reaction kinetics of a b 0 a method to find the exact reaction rate density function of an a b 0 reaction on a finite 1d bar is derived we consider the case where the a and b species diffuse at the same rate have equal initial numbers with the initial location of either species on the bar being arbitrary the extension of the method to study a b 0 on fractals is also shown additionally we introduce the notion of a continual in time reaction front and a ceasing in time reaction front and provide a physical criterion for when either occurs

numerical solution for the variable order linear cable equation with bernstein polynomials abstract in this paper bernstein polynomials method is proposed for the numerical solution of a class of variable order fractional linear cable equation in this paper we adopted bernstein polynomials basis defined on the interval 0 r to solve the equations defined on the section 0 x 0 t the main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of bernstein polynomials with the operational matrixes the initial equation is transformed into the products of several dependent matrixes which can also be viewed as the system of linear equations after dispersing the variable by solving the linear system of algebraic equations the numerical solutions are acquired only a small number of bernstein polynomials are needed to obtain a satisfactory result numerical examples are provided to show that the method is computationally efficient

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

computer simulation of geometric stable distributions we present a new method for computer simulation of strictly geometric stable random variables the method is based on their representation as a product of two independent random variables with explicit distribution functions coupled with the inversion method we also extend the method to the multivariate case by deriving new representation and simulation algorithm for multivariate linnik distribution

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

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

systematic derivation of reaction diffusion equations with distributed delays and relations to fractional reaction diffusion equations and hyperbolic transport equations application to the theory of neolithic transition we introduce a general method for the systematic derivation of nonlinear reaction diffusion equations with distributed delays we study the interactions among different types of moving individuals atoms molecules quasiparticles biological organisms etc the motion of each species is described by the continuous time random walk theory analyzed in the literature for transport problems whereas the interactions among the species are described by a set of transformation rates which are nonlinear functions of the local concentrations of the different types of individuals we use the time interval between two jumps the transition time as an additional state variable and obtain a set of evolution equations which are local in time in order to make a connection with the transport models used in the literature we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so called generalized master equations the method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations hyperbolic reaction diffusion equations and delay differential reaction diffusion equations thus in the analysis of a given problem we can fit to the data the type of reaction diffusion equation and the corresponding physical and kinetic parameters the method is illustrated as a test case by the study of the neolithic transition we introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction diffusion equation for the population density we derive a delay evolution equation for the rate of advance of agriculture which illustrates an application of our analysis

relationship between time reversal and linear equalization in digital communications iterative time reversal has been suggested as both an efficient method of creating a spatio temporal focus and for use in telecommunications as a form of equalization in this paper the equivalence of a passive i e via computation iterative time reversal to the moore penrose pseudo inverse of the propagation matrix is shown in the context of communications however any received signal is corrupted by noise therefore a regularization term is introduced to the iterative equations causing convergence to the canonical minimum mean squared error linear equalizer hence a relationship between time reversal and equalization is demonstrated

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

kinetic equations for reaction subdiffusion systems derivation and stability analysis we derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by vlad and ross phys rev e 66 061908 2002 reaction and diffusion processes are separable in a typical reaction diffusion system and their combined influence on the evolution of the density of a species is a simple sum our derivation shows that this is no longer true for subdiffusive entities undergoing reactions the strong memory effects in the transport process i e the non markovian nature of subdiffusion results in a nontrivial combination of reactions and spatial dispersal which we discuss in detail we carry out a linear stability analysis of the derived reaction subdiffusion system to understand the effects of memory on pattern formation we find that the turing instability persists in the subdiffusive system however the memory modifies the turing threshold and the characteristics of the band of unstable modes close to this threshold

optimal focusing by spatio temporal inverse filter i basic principles a focusing technique based on the inversion of the propagation operator relating an array of transducers to a set of control points inside a medium was proposed in previous work tanter et al j acoust soc am 108 223 234 2000 and is extended here to the time domain as the inversion of the propagation operator is achieved both in space and time this technique allows calculation of the set of temporal signals to be emitted by each element of the array in order to optimally focus on a chosen control point this broadband inversion process takes advantage of the singular value decomposition of the propagation operator in the fourier domain the physical meaning of this decomposition is explained in a homogeneous medium in particular a definition of the number of degrees of freedom necessary to define the acoustic field generated by an array of limited aperture in a focal plane of limited extent is given this number corresponds to the number of independent signals that can be created in the focal area both in space and time in this paper this broadband inverse focusing technique is compared in homogeneous media with the classical focusing achieved by simple geometrical considerations but also with time reversal focusing it is shown that even in a simple medium slight differences appear between these three focusing strategies in the companion paper aubry et al j acoust soc am 110 48 58 2001 the three focusing techniques are compared in heterogeneous absorbing or complex media where classical focusing is strongly degraded the strong improvement achieved by the spatio temporal inverse filter technique emphasizes the great potential of multiple channel systems having the ability to apply completely different signal waveforms on each transducer of the array the application of this focusing technique could be of great interest in various ultrasonic fields such as medical imaging nondestructive testing and underwater acoustics

experimental demonstration of the utility of pressure sensitivity kernels in time reversal pressure sensitivity kernels were recently applied to time reversal acoustics in an attempt to explain the enhanced stability of the time reversal focal spot raghukumar et al j acoust soc am 124 98 112 2008 the theoretical framework developed was also used to derive optimized source functions closely related to the inverse filter the use of these optimized source functions results in an inverse filter like focal spot which is more robust to medium sound speed fluctuations than both time reversal and the inverse filter in this paper the theory is applied to experimental data gathered during the focused acoustic fields experiment conducted in 2005 north of elba island in italy sensitivity kernels are calculated using a range independent sound speed profile for a geometry identical to that used in the experiment and path sensitivities are identified with observed arrivals the validity of the kernels in tracking time evolving green s functions is studied along with limitations that resul

rytov approximation of tomographic receptions in weakly range dependent ocean environments an approximation method based on the born and rytov approximation is presented for the wave theoretic prediction of acoustic arrival patterns associated with long range pulse propagation in weakly range dependent ocean environments the environment is considered as a perturbation of a range independent background state and normal mode theory is used for the representation of the background green s function using the born and rytov approximations the perturbed green s function corresponding to the range dependent environment is expressed for each frequency within the source bandwidth in terms of the background green s function and the medium sound speed perturbation the actual arrival pattern in the time domain is then computed through the inverse fourier transform using the normal mode representation closed form expressions for the first and second born and rytov approximations are derived generalizing previous range independent results and indicating that the effect of range dependence on the

local discontinuous galerkin methods for the time tempered fractional diffusion equation abstract in this article we consider high order discrete schemes for solving the time tempered fractional diffusion equation we present a semi discrete scheme by using the local discontinuous galerkin ldg discretization in the spatial variable we prove that the semi discrete scheme is unconditionally stable in l2 norm and convergence with optimal convergence rate o h k 1 where h is the spatial step size we develop a class of fully discrete ldg schemes by combining the weighted and shifted lubich difference operators with respect to the time variable and establish the error estimates finally numerical experiments are presented to verify the theoretical results

numerical analysis for a variable order nonlinear cable equation in this paper a variable order nonlinear cable equation is considered a numerical method with first order temporal accuracy and fourth order spatial accuracy is proposed the convergence and stability of the numerical method are analyzed by fourier analysis we also propose an improved numerical method with second order temporal accuracy and fourth order spatial accuracy finally the results of a numerical example support the theoretical analysis

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

g majorization with applications to matrix orderings abstract a vector x is said to g majorize a vector y if y lies in the convex hull of the orbit of x under a group g the present paper contains a straightforward account with two important statements equivalent to g majorization in certain cases for example when g is a finite reflection group one equivalent condition reduces to a finite set of linear inequalities representing a cone ordering in the fundamental region of the group the other condition is that every convex g invariant function of y is less than the same function of x upper and lower weak majorizations gw majorizations are introduced by combining g with a second ordering compatible with it the results are applied where possible to matrix orderings where a g b is g majorization when g is a subgroup of the orthogonal group o n acting by congruence i e g b qbq t with q the matrix representation for an element of o n when g is the symmetric group of permutation matrices this defines a new ordering and generalizes a proposition of kiefer in the design of experiments

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

front propagation in hyperbolic fractional reaction diffusion equations from the continuous time random walk scheme and assuming a l evy waiting time distribution typical of subdiffusive transport processes we study a hyperbolic reaction diffusion equation involving time fractional derivatives the linear speed selection of wave fronts in this equation is analyzed when the reaction diffusion dimensionless number and the fractional index satisfy a certain condition we find fronts exhibiting an unphysical behavior they travel faster in the subdiffusive than in the diffusive regime

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

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

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

conditions for optimality in experimental designs abstract the purpose of this paper is to establish some results on optimal criteria in experimental designs some relationships between optimality criteria are shown in particular we extend results on the p criteria we prove the yeh biometrika 73 1986 701 conjecture that gives a necessary and sufficient condition for a design to be universally optimal we also give a similar result based on the eigenvalues of the information matrix

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

conditions for universal optimality of block designs summary kiefer 1975 introduced the criterion of universal optimality and provided a sufficient condition for a design being universally optimal the condition requires maximum trace and complete symmetry from the information matrix of a design in many instances this condition cannot be achieved in this paper kiefer s condition is generalized and applications are found in the justification of universal optimality over the class of binary block designs

front propagation in reaction dispersal models with finite jump speed the continuous time random walk and the hamilton jacobi method are used to reach analytical expressions for the speed of traveling fronts in reaction dispersal models in this work the waiting time and jump length are assumed to be coupled random variables the jump length for any jump is selected according to the waiting time at the end of the previous jump and in consequence jumps of finite speed are performed we study the effect of finite jump speed of the particles on the speed of the traveling fronts and find that in the parabolic and hyperbolic limits it can exceed the jump speed of the particles we report analytical expressions for different probability distribution functions finally we introduce the possibility that several particle speeds are allowed so different dispersal mechanisms can be considered simultaneously

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

time reversal and the inverse filter to focus ultrasonic waves in an unknown inhomogeneous medium using a phased array one has to calculate the optimal set of signals to be applied on the transducers of the array in the case of time reversal mirrors one assumes that a source is available at the focus providing the green s function of this point in this paper the robustness of this time reversal method is investigated when loss of information breaks the time reversal invariance it arises in dissipative media or when the field radiated by the source is not entirely measured by the limited aperture of a time reversal mirror however in both cases linearity and reciprocity relations ensure time reversal to achieve a spatiotemporal matched filtering nevertheless though it provides robustness to this method no constraints are imposed on the field out of the focus and sidelobes may appear another approach consists of measuring the green s functions associated to the focus but also to neighboring points thus the whole information characterizing the medium is known and the inverse source problem can be solved a matrix formalism of the propagation operator is introduced to compare the time reversal and inverse filter techniques moreover experiments investigated in various media are presented to illustrate this comparison

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