stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

diffusion on loopless critical percolation cluster in this paper we calculate the dynamical exponents of diffusion dw df and ds on a percolation cluster at pc with no loops for the square and simple cubic lattices by the method of spectral analysis of the transition probability matrix results show that ds varies significantly with the spatial dimension unlike in conventional percolation but the alexander orbach scaling relation ds 2df dw still holds thus it rules out the possibility that this scaling relation fails for all loopless fractals because of trapping

branched polymer approach to the structure of lattice animals and percolation clusters the authors treat the percolation problem as a solution of randomly branched chains and argue that such chains are swollen by a three body repulsion acting only within a connected cluster this type of repulsion should not be subject to the screening effects which inhibit ordinary repulsions the repulsion becomes relevant below six dimensions they estimate its effect on the size of a cluster in analogy with flory s argument for the swelling of linear chains above six dimensions the cluster size scales as mass to the 1 4 power as with isolated branched chains in high dimensions

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

the fractal dimension and other percolation exponents in four and five dimensions for percolation the fractal dimension df which is identical to the magnetic field scaling power yh has never been calculated for hypercubic lattices of dimension d 5 percolation is studied for systems of high dimensionality using the method of large cell monte carlo position space renormalisation group the estimate df yh 3 69 or 0 02 d 5 and 3 12 or 0 02 d 4 is obtained the thermal scaling power yt 1 nu where nu is the correlation length exponent is calculated yt 1 nu 0 51 or 0 05 d 5 and nu 0 64 or 0 02 d 4 the results are compared with the epsilon expansions of df and nu

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

markov chain analysis of random walks in disordered media we study the dynamical exponents d w and d s for a particle diffusing in a disordered medium modeled by a percolation cluster from the regime of extreme disorder i e when the percolation cluster is a fractal at p p c to the lorentz gas regime when the cluster has weak disorder at p p c and the leading behavior is standard diffusion a new technique of relating the velocity autocorrelation function and the return to the starting point probability to the asymptotic spectral properties of the hopping transition probability matrix of the diffusing particle is used and the latter is numerically analyzed using the arnoldi saad algorithm we also present evidence for a new scaling relation for the second largest eigenvalue in terms of the size of the cluster ln lambda max sim s d w d f which provides a very efficient and accurate method of extracting the spectral dimension d s where d s 2d f d w

markov matrix analysis of random walks in disordered continuous media abstract we study by a markov matrix analysis of the equivalent random walks the dynamic properties of continuous media consisting of both correlated and uncorrelated equal size spheres we employ a blind ant random walk model using the rule that a walker jumps among centers of the directly connected spherical particles on an infinite network the dominant eigenvalues and eigenvectors of the transition probability matrix of the random walks are calculated yielding estimates of the spectral dimension d s and the fractal dimension d w of random walks on the continuous network we find that for the present model the estimates are very close to the corresponding lattice percolation values though only after the finite size effects have been carefully taken into account we also show that the finite size scaling of the largest nontrivial eigenvalues holds for our model with the same exponents as for the lattice percolation

conductivity exponents from the analysis of series expansions for random resistor networks there has been considerable controversy in recent years over the value of the conductivity exponent t this exponent can be deduced from series expansions via the scaling relations t zeta d 2 nu where zeta is deduced from differences between the exponents of the resistive gamma r percolative gamma p and conductive gamma c susceptibility the author finds that allowance for non analytic confluent corrections to scaling and the use of recent pc estimates leads to estimates for gamma r gamma p and gamma c that are somewhat different to those of fisch and harris 1978 however the differences between these exponents do not change significantly moreover the change in accepted estimates of nu in the last five years cancels some of this remaining discrepancy and she concludes using the relation zeta gamma r gamma p that t 1 31 d 2 t 2 04 d 3 t 2 39 d 4 t 2 72 d 5 with an error of about or 0 10 in each case the d 2 estimate is in significantly better agreement with those of other methods than that of fisch and harris

optimal block type decodable encoders for constrained systems a constrained system is presented by a finite state labeled graph for such systems we focus on block type decodable encoders comprising three classes known as block block decodable and deterministic encoders franaszek 1968 gives a sufficient condition which guarantees the equality of the optimal rates of block decodable and deterministic encoders for the same block length we introduce another sufficient condition called the straight line condition which yields the same result run length limited rll d k and maximum transition run mtr j k constraints are shown to satisfy both conditions in general block type decodable encoders are constructed by choosing a subset of states of the graph to be used as encoder states such a subset is known as a set of principal states for each type of encoder and each block length a natural problem is to find a set of principal states which maximizes the code rate we show how to compute the asymptotically optimal sets of principal states for deterministic encoders and how they are related to the case of large but finite block lengths we give optimal sets of principal states for mtr j k block type decodable encoders for all codeword lengths finally we compare the code rate of nonreturn to zero inverted nrzi encoders to that of corresponding nonreturn to zero nrz and signed nrzi encoders

the inverse spectral problem for indefinite strings motivated by the study of certain nonlinear wave equations in particular the camassa holm equation we introduce a new class of generalized indefinite strings associated with differential equations of the form begin aligned u z u omega z 2 u upsilon end aligned on an interval 0 l where omega is a real valued distribution in h 1 mathrm loc 0 l upsilon is a non negative borel measure on 0 l and z is a complex spectral parameter apart from developing basic spectral theory for these kinds of spectral problems our main result is an indefinite analogue of m g krein s celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings

amplitude ratios and β estimates from general dimension percolation moments low concentration series are generated for moments of the percolation cluster size distribution gamma j sj 1 s is the number of sites on a cluster for j 2 8 and general dimensionality d these diverge at pc as gamma j approximately aj pc p gamma j with gamma j gamma j gamma j 2 delta where delta gamma beta is the gap exponent the series yield new accurate values for delta and beta delta 2 23 or 0 05 2 10 or 0 04 2 03 or 0 05 and beta 0 44 or 0 15 0 66 or 0 09 0 83 or 0 08 at d 3 4 5 in addition ratios of the form ajak aman with j k m n are shown to be universal new values for some of these ratios are evaluated from the series from the epsilon expansion epsilon 6 d and exactly in d 1 and on the bethe lattice the results are in excellent agreement with each other for all dimensions results for different lattices at d 2 3 agree very well these amplitude ratios are much better behaved than other ratios considered in the past and should thus be more useful in characterising percolating systems

the inverse spectral transform for the conservative camassa holm flow with decaying initial data we establish the inverse spectral transform for the conservative camassa holm flow with decaying initial data in particular it is employed to prove the existence of weak solutions for the corresponding cauchy problem

direct and inverse spectral theory of singular left definite sturm liouville operators abstract we discuss direct and inverse spectral theory of self adjoint sturm liouville relations with separate boundary conditions in the left definite setting in particular we develop singular weyl titchmarsh theory for these relations consequently we apply de branges subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure the results can be applied to solve the inverse spectral problem associated with the camassa holm equation

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

tradeoff functions for constrained systems with unconstrained positions we introduce a new method for analyzing and constructing combined modulation and error correcting codes eccs in particular codes that utilize some form of reverse concatenation and whose ecc decoding scheme requires easy access to soft information we expand the work of immink and wijngaarden and also of campello marcus new and wilson in which certain bit positions in the modulation code are deliberately left unconstrained for the ecc parity bits in the sense that such positions can take on either bit value without violating the constraint our method of analysis involves creating a single graph that incorporates information on these unconstrained positions directly into the constraint graph without any assumptions of periodicity or sets of unconstrained positions and is thus completely general we establish several properties of the tradeoff function that relates the density of unconstrained positions to the maximum code rate in particular the tradeoff function is shown to be concave and continuous algorithms for computing lower and upper bounds for this function are presented we also show how to compute the maximum possible density of unconstrained positions and give explicit values for the runlength limited rll d k and maximum transition run mtr j k constraints

exact series studies of self avoiding walks on two dimensional critical percolation clusters the author presents results of exact series studies of self avoiding walks on percolation clusters on the square lattice performed very close to the percolation threshold and estimates the critical exponents nu and gamma defined by the disorder averages of the mean square radius of gyration or the end to end distance and the number of self avoiding walks respectively the self avoiding walk configurations are enumerated exactly up to 35 steps the result for nu indicates a large increase compared with the self avoiding walks on fully occupied lattice the result for gamma however indicates a behaviour similar to that on a fully occupied lattice the result for nu is completely different from that of lee nakanishi and kim 1989

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

topological loop intermittency in the three dimensional turbulence a new topological or loop type of intermittency is studied in three dimensional turbulence the spectral or fracton dimension ds 7 4 7 1 95 for this turbulence i e all states are localized this intermittency is different both from kolmogorov type unlocalized since ds 2 and from alexander orbach type strong localized helical fractons with ds 43 the corresponding energy spectral exponent is obtained to be equal to 1 7 1 65 these results are in good agreement with recent numerical spectral dimension and laboratory turbulent energy spectrum data the anomaly in the experimentally observed fractal dimension of clouds surface d 2 35 can be also explained in the framework of this intermittency phenomenon the results have been obtained by a renormalization of the noise like 1k energy spectrum in isotropic incompressible turbulence with taking into account of spontaneous parity breaking by virtual pairs of helical excitations

scaling at the percolation threshold above six dimensions the fractal dimensionality of the infinite cluster at the percolation threshold for dimensionalities d 6 is shown to be d 4 rather than the naive finite size scaling prediction d d 2 similarly the conductivity of a sample of size l scales as l d rather than l 6 this anomalous behaviour is related to a dangerous irrelevant variable associated with the probability to have vertices of three bonds the crossover to the homogeneous behaviour occurs at length scales which are short compared with the correlation length the links and blobs picture is confirmed for d 6 and the size of the latter is estimated

a limit point criterion with applications to nonselfadjoint equations a unified approach is given which allows one to construct examples of all three cases resulting from a generalization of the classical weyl limit point limit circle analysis for differential equations as well as for difference equations

constrained systems with unconstrained positions we develop methods for analyzing and constructing combined modulation error correcting codes ecc codes in particular codes that employ some form of reversed concatenation and whose ecc decoding scheme requires easy access to soft information e g turbo codes low density parity check ldpc codes or parity codes we expand on earlier work of wijngaarden and immink 1998 2001 immink 1999 and fan 1999 in which certain bit positions are reserved for ecc parity in the sense that the bit values in these positions can be changed without violating the constraint earlier work has focused more on block codes for specific modulation constraints while our treatment is completely general we focus on finite state codes for maximum transition run mtr constraints we 1 obtain some improved constructions for mtr codes based on short block lengths 2 specify an asymptotic lower bound for mtr constraints which is tight in very special cases for the maximal code rate achievable for an mtr code with a given density of unconstrained positions and 3 show how to compute the capacity of the set of sequences that satisfy a completely arbitrary constraint with a specified set of bit positions unconstrained

a general approach to approximation problems of the bernstein type the bernstein approximation problem is treated in a general setting of banach function spaces an abstract theory is developed which includes as special cases the solutions of the bernstein problem given by mergelyan and pollard it also provides a unified setting for related results from polynomial and trigonometric approximation theory and yields as corollaries a number of new results

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

finite size analysis of eigenvalue spectrum for random walks on a critical percolation cluster in four dimensions we study by markov chain analysis the random walks on a critical percolation cluster embedded in a four dimensional hypercubic lattice we calculate the number of dominant eigenvalues of the transition probability matrix and estimate the spectral and fractal dimensions ds and dw of random walks from the eigenvalues and their distribution the estimates of ds and dw obtained from the data for a given size s of the percolation cluster exhibit some s dependence extrapolating the results to s limit we obtain ds 1 330 0 010 close to the previous result by other methods and a new result dw 4 50 0 15 these values are also confirmed by direct monte carlo simulations of random walks on a percolation cluster

on the relationship between the critical exponents of percolation conductivity and static exponents of percolation the author argues that the critical exponent t of random conductance networks near the percolation threshold is given by t d 1 nu for low dimensionalities and t 1 beta for high dimensionalities where nu is the correlation length exponent beta the backbone exponent and d is dimensionality the author argues that what separates the two regimes is a critical fractal dimensionality dl which equals 2 the author also argues that dl is also a critical fractal dimensionality for fractals such as lattice animals and diffusion limited aggregates the result for low dimensionalities has been also obtained by aharony and stauffer by a different argument

block cluster theory of site percolation on the four and five dimensional ordinary hypercubic lattices site percolation on the ordinary hypercubic lattice in four and five dimensions is studied using block cluster theory the exact analytic renormalisation group equations of block cluster theory are obtained by a computer algorithm they are solved numerically and yield for the susceptibility exponent gamma 1 277 d 4 gamma 1 031 d 5 and for the correlation length exponent nu 0 566 d 5

diffusion on a dla cluster in two and three dimensions we present a much improved calculation of the dynamic critical exponents of diffusion dw ds and the combination df dw for the dla diffusion limited aggregate on square and simple cubic lattices using two independent approaches one based on the exact enumeration for the displacement and velocity autocorrelation and the other on the eigenspectrum of random walks on the dla the two methods give consistent results which however clearly rule out the scaling relation ds 2df dw first proposed by alexander and orbach 1982 similarly to the case of the eden tree reported previously

effect of loops on the vibrational spectrum of percolation network abstract we study the effects of adding loops to a critical percolation cluster on the diffusional and equivalently scalar elastic properties of the fractal network from the numerical calculations of the eigenspectrum of the transition probability matrix we find that the spectral dimension ds and the walk dimension dw change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops and we find that the additional inclusion of successively smaller loops only change these exponents little if at all this suggests that there is a new universality class associated with the loop enhanced percolation problem

on the isospectral problem of the dispersionless camassa holm equation abstract we discuss direct and inverse spectral theory for the isospectral problem of the dispersionless camassa holm equation where the weight is allowed to be a finite signed measure in particular we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite the results are applied to deduce several facts for the dispersionless camassa holm equation in particular we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by mckean

accessible external perimeters of percolation clusters the authors introduce a new definition of a cluster s accessible external perimeter which is different from the usual definition of the hull this external perimeter can be measured by probing adsorbent particles coming from the cluster s exterior the size of these particles determines which perimeter sites are accessible from the outside and hence one can define different external or accessible perimeters according to this size using a monte carlo simulation they measured the fractal dimensions of these accessible perimeters de for percolation clusters at the threshold on the square and the triangular lattices the fractal dimension of the accessible perimeter is found to depend on the size of the probing particles for several sizes which are larger than some lattice dependent threshold they find de approximately 4 3 which is significantly smaller than the hull dimension dh 7 4 found with smaller probing particles

where two fractals meet the scaling of a self avoiding walk on a percolation cluster the scaling properties of self avoiding walks on a d dimensional diluted lattice at the percolation threshold are analyzed by a field theoretical renormalization group approach to this end we reconsider the model of y meir and a b harris phys rev lett 63 2819 1989 and argue that via renormalization its multifractal properties are directly accessible while the former first order perturbation did not agree with the results of other methods our analytic result gives an accurate description of the available mc and exact enumeration data in a wide range of dimensions 2 ensuremath leqslant d ensuremath leqslant 6

scattering and inverse scattering for a left definite sturm liouville problem this work develops a scattering and an inverse scattering theory for the sturm liouville equation u qu wu where w may change sign but q 0 thus the left hand side of the equation gives rise to a positive quadratic form and one is led to a left definite spectral problem the crucial ingredient of the approach is a generalized transform built on the jost solutions of the problem and hence termed the jost transform and the associated paley wiener theorem linking growth properties of transforms with support properties of functions one motivation for this investigation comes from the camassa holm equation for which the solution of the cauchy problem can be achieved by the inverse scattering transform for formula

the transport exponent in percolation models with additional loops abstract several percolation models with additional loops were studied the transport exponents for these models were estimated numerically by means of a transfer matrix approach it was found that the transport exponent has a drastically changed value for some of the models this result supports some previous numerical studies on the vibrational properties of similar models with additional loops

time varying maximum transition run constraints maximum transition run mtr constrained systems are used to improve detection performance in storage channels recently there has been a growing interest in time varying mtr tmtr systems after such codes were observed to eliminate certain error events and thus provide high coding gain for enpr4 channels for n 2 3 in this work tmtr constraints parameterized by a vector whose coordinates specify periodically the maximum runlengths of 1 s ending at the positions are investigated a canonical way to classify such constraints and simplify their minimal graph presentations is introduced it is shown that there is a particularly simple presentation for a special class of tmtr constraints and explicit descriptions of their characteristic equations are derived new upper bounds on the capacity of tmtr constraints are established and an explicit linear ordering by capacity of all tight tmtr constraints up to period 4 is given for mtr constrained systems with unconstrained positions it is shown that the set of sequences restricted to the constrained positions yields a natural tmtr constraint using tmtr constraints a new upper bound on the tradeoff function for mtr systems that relates the density of unconstrained positions to the maximum code rates is determined