paley wiener spaces with vanishing conditions and painleve vi transcendents abstract we modify the classical paley wiener spaces pw x of entire functions of finite exponential type at most x 0 which are square integrable on the real line via the additional condition of vanishing at finitely many complex points z 1 z n we compute the reproducing kernels and relate their variations with respect to x to a krein differential system whose coefficient which we call the function and solutions have determinantal expressions arguments specific to the case where the trivial zeros z 1 z n are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non linear first order differential equations a computation having this non linear system at his start obtains quasi algebraic and among them rational painleve transcendents of the sixth kind as certain quotients of such functions

exclusion and persistence in deterministic and stochastic chemostat models abstract we first introduce and analyze a variant of the deterministic single substrate chemostat model in this model microbe removal and growth rates depend on biomass concentration with removal terms increasing faster than growth terms using a comparison principle we show that persistence of all species is possible in this scenario then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation in particular we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence on the other hand we also establish some stochastic persistence results

stochastic perturbation of dynamical systems the weak convergence of measures in this paper we consider the asymptotic behaviour of randomly perturbed discrete dynamical systems we treat this problem by examining the evolution of the corresponding sequences of distributions we show that an average contractive property is sufficient to ensure the weak convergence of the sequence of distributions to a unique stationary measure

diffusion on multifractals abstract we introduce a class of random fields with variable mean square regularity order defined on multifractal domains the local singularity order of these random fields then depends on the initial variable mean square regularity order and on the variable local singularity exponent of the multifractal measure defining the local dimension of the domain considered the theory is developed in a generalized framework through the covariance factorization using the tools of reproducing kernel hilbert spaces and fractional sobolev spaces of variable order

huber fractal image coding based on a fitting plane recently there has been significant interest in robust fractal image coding for the purpose of robustness against outliers however the known robust fractal coding methods hfic and lad fic etc are not optimal since besides the high computational cost they use the corrupted domain block as the independent variable in the robust regression model which may adversely affect the robust estimator to calculate the fractal parameters depending on the noise level this paper presents a huber fitting plane based fractal image coding hfpfic method this method builds huber fitting planes hfps for the domain and range blocks respectively ensuring the use of an uncorrupted independent variable in the robust model on this basis a new matching error function is introduced to robustly evaluate the best scaling factor meanwhile a median absolute deviation mad about the median decomposition criterion is proposed to achieve fast adaptive quadtree partitioning for the image corrupted by salt pepper noise in order to reduce computational cost the no search method is applied to speedup the encoding process experimental results show that the proposed hfpfic can yield superior performance over conventional robust fractal image coding methods in encoding speed and the quality of the restored image furthermore the no search method can significantly reduce encoding time and achieve less than 2 0 s for the hfpfic with acceptable image quality degradation in addition we show that combined with the mad decomposition scheme the hfp technique used as a robust method can further reduce the encoding time while maintaining image quality

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

fractal image denoising over the past decade there has been significant interest in fractal coding for the purpose of image compression however applications of fractal based coding to other aspects of image processing have received little attention we propose a fractal based method to enhance and restore a noisy image if the noisy image is simply fractally coded a significant amount of the noise is suppressed however one can go a step further and estimate the fractal code of the original noise free image from that of the noisy image based upon a knowledge or estimate of the variance of the noise assumed to be zero mean stationary and gaussian the resulting fractal code yields a significantly enhanced and restored representation of the original noisy image the enhancement is consistent with the human visual system where extra smoothing is performed in flat and low activity regions and a lower degree of smoothing is performed near high frequency components e g edges of the image we find that for significant noise variance spl sigma spl ges 20 the fractal based scheme yields results that are generally better than those obtained by the lee filter which uses a localized first order filtering process similar to fractal schemes we also show that the lee filter and the fractal method are closely related

iterated fuzzy set systems a new approach to the inverse problem for fractals and other sets abstract images with grey or colour levels admit a natural representation in terms of fuzzy sets but without the usual probabilistic interpretation of the latter we introduce a fuzzy set approach which incorporates in part the technique of iterated function systems ifs for the construction analysis and or approximation of typically fractal sets and images the method represents a significant departure from ifs especially in the interpretation of the resulting image the introduction of grey level maps i 0 1 0 1 associated with the contractive maps w i of the ifs affords much greater flexibility in the generation of images as well as in the inverse problem

approximations of fractal sets abstract five approximations of fractal sets a are compared the first three were described by hutchinson barnsley demko and williams respectively two new other approximations are still efficient in all circumstances for any algorithm error estimates in the approximations are found numerical comparisons are also done through classical examples of attractors in the plane

recognition of an organism from fragments of its complete genome this paper considers the problem of matching a fragment to an organism using its complete genome our method is based on the probability measure representation of a genome we first demonstrate that these probability measures can be modeled as recurrent iterated function systems rifs consisting of four contractive similarities our hypothesis is that the multifractal characteristics of the probability measure of a complete genome as captured by the rifs is preserved in its reasonably long fragments we compute the rifs of fragments of various lengths and random starting points and compare with that of the original sequence for recognition using the euclidean distance a demonstration on five randomly selected organisms supports the above hypothesis

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

on the hausdorff dimension of some graphs consider the functions wb x b cn 1 bnx on 1 0n n oo where b 1 0 0 such that if b is large enough then the hausdorff dimension of the graph of wb is bounded below by 2a c ln b we also show that if a function f is convex lipschitz of order a then the graph of f has a finite measure with respect to hausdorff s measure in dimension 2 a the convex lipschitz functions of order a include zygmund s class a our analysis shows that the graph of the classical van der waerdentagaki nowhere differentiable function has a finite measure with respect to h t t in 1 t we consider the hausdorff dimension of the graphs of various continuous functions we introduce a new geometric property of a function convex lipschitz of some order and obtain an upper bound on the dimension of a graph with this property in particular our analysis includes functions of the form

on computing jacobi matrices associated with recurrent and mobius iterated function systems a new technique for the computation of jacobi matrices associated with measures possessing scale invariance properties is described invariance is imposed via linear transformations and by mobius transformations in psl 2 r this technique is proven to be numerically stable and apt to compute jacobi matrices of large orders

a mathematical model of the chemostat with a general class of functions describing nutrient uptake a model of the chemostat involving n microorganisms competing for a single essential growth limiting substrate is considered instead of assuming the familiar michaelis menten kinetics for nutrient uptake a general class of functions is used which includes all monotone increasing uptake functions but which also allows uptake functions that describe inhibition by the substrate at high concentrations the qualitative behaviour of this generalized model is determined analytically it is shown that the behaviour depends intimately upon certain parameters provided that all the parameters are distinct which is a biologically reaonable assumption at most one competitor survives the substrate and the surviving competitor if one exists approach limiting values thus there is competitive exclusion however unlike the standard model in certain cases the outcome is initial condition dependent

similarity and fractality in the modelling of roughness by a multilevel profile with hierarchical structure some modern scaling and fractal approaches to the characterization of surface roughness are under consideration a new mathematical model of a multilevel profile suitable for flexible simulation of roughness is presented as a development from these approaches the iterative construction of the model is described it is shown that the profile is a hierarchical structure with self affine and fractal features its geometrical properties are studied both analytically and numerically and some of them are proved to be of the renormalization type the profile contour length can be finite or infinite depending on the chosen values of the structural parameters some characteristics including the bearing curve and the upper volume function of the profile and the corresponding prestructures are calculated

an approach to the branching rule from sl2n c to sp2n c via littelmann s path model abstract we propose a conjecture describing the branching rule in terms of littelmann s path model from the special linear lie algebra sl 2 n c of type a 2 n 1 to the symplectic lie algebra of type c n embedded as the fixed point subalgebra of the diagram automorphism of sl 2 n c moreover we prove the conjecture in certain cases and also provide some supporting examples in addition we show that the branching coefficients can be obtained explicitly by using the inverse kostka matrix and path models for tensor products of symmetric powers of the defining or natural representation c 2 n of sl 2 n c

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

refinable functions functionals and iterated function systems invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage that although usually considered in different contexts are deeply linked we outline these links in particular with respect to existence and regularity of these objects

scaling range and cutoffs in empirical fractals fractal structures appear in a vast range of physical systems a literature survey including all experimental papers on fractals which appeared in the six physical review journals a char21 e and letters during the 1990s shows that experimental reports of fractal behavior are typically based on a scaling range ensuremath delta that spans only 0 5 char21 2 decades this range is limited by upper and lower cutoffs either because further data are not accessible or due to crossover bends focusing on spatial fractals a classification is proposed into a aggregation b porous media c surfaces and fronts d fracture and e critical phenomena most of these systems except for class e involve processes far from thermal equilibrium the fact that for self similar fractals in contrast to the self affine fractals of class c there are hardly any exceptions to the finding of ensuremath delta l 2 decades raises the possibility that the cutoffs are due to intrinsic properties of the measured systems rather than the specific experimental conditions and apparatus to examine the origin of the limited range we focus on a class of aggregation systems in these systems a molecular beam is deposited on a surface giving rise to nucleation and growth of diffusion limited aggregation like clusters scaling arguments are used to show that the required duration of the deposition experiment increases exponentially with ensuremath delta furthermore using realistic parameters for surfaces such as al 111 it is shown that these considerations limit the range of fractal behavior to less than two decades in agreement with the experimental findings it is conjectured that related kinetic mechanisms that limit the scaling range are common in other nonequilibrium processes that generate spatial fractals

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

fractal models of elastic perfectly plastic contact of rough surfaces based on the cantor set abstract the objective of this study was to formulate discrete and continuous models to describe the elastic perfectly plastic deformation of two rough surfaces in contact the two surfaces in contact are assumed to exhibit fractal behavior and are modeled as an effective fractal surface compressed into a smooth rigid substrate the rough self affine fractal structure of the effective surface is approximated using a cantor set representation both of the proposed models admit analytical solution for the cases when the plastic deformation is volume conserving or not results are presented that illustrate the effects that volume conservation and initial surface structure have on the elastic perfectly plastic deformation process the results from the continuous model are compared with the results obtained from the discrete model and also with existing experimental load displacement results for the deformation of a ground steel surface

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

missing moment and perturbative methods for polynomial iterated function systems abstract an iterated function system ifs over a compact metric space x is defined by a set of contractive maps wi x x i 1 n with associated nonzero probabilities pi 0 p i 1 the parallel action of the maps defines a unique compact invariant attractor set a x which supports an invariant measure and which is balanced with respect to the pi for linear w i on x r the invariance of yields a relation between the moments gn nd which permits their recursive computation from the initial value g0 1 for nonlinearwi however the moment relations are incomplete and do not permit a recursive computation this paper describes two methods of obtaining accurate estimates of the moments when the ifs maps wi are polynomials i application of the necessary hausdorff conditions on the gi to obtain convergent upper and lower bounds and ii a perturbation expansion approach the methods are applied to some model problems

on local fractal functions in besov and triebel lizorkin spaces abstract within the new concept of a local iterated function system local ifs we consider a class of attractors of such ifss namely those that are graphs of functions these new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature for a class of local fractal functions we derive explicit conditions for them to be elements of besov and triebel lizorkin spaces these two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical function spaces in the sense of equivalent norms the conditions we derive provide immediate information about inclusion of local fractal functions in for instance lebesgue sobolev slobodeckij holder bessel potential and local hardy spaces

applications of minor summation formula i littlewood s formulas abstract the first and the third authors obtained a minor summation formula of pfaffian which expresses a weighted sum of minors of any rectangular matrix in terms of a single pfaffian in this paper as an application of this minor summation formula we give a new proof of littlewood s formulas with additional parameters and generalize them for the classical groups

on the scaling ranges of detrended fluctuation analysis for long term memory correlated short series of data abstract we examine the scaling regime for the detrended fluctuation analysis dfa the most popular method used to detect the presence of long term memory in data and the fractal structure of time series first the scaling range for dfa is studied for uncorrelated data as a function of time series length l and the correlation coefficient of the linear regression r 2 at various confidence levels next a similar analysis for artificial short series of data with long term memory is performed in both cases the scaling range is found to change linearly both with l and r 2 we show how this dependence can be generalized to a simple unified model describing the relation l r 2 h where h 1 2 h 1 stands for the hurst exponent of the long range autocorrelated signal our findings should be useful in all applications of dfa technique particularly for instantaneous local dfa where a huge number of short time series has to be analyzed at the same time without possibility of checking the scaling range in each of them separately

multifractal analysis of hang seng index in hong kong stock market abstract in this paper the daily hang seng index in hong kong stock market is analyzed by multifractal the correlation of the parameters of the multifractal spectra with the variation of close return z is studied statistically it is found that the amount of the variation of return is correlated with the amount of of that day the increase or decrease of the return is related to the positive or negative value of f the gain probability and the increasing day s probability can be higher than 70 at the larger f region and be lower than 20 at the more negative region

fractal analysis of measure representation of large proteins based on the detailed hp model abstract the notion of measure representation of protein sequences is introduced based on the detailed hp model multifractal analysis and detrended fluctuation analysis are then performed on the measure representations of a large number of long protein sequences it is concluded that these protein sequences are not completely random sequences through the measure representations and the values of the d q spectra and related c q curves the values of the exponent from the detrended fluctuation analysis show that the k strings with the ordering in the measure representation exhibit strong long range correlation for substrings with length k 5 the d q spectra of all proteins studied are multifractal like and sufficiently smooth for the c q curves to be meaningful the c q curves of all proteins resemble a classical phase transition at a critical point an ifs model is found to simulate the measure representation of protein sequences very well from the estimated values of parameters in the ifs model we think the non polar residues and uncharged polar residues play a more important role than other kinds of residues in the protein folding process

the parafermion fock space and explicit mathfrak so 2n 1 representations the defining relations triple relations of n pairs of parafermion operators f j j 1 n are known to coincide with a set of defining relations for the lie algebra in terms of 2n generators with the common hermiticity conditions this means that the parafermions of order p correspond to a finite dimensional unitary irreducible representation w p of with highest weight although the dimension and the character of w p is known by classical formulae there is no explicit basis of w p available in which the parafermion operators have a natural action in this paper we construct an orthogonal basis for w p and present the explicit actions of the parafermion generators on these basis vectors we use group theoretical techniques in which the subalgebra of plays a crucial role a set of gelfand zetlin patterns of will be used to label the basis vectors of w p and also in the explicit action matrix elements certain clebsch gordan coefficients are essential

on the fractal dimension of self affine profiles one dimensional profiles f x can be characterized by a minkowski bouligand dimension d and by a scale dependent generalized roughness w f epsilon this roughness can be defined as the dispersion around a chosen fit to f x in an epsilon scale it is shown that d limepsilon to 0 2 in w f epsilon in epsilon holds for profiles nowhere differentiable this establishes a close connection between the roughness and the fractal dimension and proves that d 2 h for self affine profiles h is the roughness or hurst exponent two numerical algorithms based on the roughness one around the local average f x epsilon usual roughness and the other around the local rms straight line a generalized roughness are discussed the estimates of d for standard self affine profiles are reliable and robust especially for the last method

the box dimensions of inhomogeneous self conformal sets with overlaps abstract we investigate the box dimensions of inhomogeneous self conformal sets with overlaps firstly we significantly generalize the result of fraser concerning upper box dimension under some mild overlap conditions secondly we give corresponding formula of lower box dimension for condense set c with special construction these results extend the bowen s equation for inhomogeneous ifs

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

a criterion for the determination of optimal scaling ranges in dfa and mf dfa abstract we develop a criterion based on a brute force algorithm to systematically determine optimal fitting regions for fluctuation functions in detrended fluctuation analysis dfa and multifractal detrended fluctuation analysis mf dfa we analyze and compare results with several artificially generated time series with known parameters as illustrative examples of this technique we show that crossover time scales can also be objectively and efficiently estimated using the introduced algorithm finally we employ the proposed methodology to study the scaling behavior of a natural time series monthly sunspot data previously filtered with empirical mode decomposition emd in this case we find results comparable with those found in the literature

an elliptic generalization of schur s pfaffian identity we present a pfaffian identity involving elliptic functions whose rational limit gives a generalization of schur s pfaffian identity for pf xj xi xj xi this identity can be regarded as the pfaffian counterpart of frobenius generalization of cauchy s determinant identity for det 1 xi xj

sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat abstract this paper studies the asymptotic behaviors of one classical chemostat model in a stochastic environment based on the feller property sharp conditions are derived for the existence of a stationary distribution by using the mutually exclusive possibilities known in 11 12 see lemma 2 4 for details which closes the gap left by the lyapunov function further we obtain a sufficient condition for the extinction of the organism based on two noise induced parameters an analogue of the feed concentration s and the break even concentration results indicate that both noises have negative effects on persistence of the microorganism

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

the parafermion fock space and explicit so 2n 1 representations the defining relations triple relations of n pairs of parafermion operators f j j 1 n are known to coincide with a set of defining relations for the lie algebra so 2n 1 in terms of 2n generators with the common hermiticity conditions this means that the parafermions of order p correspond to a finite dimensional unitary irreducible representation w p of so 2n 1 with highest weight p 2 p 2 p 2 although the dimension and the character of w p is known by classical formulae there is no explicit basis of w p available in which the parafermion operators have a natural action in this paper we construct an orthogonal basis for w p and present the explicit actions of the parafermion generators on these basis vectors we use group theoretical techniques in which the u n subalgebra of so 2n 1 plays a crucial role a set of gelfand zetlin patterns of u n will be used to label the basis vectors of w p and also in the explicit action matrix elements certain u n clebsch gordan coefficients are essential

multifractal analysis of evolving noise associated with unstable plastic flow plastic flow instability attracts increasing interest as a self organization phenomenon showing various dynamical regimes including deterministic chaos and self organized criticality the analysis of the associated nonrandom noise drastic jumps of the mechanical stress however confronts the variation of the noise average parameters due to the evolution of the dislocation microstructure the present paper examines some limitations of the multifractal approach to the study of the evolving noise the applicability of the multifractal analysis to practical situations is proven using the example of discontinuous deformation curves observed under conditions of the portevin le ch telier effect in an a1mg alloy as well as model signals generated by stretching multifractal cantor sets it is found that the smooth trends in the stress serration parameters may narrow the range of the scale invariant behavior associated with the multifractal structure but do not essentially mask it

a fast encoding algorithm for fractal image compression using the dct inner product in this paper a fast encoding algorithm is developed for fractal image compression at each search entry in the domain pool the mean square error mse calculations of the given range block and the eight dihedral symmetries of the domain block are obtained simultaneously in the frequency domain in which the redundant computations are all eliminated in the new encoding algorithm it is shown in software simulation that the encoding time is about six times faster than that of the baseline method with almost the same psnr for the retrieved image the fast algorithm is performed to deal with the eight dihedral symmetries at each search entry therefore it can be applied to various enhanced algorithms which are equipped with quadtree classification and other mechanisms

moments of balanced measures on julia sets by a theorem of s demko there exists a balanced measure on the julia set of an arbitrary nonlinear rational transformation on the riemann sphere it is proved here that if the transformation admits an attractive or indifferent cycle then there is a point with respect to which all the moments of a balanced measure exist moreover these moments can be calculated exactly an explicit balanced measure is exhibited in an example where the julia set is the whole sphere and for which the moments with respect to any point do not all exist 1 introduction the julia set for a nonlinear rational transformation on the riemann sphere is the closure of the set of all repulsive cycles of the transformation br fa j the location or distribution of the set is of interest in the theory of iterated maps and it may be characterized by an associated invariant measure in this paper we extend previous results by showing that when there is an attractive or indifferent cycle certain moments of such a measure exist and can be calculated exactly from the rational transformation this means for example that when the julia set is contained strictly in the real line all of the orthogonal polynomials on the set can be calculated as can the entries of the corresponding infinite dimensional jacobi matrices the latter are of the type which have recently been shown to furnish model schrodinger equations for crystals with almost periodic potentials bbm this leads us to the following speculation which provides one reason for our interest in the present topic we note first the frequent occurrence in the physical universe of objects steady state systems which are fractals from some point of

vicious walkers friendly walkers and young tableaux ii with a wall we derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of young tableaux and combinatorial descriptions of symmetric functions for the problem of n friendly walkers we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall

fractal dimensions and singularities of the weierstrass type functions a new type of fractal measures k s 1 s 2 defined on the subsets of the graph of a continuous function is introduced the k dimension defined by this measure is closer to the hausdorff dimension than the other fractal dimensions in recent literatures for the weierstrass type functions defined by w x 0 i g i x where 1 0 1 and g is an almost periodic lipschitz function of order greater than it is shown that the k dimension of the graph of w equals to 2 this conclusion is also equivalent to certain rate of the local oscillation of the function some problems on the knot points and the nondifferentiability of w are also discussed

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

linear elastic contact of the weierstrass profile a contact problem is considered in which an elastic half plane is pressed against a rigid fractally rough surface whose profile is defined by a weierstrass series it is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension the contact area consists of a set of fractal character for all values of the geometric and loading parameters a solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n 1 and that at scale n recursive numerical integration of this relation yields the contact area as a function of scale an analytical solution to the same problem appropriate at large n is constructed following a technique due to archard this is found to give a very good approximation to the numerical results even at small n except for cases where the dimensionless applied load is large the contact area is found to decrease continuously with n tending to a power law behaviour at large n which corresponds to a limiting fractal dimension of 2 d where d is the fractal dimension of the surface profile however it is not a simple fractal in the sense that it deviates from the power law form at low n at which there is also a dependence on the applied load contact segment lengths become smaller at small scales but an appropriately normalized size distribution tends to a limiting function at large n the authors dedicate this paper to the memory of dr j f archard 1918 1989

inverse and approximation problem for two dimensional fractal sets the geometry of fractals is rich enough that they have extensively been used to model natural phenomena and images iterated function systems ifs theory provides a convenient way to describe and classify deterministic fractals in the form of a recursive definition as a result it is conceivable to develop image representation schemes based on the ifs parameters that correspond to a given fractal image in this paper we consider two distinct problems an inverse problem and an approximation problem the inverse problem involves finding the ifs parameters of a signal that is exactly generated via an ifs we make use of the wavelet transform and of the image moments to solve the inverse problem the approximation problem involves finding a fractal ifs generated image whose moments match either exactly or in a mean squared error sense a range of moments of the original image the approximating measures are generated by an ifs model of a special form and provide a general basis for the approximation of arbitrary images experimental results verifying our approach will be presented

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

diagonally and antidiagonally symmetric alternating sign matrices of odd order we study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices dasasms of fixed odd order by introducing a case of the six vertex model whose configurations are in bijection with such matrices the model involves a grid graph on a triangle with bulk and boundary weights which satisfy the yang baxter and reflection equations we obtain a general expression for the partition function of this model as a sum of two determinantal terms and show that at a certain point each of these terms reduces to a schur function we are then able to prove a conjecture of robbins from the mid 1980 s that the total number of 2n 1 x 2n 1 dasasms is prod i 0 n 3i n i and a conjecture of stroganov from 2008 that the ratio between the numbers of 2n 1 x 2n 1 dasasms with central entry 1 and 1 is n n 1 among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980 s that for odd order dasasms is the last to have been proved

identities for classical group characters of nearly rectangular shape abstract we derive several identities that feature irreducible characters of the general linear the symplectic the orthogonal and the special orthogonal groups all of the identities feature characters that are indexed by shapes that are nearly rectangular by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others as applications we prove new results in plane partitions and tableaux enumeration including new refinements of the bender knuth and macmahon ex conjectures

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

generalized dimensions entropies and liapunov exponents from the pressure function for strange sets for conformal mixing repellers such as julia sets and nonlinear one dimensional cantor sets we connect the pressure of a smooth transformation on the repeller with its generalized dimensions entropies and liapunov exponents computed with respect to a set of equilibrium gibbs measures this allows us to compute the pressure by means of simple numerical algorithms our results are then extended to axiom a attractors and to a nonhyperbolic invariant set of the line in this last case we show that a first order phase transition appears in the pressure