towards a physics on fractals differential vector calculus in three dimensional continuum with fractal metric abstract one way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric on this way different definitions of the fractal metric were suggested to account for the essential fractal features in this work we develop the metric differential vector calculus in a three dimensional continuum with a non euclidean metric the metric differential forms and laplacian are introduced fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the moisil teodoresco operator which has been introduced and partially developed in this paper the relations between the metric and conventional operators are revealed it should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric this offers a novel tool to study physics on fractals

equations of motion in a non integer dimensional space equations of motion are derived for a fractional dimensional system of n spatial coordinates to be used as an effective description of anisotropic and confined systems an existing measure theoretic approach is extended to multiple variables and different degrees of confinement in orthogonal directions and comparisons are made with the analytic continuation of gaussian integrals this is applied to the variational principle and equations of motion for a field described by a lagrange density are found a specific example is looked at in schrodinger wave mechanics particularly in three coordinate systems

root properties and convergence rates of median filters median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to additional passes of the median filter in this paper the theory is developed both for determining the cardinality of the root signal space of arbitrary window width filters applied to signals with any number of quantization levels and for counting or estimating the number of passes required to produce a root for binary signals

alpha trimmed means and their relationship to median filters suppose that x is a finite set of n numbers the trimmed mean of x is obtained by sorting x into ascending order removing trimming a fixed fraction alpha 0 leq alpha leq 0 5 from the high and low ends of the sorted set and computing the average of the remaining values when applied to a sliding window of length lw the trimming process is called trim filtering for 0 5 the trimmed mean of a set is the median of the set and the filtering operation is called median filtering repeated application of a median filter to the output of a previous median filter of the same length lw eventually produces a signal which is invariant to median filtering this final signal is called the root signal this paper explains the relationship between trimmed means and median filters derives a simple straightforward and fast algorithm for applying a median filter and provides a new explanation of the convergence of repeated median filtering to the root signal the latter result incorporates an approach which permits generalization of the associated concepts to a larger class of index map filters

a training framework for stack and boolean filtering fast optimal design procedures and robustness case study a training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks the targeted families of nonlinear filters are the boolean filters and stack filters the main merit of this framework at the implementation level is perhaps the absence of constraining models making it nearly universal in terms of application areas we develop fast procedures to design optimal or close to optimal filters based on some representative training set furthermore the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution symmetry constraints are imposed on the data and consequently on the resulting optimal solutions for improved performance and ease of implementation the case study is dedicated to natural images the properties of optimal boolean and stack filters when the desired signal in the training set is the image of a natural scene are analyzed specifically the effect of changing the desired signal using various natural images and the characteristics of the noise the probability distribution function the mean and the variance is analyzed elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets a remarkably low sensitivity and consequently a good generalization power of boolean and stack filters are revealed boolean based filters are thus shown to be not only suitable for image restoration but also robust making it possible to build libraries of optimal filters which are suitable for a set of applications

optimal weighted median filtering under structural constraints a new expression for the output moments of weighted median filtered data is derived the noise attenuation capability of a weighted median filter can now be assessed using the l vector and m vector parameters in the new expression the second major contribution of the paper is the development of a new optimality theory for weighted median filters this theory is based on the new expression for the output moments and combines the noise attenuation and some structural constraints on the filter s behavior in certain special cases the optimal weighted median filter can be obtained by merely solving a set of linear inequalities this leads in some cases to closed form solutions for optimal weighted median filters some applications of the theory developed in this paper in 1 d signal processing and image processing are discussed throughout the analysis some striking similarities are pointed out between linear fir filters and weighted median filters

a simple empirical formula of origin intensity factor in singular boundary method for two dimensional hausdorff derivative laplace equations with dirichlet boundary abstract this paper presents a simple empirical formula of origin intensity factor in singular boundary method sbm solution of hausdorff derivative laplace equations the sbm with the empirical formula is mathematically more simple and computationally more efficient than using the other techniques for origin intensity factor numerical experiments simulate the steady heat conduction through fractal media governed by the hausdorff laplace equation and show the efficiency and reliability benefits of the present sbm empirical formula

local fractional fokker planck equation we propose a new class of differential equations which we call local fractional differential equations they involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time a local fractional analog of the fokker planck equation has been derived starting from the chapman kolmogorov condition we solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

adaptive trimmed mean filters for image restoration an adaptive smoothing filter is proposed for reducing noise in digital signals of any dimensionality the adaptive procedure is based on the selection of an appropriate inner or outer trimmed mean filter according to local measurements of the tail behavior impulsivity of the noise process the set of trimmed means used provides robustness against a wide range of noise possibilities ranging from very shallow tailed to very heavy tailed a monte carlo analysis using a family of generalized exponential distributions supports the choice of the trimmed mean selected for measured values of an impulsivity statistic the assumption underlying the definition of the filter is that the signal to be filtered is locally smoothly varying and that the noise process is uncorrelated and derives from an unknown unimodal symmetric distribution for image processing applications a second statistic is used to mark the location of abrupt intensity changes or edges in the vicinity of an edge the trend preserving median filter is used since the impulsivity and edge statistics used in defining the adaptive filter are both functions of order statistics the extra computation required for their calculation is minimal examples are provided of the filter as applied to images corrupted by a variety of noises

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

some statistical properties of median filters some statistical properties of median filters are analyzed it is shown that median filters can remove impulsive plus gaussian white noise better than hanning filters when the amplitude of the impulses is large or the energy of the gaussian noise is relatively low it is also shown that unlike linear filters median filters preserve sharp changes in signals when the noise energy is not too high

on second moment properties of median filtered sequences of independent data in this paper we present the derivation of the bivariate distribution function for median filtered sequences of independent arbitrary second order random variables this result is then used to qualitatively analyze second moment properties of median filtered data the results hint towards a low pass characteristic of the median filters and a low dependency of the output spectrum on input alphabet size and distribution

a very singular solution of the porous media equation with absorption turns out to play an important role in the description of both the short and the long time behaviour of positive solutions of eq 1 1 for the short time behaviour it provides together with the one parameter family of singular solutions i of problem i a complete description of the possible types of singular behaviour at the point 0 o 121 for the long time behaviour of solutions u x t of the cauchy problem for 1 1 1 2 it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 co t p u x t w x t 0 ast co uniformly in r 7 81 in this paper we shall show that if

adaptive multistage median filter an adaptive multistage median filter is proposed for preserving the details of images it is shown that this filter can smooth noise more efficiently than the multistage median filter

some statistical properties of alpha trimmed mean and standard type m filters the statistical properties of two classes of filters generalizing the median filter are considered the two classes are the alpha trimmed mean alpha tm filter and the standard type m stm filter both of which are special cases of the l filter results are developed to quantify the white noise suppression and edge preservation characteristics of the filters by considering their output sequence error statistics it is shown that the alpha tm and stm filters perform better than the running mean and median filters in white noise suppression while they can be designed to be comparable to the median filter in edge preservation in the presence of noise

thermal waves in absorbing media abstract we discuss the existence of travelling wave solutions with interfaces for the nonlinear heat equation with absorption u t a u m xx bu n with a b 0 and m n r several situations occur depending on the relative strength of the diffusion and absorption terms reflected by their exponents m and n we characterize the existence of finite travelling waves in terms of m and n show their uniqueness up to translations in space and time and derive their velocity from the wave profile near the interface or front

steerable weighted median filters a filter is steerable if transformed i e rotated scaled etc versions of its impulse response can be expressed as linear combinations of a fixed set of basis functions steerability is important for numerous image processing applications however it is a property presently shared only by a specific class of linear filters on the other hand several classes of nonlinear filters such as weighted median filters wmfs may offer certain advantages over linear filters such as robustness and edge preserving capabilities in this paper the concept of steerability is extended to encompass wmfs it will be shown that in general a steerable wmf design technique needs to be capable of handling negative weights although methods that allow the design of wmfs admitting negative weights have already been proposed such methods do not necessarily produce filters that are steerable as opposed to the approach presented in this work experimental results illustrate the applicability of steerable wmfs in two applications namely edge detection and orientation analysis

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

common aspects of q deformed lie algebras and fractional calculus fractional calculus and q deformed lie algebras are closely related both concepts expand the scope of standard lie algebras to describe generalized symmetries a new class of fractional q deformed lie algebras is proposed which for the first time allows a smooth transition between different lie algebras for the fractional harmonic oscillator the corresponding fractional q number is derived it is shown that the resulting energy spectrum is an appropriate tool to describe e g the ground state spectra of even even nuclei in addition the equivalence of rotational and vibrational spectra for fractional q deformed lie algebras is shown and the b e2 values for the fractional q deformed symmetric rotor are calculated a first interpretation of half integer representations of the fractional rotation group is given in terms of a description of k 1 2 band spectra of odd even nuclei

generalized multistage median filters as an extension of the multistage median filters a class of r filters is presented some properties of these filters are discussed it is shown that the filters can smooth noise and preserve details of images efficiently

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

nonlinear mean filters in image processing the use of nonlinear means in image processing is introduced the properties of these means in the presence of different types of noise are investigated it is shown that nonlinear filters based on these means behave well for both additive and impulse noise their performance in the presence of signal dependent noise is satisfactory they preserve the edges better than linear filters and they reject the noise better than median filters

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

a new class of detail preserving filters for image processing a new class of median type filters for image processing is proposed in the filters linear fir substructures are used in conjunction with the median operation the root signals and noise attenuation properties of the fir median hybrid filters are analyzed and compared to representative edge preserving filtering operations the concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the fir median hybrid filters in the multilevel filters there exists a tradeoff between noise attenuation and detail preservation the analysis and examples indicate that fir median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the k nearest neighbor averaging filters

wavelet transform and scale space filtering of fractal images in this correspondence we consider wavelet transform and scale space filtering as special cases of general scale shift mapping ssm then the ssm is used as a tool for characterizing the geometrical complexity of uniform fractals this tool can reveal the construction rule of fractals and estimate fractal dimension finally we use the results obtained from scale shift mapping to estimate iterated function system ifs codes of a class of fractal images some examples have been illustrated to demonstrate its good performance

edge sensitive image restoration using order constrained least squares methods in this paper we consider a novel technique for the restoration of noise corrupted images using order constrained least squares methods the restoration process uses a moving cross shaped filter window within which two operations are combined the first operation consists of simple hypothesis tests for the presence of an edge of some minimal height crossing the center of the window the second operation computes the window output as the order constrained least squares fit of the windowed values if an edge is deemed to be present or simply the average if no edge is present the new technique is applied to some actual noise corrupted images and the results are compared to the results of applying similarly configured median and averaging filters some computational considerations and comparisons are discussed at the end of the paper

genetic based fuzzy hybrid multichannel filters for color image restoration abstract on the design of multichannel filters especially in color image restoration it is not easy to simultaneously achieve three objectives noise attenuation chromaticity retention and edges or details preservation in this paper we propose a new class of multichannel filters called genetic based fuzzy hybrid multichannel gfhm filters to reach these three objectives simultaneously the design of gfhm filters is mainly based on human concept heuristic rules and genetic algorithms because the human concept can be readily and efficiently expressed by fuzzy implicative rules gfhm filters can take the useful characteristics of filtering behavior of three filters a vector median a vector directional and an identity filter since genetic algorithms possess the global searching capability for an optimal solution they are able to effectively optimize gfhm filters to improve the filtering performance in color image restoration applications extensive simulation results illustrate that gfhm filters not only achieve these three objectives but also possess the robust and the adaptive capability moreover these simulation results also demonstrate that the performance of gfhm filters outperforms that of other proposed filtering techniques

fractional dirac bracket and quantization for constrained systems so far it is not well known how to deal with dissipative systems there are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem on the other hand it is well known that the fractional formalism is a powerful alternative when treating dissipative problems in this paper we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way we believe that by using the extended dirac bracket definition it will be possible to analyze more deeply gauge theories starting with second class systems

anomalous diffusion modeling by fractal and fractional derivatives this paper makes an attempt to develop a fractal derivative model of anomalous diffusion we also derive the fundamental solution of the fractal derivative equation for anomalous diffusion which characterizes a clear power law this new model is compared with the corresponding fractional derivative model in terms of computational efficiency diffusion velocity and heavy tail property the merits and distinctions of these two models of anomalous diffusion are then summarized

self similar solutions satisfying or not the equation of the interface abstract we study the existence of self similar solutions for the porous medium equation with reaction and convection u t u m 1 u x x u n 1 u x ku p in r 0 where m n 1 0 0 we are in particular interested in compactly supported self similar solutions satisfying some good equation at the interface the same equation appearing in the pure diffusion case we prove that there exist such self similar solutions only if k 1 4n an infinity of solutions with bad behaviour at least at one interface also exist there exist no self similar solutions with support arbitrarily small we complete the study by considering the case 0

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

global travelling waves in reaction convection diffusion equations we study the existence and properties of solutions in travelling wave form u x t x st defined for every z x st r for the reaction convection diffusion equationut a um xx b un x kupforx r t 0 with a m n 0 b k p r in the reaction case k 0 we prove that there exist travelling waves vanishing for z if and only if b 0 and min 1 n m p2 max 1 n p max 1 n moreover if m p 2n there exists a minimal velocity s a b k m n p 0 for which there are travelling waves only with s s while in the case m p 2n there are travelling waves only when 4amk b2n and for every velocity s 0 some properties of the function s are established all the waves are decreasing in their support and waves having bounded support from the right exist if and only if m min 1 n also the absorption case k 0 is treated where we find that for different values of the parameters there exists a unique travelling wave for every velocity s r but for some case where only negative velocities exist the cases b 0 or k 0 are well known in the literature

a possible deformed algebra and calculus inspired in nonextensive thermostatistics we present a deformed algebra related to the q exponential and the q logarithm functions that emerge from nonextensive statistical mechanics we also develop a q derivative and consistently a q integral for which the q exponential is an eigenfunction the q derivative and the q integral have a dual nature that is also presented

detail preserving ranked order based filters for image processing the theoretical analysis of multistage median filters is developed it is shown that multistage median filters are a combination of max median and min median filters since multistage median filters belong to the class of two dimensional stack filters they have threshold decomposition attributes making their theoretical analysis simple statistical threshold decomposition is applied to derive the statistical characteristics of these filters and the results are used to evaluate the performance of these two types of multistage filters finally a quantitative and qualitative comparison of the multistage filters and of other efficient detail preserving filters is presented the comparisons are made using the mean squared error and the mean absolute error criteria

spl lscr m s filters for image restoration applications a new filtering architecture is proposed generalizing some previously introduced multilevel median filters an efficient design procedure for the new filtering architecture is demonstrated for image restoration application simulation results show a good noise rejection performance combined with a fine detail preservation capability

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

variational approach and deformed derivatives recently we have demonstrated that there exists a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely the tsallis framework and the kaniadakis scenario with a local form of fractional derivative operators for fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure here in this paper we present an extension of the traditional calculus of variations for systems containing deformed derivatives embedded into the lagrangian and the lagrangian densities for classical and field systems the results extend the classical euler lagrange equations and the hamiltonian formalism the resulting dynamical equations seem to be compatible with those found in the literature specially with mass dependent and with nonlinear equations for systems in classical and quantum mechanics examples are presented to illustrate applications of the formulation also the conserved noether current is worked out

discussing the extension and applications of a variational approach with deformed derivatives one has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives both in the lagrangian of systems of particles and in the lagrangian density of field theoretic models classical euler lagrange equations and the hamiltonian formalism have been reassessed in this approach whenever applied to a number of physical systems the resulting dynamical equations come out to be the correct ones found in the literature especially with mass dependent and with nonlinear equations for classical and quantum mechanical systems in the present contribution one extends the variational approach including a piecewise form of deformed derivatives to study higher order dissipative systems and to obtain as an option deformed equations as well applications to concrete situations are contemplated such as an accelerated point charge this is the problem of the abraham lorentz dirac force stochastic dynamics like the langevin the advection convection reaction and fokker planck equations the korteweg de vries equation the landau lifshitz gilbert problem and the caldirola kanai hamiltonian and heat transfer equation of the fourier and non fourier types by considering these different examples it is shown that the formulation proposed in this paper may be a simple but promising path for dealing for example with dissipative nonlinear stochastic systems and the anomalous heat transfer problem by adopting a variational approach one has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives both in the lagrangian of systems of particles and in the lagrangian density of field theoretic models classical euler lagrange equations and the hamiltonian formalism have been reassessed in this approach whenever applied to a number of physical systems the resulting dynamical equations come out to be the correct ones found in the literature especially with mass dependent and with nonlinear equations for classical and quantum mechanical systems in the present contribution one extends the variational approach including a piecewise form of deformed derivatives to study higher order dissipative systems and to obtain as an option deformed equations as well applications to concrete situations are contemplated such as an accelerated point charge this is the problem of the abraham lorentz dirac force stochastic dynamics like the langevin the advection convection reaction and

stack filters and the mean absolute error criterion a method to determine the stack filter which minimizes the mean absolute error between its output and a desired signal given noisy observations of this desired signal is presented specifically an optimal window width b stack filter can be determined with a linear program with o b2 sup b variables this algorithm is efficient since the number of different inputs to a window width b filter is m sup b if the filter has m valued input and the number of stack filters grows faster than 2 raised to the 2 sup b 2 power it is shown that optimal stack filtering under the mean absolute error criterion is analogous to optimal linear filtering under the mean squared error criterion both linear filters and stack filters are defined by superposition properties both classes are implementable and both have tractable procedures for finding the optimal filter under an appropriate error criterion

quantum field theory on fractal spacetime a new regularisation method spacetime is modelled as a fractal subset of rn analysis on homogeneous sets with non integer hausdorff dimensions is applied to the low order perturbative renormalisation of quantum electrodynamics this new regularisation method implements the dirac matrices and tensors in r4 without difficulties is gauge invariant covariant and differs from dimensional regularisation in some aspects

adaptive fuzzy multilevel median filter a modified multilevel median filter mlmf incorporating the fuzzy associative memory fam system is proposed in this correspondence the modified method the adaptive fuzzy multilevel median filter afmmf overcomes some disadvantages common to most of the conventional mlmf experimental results demonstrated that the performance of the afmmf is superior to that of conventional mlmf in dealing with short line like noise while preserving edges

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

combination median filter a detail and structure preserving smoothing filter is introduced it is called the combination median filter as it uses directional median multilevel median and median filters to smooth different regions of the image the decision about the region is made by robust dixon s r test which is well known in statistics for outlier detection the threshold value of dixon s test can be kept constant as a result the filtering algorithm operates like a quasi nonadaptive filter and no computation of local statistics is involved some properties of the filter as well as detailed experimental results that demonstrate its superior performance are presented

on calculus of local fractional derivatives abstract local fractional derivative lfd operators have been introduced in the recent literature chaos 6 1996 505 513 being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions in the present paper we prove leibniz rule chain rule for lfd operators generalization of directional lfd and multivariable fractional taylor series to higher orders have been presented

fractional calculus as a macroscopic manifestation of randomness we generalize the method of van hove physica amsterdam 21 517 1955 so as to deal with the case of nonordinary statistical mechanics that being phenomena with no time scale separation we show that in the case of ordinary statistical mechanics even if the adoption of the van hove method imposes randomness upon hamiltonian dynamics the resulting statistical process is described using normal calculus techniques on the other hand in the case where there is no time scale separation this generalized version of van hove s method not only imposes randomness upon the microscopic dynamics but it also transmits randomness to the macroscopic level as a result the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus

similarity solutions of the porous medium equation with sign changes with m 1 here u is a function of x and t for nonnegative u eq 1 1 arises in the theory of gas flow through a one dimensional porous medium and an extensive theory has been developed over the last decades see e g l 3 151 more recently 4 5 12 131 applied and pure mathematicians have also become interested in solutions of 1 1 with sign changes the most striking property of 1 1 is that solutions may have compact support whose free boundaries or interfaces move outwards with a finite possibly zero speed proportional to the one sided x derivative of 1 1 at the free boundary taken from the interior of the support this is usually called the interface condition the expression between brackets in l l i e 1 1 ux l m iu u which we shall think of as theflux because of the divergence form of l l is continuous with respect to x in zeros of u if it is nonzero we shall call this the sign change condition we observe that for u bounded away from zero and infinity eq 1 1 is uniformly parabolic and local smoothness follows from standard regularity theory for the development of the theory of 1 1 similarity solutions play an important role essentially these are solutions whose profiles remain the same as t varies in the nonnegative case they were studied extensively in 7 91 the most familiar of them are of course the burenblatt pattle solutions given by

on the local fractional derivative abstract we present the necessary conditions for the existence of the kolwankar gangal local fractional derivatives kg lfd and introduce more general but weaker notions of lfds by using limits of certain integral averages of the difference quotient by applying classical results due to stein and zygmund 1965 16 we show that the kg lfd is almost everywhere zero in any given intervals we generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing

representation of digital image by fuzzy neural network a multilayer feedforward fuzzy neural network fnn by which the predetermined fuzzy system can be realized is constructed to express a given two dimensional 2 d digital image it is shown that such a network is universal approximator the fnn approach provides us with the representation model of the 2 d discrete image in noise environment some given fuzzy numbers are employed to describe gray levels of a digital image and the deviation of corrupted image to its noise free one a class of fuzzy rules for removing impulse noise in degraded image are presented the corresponding fnn may work as a filter which improves the performances of median and rank conditioned rank selection filters especially such a filter may to a greatest extent maintain the fine structure of the image under the minimum mean absolute error criterion the membership functions of fuzzy numbers are adaptively adjusted and an optimal fnn for image representation is derived in noise environment simultaneously some real image examples are systemically analyzed to show that the fnn may result in higher quality global restoration

double window hodges lehman d filter and hybrid d median filter for robust image smoothing the double window hodges lehman filter dwd filter and a hybrid d median filter hdm filter for robust image smoothing are proposed an adaptive mixture of the median and the d filter the hdm filter first makes decisions about the presence of edges on the basis of a two way classification of pixels near and around the pixel to be filtered subsequently straightforward d filtering is used in the absence of edges and median filtering is used in the presence of edges the dwd filter uses two windows and d filtering the smaller window is used to preserve the details then the larger window to provide for sufficient smoothing detailed simulation results show that the hdm filter while retaining all the good properties of the dwd filter consistently performs better in terms of signal to noise ratio than the dwd filter and a number of other filters including the median filter the dwd filter is shown to have simpler structure although not necessarily lesser computational complexity

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

nonlinear smoothing filters based on rank estimates of location a class of nonlinear filters that are based on the rank estimates r estimates of location parameters in statistical theory is introduced it is shown how moving window rank filters r filters can be defined starting from rank estimates of location these filters utilize the relative ranks of the observations in each window to produce an output value the idea of rank winsorization is extended to that of averaging only observations which lie within small temporal neighborhoods this leads to a definition of the class of generalized wilcoxon gw filters which are parameterized by three parameters namely the degrees of temporal and rank winsorization and the degree of averaging the gw filters can be defined to have desirable characteristics of edge preservation detail retention and impulse rejection in addition to the property of gaussian noise smoothing performance characteristics of these filters are considered through analysis and simulation the filters show that all three well known classes of robust location estimates the l m and r estimates can be applied to nonlinear smoothing of signals

a robust approach to image enhancement based on fuzzy logic in this paper we propose a robust approach to image enhancement based on fuzzy logic that addresses the seemingly conflicting goals of image enhancement i removing impulse noise ii smoothing out nonimpulse noise and iii enhancing or preserving edges and certain other salient structures we derive three different filters for each of the above three tasks using the weighted or fuzzy least squares ls method and define the criteria for selecting each of the three filters the criteria are based on the local context and they constitute the antecedent clauses of the fuzzy rules the overall result of the fuzzy rule based system is the combination of the results of the individual filters where each result contributes to the degree that the corresponding antecedent clause is satisfied this approach gives us a powerful and flexible image enhancement paradigm results of the proposed method on several types of images are compared with those of other standard techniques

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

a new definition of fractional derivative we give a new definition of fractional derivative and fractional integral the form of the definition shows that it is the most natural definition and the most fruitful one the definition for email protected email protected 1 coincides with the classical definitions on polynomials up to a constant further if a 1 the definition coincides with the classical definition of first derivative we give some applications to fractional differential equations

regularity of interfaces in diffusion processes under the influence of strong absorption we present a method of analysis which allows us to establish the interface equation and to prove lipschitz continuity of interfaces and solutions which appear in a large class of nonlinear parabolic equations and conservation laws posed in one space dimension its main feature is intersection comparison with travelling waves the method is explained on the following study case we consider the cauchy problem for the diffusion absorption model u t left u m right xx u p quad u ge 0 in the range of parameters m 1 0 1 where the support of the solution expands with time and the motion is governed by darcy s law in the strong absorption range there might appear shrinking interfaces and the interface evolution obeys a different mechanism previous methods have failed to provide an adequate analysis of the interface motion and regularity in such a situation

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

comments on theoretical analysis of the max median filter by g r arce and m p mclaughlin in the original paper ibid vol assp 35 no 1 p 60 9 jan 1987 an approach the max median filter to remove noise from images without distorting geometrical information was introduced the proposed filter structure due to its asymmetrical nature is able to preserve only the subtle signal structures that are larger in intensity than the local background level the commenters suggest another filter structure that has the ability to preserve both positive and negative geometrical information with respect to the background level