hypergeometric functions on complex matrix space the authors had presented the general foundations for a theory of hypergeometric functions of matrix argument over real division algebras here they further develop the fine structure of these functions over the complex field including series expansions integral representations asymptotics differential equations addition formulas multiplication formulas summation theorems transformation properties etc especially important are the operator valued hypergeometric functions required for nonspherical expansions such as addition formulas by the noncommutativity of matrix multiplication

survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps a fractional equation approach we calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps i e traps that disappear in the course of their motion our calculation is based on a fractional reaction subdiffusion equation derived from a continuous time random walk model of the system contrary to an earlier method valid only in one dimension d 1 the equation is applicable in any euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport the irreversible evanescence reaction and the dimension in which both the traps and the target are embedded explicit results for the survival probability of the target are obtained for a density t of traps which decays i exponentially and ii as a power law in the former case the target has a finite asymptotic survival probability in all integer dimensions whereas in the latter case there are several regimes where the values of the decay exponent for t and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one two and three dimensions

microscopic approach to nonlinear reaction diffusion the case of morphogen gradient formation we develop a microscopic theory for reaction diffusion rd processes based on a generalization of einstein s master equation with a reactive term and show how the mean field formulation leads to a generalized rd equation with nonclassical solutions for the nth order annihilation reaction a a a a 0 we obtain a nonlinear reaction diffusion equation for which we discuss scaling and nonscaling formulations we find steady states with solutions either exhibiting long range power law behavior showing the relative dominance of subdiffusion over reaction effects in constrained systems or conversely solutions that go to zero a finite distance from the source i e having finite support of the concentration distribution describing situations in which diffusion is slow and extinction is fast theoretical results are compared with experimental data for morphogen gradient formation 2012 american physical society

reaction subdiffusion and reaction superdiffusion equations for evanescent particles performing continuous time random walks starting from a continuous time random walk ctrw model of particles that may evanesce as they walk our goal is to arrive at macroscopic integro differential equations for the probability density for a particle to be found at point r at time t given that it started its walk from r0 at time t 0 the passage from the ctrw to an integro differential equation is well understood when the particles are not evanescent depending on the distribution of stepping times and distances one arrives at standard macroscopic equations that may be normal diffusion or anomalous subdiffusion and or superdiffusion the macroscopic description becomes considerably more complicated and not particularly intuitive if the particles can die during their walk while such equations have been derived for specific cases e g for location independent exponential evanescence we present a more general derivation valid under less stringent constraints than those found in the current literature

restart could optimize the probability of success in a bernoulli trial the recently noticed ability of restart to reduce the expected completion time of first passage processes allows appealing opportunities for performance improvement in a variety of settings however complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency here we show that restart may have profound consequences on the splitting probabilities of a bernoulli like first passage process i e of a process which can end with one of two outcomes particularly intriguing in this respect is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way we reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first passage under restart

persistent random motion with maximally correlated fluctuations how often should a random walker change its direction of motion in order to maximize correlation in velocity fluctuations over a finite time interval we address this optimal diffusion problem in the context of the one dimensional persistent random walk where we evaluate the correlation and mutual information in velocity trajectories as a function of the persistence level and the observation time we find the optimal persistence level corresponds to the average number of direction reversals asymptotically scaling as the square root of the observation time this square root scaling law makes the relative growth between the average number of direction reversals and the persistence length invariant with respect to changes in the overall time duration of the random walk

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

on the invariance of spatially inhomogeneous relaxation processes we consider a general relaxation model based on an ensemble of particles propagating randomly in a general space the particles trajectories are independent and identically distributed copies of an arbitrary random spatial curve and each particle has its own parameter triplet initiation epoch velocity and relaxation coefficient the relaxation rate of each particle is dynamic depending on its spatial location and quantified by an arbitrary spatial landscape function we study the relaxation rate of the entire ensemble and characterize the class of poissonian randomizations of the particles parameters that render the ensemble s relaxation rate invariant with respect to both the particles trajectories and the landscape function the invariant relaxation rates turn out to be governed by power laws and display a statistical behavior analogous to anomalous diffusion applications range from chemical reactions to target search and intelligence sifting

geometric theory for weibull s distribution weibull s distribution is the principal phenomenological law of relaxation in the physical sciences and spans three different relaxation regimes subexponential stretched exponential exponential and superexponential the probabilistic theory of extreme value statistics asserts that the linear scaling limits of minima of ensembles of positive valued random variables which are independent and identically distributed are universally governed by weibull s distribution however this probabilistic theory does not take into account spatial geometry which often plays a key role in the physical sciences in this paper we present a general and versatile model of random reactions in random environments and establish a geometry based theory for the universal emergence of weibull s distribution

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

evanescent continuous time random walks we study how an evanescence process affects the number of distinct sites visited by a continuous time random walker in one dimension we distinguish two very different cases namely when evanescence can only occur concurrently with a jump and when evanescence can occur at any time the first is characteristic of trapping processes on a lattice whereas the second is associated with spontaneous death processes such as radioactive decay in both of these situations we consider three different forms of the waiting time distribution between jumps namely exponential long tailed and ultraslow

bessel functions associated with representations of formally real jordan algebras abstract we study bessel functions on the symmetric cone associated with a special formally real jordan algebra we extend the classical results of radial fourier analysis and we prove an asymptotic formula

universal self similarity of propagating populations this paper explores the universal self similarity of propagating populations the following general propagation model is considered particles are randomly emitted from the origin of a d dimensional euclidean space and propagate randomly and independently of each other in space all particles share a statistically common yet arbitrary motion pattern each particle has its own random propagation parameters emission epoch motion frequency and motion amplitude the universally self similar statistics of the particles displacements and first passage times fpts are analyzed statistics which are invariant with respect to the details of the displacement and fpt measurements and with respect to the particles underlying motion pattern analysis concludes that the universally self similar statistics are governed by poisson processes with power law intensities and by the fr echet and weibull extreme value laws

target problem with evanescent subdiffusive traps we calculate the survival probability of a stationary target in one dimension surrounded by diffusive or subdiffusive traps of time dependent density the survival probability of a target in the presence of traps of constant density is known to go to zero as a stretched exponential whose specific power is determined by the exponent that characterizes the motion of the traps a density of traps that grows in time always leads to an asymptotically vanishing survival probability trap evanescence leads to a survival probability of the target that may go to zero or to a finite value indicating a probability of eternal survival depending on the way in which the traps disappear with time

universal generation of statistical self similarity a randomized central limit theorem a universal mechanism for the generation of statistical self similarity i e fractality in the context of random processes is established we consider a generic system which superimposes independent stochastic signals producing a system output all signals share a common statistical signal pattern yet each signal has its own transmission parameters amplitude frequency and initiation epoch we characterize the class of parameter randomizations yielding statistically self similar outputs in a universal fashion i e for whatever signals fed into the system statistically self similar outputs with finite variance further display i anomalous diffusion behavior characterized by power law temporal variance growth and ii 1 f noise behavior characterized by power law power spectra the mechanism presented is a randomized central limit theorem for fractal statistics of random processes

special functions of matrix argument i algebraic induction zonal polynomials and hypergeometric functions hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis multivariate statistics quantum physics molecular chemistry and number theory this paper presents a general theory of such functions for real division algebras these functions which generalize the classical hypergeometric functions are defined by infinite series on the space s s n f of all n x n hermitian matrices over the division algebra f the theory depends intrinsically upon the representation theory of the general linear group g gl n f of invertible n x n matrices over f and the theme of this work is the full exploitation of the inherent group theory the main technique is the use of the method of algebraic induction to realize explicitly the appropriate representations of g to decompose the space of polynomial functions on s and to describe the zonal polynomials from which the hypergeometric functions are constructed detailed descriptions of the convergence properties of the series expansions are given and integral representations are provided future papers in this series will develop the fine structure of these functions 0 introduction we begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument by fine structure we allude to the analogues of such classical results as series expansions integral formulas asymptotics differential equations summation formulas addition theorems composition formulas and recurrence relations this first paper in which we simultaneously treat real complex and quaternionic analysis is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras not only as a framework for the body of detailed results to follow in later papers but also to clarify the representationtheoretic foundation for such a theory although these hypergeometric functions are of interest on purely analytic grounds they arise in a wide range of applications indeed various classes of received by the editors may 19 1986 1980 malhema1zics s ject claszficain 1985 ren primary 22e30 22e45 33a75 43a85 43a90 62h10 secondary 20g20 32a07 32m15 44a10 62e15 key w and phrases generalized hypergeometric functions zonal polynomials representation theory algebraic induction multivariate statistics general linear group generalized gamma functions pochhammer symbols laplace transforms maximal compact subgroup invariant polynomials positive cones symmetric spaces schur functions special functions of matrix argument the first author is on leave from the university of wyoming the second author was partially supported by the national science foundation under grant mcs 8403381 and by the research council of the university of north carolina b1987 american mathematical society 0002 9947 87 1 00 25 per page

pattern formation on networks with reactions a continuous time random walk approach we derive the generalized master equation for reaction diffusion on networks from an underlying stochastic process the continuous time random walk ctrw the nontrivial incorporation of the reaction process into the ctrw is achieved by splitting the derivation into two stages the reactions are treated as birth death processes and the first stage of the derivation is at the single particle level taking into account the death process while the second stage considers an ensemble of these particles including the birth process using this model we have investigated different types of pattern formation across the vertices on a range of networks importantly the ctrw defines the laplacian operator on the network in a non ad hoc manner and the pattern formation depends on the structure of this laplacian here we focus attention on ctrws with exponential waiting times for two cases one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree this results in nonsymmetric and symmetric ctrw laplacians respectively in the case of symmetric laplacians pattern formation follows from the turing instability however in nonsymmetric laplacians pattern formation may be possible with or without a turing instability

representations of the affine transformation groups acting simply transitively on siegel domains let g be the split solvable lie group acting simply transitively on a siegel domain d we consider irreducible unitary representations of g realized on hilbert spaces of holomorphic functions on d we determine all such hilbert spaces by connecting them with positive riesz distributions on the dual cone and describe them through the fourier laplace transform moreover we classify the representations of g by making use of the orbit method

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

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

extreme values and the level crossing problem an application to the feller process we review the question of the extreme values attained by a random process we relate it to level crossings to one boundary first passage problems as well as to two boundaries escape problems the extremes studied are the maximum the minimum the maximum absolute value and the range or span we specialize in diffusion processes and present detailed results for the wiener and feller processes

universal statistics and control of random transport processes a general model of random transport is considered independent agents are stochastically emitted into a space with general topology the agents motion in space is governed by a general random transport mechanism the agents are detected only when in a target zone which is an arbitrary subset of space we explore emission intensities which render the agents target zone statistics universal i e invariant with respect to the underlying spatial topology and transport mechanism the results established determine how to universally control the target zone statistics via the emission intensity and show that the resulting universal statistics are governed by power laws

operator valued bessel functions holomorphic discrete series and harmonic representations of u p q abstract this paper presents representation theoretic applications of the theory of operator valued bessel functions which was developed in the first paper of this series using the reduced bessel function k we study the harmonic representation and holomorphic discrete series for the group u p q of biholomorphic automorphisms of the siegel domains of type ii

brownian gas models for extreme value laws in this paper we establish one dimensional brownian gas models for the extreme value laws of gumbel weibull and frechet a gas model is a countable collection of independent particles governed by common diffusion dynamics the extreme value laws are the universal probability distributions governing the affine scaling limits of the maxima and minima of ensembles of independent and identically distributed one dimensional random variables using the recently introduced concept of stationary poissonian intensities we construct two gas models whose global statistical structures are stationary and yield the extreme value laws a linear brownian motion gas model for the gumbel law and a geometric brownian motion gas model for the weibull and frechet laws the stochastic dynamics of these gas models are studied in detail and closed form analytical descriptions of their temporal correlation structures their topological phase transitions and their intrinsic first passage time fluxes are presented

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

pascal principle for diffusion controlled trapping reactions in this paper we analyze the long time behavior of the survival probability p a t of an a particle that performs lattice random walk in the presence of randomly moving traps b we show that for both perfect and imperfect trapping reactions for arbitrary spatial dimension d and for a rather general class of random walks p a t is less than or equal to the survival probability of an immobile target a in the presence of randomly moving traps

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed

optimal search strategies of space time coupled random walkers with finite lifetimes we present a simple paradigm for detection of an immobile target by a space time coupled random walker with a finite lifetime the motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed we call these walkers mortal creepers a mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate m while still alive the creeper has a finite mean frequency of change of the direction of motion in particular we consider the efficiency of the target search process characterized by the probability that the creeper will eventually detect the target analytic results confirmed by numerical results show that there is an m dependent optimal frequency opt that maximizes the probability of eventual target detection we work primarily in one dimensional d 1 domains and examine the role of initial conditions and of finite domain sizes numerical results in d 2 domains confirm the existence of an optimal frequency of change of direction thereby suggesting that the observed effects are robust to changes in dimensionality in the d 1 case explicit expressions for the probability of target detection in the long time limit are given in the case of an infinite domain we compute the detection probability for arbitrary times and study its early and late time behavior we further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time we also consider a version of the standard target problem in which many creepers start at random locations at the same time

exploration and trapping of mortal random walkers exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature and yet a finite lifetime is a frequent occurrence and its effects on a number of random walk properties may be profound for instance whereas the average number of distinct sites visited by an immortal walker grows with time without bound that of a mortal walker may depending on dimensionality and rate of evanescence remain finite or keep growing with the passage of time this number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps if the traps are immortal the survival probability will vanish with increasing time however if the traps are evanescent the target may be spared a certain death we analytically calculate a number of basic and broadly used quantities for evanescent random walkers

a unified and universal explanation for levy laws and 1 f noises levy laws and 1 f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals the signals are considered to follow statistically a common yet arbitrary generic signal pattern which may be either stationary or dissipative each signal is considered to have its own random transmission amplitude and frequency we characterize the amplitude frequency randomizations which render the system output s stationary law and power spectrum universal i e independent of the underlying generic signal pattern the classes of universal stationary laws and power spectra are shown to coincide respectively with the classes of levy laws and 1 f noises thus providing a unified and universal explanation for the ubiquity of these classes of anomalous statistics in various fields of science and engineering

arboricity and subgraph listing algorithms in this paper we introduce a new simple strategy into edge searching of a graph which is useful to the various subgraph listing problems applying the strategy we obtain the following four algorithms the first one lists all the triangles in a graph g in o a g m time where m is the number of edges of g and a g the arboricity of g the second finds all the quadrangles in o a g m time since a g is at most three for a planar graph g both run in linear time for a planar graph the third lists all the complete subgraphs k l of order l in o la g l 2 m time the fourth lists all the cliques in o a g m time per clique all the algorithms require linear space we also establish an upper bound on a g for a graph g a g leqq lceil 2m n 1 2 rceil where n is the number of vertices in g

stability in p5 and banner free graphs abstract we prove polynomial solvability of the maximum stable set problem in p 5 free banner free p 5 b free graphs where p 5 is the simple path on five vertices and a banner is the graph with vertices a b c d e and edges a b b c b d c e d e

generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains the ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena these models often involve fractional derivatives the important physical extension of this work to processes occurring in growing materials has proven highly nontrivial here we derive evolution equations for modeling subdiffusive transport in a growing medium the derivation is based on a continuous time random walk the concise formulation of these evolution equations requires the introduction of a new comoving fractional derivative the implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane

weighted laplace transforms and bessel functions on hermitian symmetric spaces this paper defines ir weighted laplace transforms on the spaces of ir covariant functions by the inverse laplace transform we define operator valued bessel functions we also study the holomorphic discrete series of the automorphism group of a siegel domain of type ii

a fractional order infectivity sir model fractional order sir models have become increasingly popular in the literature in recent years however unlike the standard sir model they often lack a derivation from an underlying stochastic process here we derive a fractional order infectivity sir model from a stochastic process that incorporates a time since infection dependence on the infectivity of individuals the fractional derivative appears in the generalised master equations of a continuous time random walk through sir compartments with a power law function in the infectivity we show that this model can also be formulated as an infection age structured kermack mckendrick integro differential sir model under the appropriate limit the fractional infectivity model reduces to the standard ordinary differential equation sir model

role of depletion on the dynamics of a diffusing forager we study the dynamics of a starving random walk in general spatial dimension d this model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources the forager depletes its environment by consuming resources and dies if it wanders too long without finding food in the exactly solvable case of one dimension we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation we also give a heuristic derivation for the averages of these two quantities we tackle the complex but ecologically relevant case of two dimensions by an approximation in which the depleted zone is assumed to always be circular and which grows incrementally each time the walk reaches the edge of this zone within this framework we derive a lower bound for the scaling of the average lifetime and number of distinct sites visited at starvation we also determine the asymptotic distribution of the number of distinct sites visited at starvation finally we solve the case of high spatial dimensions within a mean field approach

survival probability of a subdiffusive particle in a d dimensional sea of mobile traps we investigate the long time behavior of the survival probability pt of a mobile particle in d dimensional continuous euclidean media doped with noninteracting mobile traps the particle is strictly subdiffusive implying that its mean square displacement grows as t with 01 initially the traps are scattered randomly and their subsequent mean square displacement grows as t with 01 instantaneous annihilation of the particle takes place upon contact with any of the traps the solution to this problem is obtained by deriving lower and upper asymptotic bounds of the survival probability and showing that they converge to one another for long times thereby unambiguously determining the long time decay of pt for d2 we find that at late times the survival probability is that of the pure target problem the problem where the particle remains immobile in agreement with previous studies for the d 1 case these decay laws remain invariant over the whole range as opposed to the dynamical crossover observed for the case of a purely diffusive particle 1 where for 2 2 d the survival probability becomes that of the so called trapping problem the problem where the particle moves in a sea of static traps this behavior implies that for sufficiently low values of 2 2 d the survival probability becomes singular in the limit 1 trappinglike for 1 and targetlike for any 1

from stochastic processes to numerical methods we have introduced a new explicit numerical method based on a discrete stochastic process for solving a class of fractional partial differential equations that model reaction subdiffusion the scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks we show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest this limiting procedure guarantees the consistency of the numerical scheme the positivity of the solution and stability results are simply obtained provided that the underlying process is well posed we also show that the method can be applied to standard reaction diffusion equations this work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations including fractional partial differential equations

on atomic structure of p5 free subclasses and maximum weight independent set problem graph decompositions play a crucial role in structural graph theory and in designing efficient graph algorithms among them clique separator decomposition a decomposition tree of the graph whose leaves have no clique separator so called atoms used by tarjan for solving various optimization problems recently received much attention in this note we first derive the atomic structure of two subclasses of p 5 free graphs where p 5 is a chordless path on five vertices these results enable us to provide efficient solutions for the maximum weight independent set problem in these classes of graphs

reaction subdiffusion equations to analyze possible generalizations of reaction diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion a ensuremath rightarrow b we derive the corresponding kinetic equations for the local a and b concentrations their form is rather unusual the parameters of the reaction influence the diffusion term in the equation for a component a a consequence of the non markovian nature of subdiffusion the equation for the product contains a term which depends on the concentration of a at all previous times our discussion shows that reaction subdiffusion equations may not resemble the corresponding reaction diffusion ones and are not obtained by a trivial change of the diffusion operator for a subdiffusion one

trapping reactions with randomly moving traps exact asymptotic results for compact exploration trapping a1b b and recombination a1b 0 reactions tr and rr involving randomly moving a and b particles which react when they meet at a certain distance b are ubiquitous in nature a few stray examples include quenching of delocalized excitations coagulation recombination of radicals charge carriers or defects or biological processes related to population survival 1 in recent years there has been much interest in the longtime behavior of these processes following a remarkable discovery 2 10 of many particle effects which induce essential departures from the conventionally expected behavior 1 a pronounced deviation from the text book predictions was found for the diffusion controlled rr in case when initially the particles of the a and b species are all distributed at random with strictly equal mean densities n0 it has been first shown 2 and subsequently proven 3 4 that as t the mean density n t follows n t a d n 0 dt 2d 4 where d is the space dimensionality a d is a constant and d5d a 1db is the sum of particles diffusion coefficients this law contradicts the decay law obtained within the smoluchowski approach sa n t 1 f bd t 1 where as t

survival of an evasive prey abstract we study the survival of a prey that is hunted by n predators the predators perform independent random walks on a square lattice with v sites and start a direct chase whenever the prey appears within their sighting range the prey is caught when a predator jumps to the site occupied by the prey we analyze the efficacy of a lazy minimal effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range otherwise the prey stays still we show that if the sighting range of such a lazy prey is equal to 1 lattice spacing at least 3 predators are needed in order to catch the prey on a square lattice in this situation we establish a simple asymptotic relation lnp ev t n v 2lnp imm t between the survival probabilities of an evasive and an immobile prey hence when the density n v of the predators is low 1 the lazy evasion strategy leads to the spectacular increase of the survival probability we also argue that a short sighting prey its sighting range is smaller than the sighting range of the predators undergoes an effective superdiffusive motion as a result of its encounters with the predators whereas a far sighting prey performs a diffusive type motion

nonlinear subdiffusive fractional equations and the aggregation phenomenon in this article we address the problem of the nonlinear interaction of subdiffusive particles we introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles we derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations we show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection diffusion transport regime this transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering we illustrate the general results through the use of the examples from cell and population biology we find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon

delayed uncoupled continuous time random walks do not provide a model for the telegraph equation while the velocity of the motion of particles in the dctrws is infinite in this paper we investigate the accuracy of the approximations to the dctrws provided by the telegraph equation we show that the diffusion equation beingthecorrectlimitofthedctrws givesbetterapproximationsin l2 normtothedctrwsthanthetelegraph equation we conclude therefore that first the dctrws do not provide any correct microscopic interpretation of the one dimensional telegraph equation and second the kinetic exact model of the telegraph equation is different from the model based on the dctrws

a single predator charging a herd of prey effects of self volume and predator prey decision making we study the degree of success of a single predator hunting a herd of prey on a two dimensional square lattice landscape we explicitly consider the self volume of the prey restraining their dynamics on the lattice the movement of both predator and prey is chosen to include an intelligent decision making step based on their respective sighting ranges the radius in which they can detect the other species prey cannot recognise each other besides the self volume interaction after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase respectively we consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey we find that the prey s sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight we characterise the dynamics in terms of the mean distance between the predator and the nearest prey it turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple ornstein uhlenbeck picture reducing the many body problem to a simple two body problem by imagining predator and nearest prey to be connected by an effective hookean bond all features of the model such as prey density and sighting ranges merge into the effective binding constant

numerical analysis and physical simulations for the time fractional radial diffusion equation we do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field based on rewriting the equation in a new form we first present two kinds of implicit finite difference schemes for numerically solving the equation then we strictly establish the stability and convergence results we prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes finally we do the physical simulations some interesting physical phenomena are revealed we verify that the long time asymptotic survival probability kt a but independent of the dimension where a is the anomalous diffusion exponent

topography of chance we present a model of multiplicative langevin dynamics that is based on two foundations the langevin equation and the notion of multiplicative evolution the model is a nonlinear mechanism transforming a white noise input to a dynamic equilibrium output using a single control an underlying convex u shaped potential function the output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories the model generates each admissible combination of the output s shape and randomness in a universal and robust fashion moreover practically all the probability distributions that are supported on the positive half line and that are commonly encountered and applied across the sciences can be reverse engineered by this model hence this model is a universal equilibrium mechanism in the context of multiplicative dynamics for the robust generation of chance the model s output in turn the properties of the produced chance the output s shape and randomness are determined with mathematical precision by the control s landscape its topography thus a topographic map of chance is established as a particular application probability distributions with power law tails are shown to be universally and robustly generated by controls on the edge of convexity convex u shaped potential functions with asymptotically linear wings

new applications of clique separator decomposition for the maximum weight stable set problem graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms clique separators in graphs were used by tarjan as a divide and conquer approach for solving various problems such as the maximum weight stable set mws problem colouring and minimum fill in the basic tool is a decomposition tree of the graph whose leaves have no clique separator so called atoms and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms we give new examples where the clique separator decomposition works well for the mws problem our results improve and extend various recently published results in particular we describe the atom structure for some new classes of graphs whose atoms are p 5 free the p 5 is the induced path with five vertices and obtain new polynomial time results for the mws problem the complexity of this problem on the class of p 5 free graphs is still unknown

number of distinct sites visited by a subdiffusive random walker the asymptotic mean number of distinct sites visited by a subdiffusive continuous time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature this number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps we present an explicit derivation for two cases in all integer dimensions so as to formally complete a tableau of results in this tableau we include the dominant as well as subdominant contributions in all integer dimensions other quantities that can be calculated from the mean number of distinct sites visited are also discussed

persistent random walk of cells involving anomalous effects and random death the purpose of this paper is to implement a random death process into a persistent random walk model which produces sub ballistic superdiffusion levy walk we develop a stochastic two velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction it is assumed that the switching rate is a decreasing function of residence running time this assumption leads to the power law for the velocity switching time distribution this describes the anomalous persistence of cell motility the longer the cell moves in one direction the smaller the switching probability to another direction becomes we derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives we show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process in the long time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a levy jump distribution we find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient monte carlo simulations confirm these bounds

first passage times in multiscale random walks the impact of movement scales on search efficiency an efficient searcher needs to balance properly the trade off between the exploration of new spatial areas and the exploitation of nearby resources an idea which is at the core of scale free levy search strategies here we study multiscale random walks as an approximation to the scale free case and derive the exact expressions for their mean first passage times in a one dimensional finite domain this allows us to provide a complete analytical description of the dynamics driving the situation in which both nearby and faraway targets are available to the searcher so the exploration exploitation trade off does not have a trivial solution for this situation we prove that the combination of only two movement scales is able to outperform both ballistic and levy strategies this two scale strategy involves an optimal discrimination between the nearby and faraway targets which is only possible by adjusting the range of values of the two movement scales to the typical distances between encounters so this optimization necessarily requires some prior information albeit crude about target distances or distributions furthermore we found that the incorporation of additional three four movement scales and its adjustment to target distances does not improve further the search efficiency this allows us to claim that optimal random search strategies arise through the informed combination of only two walk scales related to the exploitative and the explorative scales respectively expanding on the well known result that optimal strategies in strictly uninformed scenarios are achieved through levy paths or equivalently through a hierarchical combination of multiple scales

operator valued bessel functions on schrodinger fock spaces and siegel domains of type ii in this paper a general theory of operator valued bessel functions on mixed schrodinger fock spaces and siegel domains of type ii is presented in general bessel functions j are associated to the partial fourier transform on a mixed schrodinger fock space j can be realized as a reduced bessel function k which is a generalized inverse laplace transform on a siegel domain of type ii the application of this theory to representation theory will be studied in the next paper

a center box method for radially symmetric solution of fractional subdiffusion equation in this paper a center box difference method is considered for the radially symmetric solution of fractional subdiffusion equation by method of order reduction the derivative boundary condition is transformed into dirichlet boundary condition and thus the geometrical singularity is successfully removed from the original problem as a matter of course a natural discretization scheme is obtained to investigate the stability and convergence of the method we define a new norm with a weight r d 1 thus the usual sobolev inequality is not suitable to the new norm therefore we prove three new sobolev like embedding inequalities which can also be applied to the other problems in polar coordinates then the scheme is proved to be unconditionally stable and convergent in maximum norm with the help of the new sobolev like embedding inequalities some illustrative examples are provided to demonstrate the theoretical results by some comparisons it can be seen that the natural discretization scheme is accurate and effective in physical simulations and it can be used to both long time and short time computation

divergent series and memory of the initial condition in the long time solution of some anomalous diffusion problems we consider various anomalous d dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry the motion of particles is described by a fractional diffusion equation their mean square displacement is given by r 2 t 0 1 resulting in normal diffusive motion if 1 and subdiffusive motion otherwise for the subdiffusive case in sufficiently high dimensions divergent series appear when the concentration or survival probabilities are evaluated via the method of separation of variables while the solution for normal diffusion problems is at most divergent as t 0 the emergence of such series in the long time domain is a specific feature of subdiffusion problems we present a method to regularize such series and in some cases validate the procedure by using alternative techniques laplace transform method and numerical simulations in the normal diffusion case we find that the signature of the initial condition on the approach to the steady state rapidly fades away and the solution approaches a single the main decay mode in the long time regime in remarkable contrast long time memory of the initial condition is present in the subdiffusive case as the spatial part 1 r describing the long time decay of the solution to the steady state is determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem the weight being dependent on the initial condition interestingly 1 r turns out to be independent of the anomalous diffusion exponent

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

paretian poisson processes many random populations can be modeled as a countable set of points scattered randomly on the positive half line the points may represent magnitudes of earthquakes and tornados masses of stars market values of public companies etc in this article we explore a specific class of random such populations we coin paretian poisson processes this class is elemental in statistical physics connecting together in a deep and fundamental way diverse issues including the poisson distribution of the law of small numbers paretian tail statistics the frechet distribution of extreme value theory the one sided levy distribution of the central limit theorem scale invariance renormalization and fractality resilience to random perturbations

mixing driven equilibrium reactions in multidimensional fractional advection dispersion systems we study instantaneous mixing driven bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation the superdiffusive nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical fickian diffusion model in particular regions where reaction rates would be zero for the fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates we also study a global metric of mixing in the system the scalar dissipation rate and compute its asymptotic scaling rates analytically the scalar dissipation rate scales asymptotically as t d where d is the number of spatial dimensions and is the fractional derivative exponent

on algorithms for p 5 gem free graphs a graph is p5 gem free when it does not contain p5 an induced path with five vertices or a gem a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path p4 as an induced subgraph we present o n2 time recognition algorithms for chordal gem free graphs and for p5 gem free graphs using a characterization of p5 gem free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees a brandstadt d kratsch on the structure of p5 gem free graphs discrete appl math 145 2005 155 166 we give linear time algorithms for the following np complete problems on p5 gem free graphs minimum coloring maximum weight stable set maximum weight clique and minimum clique cover