scale free animal movement patterns levy walks outperform fractional brownian motions and fractional levy motions in random search scenarios the movement patterns of a diverse range of animals have scale free characteristics these characteristics provide necessary but not sufficient conditions for the presence of movement patterns that can be approximated by levy walks nevertheless it has been widely assumed that the occurrence of scale free animal movements can indeed be attributed to the presence of levy walks this is in part because it is known that the super diffusive properties of levy walks can be advantageous in random search scenarios when searchers have little or no prior knowledge of target locations however fractional brownian motions fbms and fractional levy motions flms are both scale free and super diffusive and so it is possible that these motions rather than levy walks underlie some or all occurrences of scale free animal movement patterns here this possibility is examined in numerical simulations through a determination of the searching efficiencies of fbm and flm searches it is shown that these searches are less efficient than levy walk searches this finding does not rule out the possibility that some animals with scale free movement patterns are executing fbm and flm searches but it does make levy walk searches the more likely possibility

continuous time modeling of random searches statistical properties and inference mathematical modeling of random searches is of great relevance in the fields of physics chemistry biology or modern ecology a large number of existing studies record the search movement at equidistant time intervals and model such time series data directly with discrete time random walks such as levy flights and correlated random walks given the increasing availability of high resolution observation data statistical inference for search paths based on such high resolution data has recently become one of the major interests and has raised an important issue of robustness of random walk models to the sampling rate as estimation results for the discrete observation data are found to be largely different at different sampling rates even when the underlying movement is supposedly independent of scale to address this issue in this paper we propose to model the continuous time search paths directly with a continuous time stochastic process for which the observer makes statistical inference based on its discrete observation as continuous time counterparts of levy flights we consider two dimensional levy processes and discuss the relevance of those models based upon advantages and limitations in terms of statistical properties and inference among the proposed models the brownian motion is most tractable in various ways while its gaussianity and infinite variation of sample paths do not well describe the reality such drawbacks in statistical properties may be remedied by employing the stable and tempered stable levy motions while those models are less tractable and cause an issue in statistical inference

asymptotic behaviour of densities of stable semigroups of measures we prove that densities of the measures in a strictly stable semigroup ht of symmetric measures on a homogeneous group if they exist have the following asymptotic behaviour mathop lim limits x to infty x q alpha cdot h 1 x k bar x where is the characteristic exponent bar x x 1 x andk is the density of the levy measure associated to the semigroup moreover if k bar x 0 a more precise description is given

a spline method for second order singularly perturbed boundary value problems we consider a difference scheme based on cubic spline in compression for second order singularly perturbed boundary value problem of the form ey p x y q x y r x y a 0 y b 1 the method is shown to have second and fourth order convergence depending on the choice of parameters 1 and 2 involved in the method the method is tested on an example and the results found to be in agreement with the theory

on the spectral side of arthur s trace formula combinatorial setup in arthur s trace formula a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the coxeter fan g m families these include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side we introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes in a companion paper the results are used to study the spectral side of the trace formula

intermittent search strategies revisited effect of the jump length and biased motion we study the kinetics of a search of a single fixed target by a large number of searchers performing an intermittent biased random walk in a homogeneous medium our searchers carry out their walks in one of two states between which they switch randomly one of these states search phase is a nearest neighbor walk characterized by the probability of stepping in a given direction i e the walks in this state are not necessarily isotropic the other relocation phase is characterized by the length of the jumps i e when in this state a walker does not perform a nearest neighbor walk within such a framework we propose a model to describe the searchers dynamics generalizing results of our previous work we have obtained and numerically evaluated analytic results for the mean number of distinct sites visited up to a maximum evolution time we have studied the dependence of this quantity on both the transition probability between the states and the parameters that characterize each state in addition to our theoretical approach we have implemented monte carlo simulations finding excellent agreement between the theoretical numerical and simulations results

on the inner product of truncated eisenstein series introduction 1 the problem 2 residues of cuspidal eisenstein series 83 exponents 4 a comparison between two groups 5 a property of the truncation operator 6 the constant terms of eisenstein series 7 the negative dual chamber 8 coefficients of the zero exponents 9 conclusion bibliography

spectral asymptotics for arithmetic quotients of sl n r so n in this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space sl n r so n in particular we obtain weyl s law with an estimation on the remainder term this extends some of the main results of duistermaat kolk and varadarajan dkv1 to this setting

levy flights in random searches we review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager and discuss its quantitative description using the theory of random walks we illustrate levy flight foraging by comparison to brownian random walks and discuss experimental observations of levy flights in biological foraging we review recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l can lead to optimal searches finally we survey the explanations put forth to account for these unexpected findings

the spectral side of arthur s trace formula the trace formula is one of the most important tools in the theory of automorphic forms it was invented in the 1950 s by selberg who mostly studied the case of hyperbolic surfaces and was later on developed extensively by arthur in the generality of an adelic quotient of a reductive group over a number field here we provide an explicit expression for the spectral side improving arthur s fine spectral expansion as a result we obtain its absolute convergence for a wide class of test functions

global convergence method for singularly perturbed boundary value problems we give uniformly convergent splines difference scheme for singularly perturbed boundary value problems eu p x u q x u f x u a 0 u b 1 1 by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration we prove the o min h2 e2 order of uniform convergence with respect to small parameter e at nodes on uniform mesh and o min h e order of uniform global convergence with respect to the approximate solution given by s x i 1n s i x h xi x where h is the heaviside function which is the approximation for the closed form of the exact solution

intertwining operators and residues i weighted characters operateurs d entrelacement normalisation groupes reels groupes p adiques representations standard les distributions j m la distribution j m x residus changements de contour les espaces 7b h ac g f s et 7b f ac g f s l application m

on simulation of tempered stable random variates various simulation methods for tempered stable random variates with stability index greater than one are investigated with a view towards practical implementation in particular cases of very small scale parameter which correspond to increments of a tempered stable levy process with a very short stepsize methods under consideration are based on acceptance rejection sampling a gaussian approximation of a small jump component and infinite shot noise series representations numerical results are presented to discuss advantages limitations and trade off issues between approximation error and required computing effort with a given computing budget an approximative acceptance rejection sampling technique baeumer and meerschaert 2009 11 is both most efficient and handiest in the case of very small scale parameter and moreover any desired level of accuracy may be attained with a small amount of additional computing effort

higher order global solution and normalized flux for singularly perturbed reaction diffusion problems in this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction diffusion two point boundary value problems the hoc higher order compact finite difference scheme developed in gracia et al 2001 4 and which is constructed on an appropriate piecewise uniform shishkin mesh has been considered to find an almost fourth order convergent solution at mesh points using these values piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined it has been shown that the global solution and the global normalized flux are also uniformly convergent moreover for the global solution the order of uniform convergence in the whole domain is optimal i e it is the same as this one obtained at mesh points whereas for the global normalized flux the uniform convergence is almost third order except at midpoints of the mesh where it is also almost fourth order theoretical error bounds have been provided along with some numerical examples which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux

analytic approach to the problem of convergence of truncated levy flights towards the gaussian stochastic process an analytic expression for characteristic function defining a truncated l evy flight is derived it is shown that the characteristic function yields results in agreement with recent simulations of truncated l evy flights by mantegna and stanley phys rev lett 73 2946 1994 with the analytic expression for the characteristic function the convergence of the l evy process towards the gaussian is demonstrated without simulations in the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures

on the spectral side of arthur s trace formula absolute convergence we derive a renement of the spectral expansion of arthur s trace formula the expression is absolutely convergent with respect to the trace norm

a uniformly convergent finite difference scheme for a singularly perturbed semilinear equation boundary value problems for singularly perturbed semilinear elliptic equations are considered special piecewise uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter numerical methods composed of standard monotone finite difference operators and these piecewise uniform meshes are shown theoretically to be uniformly with respect to the singular perturbation parameter convergent numerical results are also presented which indicate that in practice the method is first order accurate

multi scale properties of random walk models of animal movement lessons from statistical inference the random search problem has long attracted continuous interest owing to its broad interdisciplinary range of applications including animal foraging facilitated target location in biological systems and human motion in this paper we address the issue of statistical inference for ordinary gaussian pareto tempered pareto and fractional gaussian random walk models which are among the most studied random walk models proposed as the best strategy in the random search problem based on rigorous analysis of the local asymptotic normality property and the fisher information we discuss some issues in unbiased joint estimation of the model parameters in particular the maximum likelihood estimation we present that there exist both theoretical and practical difficulties in more realistic tempered pareto and fractional gaussian random walk models from a statistical standpoint we discuss our findings in the context of individual animal movement and show how our results may be used to facilitate the analysis of movement data and to improve the understanding of the underlying stochastic process

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

higher order fractional stable motion hyperdiffusion with heavy tails we introduce the class of higher order fractional stable motions that can exhibit hyperdiffusive spreading with heavy tails we define the class as a generalization of higher order fractional brownian motion as well as a generalization of linear fractional stable motions higher order fractional stable motions are self similar with hurst index larger than one and non gaussian stable marginals with infinite variance and have stationary higher order increments we investigate their sample path properties and asymptotic dependence structure on the basis of codifference in particular by incrementing or decrementing sample paths once under suitable conditions the diffusion rate can be accelerated or decelerated by one order with a view towards simulation study we provide a ready for use sample path simulation recipe at discrete times along with error analysis the proposed simulation scheme requires only elementary numerical operations and is robust to high frequency sampling irregular spacing and super sampling

on a family of distributions obtained from eisenstein series i application of the paley wiener theorem introduction this is the first of two papers aimed at finding explicit formulas for certain distributions the distributions are obtained from eisenstein series and are important ingredients in the trace formula in this paper we shall resolve some analytic difficulties that center around an interchange of two limits the next paper will be devoted to the calculations which will culminate in the formulas let g be a reductive algebraic group defined over q as usual we will write g a for the intersection of the kernels of the maps

on finite truncation of infinite shot noise series representation of tempered stable laws abstract tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long range gaussian behaviors to stable processes infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed in this paper we derive series representations for the tempered stable laws of increasing practical interest through the thinning rejection and inverse levy measure methods we make a rigorous comparison among those representations including the existing one due to imai and kawai 29 and rosinski 2007 3 in terms of the tail mass of levy measures which can be simulated under a common finite truncation scheme the tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws we prove that the representation via the inverse levy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations numerical results are presented to support our theoretical analysis

oscillation of delay parabolic differential equations with several coefficients some new sufficient conditions are established for the oscillation of delay parabolic differential equations of the form u x t t a t u i 1n pi x t u x t i j 1m qj x t u x t j x t t0 g where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn with three different boundary conditions our results improve the well known oscillation result of e when n m 1 an example is considered to illustrate our main results

absolute convergence of the twisted trace formula we show that the distributions occurring in the geometric and spectral side of the twisted arthur selberg trace formula extend to non compactly supported test functions the geometric assertion is modulo a hypothesis on root systems proven when the group is split the result extends the work of finis lapid and muller spectral side to the twisted setting we use the absolute convergence to give a geometric interpretation of sums of residues of certain rankin selberg l functions