subdiffusive master equation with space dependent anomalous exponent and structural instability we derive the fractional master equation with space dependent anomalous exponent we analyze the asymptotic behavior of the corresponding lattice model both analytically and by monte carlo simulation we show that the subdiffusive fractional equations with constant anomalous exponent in a bounded domain 0 l are not structurally stable with respect to the nonhomogeneous variations of parameter in particular the gibbs boltzmann distribution is no longer the stationary solution of the fractional fokker planck equation whatever the space variation of the exponent might be we analyze the random distribution of in space and find that in the long time limit the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events we show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a black swan the low probability event of the small value of the anomalous exponent that completely dominates the long time behavior of subdiffusive systems

first passage and risk evaluation under stochastic volatility we solve the first passage problem for the heston random diffusion model we obtain exact analytical expressions for the survival and the hitting probabilities to a given level of return we study several asymptotic behaviors and obtain approximate forms of these probabilities which prove among other interesting properties the nonexistence of a mean first passage time one significant result is the evidence of extreme deviations which implies a high risk of default when certain dimensionless parameter related to the strength of the volatility fluctuations increases we confront the model with empirical daily data and we observe that it is able to capture a very broad domain of the hitting probability we believe that this may provide an effective tool for risk control which can be readily applicable to real markets both for portfolio management and trading strategies

survival probability of a particle in a sea of mobile traps a tale of tails we study the long time tails of the survival probability p t of an a particle diffusing in d dimensional media in the presence of a concentration p of traps b that move subdiffusively such that the mean square displacement of each trap grows as t with 0 1 starting from a continuous time random walk description of the motion of the particle and of the traps we derive lower and upper bounds for p t and show that for 2 d 2 these bounds coincide asymptotically thus determining asymptotically exact results the asymptotic decay law in this regime is exactly that obtained for immobile traps this means that for sufficiently subdiffusive traps the moving a particle sees the traps as essentially immobile and lifshitz or trapping tails remain unchanged for 2 d 2 and d 2 the upper and lower bounds again coincide leading to a decay law equal to that of a stationary particle thus in this regime the moving traps see the particle as essentially immobile for d 2 however the upper and lower bounds in this regime no longer coincide and the decay law for the survival probability of the a particle remains ambiguous

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

subspace clustering of categorical and numerical data with an unknown number of clusters in clustering analysis data attributes may have different contributions to the detection of various clusters to solve this problem the subspace clustering technique has been developed which aims at grouping the data objects into clusters based on the subsets of attributes rather than the entire data space however the most existing subspace clustering methods are only applicable to either numerical or categorical data but not both this paper therefore studies the soft subspace clustering of data with both of the numerical and categorical attributes also simply called mixed data for short specifically an attribute weighted clustering model based on the definition of object cluster similarity is presented accordingly a unified weighting scheme for the numerical and categorical attributes is proposed which quantifies the attribute to cluster contribution by taking into account both of intercluster difference and intracluster similarity moreover a rival penalized competitive learning mechanism is further introduced into the proposed soft subspace clustering algorithm so that the subspace cluster structure as well as the most appropriate number of clusters can be learned simultaneously in a single learning paradigm in addition an initialization oriented method is also presented which can effectively improve the stability and accuracy of k means type clustering methods on numerical categorical and mixed data the experimental results on different benchmark data sets show the efficacy of the proposed approach

quasi steady state analysis of two dimensional random intermittent search processes we use perturbation methods to analyze a two dimensional random intermittent search process in which a searcher alternates between a diffusive search phase and a ballistic movement phase whose velocity direction is random a hidden target is introduced within a rectangular domain with reflecting boundaries if the searcher moves within range of the target and is in the search phase it has a chance of detecting the target a quasi steady state analysis is applied to the corresponding chapman kolmogorov equation this generates a reduced fokker planck description of the search process involving a nonzero drift term and an anisotropic diffusion tensor in the case of a uniform direction distribution for which there is zero drift and isotropic diffusion we use the method of matched asymptotics to compute the mean first passage time mfpt to the target under the assumption that the detection range of the target is much smaller than the size of the domain we show that an optimal search strategy exists consistent with previous studies of intermittent search in a radially symmetric domain that were based on a decoupling or moment closure approximation we also show how the decoupling approximation can break down in the case of biased search processes finally we analyze the mfpt in the case of anisotropic diffusion and find that anisotropy can be useful when the searcher starts from a fixed location

a new expansion of the confluent hypergeometric function in terms of modified bessel functions abstract an asymptotic expansion of the confluent hypergeometric function u a b x for large positive 2a b is given in terms of modified bessel functions multiplied by buchholz polynomials a family of double polynomials in the variables b and x with rational coefficients

subdiffusion in a bounded domain with a partially absorbing reflecting boundary the exit time of a subdiffusive process from a bounded domain with a partially absorbing reflecting boundary is considered the short time and long time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the laplace operator eigenfunctions rotation invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations the asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane in a chemical reactor filled with catalytic grains of finite reactivity or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

fractional cable equation models for anomalous electrodiffusion in nerve cells finite domain solutions in recent work we introduced fractional nernst planck equations and related fractional cable equations to model electrodiffusion of ions in nerve cells with anomalous subdiffusion along and across the nerve cells this work was motivated by many computational and experimental studies showing that anomalous diffusion is ubiquitous in biological systems with binding crowding or trapping for example recent experiments have shown that anomalous subdiffusion occurs along the axial direction in spiny dendrites due to trapping by the spines we modeled the subdiffusion in two ways leading to two fractional cable equations and presented fundamental solutions on infinite and semi infinite domains here we present solutions on finite domains for mixed robin boundary conditions the finite domain solutions model passive electrotonic properties of spiny dendritic branch segments with ends that are voltage clamped sealed or killed the behavior of the finite domain solutions is similar for both fractional cable

poset block equivalence of integral matrices given square matrices b and b with a poset indexed block structure for which an ij block is zero unless i j when are there invertible matrices u and v with this required zero block structure such that ubv b we give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain r as one application when r is a field we classify such matrices up to similarity by matrices respecting the block structure we also give complete invariants for equivalence under the additional requirement that the diagonal blocks of u and v have determinant 1 the invariants involve an associated diagram the k web of r module homomorphisms the study is motivated by applications to symbolic dynamics and c algebras

analytical representations of the spread harmonic measure density we study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface for euclidean domains in which brownian motion can be split into independent lateral and transverse displacements we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior this analysis is applicable to slab domains general cylindrical domains and a half space we investigate the spreading effect due to multiple reflections on the surface and the underlying role of finite reactivity we discuss further extensions and applications of analytical results to describe laplacian transfer phenomena such as permeation through semipermeable membranes secondary current distribution on partially blocking electrodes and surface relaxation in nuclear magnetic resonance

survival probability of diffusion with trapping in cellular neurobiology the problem of diffusion with absorption and trapping sites arises in the theory of molecular signaling inside and on the membranes of biological cells in particular this problem arises in the case of spine dendrite communication where the number of calcium ions modeled as random particles is regulated across the spine microstructure by pumps which play the role of killing sites while the end of the dendritic shaft is an absorbing boundary we develop a general mathematical framework for diffusion in the presence of absorption and killing sites and apply it to the computation of the time dependent survival probability of ions we also compute the ratio of the number of absorbed particles at a specific location to the number of killed particles we show that the ratio depends on the distribution of killing sites the biological consequence is that the position of the pumps regulates the fraction of calcium ions that reach the dendrite

the numerical computation of the confluent hypergeometric functionu a b z an algorithm is given for the computation of the confluent hypergeometric functionu a b z for real values ofa b andz z 0 algol 60 procedures are given the computations are based on a miller algorithm and on asymptotic expansions

partially reflected diffusion the radiation reactive or robin boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries the underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary however the relation between the reaction constant in the robin boundary condition and the reflection probability is not well defined in this paper we define the partially reflected process as a limit of the markovian jump process generated by the euler scheme for the underlying ito dynamics with partial boundary reflection trajectories that cross the boundary are terminated with probability p sqrt delta t and otherwise are reflected in a normal or oblique direction we use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the fokker planck equation in a half space wi

default risk modeling beyond the first passage approximation extended black cox model we develop a generalization of the black cox structural model of default risk the extended model captures uncertainty related to firm s ability to avoid default even if company s liabilities momentarily exceeding its assets diffusion in a linear potential with the radiation boundary condition is used to mimic a company s default process the exact solution of the corresponding fokker planck equation allows for derivation of analytical expressions for the cumulative probability of default and the relevant hazard rate obtained closed formulas fit well the historical data on global corporate defaults and demonstrate the split behavior of credit spreads for bonds of companies in different categories of speculative grade ratings with varying time to maturity introduction of the finite rate of default at the boundary improves valuation of credit risk for short time horizons which is the key advantage of the proposed model we also consider the influence of uncertainty in the initial distance to the default barrier on the outcome of the model and demonstrate that this additional source of incomplete information may be responsible for nonzero credit spreads for bonds with very short time to maturity

residence times and other functionals of reflected brownian motion we study the residence and local times for a brownian particle confined by reflecting boundaries and propose a general solution to the problem of finding the related probability distribution its fourier transform characteristic function and laplace transform survival probability are obtained in a compact matrix form involving the laplace operator eigenbasis explicit combinatorial relations are derived for the moments and the probability distribution is shown to be nearly gaussian when the exploration time is long enough when the eigenbasis or a part of it is known the numerical computation of the residence time distributions is straightforward and accurate the present approach can also be applied to investigate other functionals of reflected brownian motion describing in particular restricted diffusion in an external field or potential e g nuclei diffusing in an inhomogeneous magnetic field theoretical results for the local times are confronted with monte carlo simulations on the unit interval disk and sphere

default risk modeling with position dependent killing abstract diffusion in a linear potential in the presence of position dependent killing is used to mimic a default process different assumptions regarding transport coefficients initial conditions and elasticity of the killing measure lead to diverse models of bankruptcy one stylized fact is fundamental for our consideration empirically default is a rather rare event especially in the investment grade categories of credit ratings hence the action of killing may be considered as a small parameter in a number of special cases we derive closed form expressions for the entire term structure of the cumulative probability of default its hazard rate and intensity comparison with historical data on aggregate global corporate defaults confirms the validity of the perturbation method for estimations of long term probability of default for companies with high credit quality on a single company level we implement the derived formulas to estimate the one year likelihood of default of enron on a daily basis from august 2000 to august 2001 three months before its default and compare the obtained results with forecasts of traditional structural models

first exit times of harmonically trapped particles a didactic review we revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one or multidimensional ornstein uhlenbeck process we start by recalling the main derivation steps of a propagator using langevin and fokker planck equations the mean exit time the moment generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed we also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing fokker planck operator we discuss several applications of first exit times such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single particle tracking experiments adhesion bond dissociation under mechanical stress characterization of active periods of trend following and mean reverting strategies in algorithmic trading on stock markets relation to the distribution of first crossing times of a moving boundary by brownian motion some extensions are described including diffusion under quadratic double well potential and anomalous diffusion

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

scaling properties and universality of first passage time probabilities in financial markets financial markets provide an ideal frame for the study of crossing or first passage time events of non gaussian correlated dynamics mainly because large data sets are available tick by tick data of six futures markets are herein considered resulting in fat tailed first passage time probabilities the scaling of the return with its standard deviation collapses the probabilities of all markets examined and also for different time horizons into single curves suggesting that first passage statistics is market independent at least for high frequency data on the other hand a very closely related quantity the survival probability shows away from the center and tails of the distribution a hyperbolic t ensuremath 1 2 decay typical of a markovian dynamics albeit the existence of memory in markets modifications of the weibull and student distributions are good candidates for the phenomenological description of first passage time properties under certain regimes the scaling strategies shown may be useful for risk control and algorithmic trading

a new distance metric for unsupervised learning of categorical data distance metric is the basis of many learning algorithms and its effectiveness usually has a significant influence on the learning results in general measuring distance for numerical data is a tractable task but it could be a nontrivial problem for categorical data sets this paper therefore presents a new distance metric for categorical data based on the characteristics of categorical values in particular the distance between two values from one attribute measured by this metric is determined by both the frequency probabilities of these two values and the values of other attributes that have high interdependence with the calculated one dynamic attribute weight is further designed to adjust the contribution of each attribute distance to the distance between the whole data objects promising experimental results on different real data sets have shown the effectiveness of the proposed distance metric

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

first exit times and residence times for discrete random walks on finite lattices in this paper we derive explicit formulas for the surface averaged first exit time of a discrete random walk on a finite lattice we consider a wide class of random walks and lattices including random walks in a nontrivial potential landscape we also compute quantities of interest for modeling surface reactions and other dynamic processes such as the residence time in a subvolume the joint residence time of several particles and the number of hits on a reflecting surface

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

subdiffusive target problem survival probability the asymptotic survival probability of a spherical target in the presence of a single subdiffusive trap or surrounded by a sea of subdiffusive traps in a continuous euclidean medium is calculated in one and two dimensions the survival probability of the target in the presence of a single trap decays to zero as a power law and as a power law with logarithmic correction respectively the target is thus reached with certainty but it takes the trap an infinite time on average to do so in dimensions higher than two a single trap may never reach the target and so the survival probability is finite and in fact does not depend on whether the traps move diffusively or subdiffusively when the target is surrounded by a sea of traps on the other hand its survival probability decays as a stretched exponential in all dimensions with a logarithmic correction in the exponent for d 2 a trap will therefore reach the target with certainty and will do so in a finite time these results may be directly related to enzyme binding kinetics on dna in the crowded cellular environment

cooperative and penalized competitive learning with application to kernel based clustering abstract competitive learning approaches with individual penalization or cooperation mechanisms have the attractive ability of automatic cluster number selection in unsupervised data clustering in this paper we further study these two mechanisms and propose a novel learning algorithm called cooperative and penalized competitive learning cpcl which implements the cooperation and penalization mechanisms simultaneously in a single competitive learning process the integration of these two different kinds of competition mechanisms enables the cpcl to locate the cluster centers more quickly and be insensitive to the number of seed points and their initial positions additionally to handle nonlinearly separable clusters we further introduce the proposed competition mechanism into kernel clustering framework correspondingly a new kernel based competitive learning algorithm which can conduct nonlinear partition without knowing the true cluster number is presented the promising experimental results on real data sets demonstrate the superiority of the proposed methods

subdiffusive behavior in a trapping potential mean square displacement and velocity autocorrelation function a theoretical framework for analyzing stochastic data from single particle tracking in viscoelastic materials and under the influence of a trapping potential is presented starting from a generalized langevin equation we found analytical expressions for the two time dynamics of a particle subjected to a harmonic potential the mean square displacement and the velocity autocorrelation function of the diffusing particle are given in terms of the time lag in particular we investigate the subdiffusive case using a power law memory kernel exact expressions for the mean square displacement and the velocity autocorrelation function are obtained in terms of mittag leffler functions and their derivatives the behaviors for short intermediate and long time lags are investigated in terms of the involved parameters finally the validity of usual approximations is examined

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

mathematical basis for a general theory of laplacian transport towards irregular interfaces the theory of laplacian transport towards and across irregular surfaces is reformulated in terms of the dirichlet to neumann operator and its spectral characteristics this permits us to obtain an exact equivalent circuit for the impedance of a working interface of arbitrary shape the important result is that only very few eigenmodes of this operator do govern the entire response of a macroscopic system this property drastically simplifies the understanding of irregular or prefractal interfaces the results can be applied in electrochemistry physiology and chemical engineering fields where exchange processes across surfaces with complex geometry are ubiquitous

categorical and numerical attribute data clustering based on a unified similarity metric without knowing cluster number most of the existing clustering approaches are applicable to purely numerical or categorical data only but not the both in general it is a nontrivial task to perform clustering on mixed data composed of numerical and categorical attributes because there exists an awkward gap between the similarity metrics for categorical and numerical data this paper therefore presents a general clustering framework based on the concept of object cluster similarity and gives a unified similarity metric which can be simply applied to the data with categorical numerical and mixed attributes accordingly an iterative clustering algorithm is developed whose outstanding performance is experimentally demonstrated on different benchmark data sets moreover to circumvent the difficult selection problem of cluster number we further develop a penalized competitive learning algorithm within the proposed clustering framework the embedded competition and penalization mechanisms enable this improved algorithm to determine the number of clusters automatically by gradually eliminating the redundant clusters the experimental results show the efficacy of the proposed approach

learning a mixture model for clustering with the completed likelihood minimum message length criterion mixture model based clustering also simply called model based clustering hereinafter consists of fitting a mixture model to data and identifying each cluster with one of its components this paper tackles the model selection and parameter estimation problems in model based clustering so as to improve the clustering performance on the data sets whose true kernel distribution functions are not in the family of assumed ones as well as with inherently overlapped clusters being tailored to clustering applications an effective model selection criterion is first proposed unlike most criteria that measure the goodness of fit of the model only to generate data the proposed one also evaluates whether the candidate model provides a reasonable partition for the observed data which enforces a model with well separated components accordingly an improved method for the estimation of mixture parameters is derived which aims to suppress the spurious estimates by the standard expectation maximization em algorithm and enforce well supported components in the mixture model finally the estimation of mixture parameters and the model selection is integrated in a single algorithm which favors a compact mixture model with both the well supported and well separated components extensive experiments on synthetic and real world data sets are carried out to show the effectiveness of the proposed approach to the mixture model based clustering

from the solutions of diffusion equation to the solutions of subdiffusive one starting with green s functions found for normal diffusion we construct exact time dependent green s functions for a subdiffusive equation with fractional time derivatives green s functions satisfy the boundary conditions involving a linear combination of fluxes and concentrations the method is particularly useful to calculate the concentration profiles in a multi part system where different kinds of transport occur in each part of it as an example we find the solution of the subdiffusive equation for the system composed of two parts with normal diffusion and subdiffusion

new semi supervised classification method based on modified cluster assumption the cluster assumption which assumes that similar instances should share the same label is a basic assumption in semi supervised classification learning and has been found very useful in many successful semi supervised classification methods it is rarely noticed that when the cluster assumption is adopted there is an implicit assumption that every instance should have a crisp class label assignment in real applications however there are cases where it is difficult to tell that an instance definitely belongs to one class and does not belong to other neighboring classes in such cases it is more adequate to assume that similar instances should share similar label memberships rather than sharing a crisp label assignment here label memberships can be represented as a vector where each element corresponds to a class and the value at the element expresses the likelihood of the concerned instance belonging to the class by adopting this modified cluster assumption in this paper we propose a new semi supervised classification method that is semi supervised classification based on class membership ssccm specifically we try to solve the decision function and adequate label memberships for instances simultaneously and constrain that an instance and its local weighted mean lwm share the same label membership vector where the lwm is a robust image of the instance constructed by calculating the weighted mean of its neighboring instances we formulate the problem in a unified objective function for the labeled unlabeled data and their lwms based on the square loss function and take an alternating iterative strategy to solve it in which each step generates a closed form solution and the convergence is guaranteed the solution will provide both the decision function and the label membership function for classification their classification results can verify each other and the reliability of semi supervised classification learning might be enhanced by checking the consistency between those two predictions experiments show that ssccm obtains encouraging results compared to state of the art semi supervised classification methods

nonspectral relaxation in one dimensional ornstein uhlenbeck processes the relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates which are typically considered to be independent of the initial condition the rates follow from the spectrum of a hermitian operator obtained by a similarity transformation of the initial fokker planck operator however some initial conditions are mapped by this similarity transformation to functions which growat infinity these cannot be expanded in terms of the eigenfunctions of a hermitian operator and show different relaxation patterns considering the exactly solvable examples of gaussian and generalized levy ornstein uhlenbeck processes oups we show that the relaxation rates belong to the hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise while for an ordinary oup initial conditions leading to nonspectral relaxation can be considered exotic for generalized oups driven by levy noise such initial conditions are the rule doi 10 1103 physrevlett 110 150602

equivariant flow equivalence for shifts of finite type by matrix equivalence over group rings let g denote a nite group in this paper by a g shift of nite type g sft we will mean a shift of nite type sft together with a continuous g action which commutes with the shift where in addition the action is free and the sft is irreducible and is non trivial contains more than one orbit we will classify these systems up to g 5ow equivalence this equivalence relation can be described in terms of g sfts skew products or suspension 5ows x 2 for example two g sfts are g 5ow equivalent if and only if there exists an orientation preserving homeomorphism between their mapping tori which commutes with the induced g actions a g sft can be presented by a nite square matrix a over z g the positive cone of the integral group ring zg w parry personal communication 2001 let ia 1 denote the nn matrix whose upper left corner is ia and which otherwise equals the in nite identity matrix let e zg be the group of nn matrices generated by basic elementary matrices those which di er from i in at most one entry which must be o diagonal over zg let w a denote the weight class of a de nition 4 1 the conjugacy class in g of the group of weights of loops based at a xed vertex we show that the weight class is an invariant of g 5ow equivalence when w a w b g we will show that g sfts presented by matrices a and b are g 5ow equivalent if and only if there are matrices u and v in e zg such that u ia 1v ib 1 theorem 6 1 the complete classi cation up to g 5ow equivalence which allows the possibility w a g has a more complicated statement theorem 6 4 in the case that g is trivial our classi cation reduces to the familiar classi cation of franks 14 by cokernel group and determinant when g is non trivial the classi cation up to e zg is much more dibcult and interesting and remains an open problem we consider these algebraic issues inxx 8 and 9 in x 8 we give the modest requisite k theory terminology and background and for the case g z 2 we give a constructive partial result theorem 8 1 and some very concrete illustrative examples examples 8 6 and 8 7 which indicate how the zg equivalence problem becomes more dibcult when g is non trivial that is zg6 z in x 9 we consider e zg equivalence of injective matrices in this case gl zg equivalence amounts to isomorphism of cokernel modules and the re nement toe zg equivalence is classi ed by k 1 zg h for an associated subgroup h ofs k 1 zg as one consequence if g is abelian and det ia is not a zerodiviso r in zg then det ia determines the g 5ow equivalence class up

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

geometric mean for subspace selection subspace selection approaches are powerful tools in pattern classification and data visualization one of the most important subspace approaches is the linear dimensionality reduction step in the fisher s linear discriminant analysis flda which has been successfully employed in many fields such as biometrics bioinformatics and multimedia information management however the linear dimensionality reduction step in flda has a critical drawback for a classification task with c classes if the dimension of the projected subspace is strictly lower than c 1 the projection to a subspace tends to merge those classes which are close together in the original feature space if separate classes are sampled from gaussian distributions all with identical covariance matrices then the linear dimensionality reduction step in flda maximizes the mean value of the kullback leibler kl divergences between different classes based on this viewpoint the geometric mean for subspace selection is studied in this paper three criteria are analyzed 1 maximization of the geometric mean of the kl divergences 2 maximization of the geometric mean of the normalized kl divergences and 3 the combination of 1 and 2 preliminary experimental results based on synthetic data uci machine learning repository and handwriting digits show that the third criterion is a potential discriminative subspace selection method which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in flda and its several representative extensions

first passage times for random walks in bounded domains we present a novel computational method of first passage times between a starting site and a target site of regular bounded lattices we derive accurate expressions for all the moments of this first passage time validated by numerical simulations their range of validity is discussed we also consider the case of a starting site and two targets in addition we present the extension to continuous brownian motion these results are of great relevance to any system involving diffusion in confined media

time averaged quadratic functionals of a gaussian process the characterization of a stochastic process from its single random realization is a challenging problem for most single particle tracking techniques which survey an individual trajectory of a tracer in a complex or viscoelastic medium we consider two quadratic functionals of the trajectory the time averaged mean square displacement msd and the time averaged squared root mean square displacement srms for a large class of stochastic processes governed by the generalized langevin equation with arbitrary frictional memory kernel and harmonic potential the exact formulas for the mean and covariance of these functionals are derived the formula for the mean value can be directly used for fitting experimental data e g in optical tweezers microrheology the formula for the variance and covariance allows one to estimate the intrinsic fluctuations of measured or simulated time averaged msd or srms for choosing the experimental setup appropriately we show that the time averaged srms has smaller fluctuations than the time averaged msd in spite of much broader applications of the latter one the theoretical results are successfully confirmed by monte carlo simulations of the langevin dynamics we conclude that the use of the time averaged srms would result in a more accurate statistical analysis of individual trajectories and more reliable interpretation of experimental data

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

kinetics of active surface mediated diffusion in spherically symmetric domains we present an exact calculation of the mean first passage time to a target on the surface of a 2d or 3d spherical domain for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion we generalize the results of benichou et al in j stat phys 142 657 2011 and consider a biased diffusion in a general annulus with an arbitrary number of regularly spaced targets on a partially reflecting surface the presented approach is based on an integral equation which can be solved analytically numerically validated approximation schemes which provide more tractable expressions of the mean first passage time are also proposed in the framework of this minimal model of surface mediated reactions we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface

fractional translational diffusion of a brownian particle in a double well potential the fractional translational diffusion of a particle in a double well potential excluding inertial effects is considered the position correlation function and its spectrum are evaluated using a fractional probability density diffusion equation based on the diffusion limit of a fractal time random walk exact and approximate solutions for the dynamic susceptibility describing the position response to a small external field are obtained the exact solution is given by matrix continued fractions while the approximate solution relies on the exponential separation of the time scales of the fast intrawell and low overbarrier relaxation processes associated with the bistable potential it is shown that knowledge of the characteristic relaxation times for normal diffusion allows one to predict accurately the anomalous relaxation behavior of the system for all relevant time scales

first passage time statistics for two channel diffusion we present rigorous results for the mean first passage time and first passage time statistics for two channel markov additive diffusion in a 3 dimensional spherical domain inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes which is shown to effect a non trivial first passage behaviour we also address the scenario of intermittent immobilisation in both cases we prove that despite the perfectly non recurrent motion of two channel markov additive diffusion in 3 dimensions the first passage statistics at long times do not display poisson like behaviour if none of the phases has a vanishing diffusion coefficient this stands in stark contrast to the standard one channel markov diffusion counterpart we also discuss the relevance of our results in the context of cellular signalling

optimization and universality of brownian search in a basic model of quenched heterogeneous media the kinetics of a variety of transport controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time the so called mean first passage time mfpt problem the occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the mfpt by reducing oversampling of space here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension in a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity the mfpt is analyzed under the constraint of conserved average dynamics that is the spatially averaged diffusivity is kept constant our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the mfpt to the target we prove that the mfpt for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality in contrast to intermittent strategies which are most profitable in low spatial dimensions the spatially inhomogeneous search performs best in higher dimensions discussing our results alongside recent experiments on single particle tracking in living cells we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

a novel attribute weighting algorithm for clustering high dimensional categorical data due to data sparseness and attribute redundancy in high dimensional data clusters of objects often exist in subspaces rather than in the entire space to effectively address this issue this paper presents a new optimization algorithm for clustering high dimensional categorical data which is an extension of the k modes clustering algorithm in the proposed algorithm a novel weighting technique for categorical data is developed to calculate two weights for each attribute or dimension in each cluster and use the weight values to identify the subsets of important attributes that categorize different clusters the convergence of the algorithm under an optimization framework is proved the performance and scalability of the algorithm is evaluated experimentally on both synthetic and real data sets the experimental studies show that the proposed algorithm is effective in clustering categorical data sets and also scalable to large data sets owning to its linear time complexity with respect to the number of data objects attributes or clusters

feature selection and kernel learning for local learning based clustering the performance of the most clustering algorithms highly relies on the representation of data in the input space or the hilbert space of kernel methods this paper is to obtain an appropriate data representation through feature selection or kernel learning within the framework of the local learning based clustering llc wu and scholkopf 2006 method which can outperform the global learning based ones when dealing with the high dimensional data lying on manifold specifically we associate a weight to each feature or kernel and incorporate it into the built in regularization of the llc algorithm to take into account the relevance of each feature or kernel for the clustering accordingly the weights are estimated iteratively in the clustering process we show that the resulting weighted regularization with an additional constraint on the weights is equivalent to a known sparse promoting penalty hence the weights of those irrelevant features or kernels can be shrunk toward zero extensive experiments show the efficacy of the proposed methods on the benchmark data sets

filling of a poisson trap by a population of random intermittent searchers we extend the continuum theory of random intermittent search processes to the case of n independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi infinite track each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state we assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process the hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target the particle is removed from the system after delivering its cargo as a further generalization of previous models we assume that up to n successive particles can find the target and deliver its cargo assuming that the rate of target detection scales as 1 n we show that there exists a well defined mean field limit n rightarrow infty in which the stochastic model reduces to a deterministic system of linear reaction hyperbolic equations for the concentrations of particles in each of the internal states these equations decouple from the stochastic process associated with filling the target with cargo the latter can be modeled as a poisson process in which the time dependent rate of filling lambda t depends on the concentration of stationary particles within the target domain hence we refer to the target as a poisson trap we analyze the efficiency of filling the poisson trap with n particles in terms of the waiting time density f n t the latter is determined by the integrated poisson rate mu t int 0 t lambda s ds which in turn depends on the solution to the reaction hyperbolic equations we obtain an approximate solution for the particle concentrations by reducing the system of reaction hyperbolic equations to a scalar advection diffusion equation using a quasi steady state analysis we compare our analytical results for the mean field model with monte carlo simulations for finite n we thus determine how the mean first passage time mfpt for filling the target depends on n and n

reaction diffusion theory in the presence of an attractive harmonic potential problems involving the capture of a moving entity by a trap occur in a variety of physical situations the moving entity being an electron an excitation an atom a molecule a biological object such as a receptor cluster a cell or even an animal such as a mouse carrying an epidemic theoretical considerations have almost always assumed that the particle motion is translationally invariant we study here the case when that assumption is relaxed in that the particle is additionally subjected to a harmonic potential this tethering to a center modifies the reaction diffusion phenomenon using a smoluchowski equation to describe the system we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions interesting features emerge depending on the relative location of the trap the attractive center and the initial placement of the diffusing particle

exact solution for the diffusion in bistable potentials we solve analytically the fokker planck equation for a one parameter family of symmetric attractive nonharmonic potentials which include double well situations the exact knowledge of the eigenfunctions and eigenvalues allows us to fully discuss the transient behavior of the probability density in particular for the bistable potentials we can give analytical expressions for the probability current over the working barrier and for the onset time which characterizes the transition from uni to bimodal probability densities

power spectra and random walks in intermittent chaotic systems abstract we investigate anomalous diffusion and the corresponding power spectra generated by iterated maps and analyze the motion in terms of the probabilistic continous time random walk approach both stationary and non stationary conditions are considered demonstrating the dependence of the mean squared displacement and of the power spectra on the initial conditions the theoretical results are corroborated by numerical calculations and excellent and agreement is obtained

infinite densities for levy walks motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion in many cases such systems exhibit strong anomalous diffusion where low order moments ensuremath langle x t q ensuremath rangle with q below a critical value q c exhibit diffusive scaling while for qg q c a ballistic scaling emerges the mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion e g the known diffusion equations and central limit theorems fail to describe both aspects in this paper we resolve this problem by resorting to the concept of infinite density using the widely applicable l evy walk model we find a general expression for the corresponding non normalized density which is fully determined by the particles velocity distribution the anomalous diffusion exponent ensuremath alpha and the diffusion coefficient k ensuremath alpha we explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data

overview of theoretical models for reaction rates there is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences in this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the national institutes of health on may 6 8 1985 this review will focus on 1 the smoluchowski model for reaction rates together with its extension by onsager 2 first passage time formalism for discrete and continuous master equations and fokker planck equations 3 the kramers model and its extensions 4 diffusion in the presence of trapping centers

multifractal properties of the harmonic measure on koch boundaries in two and three dimensions the multifractal properties of the harmonic measure on quadratic and cubic koch boundaries are studied with the help of a new fast random walk algorithm adapted to these fractal geometries the conjectural logarithmic development of local multifractal exponents is guessed for regular fractals and checked by extensive numerical simulations this development allows one to compute the multifractal exponents of the harmonic measure with high accuracy even with the first generations of the fractal in particular the information dimension in the case of the concave cubic koch surface embedded in three dimensions is found to be slightly higher than its value d1 2 for a smooth boundary

an optimization algorithm for clustering using weighted dissimilarity measures one of the main problems in cluster analysis is the weighting of attributes so as to discover structures that may be present by using weighted dissimilarity measures for objects a new approach is developed which allows the use of the k means type paradigm to efficiently cluster large data sets the optimization algorithm is presented and the effectiveness of the algorithm is demonstrated with both synthetic and real data sets

maximum of a fractional brownian motion probabilities of small values let b t b 0 0 be a fractional brownian motion i e a gaussian process with the structure function e b t b s 2 t s 0 2 we study the logarithmic asymptotics of pt p b t 1 t t as t where is either the interval 0 1 or a bounded region that contains a vicinity of 0 for the case of multidimensional time it is shown that ln pt d ln t 1 o 1 where d is the dimension of zeroes of b t in the former case and the dimension of time in the latter

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

escape problem under stochastic volatility the heston model we solve the escape problem for the heston random diffusion model from a finite interval of span l we obtain exact expressions for the survival probability which amounts to solving the complete escape problem as well as for the mean exit time we also average the volatility in order to work out the problem for the return alone regardless of volatility we consider these results in terms of the dimensionless normal level of volatility a ratio of the three parameters that appear in the heston model and analyze their form in several asymptotic limits thus for instance we show that the mean exit time grows quadratically with large spans while for small spans the growth is systematically slower depending on the value of the normal level we compare our results with those of the wiener process and show that the assumption of stochastic volatility in an apparently paradoxical way increases survival and prolongs the escape time we finally observe that the model is able to describe the main exit time statistics of the dow jones daily index

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

enhanced soft subspace clustering integrating within cluster and between cluster information while within cluster information is commonly utilized in most soft subspace clustering approaches in order to develop the algorithms other important information such as between cluster information is seldom considered for soft subspace clustering in this study a novel clustering technique called enhanced soft subspace clustering essc is proposed by employing both within cluster and between class information first a new optimization objective function is developed by integrating the within class compactness and the between cluster separation in the subspace based on this objective function the corresponding update rules for clustering are then derived followed by the development of the novel essc algorithm the properties of this algorithm are investigated and the performance is evaluated experimentally using real and synthetic datasets including synthetic high dimensional datasets uci benchmarking datasets high dimensional cancer gene expression datasets and texture image datasets the experimental studies demonstrate that the accuracy of the proposed essc algorithm outperforms most existing state of the art soft subspace clustering algorithms

clustering objects on subsets of attributes with discussion read before the royal statistical society at a meeting organized by the research section on wednesday may 5th 2004 professor j t kent in the chair summary a new procedure is proposed for clustering attribute value data when used in conjunction with conventional distance based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously the relevant attribute subsets for each individual cluster can be different and partially or completely overlap with those of other clusters enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed applications in different domains including gene expression arrays are presented