on relating two approaches to fractional calculus abstract in this paper two methods are discussed for defining fractional powers of the nth order ordinary differential expression l x a 1 dx a 2 d x a n dx a n 1 d d dx where each ai i 1 2 n 1 is a complex number and x is real and positive an important role is played by the number m i 1 n 1 a i n when m 0 spectral theory is used to obtain a suitable formula for l c while the case m 0 considered in a previous paper by a c mcbride proc london math soc 3 45 1982 519 546 is dealt with by expressing l in terms of erdelyi kober operators we give accounts of each approach and go on to consider how the results for m 0 can be obtained as limiting cases of the results for m 0

the 2d transverse ising model at t 0 a finite size rescaling transformation approach the critical behaviour of the spin 4 ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite size rescaling transformation this technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size in a generalisation of the so called phenomenological renormalisation group the critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy but calculations on larger clusters are limited by computer time and storage

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

real space renormalisation group study of selective site percolation on triangular lattice the selective site percolation is studied on three kinds of triangular lattice tc tk and th lattices by using renormalisation groups the recursion relations on occupation probabilities are solved to give global phase diagrams with scaling factor b square root 3 2 and 3 for the tc lattice and with b square root 3 and square root 7 for tk and th lattices respectively the phase boundaries are obtained correctly with the flow direction towards a single isotropic fixed point this result is in accordance with the anisotropic bond result and consistent with the universality hypothesis in the extreme anisotropy limit of site preference these boundaries give good estimates of the critical concentration pc of chain kagome and honeycomb lattices especially in the chain limit for the tc lattice the critical exponents of both the correlation length v 1 and the dimensionality crossover phi 1 are reproduced exactly the dependence of total pc on site selectivity is also discussed

distributional fractional powers and riesz fractional integrals abstract a dual space fractional power theory is developed which follows naturally from the frechet space fractional power theory considered by s e schiavone proc roy soc edinburgh sect a 112 1989 237 247 properties of such dual space fractional powers are derived in addition an application to the riesz fractional integral associated with the three space dimensional wave equation facilitates the solution of a generalised cauchy problem conditions are established for the existence of a weak solution to this problem

correlated percolation exact bethe lattice analyses certain classes of correlated site percolation problems or correlated spreading phenomena on bethe lattices are analysed exactly the author s analysis of percolation of e g occupied sites requires that spreading of clusters of occupied sites is determined by a finite number of conditional probabilities a condition specifying the percolation threshold is provided as well as expressions for the percolation probability and average cluster size previous results for random and nearest neighbour ising model distributions are recovered as special cases results are illustrated with examples for equilibrium and non equilibrium distributions the latter obtained via irreversible cooperative filling the author also discuss two phase percolation for distributions with no occupied nn pairs of sites correlated bond percolation and other problems

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

finite size scaling analysis of the dilute baxter wu model the finite size scaling method is used to study the critical properties of the spin 1 baxter wu model as function of the fugacity z of the vacant spin zero sites for z 0 the thermal exponent converges very quickly to the exact value yt 3 2 for z 0 yt monotonically increases beyond the value 2 this increase is interpreted as indicating a first order transition out of several possible renormalisation group flows the results seem to favour the one in which the critical baxter wu hamiltonian flows to the fixed point of the 4 state potts model with the amplitude of the marginal operator equal to zero

some analytical consequences of the inverse relation for the potts model the authors present some analytical results for the potts model using the symmetry group acting on its parameters generated by the inverse relation and other symmetries of this model in particular they find the critical manifolds and study the relationship between this group and the lee yang singularities in the complex plane for those cases where the symmetry group is finite they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials

percolation transitions in two dimensions we investigate bond and site percolation models on several two dimensional lattices numerically by means of transfer matrix calculations and monte carlo simulations the lattices include the square triangular honeycomb kagome and diced lattices with nearest neighbor bonds and the square lattice with nearest and next nearest neighbor bonds results are presented for the bond percolation thresholds of the kagome and diced lattices and the site percolation thresholds of the square honeycomb and diced lattices we also include the bond and site percolation thresholds for the square lattice with nearest and next nearest neighbor bonds we find that corrections to scaling behave according to the second temperature dimension xt2 4 predicted by the coulomb gas theory and the theory of conformal invariance in several cases there is evidence for an additional term with the same exponent but modified by a logarithmic factor only for the site percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent the amplitude of the power law correction associated with xt2 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems

renormalisation group approach to delaunay percolation networks with topological disorder a renormalisation group approach is developed for delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space considering various renormalisation transformations for two and three dimensional delaunay lattices the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent the fixed points for the two dimensional bond lattice and the three dimensional site lattice are 0 3229 and 0 1443 respectively which are in excellent agreement with results of monte carlo simulations the fixed point for the two dimensional site lattice gives the value 1 2 for the critical percolation probability which is equal to the known result for a fully triangulated lattice

branched polymers exact enumeration study of three dimensional lattice animals classified by valence distribution the partial generating function method is applied to the enumeration of lattice animals classified according to their valence distribution this detailed information allows one to study the effects of the number and nature of branch points on the properties of branched polymers several series analysis techniques are applied to the data particular attention is given to the effects of correction to scaling terms

a distributional theory of fractional calculus in this paper a theory of fractional calculus is developed for certain spaces d p of generalised functions the theory is based on the construction of fractionalpowers of certain simple differential and integral operators with the parameter suitably restricted these fractional powers are shown to coincide with the riemann liouville and weyl operators of fractional integration and differentiation standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators see 6 7 some spectral properties are also obtained

survival exponential entropies the multivariate survival function of a random vector x is used to define a broad class of entropy measures several properties of the proposed class are studied and explicit expressions of the measures derived for specific probabilistic models the cumulative residual entropy introduced by rao et al and wang et al is a particular case of the proposed class of measures

correlated percolation and a sublattice dilute potts model at t 0 abstract it is shown that a sublattice dilute q state potts model at t 0 is equivalent to a correlated percolation which favours subgraphs with larger numbers of disconnected finite clusters the syozi model at t is equivalent to the above model with q 2 in the same limit and thus corresponds to the same correlated percolation a table is listed which reflects the influence of correlation on critical probabilities for such correlated percolation processes on some lattices

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

position space renormalisation group for isolated polymer chains we develop a cell position space renormalisation group psrg with which we study the scaling properties of isolated polymer chains we model a chain by a self avoiding walk constrained to a lattice for rescaling factors b s 6 we calculate recursion relations analytically on the square lattice with several different choices for the psrg weight function we also calculate implicit cell to cell transformations in which a cell of size b is rescaled to a cell of size b the results of these psrgs improve both as b increases and as b b 1 in addition we construct a true infinitesimal psrg transformation which appears to become exact as the dimensionality d approaches 1 we obtain the closed form expression for the correlation length exponent u d l d in d the flory formula deviates from this already at first order in d 1 we also develop a constant fugacity monte carlo method which enables us to simu late in an unbiased way within the grand canonical ensemble chains of up to lo3 bonds with this method we extend the psrg to larger cells b s 150 on the square lattice our numerical method provides high statistical accuracy for all cell sizes however in the range of b we study the asymptotic behaviour of our results appears to depend on the choice of weight function one weight function provides smooth behaviour as a function of b and with it we extrapolate to find u 0 756 0 004 further work is required to resolve the apparent anomalies in the results based on the other weight functions

a fractional power approach to fractional calculus abstract a theory of fractional powers of operators on an arbitrary frechet space is discussed as special cases multivariable fractional integrals and derivatives defined on certain spaces of test functions and generalised functions are obtained in particular properties of the two dimensional riesz fractional integral are determined and used to solve a distributional initial value problem involving the wave operator

arbitrary powers of d alembertians and the huygens principle by means of some reasonable rules the operators that can represent arbitrary powers of the d alembertian and their corresponding green s functions are defined it is found which powers lead to the validity of huygens principle the specially interesting case of powers that are half an odd integer in spaces of odd dimensionality obey huygens principle and can be expressed as iterated d alembertians of the retarded potential are discussed arbitrary powers of the laplacian operator as well as their corresponding green s functions are also discussed

linear defects in two dimensional systems a finite size investigation finite size scaling theory is employed to study non universal and crossover critical behaviour in two dimensional systems with linear defects the finite size method is shown to reproduce very accurately the non universality as known to the present in the case of the ising model the calculations show that the non universal behaviour of all the models investigated in the ising universality class can be described approximately by a single universal curve

fractional diffusions with time varying coefficients this paper is concerned with the fractionalized diffusion equations governing the law of the fractional brownian motion bh t we obtain solutions of these equations which are probability laws extending that of bh t our analysis is based on mcbride fractional operators generalizing the hyper bessel operators l and converting their fractional power l into erdelyi kober fractional integrals we study also probabilistic properties of the random variables whose distributions satisfy space time fractional equations involving caputo and riesz fractional derivatives some results emerging from the analysis of fractional equations with time varying coefficients have the form of distributions of time changed random variables

examples of phase equilibrium in lattice models the general z4 and z5 models on the simple quadratic lattice and the pair triplet ising model on the triangular lattice are employed to illustrate an extension of the finite size scaling technique which obtains the qualitative structure of the phase equilibrium surface in terms of multiple phase coexistence

threshold and scaling in percolation with restricted valence monte carlo calculations have been carried out to study the problem of percolation with restricted valence on the square and simple cubic lattices the results are in full agreement with the expectations there is no transition when the restriction nu is equal to two while even for nu 3 the transition occurs and the correlation length exponent determined via finite size scaling is within the numerical accuracy the same as that in the unrestricted random percolation the maximum random occupancy as a function of the restriction is also determined

computer simulation of a model for irreversible gelation a random mixture of bifunctional and tetrafunctional units is placed on a simple cubic lattice permanent bonds between these units are then formed by the random motion of active centres radicals resulting in a model for the gelation of polyacrylamide and similar processes the largest of the clusters formed by this kinetic percolation process is identified with the gel fraction the mean cluster size with the weight average degree of polymerisation the critical exponent gamma of the mean cluster size is roughly the same as for random percolation that is different from that of the classical theory of flory 1941 and stockmayer 1943 but the corresponding amplitude ratio that is the ratio of average molecular weight on both sides of the gel point differs strongly from its value in random percolation thus this kinetic gelation model seems to belong to a universality class of its own different from both that of random percolation and that of classical gelation theories

on quantizer distortion and the upper bound for exponential entropy a sharp upper bound is derived for the exponential entropy in the class of absolutely continuous distributions with specific standard deviation and an exact description of the extremal distributions this result is interpreted as determining the least favorable cases for certain methods of quantization of analog sources it is known that for a large class of quantizers both zero memory and vector the rth power distortion as well as some other distortion criteria are bounded below by a constant depending on r multiplied by a certain integral of the source s probability density it is pointed out that this bound can be rewritten in terms of the exponential entropy the exponential entropy measures the quantitative extent or range of the source distribution this fact gives a physical interpretation of the indicated limits of quantizer performance further elucidated by the main result

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

ac response near percolation threshold transfer matrix calculation in 2d the complex admittivity of a network of resistors and capacitors is calculated at pc on a square lattice using a transfer matrix the loss tangent shows a region of anomalous frequency scaling with t s finite size scaling reveals two crossover frequencies as expected from duality

the relative information generating function abstract the aim of the paper is to introduce the relative information generating function whose derivatives give well known statistical indices such as the kullback leibler divergence between two probability distributions and watanabe s measure of interdependence it contains golomb s generating function as a particular case and takes simple forms for the binomial and the poisson distributions neither of which fits into golomb s model besides its unifying role the relative information generating function suggests new information indices as for instance the standard deviation of the variation of information

on phase diagrams for directed percolation problems the authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which differing from previous phenomenological or small cell real space renormalisation groups is sensitive to the existence of an infinite cluster along any lattice direction this enables them to explore new aspects of the phase diagram for the problem which were not accessible through existing treatments in particular the crossover exponent between one and two dimensional behaviour is shown to be exactly one for directed percolation numerical estimates are in agreement with known inequalities

anisotropic bond percolation by position space renormalisation group we treat a percolation model with anisotropic bond occupation probabilities in order to study the crossover behaviour of the effective spatial dimension of the system previous position space renormalisation group psrg studies of this problem on the square lattice show a crossover from two dimensional to pseudo one dimensional critical behaviour to investigate the possible reasons for this surprising result which is the opposite of the behaviour observed in thermal critical phenomena we first develop alternative psrg schemes we study the predictions of these groups for asymptotically large rescaling parameters where the approximations in the psrg are thought to become negligible from a consideration of the approximations involved we are led to a decimation transformation that uses an anisotropic cluster the results of this method indicate that anisotropic percolation is in the same universality class as isotropic percolation in complete analogy with thermal critical phenoma our results indicate several peculiar features both of this problem and of the psrg methods

the packing of two species of polygons on the square lattice we decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons we end up with a 24 vertex model which is known in the literature as the fully packed double loop model fpl2 in the particular case in which the fugacities of the polygons are the same the model admits an exact solution the solution is obtained using coordinate bethe ansatz and provides a closed expression for the free energy in particular we find the free energy of the four colouring model and the double hamiltonian walk and recover the known entropy of the ice model when both fugacities are set equal to 2 the model undergoes an infinite order phase transition

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

an efficient numerical scheme to solve fractional diffusion wave and fractional klein gordon equations in fluid mechanics abstract the numerous applications of time fractional partial differential equations in different fields of science especially in fluid mechanics necessitate the presentation of an efficient numerical method to solve them in this paper galerkin method and operational matrix of fractional riemann liouville integration for shifted legendre polynomials has been applied to solve these equations some definitions for fractional calculus along with some basic properties of shifted legendre polynomials have also been put forth when approximations are substituted into the fractional partial differential equations a set of algebraic equations would be resulted the convergence of the suggested method was also depicted in the end the linear time fractional klein gordon equation dissipative klein gordon equations and diffusion wave equations were utilized as three examples so as to study the performance of the numerical scheme

critical behaviour of the 1d q state potts model with long range interactions the critical behaviour of the one dimensional q state potts model with long range interactions decaying with distance r as r 1 sigma has been studied in the wide range of parameters 0 sigma or 1 and 1 16 or q or 64 a transfer matrix has been constructed for a truncated range of interactions for integer and continuous q and finite range scaling has been applied results for the phase diagram and the correlation length critical exponent are presented

analytic properties of the partition function for statistical mechanical models the author explains with examples the relationship between zeros of the partition function analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models it is shown how to write down a polynomial representation of the partition function for any model with a two sheet largest eigenvalue the staggered ice model representation of the q state potts model on a sequence of semi infinite strips is solved and it is shown that for 0 q 4 the potts model result is only projected out by special boundary conditions the non potts part of the result is obtained for a 4 infinity strip and gives baxter s antiferromagnetic critical curve 1982

boundary conditions and scaling functions of percolation models we use a histogram monte carlo simulation method to calculate the scaling functions of the existence probability ep and the percolation probability p of the site percolation model on square lattices with free and periodic boundary conditions we find that different boundary conditions give quite different scaling functions near the critical region however they give the consistent critical point critical exponents and the thermodynamic order parameter from renormalization group calculations similar results are found for other percolation models the implications of our calculated results for some theoretical problems of current interest are discussed

phase equilibria and the scaling transformation it is argued that the scaling transformation which is currently being used to obtain numerical estimates of critical exponents cannot safely be employed without prior knowledge that the points in question are second order transition points the transformation when defined on finite or semi infinite systems will not distinguish between order disorder critical points and any other point on the coexistence surface this in turn means that the full phase diagram of the equilibrium surface can be found using the scaling transformation the three state potts model is used to illustrate this

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

anisotropic bond percolation for some 2d lattices the anisotropic bond percolation on the triangular square and honeycomb lattices is studied uniformly the author uses a longitudinal cell to discuss the triangular lattice and then from this obtains the square and honeycomb lattices the fixed points the critical surfaces the flow diagrams and the correlation exponents are given the results are satisfactory

finite size scaling in the 2 1 d ising model hamiltonian eigenvalues are calculated for the 2 1 d ising model on lattices up to 5 5 sites finite size scaling techniques are used to estimate the critical coupling xc 0 329 or 0 001 and index v 0 635 or 0 005 the prospects are discussed for further applications of these methods in three and four dimensions

minimum entropy of error principle in estimation abstract the principle of minimum error entropy estimation as found in the work of weidemann and stear is reformulated as a problem of finding optimum locations of probability densities in a given mixture such that the resulting differential entropy is minimized new results concerning the entropy lower bound are derived continuity of the entropy and attaining the minimum entropy are proved in the case where the mixture is finite some other examples and situations in particular that of symmetric unimodal densities are studied in more detail

duality with real space renormalization and its application to bond percolation we obtain the exact solution of the bond percolation thresholds with inhomogeneous probabilities on the square lattice our method is based on the duality analysis with real space renormalization which is a profound technique invented in the spin glass theory our formulation is a more straightforward way than that of a very recent study on the same problem r m ziff et al j phys a math theor 45 494005 2012 the resultant generic formulas from our derivation can give several estimations for the bond percolation thresholds on other lattices rather than the square lattice

ising like critical phenomena in two dimensional percolation numerical evidence a recent analytic theory of two dimensional isotropic percolation indicates that the singular behaviour near pc is identical to that appropriate for the associated dilute ising critical point a specific prediction supported by recent series expansion studies is that the mean number of clusters k p presents a singularity of the form k p approximately mod p pc mod 2ln mod ln mod p pc rather than the accepted form k p approximately mod p pc mod 2 alpha with alpha 2 3 a novel numerical and renormalisation group finite size scaling analysis of the nature of the singularity in k p is presented in support of the new theory which implies the absence of a separate universality class for two dimensional percolation processes this study is consistent with the effective values of the exponents alpha 0 and nu 1

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

on jessen s inequality for convex functions ii on etablit quelques resultats lies a l integralite de b jessen pour les fonctions convexes

polymer models and generalized potts kasteleyn models the well established relation between potts models withv spin values and random cluster models with intracluster bonding favored over intercluster bonding by a factorv is explored but with the random cluster model replaced by a much generalized polymer model implying a corresponding generalization of the potts model the analysis is carried out in terms a given defined functionr an entropy free energy density for the polymer model in the casev 1 expressed as a function of the density of units the aim of the analysis is to determine the analogrv ofr for general nonnegativev in terms ofr and thence to determine the critical value of density vg at which gelation occurs this critical value is independent ofv up to a valuevp the potts critical value what is principally required ofr is that it should show a certain given concave convex behavior although differentiability and another regularizing condition are required for complete conclusions under these conditions the unique evaluation ofrv in terms ofr is given in a form known to hold for integralv but not previously extended the analysis is carried out in terms of the legendre transforms of these functions in terms of which the phenomena of criticality gelation and potts criticality appear very transparently andvp is easily determined the value ofvp is 2 under mild conditions onr special interest attaches to the functionr0 which is shown to be the greatest concave minorant ofr the naturalness of the approach is demonstrated by explicit treatment of the first shell model

multiple multiindex mittag leffler functions and relations to generalized fractional calculus the classical mittag leer m l functions have already proved their eciency as solutions of fractional order differential and integral equations and thus have become important elements of the fractional calculus theory and applications in this paper we introduce analogues of these functions depending on two sets of multiple m tuple m 2i s an integer indices the hint for this comes from a paper by dzrbashjan izv an arm ssr 13 3 1960 21 63 related to the case m 2 we study the basic properties and the relations of the multiindex m l functions with the operators of the generalized fractional calculus corresponding generalized operators of integration and dierention of the so called gelfond leontiev type as well as borel laplace type integral transforms are also introduced and studied c 2000 elsevier science b v all rights reserved msc 26a33 33c60 33e30 44a15 44a30

development of two new mean codeword lengths two new mean codeword lengths l a b and l b are defined and it is shown that these lengths satisfy desirable properties as a measure of typical codeword lengths consequently two new noiseless coding theorems subject to kraft s inequality have been proved further we have shown that the mean codeword lengths l 1 1 a b and l 1 1 b for the best one to one code not necessarily uniquely decodable are shorter than the mean codeword length l u d a b and l u d b respectively for the best uniquely decodable code by no more than log dlog dn 3 for d 2 moreover we have studied tighter bounds of l a b

phenomenological renormalisation of monte carlo data for percolation the accuracy of a phenomenological renormalisation which is based on monte carlo data is tested by investigating site percolation in a simple cubic lattice the method appears to be very accurate and can yield precise estimates of the quantities of interest with small to moderate lattice sizes the site percolation threshold of the lattice is predicted to be 0 3115 or 0 0005 if nu is the exponent of correlation length the author finds beta nu approximately 0 48 or 0 01 for the exponent of percolation probability and beta b nu approximately 1 1 or 0 03 for the backbone exponent in agreement with most accurate data currently available an analysis of the data at the percolation threshold based on the standard finite size scaling method also supports the results

some properties of generalized exponential entropies with applications to data compression abstract the scale of exponential entropies of order is introduced it unifies a number of notions of generalized differential entropies for continuous distributions the properties of this scale are derived from results in the theory of means we show that the scale can represent the extent of a probability distribution i e its degree of concentration with regard to a reference measure this fact has several consequences that are of interest in the theory of data or signal compression the aspects of data compression considered are the intrinsic extent of a probability distribution introduced by l l campbell and the properties of entropy series we investigate specially the exponential families of distributions in particular the miller thomas or generalized gaussian family of distributions

fractional klein gordon equations and related stochastic processes this paper presents finite velocity random motions driven by fractional klein gordon equations of order alpha in 0 1 a key tool in the analysis is played by the mcbride s theory which converts fractional hyper bessel operators into erdelyi kober integral operators special attention is payed to the fractional telegraph process whose space dependent distribution solves a non homogeneous fractional klein gordon equation the distribution of the fractional telegraph process for alpha 1 coincides with that of the classical telegraph process and its driving equation converts into the homogeneous klein gordon equation fractional planar random motions at finite velocity are also investigated the corresponding distributions obtained as well as the explicit form of the governing equations fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations b n alpha with uniformly distributed orientation are considered the parameter n of b n alpha is itself a random variable with fractional poisson distribution so that fractionality acts as a subsampling of the changes of direction finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained extensions to n dimensional fractional random flights are envisaged as well as the fractional counterpart of the euler poisson darboux equation to which our theory applies

approximation schemes for finite lattices two algorithms are formulated for systematically truncating the set of basis states required in lattice hamiltonian calculations thus allowing larger lattice abort sizes to be treated they are tested for the case of the 1 1 dimensional ising model it is shown that smoothly convergent finite lattice sequences are obtained which can successfully the extrapolated to the bulk limit the authors point out the relevance of the technique for lattice calculations in higher dimensions

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

about fractional integrals in the space of locally integrable functions abstract in this paper we study the classical fractional integrals of riemann liouville and weyl considering them as operators defined in the largest functional space where these operators can be defined we describe the fractional derivatives and integrals of riemann liouville as the fractional powers of suitable non negative operators and consequently we get important properties directly this is not the case for the fractional integral of weyl however we can obtain some interesting properties for these operators

multiple phase coexistence and the scaling transformation it is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible the method is illustrated by an application to the three four and five state potts models and even in its lowest order of approximation is able to distinguish the difference in the zero field critical behaviour between the five state and the three and four state models

anisotropic generalization of stinchcombe s solution for the conductivity of random resistor networks on a bethe lattice our study is based on the work of stinchcombe 1974 j phys c solid state phys 7 179 and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic bethe lattice the structure of the bethe lattice is assumed to represent the normal directions of the regular lattice we calculate the anisotropic conductivity as an expansion in powers of the inverse coordination number of the bethe lattice the expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold we make a comparison of our analytical results with those of bernasconi 1974 phys rev b 9 4575 for the regular lattice

the exact probability distribution of a two dimensional random walk a calculation is made of the exact probability distribution of the two dimensional displacement of a particle at timet that starts at the origin moves in straight line paths at constant speed and changes its direction after exponentially distributed time intervals where the lengths of the straight line paths and the turn angles are independent the angles being uniformly distributed this random walk is the simplest model for the locomotion of microorganisms on surfaces its weak convergence to a wiener process is also shown

some new fuzzy entropy formulas the purpose of this paper is twofold firstly a general conclusion about fuzzy entropy induced by distance measure is presented based on the axiom definitions of fuzzy entropy and distance measure secondly some fuzzy entropy formulas which relate to the fuzzy entropy formula defined by de luca and termini inform control 20 1972 301 are given

bounds on the entropy series upper bounds on the entropy of a countable integer valued random variable are furnished in terms of the expectation of the logarithm function in particular an upper bound is derived that is sharper than that of p elias ibid vol it 21 no 2 p 194 203 1975 for all values of e sub p log bounds that are better only for large values of e sub p than the previous known upper bounds are also provided

monte carlo method for critical systems in infinite volume the planar ising model in this paper we propose a monte carlo method for generating finite domain marginals of critical distributions of statistical models in infinite volume the algorithm corrects the problem of the long range effects of boundaries associated to generating critical distributions on finite lattices it uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of holographic boundary condition that encodes the presence of an infinite volume beyond it we check the quality of the distribution obtained in the case of the planar ising model by comparing various observables with their infinite plane prediction we accurately reproduce planar two three and four point of spin and energy operators we also define a lattice stress energy tensor and numerically obtain the associated conformal ward identities and the ising central charge

percolation in two dimensional quasicrystals by monte carlo simulations the authors examine whether or not two dimensional quasi periodic lattices belong to the same universality class as periodic lattices and they study bond and site percolation in penrose tiling and its dual lattice by making use of monte carlo simulations for the sake of comparison they also investigate percolation in periodic systems such as a square kagome and dice lattice for all these lattices they evaluate several critical exponents such as alpha related to the total number of clusters beta related to the percolation strength gamma related to the mean cluster size nu related to the correlation length tau related to cluster sizes and the fractal dimension d their results indicate that universality holds in two dimensional lattices with or without periodicity where the coordination number could be either single valued or multi valued

anisotropic cubic lattice potts ferromagnet renormalisation group treatment the choice of a convenient self dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q state potts ferromagnet criticality the exact critical frontier and dimensionality crossover exponent hf as well as the expected universality behaviour renormalisation flow sense are recovered for any linear scaling factor b and all values of q q or 4 the b 2 and b 3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den nijs conjecture the same calculation is performed for all values of b for the exponent nu d 1 associated with the one dimensional limit and the exact result nu d 1 1 is recovered in the limit b to infinity