sandor s inequality for sugeno integrals abstract in this paper we prove a fuzzy integral inequality for convex functions our results improve recent results that appear in literature some examples are given to illustrate our theorems

beyond the poisson renewal process a tutorial survey after sketching the basic principles of renewal theory and recalling the classical poisson process we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of mittag leffler and of wright type we compare these three processes with each other

steffensen type inequalities for fuzzy integrals we provide several steffensen type inequalities for the sugeno integral the inequalities are of the form a b f g d a a a f d b or a b f g d a b b f d b where a b are constants is a fuzzy measure on r g a b 0 1 f a b r is nonincreasing or nondecreasing and b a a b g d we show that some of our sufficient conditions are also necessary

particle tracking for fractional diffusion with two time scales abstract previous work y zhang m m meerschaert b baeumer particle tracking for time fractional diffusion phys rev e 78 2008 036705 showed how to solve time fractional diffusion equations by particle tracking this paper extends the method to the case where the order of the fractional time derivative is greater than one a subordination approach treats the fractional time derivative as a random time change of the corresponding cauchy problem with a first derivative in time one novel feature of the time fractional case of order greater than one is the appearance of clustering in the operational time subordinator which is non markovian solutions to the time fractional equation are probability densities of the underlying stochastic process the process models movement of individual particles the evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations or langevin equations monte carlo simulation yields particle location and the ensemble density approximates the solution to the variable coefficient time fractional diffusion equation in one or several spatial dimensions the particle tracking code is validated against inverse transform solutions in the simplest cases further applications solve model equations for fracture flow and upscaling flow in complex heterogeneous porous media these variable coefficient time fractional partial differential equations in several dimensions are not amenable to solution by any alternative method so that the grid free particle tracking approach presented here is uniquely appropriate

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

the sugeno integral with respect to α preinvex functions abstract in this paper we establish the left hand side of the hermite hadamard inequality for preinvex functions and illustrate that the hermite hadamard inequality for preinvex functions is not valid for the sugeno integral moreover we give several upper bounds for the sugeno integral related to preinvex functions some examples are also provided to show the validity of our results

a markov type inequality for seminormed fuzzy integrals in this paper we present a markov type inequality for seminormed fuzzy integrals and its connections with chebyshev s inequality and other fundamental properties of the classical integral

chebyshev type inequalities for general fuzzy integrals abstract in this article we first introduce a class of binary operation called g seminorm which generalizes the concept of t seminorm then we use the g seminorm to define a class of sugeno like integral finally we establish some new chebyshev type inequalities for this kind of integral

estimating entropy rates with bayesian confidence intervals the entropy rate quantifies the amount of uncertainty or disorder produced by any dynamical system in a spiking neuron this uncertainty translates into the amount of information potentially encoded and thus the subject of intense theoretical and experimental investigation estimating this quantity in observed experimental data is difficult and requires a judicious selection of probabilistic models balancing between two opposing biases we use a model weighting principle originally developed for lossless data compression following the minimum description length principle this weighting yields a direct estimator of the entropy rate which compared to existing methods exhibits significantly less bias and converges faster in simulation with monte carlo techinques we estimate a bayesian confidence interval for the entropy rate in related work we apply these ideas to estimate the information rates between sensory stimuli and neural responses in experimental data shlens kennel abarbanel chichilnisky in preparation

the entropy production paradox for fractional master equations abstract time fractional evolution equations for probability distributions provide a means to describe an important class of stochastic processes their solutions show features which are essential in modeling a variety of phenomena in real world applications one aspect which has been observed in time fractional diffusion equations shows a surprising and unexpected behavior of the entropy production rate induced by these equations the entropy production rate increases as one moves away from the fully irreversible case corresponding to classical diffusion this rate is analyzed for a new class of systems with state spaces that are finite and denumerable we find that the entropy production paradox reemerges nonetheless but in a new and unexpected form

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

random walk access times on partially disordered complex networks an effective medium theory an analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices considering transition rates f that are different for steps across lattice bonds from the rates f across network shortcuts the theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter access times for random walks on these partially disordered structures are compared to those on small world networks which on average appear to provide the most effective means of decreasing access times uniformly across the network

estimating information rates with confidence intervals in neural spike trains information theory provides a natural set of statistics to quantify the amount of knowledge a neuron conveys about a stimulus a related work kennel shlens abarbanel chichilnisky 2005 demonstrated how to reliably estimate with a bayesian confidence interval the entropy rate from a discrete observed time series we extend this method to measure the rate of novel information that a neural spike train encodes about a stimulus the average and specific mutual information rates our estimator makes few assumptions about the underlying neural dynamics shows excellent performance in experimentally relevant regimes and uniquely provides confidence intervals bounding the range of information rates compatible with the observed spike train we validate this estimator with simulations of spike trains and highlight how stimulus parameters affect its convergence in bias and variance finally we apply these ideas to a recording from a guinea pig retinal ganglion cell and compare results to a simple linear decoder

an inequality related to minkowski type for sugeno integrals an inequality related to minkowski type for the sugeno integral on abstract spaces is studied in a rather general form some previous results on chebyshev type inequality obtained by the authors are generalized several examples are given to illustrate the validity of this inequality the conditions such that this inequality becomes an equality are also discussed finally conclusions and some problems for further investigations are included

subordinated stochastic processes with aged operational time in this paper subordinated stochastic processes are considered where the renewal process acting as the operational time it is assumed that the observation of the process begins at a certain time after the start of the renewal process a recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process two point correlation functions for certain subordinated stochastic processes particularly for the generalized ornstein uhlenbeck process were calculated a model of relaxation in a disordered medium with traps and obstacles is proposed

on stolarsky inequality for sugeno and choquet integrals recently flores franulic roman flores and chalco cano proved the stolarsky type inequality for sugeno integral with respect to the lebesgue measure l the present paper is devoted to generalize this result by relaxing some of its requirements moreover stolarsky inequality for choquet integral is added too

hermite hadamard inequality for fuzzy integrals in this paper we prove a hermite hadamard type inequality for fuzzy integrals some examples are given to illustrate the results

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

effective medium approximation for lattice random walks with long range jumps we consider the random walk on a lattice with random transition rates and arbitrarily long range jumps we employ bruggeman s effective medium approximation ema to find the disorder averaged coarse grained dynamics the ema procedure replaces the disordered system with a cleverly guessed reference system in a self consistent manner we give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion in the case of a power law scaling between transition rates and distance lattice variants of l evy flights emerge as the effective medium and the problem is solved analytically bearing the effective anomalous diffusivity finally we discuss several example distributions and demonstrate very good agreement with numerical simulations

extensions of effective medium theory of transport in disordered systems the effective medium theory of transport in disordered systems whose basis is the replacement of spatial disorder by temporal memory is extended in several practical directions restricting attention to a one dimensional system with bond disorder for specificity a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder it is shown how to use the memory functions in the laplace domain forms in which they first appear and in the time domain forms which are obtained via numerical inversion algorithms to address time evolution of the system beyond the asymptotic domain of large times normally treated an analytic but approximate procedure is provided to obtain the memories in addition to the inversion algorithm good agreement of effective medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit as expected the use of ensemble averages is studied for normal as well as correlation observables the effect of size on effective medium theory is explored and it is shown that even in the asymptotic limit finite size corrections develop to the well known harmonic mean prescription for finding the effective rate a percolation threshold is shown to arise even in one dimension for finite but not infinite systems at a concentration of broken bonds related to the system size spatially long range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories in spite of the fact that the original rates possess nearest neighbor character pausing time distributions in continuous time random walks corresponding to the effective medium memories are calculated

dynamics of reaction diffusion systems in a subdiffusive regime in this paper we examine the dynamics of reaction diffusion systems with fractional time derivatives it is shown that in these conditions diffusion is anomalous in the sense that the mean square displacement r 2 ensuremath sim t ensuremath gamma where ensuremath gamma l1 a situation known as subdiffusion we study the conditions for the appearance of a diffusion driven instability and show that the restrictive conditions for a turing instability are relaxed this implies that systems whose kinetics are not of the activator inhibitor kind can have a turing instability and a modulated final state we demonstrate our results with numerical calculations in two dimensions using a generic turing model

non markovian diffusion equations and processes analysis and simulations abstract in this paper we introduce and analyze a class of diffusion type equations related to certain non markovian stochastic processes we start from the forward drift equation which is made non local in time by the introduction of a suitable chosen memory kernel k t the resulting non markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l t we then consider the subordinated process y t x l t where x t is a markovian diffusion the corresponding time evolution of the marginal density function of y t is governed by a non markovian fokker planck equation which involves the memory kernel k t we develop several applications and derive the exact solutions we consider different stochastic models for the given equations providing path simulations

an inverse problem for a generalized fractional diffusion we propose a method for determining the solution and source term of a generalized time fractional diffusion equation the method is based on selecting a bi orthogonal basis of l 2 space corresponding to a nonself adjoint boundary value problem uniqueness is proven and an existence result is obtained for smooth initial and final conditions the asymptotic behavior of the generalized mittag leffler function is used to relax the smoothness requirement on these conditions

on chebyshev type inequalities for generalized sugeno integrals abstract we give the necessary and sufficient conditions guaranteeing the validity of chebyshev type inequalities for generalized sugeno integrals in the case of functions belonging to a much wider class than the comonotone functions for several choices of operators we characterize the classes of functions for which the chebyshev type inequality for the classical sugeno integral is satisfied

anomalous subdiffusion with multispecies linear reaction dynamics we have introduced a set of coupled fractional reaction diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics the model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics the effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms one consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species the general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail we have carried out monte carlo simulations corresponding to these special cases and we find excellent agreement with theory

continuous time random walk galilei invariance and relation for the nth moment we consider a decoupled continuous time random walk model with a generic waiting time probability density function pdf for the force free case we derive an integro differential diffusion equation which is related to the galilei invariance for the probability density we also derive a general relation which connects the nth moment in the presence of any external force to the second moment without external force i e it is valid for any waiting time pdf this general relation includes the generalized second einstein relation which connects the first moment in the presence of any external force to the second moment without any external force these expressions for the first two moments are verified by using several kinds of the waiting time pdf moreover we present new anomalous diffusion behaviours for a waiting time pdf given by a product of power law and exponential function

estimating the entropy rate of spike trains via lempel ziv complexity normalized lempel ziv complexity which measures the generation rate of new patterns along a digital sequence is closely related to such important source properties as entropy and compression ratio but in contrast to these it is a property of individual sequences in this article we propose to exploit this concept to estimate or at least to bound from below the entropy of neural discharges spike trains the main advantages of this method include fast convergence of the estimator as supported by numerical simulation and the fact that there is no need to know the probability law of the process generating the signal furthermore we present numerical and experimental comparisons of the new method against the standard method based on word frequencies providing evidence that this new approach is an alternative entropy estimator for binned spike trains

traversal times for random walks on small world networks we study the mean traversal time for a class of random walks on newman watts small world networks in which steps around the edge of the network occur with a transition rate that is different from the rate for steps across small world connections when f f the mean time to traverse the network exhibits a transition associated with percolation of the random graph i e small world part of the network and a collapse of the data onto a universal curve this transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps we develop a simple self consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition as is characteristic of most effective medium theories

the permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems abstract permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take as shannon entropy quantifies the diversity of values we show that the metric and permutation entropy rates measures of new disorder per new observed value are equal for ergodic finite alphabet information sources discrete time stationary stochastic processes with this result we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n dimensional intervals this result generalizes a previous one for piecewise monotone interval maps on the real line c bandt g keller b pompe entropy of interval maps via permutations nonlinearity 15 2002 1595 1602 at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits the case of non ergodic finite alphabet sources is also studied and an inequality developed finally the equality of permutation and metric entropy rates is extended to ergodic non discrete information sources when entropy is replaced by differential entropy in the usual way

phase transitions in tumor growth v what can be expected from cancer glycolytic oscillations abstract experimental evidence confirms the existence of glycolytic oscillations in cancer which allows it to self organize in time and space far from thermodynamic equilibrium and provides it with high robustness complexity and adaptability a kinetic model is proposed for hela tumor cells grown in hypoxia conditions it shows oscillations in a wide range of parameters two control parameters glucose and inorganic phosphate concentration were varied to explore the phase space showing also the presence of limit cycles and bifurcations the complexity of the system was evaluated by focusing on stationary state stability and lempel ziv complexity moreover the calculated entropy production rate was demonstrated behaving as a lyapunov function

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

brownian walkers within subdiffusing territorial boundaries inspired by the collective phenomenon of territorial emergence whereby animals move and interact through the scent marks they deposit we study the dynamics of a 1d brownian walker in a random environment consisting of confining boundaries that are themselves diffusing anomalously we show how to reduce in certain parameter regimes the non markovian many body problem of territoriality to the analytically tractable one body problem studied here the mean square displacement msd of the 1d brownian walker within subdiffusing boundaries is calculated exactly and generalizes well known results when the boundaries are immobile furthermore under certain conditions if the boundary dynamics are strongly subdiffusive we show the appearance of an interesting nonmonotonicity in the time dependence of the msd giving rise to transient negative diffusion

generalized continuous time random walks master equations and fractional fokker planck equations continuous time random walks which generalize random walks by adding a stochastic time between jumps provide a useful description of stochastic transport at mesoscopic scales the continuous time random walk model can accommodate certain features such as trapping which are not manifest in the standard macroscopic diffusion equation the trapping is incorporated through a waiting time density and a fractional diffusion equation results from a power law waiting time a generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force here we have derived the master equations for continuous time random walks with space and time dependent forcing for two cases when the force is evaluated at the start of the waiting time and at the end of the waiting time the differences persist in low order spatial continuum approximations however the two processes are shown to be governed by the same fokker planck equations in the diffusi

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

on the jensen type inequality for generalized sugeno integral we prove necessary and sufficient conditions for the validity of jensen type inequalities for generalized sugeno integral our proofs make no appeal to the continuity of neither the fuzzy measure nor the operators for several choices of operators we characterize the classes of functions for which the corresponding inequalities are satisfied

efficient numerical solution of the time fractional diffusion equation by mapping from its brownian counterpart the solution of a caputo time fractional diffusion equation of order 0 1 is expressed in terms of the solution of a corresponding integer order diffusion equation we demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem in the context of an n point finite difference time discretisation the mapping allows for an improvement in time computational complexity from o n 2 to o n given a precomputation of o n 1 ln n the mapping is applied successfully to the least squares fitting of a fractional advection diffusion model for the current in a time of flight experiment resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes a mapping between fractional diffusion and normal diffusion is explored numerically an efficient solution method for fractional diffusion equations is proposed a benchmark fractional order model fit for scher montroll transport is presented

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

interpreting non random signatures in biomedical signals with lempel ziv complexity abstract lempel ziv complexity lz j ziv a lempel on the complexity of finite sequences ieee trans inform theory 22 1976 75 81 and its variants have been used widely to identify non random patterns in biomedical signals obtained across distinct physiological states non random signatures of the complexity measure can occur under nonlinear deterministic as well as non deterministic settings surrogate data testing have also been encouraged in the past in conjunction with complexity estimates to make a finer distinction between various classes of processes in this brief letter we make two important observations 1 non gaussian noise at the dynamical level can elude existing surrogate algorithms namely phase randomized surrogates ft amplitude adjusted fourier transform aaft and iterated amplitude adjusted fourier transform iaaft thus any inference nonlinear determinism as an explanation for the non randomness is incomplete 2 decrease in complexity can be observed even across two linear processes with identical auto correlation functions the results are illustrated with a second order auto regressive process with gaussian and non gaussian innovations ar 2 processes have been used widely to model several physiological phenomena hence their choice the results presented encourage cautious interpretation of non random signatures in experimental signals using complexity measures

on the distribution function of the complexity of finite sequences investigations of complexity of sequences lead to important applications such as effective data compression testing of randomness discriminating between information sources and many others in this paper we establish formulae describing the distribution functions of random variables representing the complexity of finite sequences introduced by lempel and ziv in 1976 it is known that this quantity can be used as an estimator of entropy we show that the distribution functions depend affinely on the probabilities of the so called exact sequences

a generalized master equation approach to modelling anomalous transport in animal movement we present some models of random walks with internal degrees of freedom that have the potential to find application in the context of animal movement and stochastic search the formalism we use is based on the generalized master equation which is particularly convenient here because of its inherent coarse graining procedure whereby a random walker position is averaged over the internal degrees of freedom we show some instances in which non local jump probabilities emerge from the coupling of the motion to the internal degrees of freedom and how the tuning of one parameter can give rise to sub super and normal diffusion at long times remarks on the relation between the generalized master equation continuous time random walks and fractional diffusion equations are also presented

hermite hadamard type inequality for sugeno integrals abstract in this paper hermite hadamard type inequality for sugeno integrals based on m convex function is studied some examples are given to illustrate the results

transport properties of random walks on scale free regular lattice hybrid networks we study numerically the mean access times for random walks on hybrid disor dered structures formed by embedding scale free networks into regular lattices considering different transition rates for steps across lattice bonds f and across network shortcuts f for fast shortcuts f f 1 and low shortcut densities traversal time data collapse onto a universal curve while a crossover behavior that can be related to the percolation threshold of the scale free network component is identified at higher shortcut densities in analogy to similar observations reported recently in newman watts small world networks further more we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution and we discuss access time distri butions as functions of the initial and final node degrees these findings are relevant in particular when considering the optimization of existing information networks by the addi tion of a small number of fast shortcut connections

on invex sets and preinvex functions abstract in this paper we consider the class of the invex function introduced by hanson we show that under certain condition an invex function defined on an invex set a is preinvex on a similarly a quasiinvex function defined on an invex set a is pre quasiinvex

hilfer prabhakar derivatives and some applications abstract we present a generalization of hilfer derivatives in which riemann liouville integrals are replaced by more general prabhakar integrals we analyze and discuss its properties furthermore we show some applications of these generalized hilfer prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations and in difference differential equations governing the dynamics of generalized renewal stochastic processes

chebyshev inequality for sugeno integrals in this paper we study chebyshev inequality for sugeno integrals on an arbitrary fuzzy measure space particularly we obtain other chebyshev type inequalities which relate the measure of a level set of the maximum and the sum of two non negative integrable functions and their integrals finally we apply our results to the case of comonotone functions

the fuzzy integral for monotone functions abstract in this paper we give some optimal upper bounds for the sugeno s integral of monotone functions more precisely we show that if g 0 0 is a continuous and strictly monotone function then the fuzzy integral value p 0 a g d with respect to the lebesgue measure verifies the following sharp inequalities a g a p p for the increasing case and b g p p for the decreasing case moreover we show that under adequate conditions these optimal inequalities provides a powerful tool for solving fuzzy integrals also some examples and application are presented

λ generalized sugeno integral and its application sugeno integral is a useful tool for many problems in several domains especially in game theory measure theory decision making probability image processing and finance in this paper we introduce the concept of generalized sugeno integral then we establish general versions of chebyshev type inequality for this kind of sugeno integral by using a monotone measure our results generalize and improve the corresponding ones in the literature finally some conclusions and problems for further investigations are given

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

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data