using a couple of recursion relations which have a singular behaviour near the yang lee edge

comparative analysis of different implementations of multiple input signature analyzers signature analysis is an accepted method of obtaining data compression for built in testing applications the author deals with a unified approach to the analysis of multiple input signature analyzers by considering them as finite state switching circuits this approach is used to investigate and compare different implementations and it is shown that there is a large range of alternatives to achieve a given characteristic polynomial particular emphasis is placed on the two most common implementations it is shown that both perform polynomial division with each input of one circuit being equivalent to a combination of inputs of the other the approach also gives a method of expressing aliasing patterns for both implementations which leads to a study of the differences with regard to certain error detection capabilities

analysis and design of linear finite state machines for signature analysis testing the authors present a theoretical investigation of the aliasing error probability aep in signature analysis testing by means of linear finite state machines lfsms the equations of the resulting markov chain model of the lfsm are solved to determine an exact expression of the aep as a function of the main lfsm features and of the relevant parameters of the testing environment this expression is used to prove criteria for the synthesis of lfsms with minimum asymptotic and transient aep a fundamental lower bound on the aep is presented which represents the performance limit of any lfsm with respect to aliasing minimization it is shown that the aep in machines realizing counters mod 2 sup k 1 is the closest to such a bound in particular periodically reaching it

boundary element solution for periodic acoustic problems abstract this work shows when using the boundary element method to solve 3d acoustic scattering problems from periodic structures the coefficient matrix can be represented as a block toeplitz matrix by exploiting the toeplitz structure the computational time and storage requirements to construct the coefficient matrix are significantly reduced to solve the linear system of equations the original matrix is embedded into a larger and more structured matrix called the block circulant matrix discrete fourier transform is then employed in an iterative algorithm to solve the block toeplitz system to demonstrate the effectiveness of the formulation for periodic acoustic problems two exterior acoustic case studies are considered the first case study examines a continuous structure to predict the noise generated by a sharp edged flat plate under quadrupole excitation directivity plots obtained using the periodic boundary element method technique are compared with numerical results obtained using a conventional boundary element model the second case study examines a discrete periodic structure to predict the acoustic performance of a sonic crystal noise barrier results for the barrier insertion loss are compared with both finite element results and available data in the literature

potts model on sierpinski carpets and their relation to hypercubic lattices of non integer dimensionality the authors study the critical frontiers and the high and low temperature asymptotic behavior of the potts model two site correlation function for several sierpinski carpets with fractal dimension lying between one and two they simulate the sierpinski carpets by means of appropriate hierarchical lattices obtaining from their basic cells the two graphs among others that arise from the anisotropic bond moving schemes they correspond to the asymptotic high and low temperature dominant terms for the hierarchical lattice two site correlation function exact result by taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the sierpinski carpets it is then possible to compare them with similar results obtained from analytical extension to d 2 of the hypercubic lattice their results indicate that these asymptotic behaviors do not always coincide suggesting that possibly they may not identify the sierpinski carpets exhibited by gefen et al 1984 as a correct implementation of the analytical extension of hypercubic lattices

coupled integral equations for sound propagation above a hard ground surface with trench cuttings a set of coupled integral equations is formulated for the investigation of sound propagation from an infinitesimal harmonic line source above a hard ground surface corrugated with cuttings two half space green s functions are employed in the formulation the first one defined for the upper half space is used to reduce the problem size and eliminate the edge effect resulting from the boundary truncation the other one for the lower half space is to simplify the representation of the neumann dirichlet map as a result the unknowns are only distributed over the corrugated part of the surface which leads to substantial reduction in the size of the final linear system the computational complexity of the neumann dirichlet map is also reduced the method is used to analyze the behavior of sound propagation above textured surfaces the impedance of which is expectedly altered the effects of number and opening of trench cuttings and the effect of source height are investigated the conclusions drawn can be used for reference in a practical problem of mitigating gun blast noise

notes on multiple input signature analysis many results regarding the probability of aliasing for multiple input compactors have been derived under error assumptions that are not very realistic for vlsi circuits recently the value of aliasing probability has been proven to tend to 2 sup k where k is the number of binary memory elements of the linear compactor this result is based on the assumption that the compactor is characterized by an irreducible polynomial and that the no error vector has a probability different from zero in these notes the above result is generalized more specifically it is proved that it is valid if any two error vectors neither of which needs to be the no error vector have probabilities of occurrence different from zero to make the error model complete the situation in which exactly one error vector has a probability different from zero is also considered for the latter type of error distributions the test lengths at which aliasing occurs are determined simple proofs for the results are provided they are based on standard linear algebra notions and well known theorems

ising model on a self avoiding walk fluctuation arguments combined with basic features of the geometry of self avoiding walks saws are shown to lead to zero transition temperature tc for the ising model on a saw the ising model on a non random fractal representing the essential geometrical features of saws is treated exactly by a decimation method it also leads to tc 0 and to ising saw critical behavior different from that on an ising chain

a boundary integral formulation for the prediction of acoustic scattering from periodic structures the scattering of sound by reflecting surfaces is an important feature in room acoustics the boundary element method has been shown in the past to be an accurate method to calculate the scattering pattern of complex surfaces however this numerical method is expensive to use on large surfaces where a large number of elements is required this paper proposes a method of reducing the number of boundary elements required to model large but periodic structures the method is based on a consideration of the relationship between the field variables at two points exactly one period apart the problem is reduced to the modeling of just one period of the structure in cases of plane wave incidence the prediction using this method agrees very well with the prediction obtained by modeling the complete structure with the point source in the near field of the surface the incidence field is no longer periodic and the assumption of a constant periodic factor does not hold in this case it is shown that the metho

absence of phase transition in the xy model on menger sponge abstract we have performed a monte carlo study of the classical xy model on a menger sponge with the wolff cluster algorithm u wolff 1989 the menger sponge is a fractal object with infinite order of ramification and fractal dimension d log 20 log 3 2 7268 from the dependence of the helicity modulus on system size and on boundary conditions we conclude that there is no phase transition in the system at any finite temperature

a note on aliasing probability for multiple input signature analyzer recently d k pradhan s k gupta and m g karpovsky 1990 affirmed that the codes generated by x alpha x beta were not reed solomon where alpha and beta were roots of the distinct primitive polynomials of degree m it was assumed that beta alpha sup 8 where s 1 and 2 sup m 1 were mutually prime the authors show that under the above assumption the respective codes are reed solomon codes

aliasing probability for multiple input signature analyzer single and multiple multiple input signature register misr aliasing probability expressions are presented for arbitrary test lengths a framework based on algebraic codes is developed for the analysis and synthesis of misr based test response compressors for bist this framework is used to develop closed form expressions for the aliasing probability of misr for arbitrary test length an error model based on q ary symmetric channel is proposed using more realistic assumptions results are presented that provide the weight distributions for q ary codes q 2 sup m where the circuit under test has m outputs these results are used to compute the aliasing probability for the misr compression technique for arbitrary test lengths this result is extended to compression using two different misrs it is shown that significant improvements can be obtained by using two signature analyzers instead of one the weight distribution of a class of codes of arbitrary length is also given

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

critical dynamics of heisenberg spins on self avoiding walk chains the dynamical exponent z for the critical spin wave dynamics of nearest neighbour interacting heisenberg spins on a self avoiding walk saw chain is estimated here using a scaling picture and also applying a real space renormalisation group technique to some quasilinear fractal lattices the results indicate z 2dt where d is the fractal dimensionality of the saw chain lattice and t is the exponent for the length of the shortest nearest neighbour connecting path of the saw

critical dynamics of the kinetic potts model on some fractals the critical dynamics of the kinetic potts model on koch curves and regular fractals is studied by means of the exact time dependent renormalization group method different critical dynamics are found on these two families of fractals it is shown that the value of the dynamic critical exponent z depends on both the potts dimensionality q and the transition rates asymmetry coefficient alpha for koch curves the scaling law of the dynamics exponent z df f q alpha v while for regular fractals z df 2f q alpha v where f q alpha characterizes the dependence of the dynamics exponent z on potts dimensionality q and the transition rates asymmetry coefficient alpha and v is the static exponent of the correlation length

quantum field theory models on fractal spacetime continuing the analysis of a previous paper the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non gaussian ultraviolet renormalization group fixed points in 4 e dimensions

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

signature analysis for multiple output circuits a scheme of a good signature analysis by a linear feedback shift register lfsr is presented it works for k output circuits even if k is greater than the register length it is built according to rules which are presented in the correspondence taking into account error models which are introduced the rules are derived from a property which is formally shown for one kind of lfsr however some of them apply to other lfsr schemes too

a renormalisation group approach to the potts model on the koch curve a three parameter decimation renormalisation is constructed and used to produce exact rg recursion relations for the potts model on the koch curve including all the nearest neighbour interactions only two trivial fixed points are obtained the authors show that no phase transition occurs at finite temperature

application of bond moving renormalization group approach to fractal lattices we investigate the application of the migdal kadanoff bond moving renormalization group rg approach to fractal lattices we find the following two results first for inhomogeneous interaction lattice models bond moving involving inequivalent bonds is unsuitable because it violates the condition 0 is the perturbation potential resulting from moving the bonds second the condition 0 does not uniquely determine the way to move bonds different choices of bond moving yield different rg recursion relations and corresponding fixed points which makes the conclusions concerning the phase transition quite uncertain

critical properties of the s4 model on diamond type hierarchial lattices we study the critical properties of the s 4 model on diamond type hierarchical lattices in the presence of an external magnetic field it is assumed that for this type of inhomogenous fractal lattice the gaussian distribution constant the four spins interaction parameter and the external magnetic field in the s4 model depend on the coordination number of the site on the fractal lattices by combining the real space renormalization group scheme with the cumulative expansion method we obtain the critical points and further calculate critical exponents according to the scaling theory the results show that on diamond type hierarchical lattices with branches m 4 of 2 bonds the critical point of the s4 model is just the gaussian fixed point and therefore critical exponents are in full agreement with those of the gaussian model and that on those with m 4 the wilson fisher fixed point as well as the gaussian fixed point is obtained the wilson fisher fixed point has a decisive influence on the critical behavior of the system and critical exponents are related to the fractal dimensionality we found that the s4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality c 2005 elsevier b v all rights reserved

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

the ising model on a saw using a renormalization group calculation the authors show that the most probable value of the static correlation function of the ising model on a saw decays as a stretched exponential with an exponent equal to the dimension d of the saw they prove that the thermal exponent vt equals d 1 on the basis of extensive monte carlo simulations they conclude that the spin spin autocorrelation function also decays as a stretched exponential they find evidence for a breakdown of dynamical scaling

the ice model and the eight vertex model on the two dimensional sierpinski gasket abstract we present the numbers of ice model configurations with boltzmann factors equal to one i n on the two dimensional sierpinski gasket s g n at stage n the upper and lower bounds for the entropy per site defined as lim v ln i n v where v is the number of vertices on s g n are derived in terms of the results at a certain stage as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy the corresponding result of the ice model on the generalized two dimensional sierpinski gasket s g b n with b 3 is also obtained and the general upper and lower bounds for the entropy per site for arbitrary b are conjectured we also consider the number of eight vertex model configurations on s g n and the number of generalized vertex models e b n on s g b n and obtain exactly e b n 2 2 b 1 b b 1 2 n b 4 b 2 it follows that the entropy per site is lim v ln e b n v 2 b 1 b 4 ln 2

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

fractals their transfer matrices and their eigen dimensional sequences this paper defines and studies a general algorithm for constructing new families of fractals in euclidean space this algorithm involves a sequence of linear interscale transformations that proceed from large to small scales the authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a variable number of addend fractals the addends relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix which they call the transfer matrix of the fractal tmf the authors first demonstrate by a series of examples then prove rigorously that the eigenvalues of our tmfs are real and positive and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of their tmf they say that these dimensions form the overall fractal s eigendimensional sequence the eigenvalues of their tmf are integers in the non random variants of the construction but are non integer in the random variants a geometrical interpretation of the eigenvalues and the eigenvectors is given their tmf have other striking and very special properties that deserve additional attention

counting spanning trees on fractal graphs and their asymptotic complexity using the method of spectral decimation and a modified version of kirchhoff s matrix tree theorem a closed form solution to the number of spanning trees on approximating graphs to a fully symmetri

spanning trees on the sierpinski gasket we present the numbers of spanning trees on the sierpinski gasket sg d n at stage n with dimension d equal to two three and four the general expression for the number of spanning trees on sg d n with arbitrary d is conjectured the numbers of spanning trees on the generalized sierpinski gasket sg d b n with d 2 and b 3 4 are also obtained

fractal dimensionality of polymer chains the authors suggest that the concept of fractal dimensionality provides a useful characterisation of the configurational properties of a single polymer from numerical studies of long polymers traced out by self avoiding walks on a planar lattice they find that the fractal dimensionality is well defined and has a constant value for most scales of length of the chain renormalisation group theory provides a theoretical basis for the concept of fractal dimensionality in polymers

bond percolation on branching koch curves with use of the exact recursion equations of the generating functions we determine the asymptotic behaviour of bond percolation on branching koch curves and obtain the critical exponents alpha beta gamma and nu the scaling law 2 alpha gamma 2 beta df nu is obtained this is similar to the scaling law for a regular lattice except that the fractal dimension df is replaced by the euclidean dimension d

can nonlinear compactors be better than linear ones this paper presents a new analytical method for estimating compaction quality based on the entropy maximization of the entropy of the signature results in an increase of compaction quality the paper studies the influence of the architecture of a compactor on the entropy growth this approach is suitable for nonlinear compactors widely known linear feedback shift registers lfsrs and the counting compactors as well useful theorems to determine entropy growth during the test have been given and proven furthermore the class of nonlinear compactors based on shift registers named cyclic feedback shift registers cfsrs which guarantee the maximal entropy growth and the aliasing probability of 2 sup n is found it is shown that they are considerably better than the simplest linear compactor feedback shift register and in many cases are quite as good as linear compactors lfsrs having primitive polynomials on the opposite side compactors based on counters are far worse than cfsrs the family of cfsrs is much greater than the already explored family of lfsrs and some cfsrs require less overhead

lower and upper bounds for the anomalous diffusion exponent on sierpinski carpets the anomalous diffusion exponent dw of random walks on a family of sierpinski carpets are studied by analytical and numerical methods we construct an effective bulk resistor and then establish the lower and upper bounds for dw where the lower bound turns out to be the same as that obtained with bond moving renormalization numerical simulations on a family of sierpinski carpets confirm our bounds and show strong dependence of dw on the lacunarity of the carpet

studies of the spectral dimension for branched koch curves the scaling behaviour of koch curves with a rather general branched structure is studied by means of the exact renormalisation method and the general formula of spectral dimension is obtained it is proved that the spectral dimension is only dependent on the topology of the curve the einstein relation between the random walk dimension fractal dimension and size scaling exponent of dc conductivity is also checked

transfer matrix solution of the ising model on the koch curve a transfer matrix approach is used to calculate the partition function of the ising model on the koch curve in zero magnetic field it is possible to obtain an exact analytic result in the presence of a field the problem becomes more difficult a procedure based on an expansion about h 0 is developed which shows that the spontaneous magnetisation vanishes identically and which makes it possible to give the zero field susceptibility as a power series in a small temperature dependent parameter

transfer matrix studies of branching fractals abstract we introduce the method of transfer matrices to study physical models defined on branching fractal lattices in this context we derive matricial recurrence relations connecting the matrices associated with successive fractal generations these relations depend only on the geometry of the fractal and may be used for any model explicit results are obtained for ferromagnetic and antiferromagnetic ising models on branching koch curves and sierpinski gasket

magnetism of internal surfaces in a fractal structure abstract we study the inhomogeneous magnetization behavior of an ising ferromagnet in sierpinski pastry shells using a real space renormalization group approach two qualitatively different regions on the fractal are distinguished the bulk and the set of internal surfaces which border the eliminated portions we obtain the spontaneous mean magnetizations for these regions as a function of the temperature for different values of js jb js and jb are the internal surface and bulk coupling constants respectively and different geometrical parameters b and l the critical exponents are obtained for the several transitions we obtain different universality classes for the bulk transitions depending on what occurs at the surfaces and a step like behavior of the magnetization as a function of the temperature of some values of b and l

potts model on a fractal lattice the authors consider the dilute potts model on the sierpinski carpet the model has a phase transition at a finite temperature and both critical and tricritical behaviour are observed for qqc where q is the number of potts states they use migdal kadanoff type recursion relations to obtain the value of qc and of critical and tricritical exponents and investigate their dependence on various geometrical characteristics of the fractal

critical behavior of the gaussian model on a diamond type hierarchical lattice with periodic and aperiodic interactions abstract the critical behavior of the gaussian models on a diamond type hierarchical lattice with periodic and aperiodic interactions according to a class of substitution sequences are studied by the exact renormalization group approach and spin rescaling for the periodic case with a two letter general fibonacci sequence given by the generating substitutions a ab b ab and the aperiodic case with the generating substitutions a ab b ba the critical behaviors are found unchanged with respect to the uniform case and belong to the same universality of the uniform case for the other aperiodic case with the generating substitutions a ab b aa the geometric fluctuation is relevant by luck s criterion and the critical phenomena of the uniform and periodic cases are found to disappear

absorbing phase transition in conserved lattice gas model on fractal lattices we study the continuous phase transition of the conserved lattice gas clg model from an active phase into an absorbing phase on two fractal lattices i e on a checkerboard fractal and on a sierpinski gasket in the clg model a particle is assumed to be active if any of the neighboring sites are occupied by a particle and inactive if all neighboring sites are empty we estimate critical exponents theta beta nu and nu perpendicular characterizing respectively the density of active particles in time the order parameter the temporal and spatial correlation lengths near the critical point and the results are confirmed by off critical scaling and finite size scaling the order parameter exponent beta on a checkerboard fractal appears to lie between the one dimensional 1d value and two dimensional 2d value of the clg model while that on a sierpinski gasket lies between the 1d and 2d values of the conserved threshold transfer process such a difference is manifested based on the intrinsic properties of the underlying fractal lattices

three dimensional analysis of a noise barrier using a quasi periodic boundary element method two dimensional 2d numerical models are often used to estimate the environmental noise attenuation of a roadside barrier the prediction of noise barrier attenuation using a 2d boundary element model assumes an infinitely long barrier with constant cross section however for barrier geometries that do not have constant cross section in the third dimension three dimensional 3d models should be used for greater accuracy of noise reduction due to the barrier the size of a numerical model and hence its computational cost can be significantly reduced using a 3d quasi periodic structure whereby the structure is truncated using a finite number of periodic sections in this study a quasi periodic model developed using the boundary element method is used to predict the acoustic performance of 3d noise barriers the convergence behavior of the quasi periodic model is discussed results from the quasi periodic model are compared with results from both a 3d analytical model and a 2d finite element model showing good agreement quasi periodic models of different noise barrier designs are developed and their acoustic performances in terms of frequency and receiver positions are discussed the quasi periodic boundary element method provides a computationally efficient tool to examine the acoustic performance of 3d noise barrier designs

aliasing probability calculations for arbitrary compaction under independently selected random test vectors this paper discusses a systematic methodology for calculating the exact aliasing probability associated with schemes that use an arbitrary finite state machine to compact the response of a combinational circuit to a sequence of independently selected random test input vectors the proposed approach identifies the strong influence of fault activation probabilities on the probability of aliasing and uses an asymmetric error model to simultaneously track the states of two fictitious compactors one driven by the response of the fault free combinational circuit and one driven by the response of the faulty combinational circuit by deriving the overall markov chain that describes the combined behavior of these two compactors we are able to calculate the exact aliasing probability for any test sequence length in particular for long enough sequences the probability of aliasing is shown to only depend on the stationary distribution of the markov chain the insights provided by our analysis are used to evaluate the testing performance of simple examples of nonlinear compactors and to demonstrate regimes where they exhibit lower aliasing probability than linear compactors with the same number of states finally by establishing connections with previous work that evaluated aliasing probability in linear compactors our analysis clarifies the role played by the entropy of the stationary distribution of the compactor states

new method of analysing self avoiding walks in four dimensions a new method based on the concept of fractal dimensionality is used to study the problem of self avoiding walks in a four dimensional lattice the authors find from monte carlo simulations that the confluent logarithmic exponent related to the end to end distance is 1 8 or 0 01 in excellent agreement with the prediction derived from the n to 0 vector model

critical dynamics of the spin exchange model in quasilinear fractal geometries comments on the dynamic critical exponents recently reported by leyvraz and jan ibid vol 19 p 603 1986 in particular that for the spin exchange kinetic ising model in one dimension z 3 this result is at variance with rigorous inequalities that z cannot be less than five for this model i e z or 5 the source of the discrepancy appears to lie in the monte carlo algorithm which subsumes a critically slow dynamical process leading to an apparently faster dynamics in the critical region in addition the author extends the finding for one dimension to the quasilinear non branching fractal koch curve concluding z 3df where df is the fractal dimension he discusses the physical factors comprising the lower bound to the dynamic exponent in one dimension z 5 he then obtains the generalised lower bound for spin exchange dynamics of the non branching koch curve z 3df dw 5df where dw is the random walk dimension

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

lacunarity and universality a new expression for the lacunarity of sierpinski carpets is proposed it turns out that the universality can be described with the new expression the fractal dimension and the connectivity

scale dependent lacunarity and fractal ising models as an alternate way of measuring the lacunarity of sierpinski carpets the authors have computed the site to site fluctuation of the quantity m n namely the total number of lattice sites within the chemical distance of n in long scales of n the resultant lacunarity agrees qualitatively with the results obtained in the past using the covering method in short scales of n however the lacunarity shows a different trend among fractals it varies monotonically with the ising correlation length exponent a trend which was sought before but not found with the previous method this applies only for the carpets of the first kind

prediction and evaluation of the scattering from quadratic residue diffusers the scattering from quadratic residue diffusers has been predicted using methods based on the helmholtz kirchhoff integral equation the methods consisted of two boundary integral methods and two more approximate solutions utilizing kirchhoff s boundary conditions the predictions were compared to measurements made in both the near and far fields an accurate boundary integral method which works below and above the well cutoff frequency has been found the limitations of the more approximate methods have been defined the approximations behind the quadratic residue diffuser s design have been tested the important assumption of a phase change local reacting admittance over each well face was found to be good over a wide range of frequencies the frequency limit of this approximation has also been defined

change in order of phase transitions on fractal lattices abstract we re examine a population model which exhibits a continuous absorbing phase transition belonging to directed percolation in 1d and a first order transition in 2d and above studying the model on sierpinski carpets of varying fractal dimensions we examine at what fractal dimension 1 d f 2 the change in order occurs as well as commenting on the order of the transitions we produce estimates for the critical points and for continuous transitions some critical exponents

a new framework for designing and analyzing bist techniques and zero aliasing compression a general framework for shift register based signature analysis is presented and a mathematical model for this framework based on coding theory is developed there are two key features of this formulation first it allows for uniform treatment of lfsr misr and multiple misr based signature analyzer in addition using this formulation a new compression scheme for multiple output cut is proposed this scheme referred to as multiinput lfsr has the potential to achieve better aliasing than other schemes such as the multiple misr scheme of comparable hardware complexity several results on aliasing are presented and certain known results are shown to be direct consequences of the formulation also developed are error models that take into account the circuit topology and the effect of faults at the outputs using these models exact closed form expressions for aliasing probability are developed a closed form aliasing expression for misr under an independent error model is provided

a fractal model for band structures just above percolation threshold a two dimensional regular fractal model is proposed to imitate the geometric texture of percolating networks just above percolation threshold the system is self similar fractal on length scales smaller than the connectedness length but becomes homogeneous non fractal square lattice on larger length scales the scaling of vibrational density of states and band structures are studied by using exact renormalisation group methods it is shown that an infinite number of narrow bands appear to have self similar structures of typical cantor sets at low frequency

frustration and correlations in fractals abstract a technique for obtaining exact thermodynamic and correlation functions of regular fractals is illustrated by a development for the ising system on the triangular sierpinsky gasket a generalised hyperscaling statement is obtained the technique is then applied to the fully frustrated antiferromagnetic case where the exact ground state entropy per spin is shown to be s 0 0 4930 this result shows that the introduction of regularly disposed holes into the triangular lattice does not relieve frustration as is commonly supposed relations completely determining the correlation functions are obtained that for the zero field pair correlation function of vertex spins at separation r is g 1 2 r g 2 r g 1 together with a relation which sets the correlation length scale exact asymptotic expressions are derived for ferromagnetic correlations in both zero and non zero field the zero field zero temperature correlation length of the fully frustrated system is shown to be 0 79 representing extremely rapid decay of correlations

random walk on self avoiding walk a model for conductivity of linear polymers random walks on self avoiding walks saws are studied using monte carlo techniques on a square lattice with nearest neighbour hopping along the chain and between saw points which are nearest neighbours on the embedding lattice the average of the square of the end to end distance for random walks of t steps on saws of length n is fitted to the scaling forms rt2 varies as ndelta tk for t or approximately ntheta where theta approximately 2 nu s k nu s being the average end to end distance exponent for saws the observed value of the exponent delta is supported by the authors real space renormalisation group result for the conductivity of saw chains the exponent k has been related to the effective fractal dimension of the saw chain

dimer coverings on the sierpinski gasket we present the number of dimer coverings nd n on the sierpinski gasket sgd n at stage n with dimension d equal to two three four or five when the number of vertices denoted as v n of the sierpinski gasket is an even number nd n is the number of close packed dimers when the number of vertices is an odd number no close packed configurations are possible and we allow one of the outmost vertices uncovered the entropy of absorption of diatomic molecules per site defined as s mathit sg d lim n to infty ln n d n v n is calculated to be ln 2 3 exactly for sg2 the numbers of dimers on the generalized sierpinski gasket sgd b n with d 2 and b 3 4 5 are also obtained exactly with entropies equal to ln 6 7 ln 28 12 ln 200 18 respectively the number of dimer coverings for sg3 is given by an exact product expression such that its entropy is given by an exact summation expression the upper and lower bounds for the entropy are derived in terms of the results at a certain stage for sgd n with d 3 4 5 as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of s mathit sg d with d 3 4 5 can be evaluated with more than a hundred significant figures accurate

competing and aperiodic interactions in the ising chain abstract we investigate the effects of the inclusion of aperiodic interactions in the ising chain with competing ferro and antiferromagnetic couplings between nearest and next nearest neighbor sites in particular we obtain the exact ground state including the spin arrangements and the residual entropies for an ising chain with extra aperiodic interactions associated with the fractal rules defined by a koch curve

a real space renormalisation group study of the ising model on self avoiding walk chains here the authors apply a small cell real space renormalisation group technique to study the critical behaviour of the nearest neighbour interacting ising model on self avoiding walk saw chains in the limit of vanishing interactions between nearest neighbours which are not connected by the walk itself the treatment reproduces exactly the critical behaviour of the linear chain ising model when all the nearest neighbour interactions are allowed the treatment gives a vanishing tc with the small cells chosen for the ising model on saws on the square and triangular lattices the study of the critical behaviour of such models indicates some interesting departures from that of the ising model on linear chains

renormalization group approach to the multi component site percolation on sierpinski carpets abstract the multi component site percolation problems on a family of sierpinski carpets scs are solved by the real space renormalization group rg method three sorts of sites on an sc lattice are distinguished and three independent kinds of occupation probabilities are assigned to them we develope the usual choice of a cell on the translationally invariant lattices and choose suitable cells to construct the rg transformation scheme based on the connectivity the non trivial percolation threshold values and correlation length exponents on such fractals are obtained

a special integral equation formulation for acoustic radiation and scattering for axisymmetric bodies and boundary conditions this paper presents a simplified model of the helmholtz integral equation for bodies with axisymmetric shape and boundary conditions by taking advantage of a body s axisymmetric properties the surface integral is reduced to a line integral along the generator of the body and an integral over the angle of revolution the integration over the angle is performed partly analytically in terms of elliptic integrals and partly numerically using a gaussian quadrature formula the analytical formulation leading to elliptic integrals deals directly with the singularities in the green s function and its normal derivative examples are shown for scattering from a single sphere and two spheres and radiation from a finite cylinder the results are compared with analytical approximate and other accepted solutions

potts model on sierpinski carpets migdal kadanoff bond moving renormalisation is used to study the q state potts model on sierpinski carpets a general approximate recursion relation including q as a parameter is given the property of bond interchanging invariance is found and used in deriving the recursion relation fixed points and critical exponents for some carpets are presented marginal fixed points instead of unstable ones are found several typical flow diagrams are also shown

the trail problem on finitely ramified fractals using an exact real space renormalisation group technique the authors show that self avoiding trails sat and self avoiding walks saw on the sierpinski gasket obey the same critical exponent nu for the correlation length and therefore belong to the same universality class on the other hand it is shown that sat on branching koch curves with maximum ramification number rmax 3 is in a different universality class from that of saw

percolation on infinitely ramified fractals we present a family of exact fractals with a wide range of fractal and fracton dimensionalities this includes the case of the fracton dimensionality of 2 which is critical for diffusion this is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal these fractals include the cases of finite and infinite ramification characterized by a ramification exponentp the infinite ramification makes the problem of percolation on these lattices a nontrivial one we give numerical evidence for a percolation transition on these fractals this transition is tudied by a real space renormalization group technique on lattices with fractal dimensionality d between 1 and 2 the critical exponents for percolation depend strongly on the geometry of the fractals

critical behaviour of the bond diluted potts model on sierpinski carpets a four parameter bond moving renormalisation is used to study the critical properties of the bond diluted q state potts model on sierpinski carpets fixed points critical exponents and a phase diagram are obtained in the authors results there is a borderline value q qt for q qt there is crossover to a new diluted fixed point this behaviour is similar to that of the diluted system on a regular lattice with translational invariance they give some values of qt for carpets with different b l where b and l are structure parameters of carpets

ising model on a 3d sierpinski gasket with a non trivial phase transition the author derives and solves exact recursion relations for a multispin interaction ising model on a 3d sierpinski gasket in addition to the high and low temperature fixed points he finds three non trivial fixed points which govern the behaviour of two new multiple of 2 phases these phases are characterised by the absence of any tetrahedron with an odd number of up spins

magnetization of a potts ferromagnet on a sierpinski carpet the authors study the inhomogeneous magnetization behaviour of a q state potts ferromagnet on a sierpinski carpet as a function of the temperature using a real space renormalization group procedure two qualitatively different regions on the carpet are distinguished a the interior of the carpet and b the set of internal walls which border the eliminated areas the spontaneous mean magnetization curves for these regions are obtained with the corresponding critical exponents beta and beta w for different values of q they find that the magnetization of region b goes to zero at the critical temperature always below the one of region a

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal