on n ary algebras branes and poly vector gauge theories in noncommutative clifford spaces in this paper poly vector valued gauge field theories in noncommutative clifford spaces are presented they are based on noncommutative but associative star products that require the use of the baker campbell hausdorff formula using these star products allows the construction of actions for noncommutative p branes branes moving in noncommutative spaces noncommutative clifford space gravity as a poly vector valued gauge theory of twisted diffeomorphisms in clifford spaces would require quantum hopf algebraic deformations of clifford algebras we proceed with the study of n ary algebras and find an important relationship among the n ary commutators of the noncommuting spacetime coordinates x1 x2 xn with the poly vector valued coordinates x123n in noncommutative clifford spaces given by x1 x2 xn n x123n the large n limit of n ary commutators of n hyper matrices leads to eguchi schild p brane actions for p 1 n a noncomutative n ary product of n functions is constructed which is a generalization of the binary star product of two functions and is associated with the deformation quantization of n ary structures and deformations of the nambu poisson brackets

a fast semi discrete kansa method to solve the two dimensional spatiotemporal fractional diffusion equation abstract fractional order diffusion equations fdes extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media real world diffusion can be multi dimensional requiring efficient numerical solvers that can handle long term memory embedded in mass transport to address this challenge a semi discrete kansa method is developed to approximate the two dimensional spatiotemporal fde where the kansa approach first discretizes the fde then the gauss jacobi quadrature rule solves the corresponding matrix and finally the mittag leffler function provides an analytical solution for the resultant time fractional ordinary differential equation numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes applications further show that the numerical method can efficiently solve two dimensional spatiotemporal fde models with either a continuous or discrete mixing measure hence this study provides an efficient and fast computational method for modeling super diffusive sub diffusive and mixed diffusive processes in large two dimensional domains with irregular shapes

average twin prime conjecture for elliptic curves let e be an elliptic curve over bbb q in 1988 n koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of e over bbb f p is prime this is an analogue of the hardy littlewood twin prime conjecture in the case of elliptic curves koblitz s conjecture is still widely open in this paper we prove that koblitz s conjecture is true on average over a two parameter family of elliptic curves one of the key ingredients in the proof is a short average distribution result of primes in the style of barban davenport halberstam where the average is taken over prime differences and over arithmetic progressions

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

pore scale modeling and continuous time random walk analysis of dispersion in porous media 1 we provide a physically based explanation for the complex macroscopic behavior of dispersion in porous media as a function of peclet number pe using a pore scale network model that accurately predicts the experimental dependence of the longitudinal dispersion coefficient dl on pe the asymptotic dispersion coefficient is only reached after the solute has traveled through a large number of pores at high pe this implies that preasymptotic dispersion is the norm even in experiments in statistically homogeneous media interpreting transport as a continuous time random walk we show that 1 the power law dispersion regime is controlled by the variation in average velocity between throats the distribution of local pe giving dl pe with 3 1 2 where is an exponent characterizing the distribution of transit times between pores 2 the crossover to a linear regime dl pe for pe pecrit 400 is due to a transition from a diffusion controlled late time cutoff to transport governed by advective movement and 3 the transverse dispersion coefficient dt pe for all pe 1

transport of conservative solutes in simulated fracture networks 2 ensemble solute transport and the correspondence to operator stable limit distributions 1 in networks where individual fracture lengths follow a fractal distribution ensemble transport of conservative solute particles at the leading plume edge exhibit characteristics of operator stable densities these densities have as their governing equations of transport either fractional order or integer order advection dispersion equations model selection depends on the identification of either multi gaussian or operator stable transport regimes which in turn depends on the power law exponent of the fracture length distribution low to moderately fractured networks with power law fracture length exponents less than or equal to 1 9 produce solute plumes that exhibit power law leading edge concentration profiles and super fickian plume growth rates for these network types a multiscaling fractional advection dispersion equation mfade provides a model of multidimensional solute transport where different rates of power law particle motion are defined along multiple directions the mfade model is parameterized by a scaling matrix to describe the super fickian growth process in which the eigenvectors correspond to primary fracture group orientations and the eigenvalues code fracture length and transmissivity the approximation of particle clouds by a multi gaussian a subset of the operator stable for densely fractured networks with finite variance fracture length distributions can be ascribed to the classical ade where fickian scaling rates pertain along orthogonal plume growth directions fracture networks show long term particle retention in low velocity fractures so that coupling of the equations of motion with retention models such as continuous time random walk or multirate mobile immobile will increase accuracy near the source particle arrival times at exit boundaries for multi gaussian plumes vary with spatial density generally arrival times are faster in sparsely fractured domains where transport is governed by a few very long fractures

transport behavior of a passive solute in continuous time random walks and multirate mass transfer 1 we consider the transport behavior of a passive solute in a heterogeneous medium modeled by continuous time random walks ctrw and linear multirate mass transfer mrmt within the ctrw framework we formulate a transport model which is formally equivalent to mrmt in both approaches the total concentration is divided into mobile and immobile parts the immobile concentration is given by the convolution in time of the mobile concentration and a memory function the memory function is a functional of the distribution of transition and trapping times for the ctrw and the mrmt approach respectively and determines the transport behavior of the solute based on different expressions for the memory function in the two frameworks we derive conditions for which both approaches describe the same transport behavior we focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime using an expansion of the laplace transform of the memory function we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the macro dispersion coefficients of the solute distribution we observe anomalous power law as well as normal fickian transport behavior depending on the exponents that dominate the trapping and transition time distributions respectively the results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions the unified transport framework presented here comprising ctrw and mrmt shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media

the cohen lenstra heuristic methodology and results in number theory great efforts have been undertaken to study the cohen lenstra probability measure on the set of all finite abelian p groups on the other hand group theorists have studied a probability measure on the set of all partitions induced by the probability that a randomly chosen n n matrix over fp is contained in a conjugacy class associated with this partitions for n this paper shows that both probability measures are identical as a consequence a multitude of results can be transferred from each theory to the other one the paper contains a survey about the known methods to study the probability measure and about the results that have been obtained so far from both communities

a probabilistic approach toward conjugacy classes in the finite general linear and unitary groups abstract the conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers probabilistic algorithms for growing random partitions according to these measures are obtained these algorithms are applied to prove group theoretic results which are typically proved by techniques such as character theory and moebius inversion among the theorems studied are steinberg s count of unipotent elements rudvalis and shinoda s work on the fixed space of a random matrix and lusztig s count of nilpotent matrices of a given rank generalizations of these algorithms based on macdonald s symmetric functions are given

canonically relativistic quantum mechanics representations of the unitary semidirect heisenberg group u 1 3 sh 1 3 born proposed a unification of special relativity and quantum mechanics that placed position time energy and momentum on equal footing through a reciprocity principle and extended the usual position time and energy momentum line elements to this space by combining them through a new fundamental constant requiring also invariance of the symplectic metric yields u 1 3 as the invariance group the inhomogeneous counterpart of which is the canonically relativistic group cr 1 3 u 1 3 sh 1 3 where h 1 3 is the heisenberg group in four dimensions this is the counterpart in this theory of the poincare group and reduces in the appropriate limits to the expected special relativity and classical hamiltonian mechanics transformation equations this group has the poincare group as a subgroup and is intrinsically quantum with the position time energy and momentum operators satisfying the heisenberg algebra the representations of the algebra are studied and casimir invariants are computed like the poincar

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

coherent states of su l 1 groups this work can be considered as a continuation of our previous study in which an explicit form of coherent states cs for all su n groups was constructed by means of representations on polynomials here we extend that approach to any su l 1 group and construct explicitly corresponding cs the cs are parametrized by dots of a coset space which is in that particular case the open complex ball cdl this space together with the projective space cpl which parametrizes the cs of the su l 1 group exhaust all complex spaces of constant curvature thus both sets of cs provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature this is why the cs of the su n and su l 1 groups are of importance in connection with quantization theory the constructed cs form an overcompleted system in the representation space and as quantum states possessing a minimum uncertainty they minimize an invariant dispersion of the quadratic casimir operator the classical limit is investigated in terms of symbols of operators the limit of the so called star commutator of the symbols generates the poisson bracket in cdl the latter plays the role of the phase space for the corresponding classical mechanics

modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit lagrangian scheme this study develops an explicit two step lagrangian scheme based on the renewal reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media the resulting 3 d anomalous transport simulator provides a flexible platform for modeling transport the first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones the mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step and the subsequent random immobile time follows a hyper exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power law memory function the second step describes well documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow applicability and implementation of the lagrangian solver are further checked against transport observed in various media results show that although the time nonlocal model parameters are predictable for transport with retention in alluvial settings the standard time nonlocal model cannot capture early arrivals retention and early arrivals observed in porous and fractured media can be efficiently modeled by our lagrangian solver allowing anomalous transport to be incorporated into 2 d 3 d models with irregular flow fields extensions of the particle tracking approach are also discussed for transport with parameters conditioned on local aquifer properties as required by transient flow and nonstationary media

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

a probabilistic approach to conjugacy classes in the finite symplectic and orthogonal groups abstract markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic as a corollary of these methods one obtains a probabilistic proof of steinberg s count of unipotent matrices and generalizations of formulas of rudvalis and shinoda

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

connectivity of joint networks with power law length distributions over the range of fracture spatial densities commonly observed in the field existing effective medium models for network connectivity yield inconsistent results because the connectivity depends on both the density and length distribution of the joints field observations indicate that joint lengths often follow a power law length distribution requiring the frequency of joint lengths to decrease with increasing length and the total area of the joints to be finite suggests approximate upper and lower bounds on the power law exponent which are consistent with observed distributions the appropriate relationship between connectivity and density is investigated numerically results motivate a new measure of fracture intensity the effective fracture density and the development of a new approach toward a closed form prediction of connectivity using measurable parameters although limited model predictions compare favorably to the conductivities of networks simulated using a geomechanical model and observed in the field

operator levy motion and multiscaling anomalous diffusion the long term limit motions of individual heavy tailed power law particle jumps that characterize anomalous diffusion may have different scaling rates in different directions operator stable motions y t tg 0 are limits of d dimensional random jumps that are scale invariant according to c h y t y mathrm ct where h is a d ifmmode times else texttimes fi d matrix the eigenvalues of the matrix have real parts 1 ensuremath alpha j with each positive ensuremath alpha j l 2 in each of the j principle directions the random motion has a different fickian or super fickian diffusion dispersion rate proportional to t 1 ensuremath alpha j these motions have a governing equation with a spatial dispersion operator that is a mixture of fractional derivatives of different order in different directions subsets of the generalized fractional operator include i a fractional laplacian with a single order ensuremath alpha and a general directional mixing measure m ensuremath theta and ii a fractional laplacian with uniform mixing measure the riesz potential the motivation for the generalized dispersion is the observation that tracers in natural aquifers scale at different super fickian rates in the directions parallel and perpendicular to mean flow

hydraulic properties of two dimensional random fracture networks following a power law length distribution 1 effective connectivity natural fracture networks involve a very broad range of fractures of variable lengths and apertures modeled in general by a power law length distribution and a lognormal aperture distribution the objective of this two part paper is to characterize the permeability variations as well as the relevant flow structure of two dimensional isotropic models of fracture networks as determined by the fracture length and aperture distributions and by the other parameters of the model such as density and scale in this paper we study the sole influence of the fracture length distribution on permeability by assigning the same aperture to all fractures in the following paper de dreuzy et al this issue we study the more general case of networks in which fractures have both length and aperture distributions theoretical and numerical studies show that the hydraulic properties of power law length fracture networks can be classified into three types of simplified model if a power law length distribution n l l a is used in the network design the classical percolation model based on a population of small fractures is applicable for a power law exponent a higher than 3 for a lower than 2 on the contrary the applicable model is the one made up of the largest fractures of the network between these two limits i e for a in the range 2 3 neither of the previous simplified models can be applied so that a simplified two scale structure is proposed for this latter model the crossover scale is the classical correlation length defined in the percolation theory above which networks can be homogenized and below which networks have a multipath multisegment structure moreover the determination of the effective fracture length range within which fractures significantly contribute to flow corroborates the relevance of the previous models and clarifies their geometrical characteristics finally whatever the exponent a the sole significant scale effect is a decrease of the equivalent permeability for networks below or at percolation threshold

counting hyperelliptic curves abstract we find a closed formula for the number hyp g of hyperelliptic curves of genus g over a finite field k f q of odd characteristic these numbers hyp g are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q 1 and q 1 as a by product we obtain a closed formula for the number of self dual curves of genus g a hyperelliptic curve is defined to be self dual if it is k isomorphic to its own hyperelliptic twist

exceptional splitting of reductions of abelian surfaces heuristics based on the sato tate conjecture and the lang trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction we prove this result for abelian surfaces with real multiplication as in previous work by charles and elkies this shows that a density 0 set of primes pertaining to the reduction of abelian varieties is infinite the proof relies on the arakelov intersection theory on hilbert modular surfaces

advection and dispersion in time and space abstract previous work showed how moving particles that rest along their trajectory lead to time nonlocal advection dispersion equations if the waiting times have infinite mean the model equation contains a fractional time derivative of order between 0 and 1 in this article we develop a new advection dispersion equation with an additional fractional time derivative of order between 1 and 2 solutions to the equation are obtained by subordination the form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

representations of the canonical group the semidirect product of the unitary and weyl heisenberg groups acting as a dynamical group on noncommutative extended phase space the unitary irreducible representations of the covering group of the poincare group define the framework for much of particle physics on the physical minkowski space where is the lorentz group while extraordinarily successful it does not provide a large enough group of symmetries to encompass observed particles with a 3 classification born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically t e qi pi t e pi qi where t e qi pi are the time energy position and momentum degrees of freedom this implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b where b is a universal constant the appropriate group of dynamical symmetries that embodies this is the canonical group 1 3 1 3 s 1 3 1 3 s 1 3 and in this theory the non commuting space 1 3 1 3 is the physical quantum space endowed with a metric that is the second casimir invariant of the canonical group t2 e2 c2b2 q2 c2 p2 b2 2i bc y bc 2 where t e qi pi i y are the generators of the algebra of 1 3 1 s 1 3 the idea is to study the representations of the canonical dynamical group using mackey s theory to determine whether the representations can encompass the spectrum of particle states the unitary irreducible representations of the canonical group contain a direct product term that is a representation of 1 3 that kalman has studied as a dynamical group for hadrons the 1 3 representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of 3 finite dimensional or 2 with degenerate 1 2 finite dimensional representations corresponding to the rest or null frames

conjugacy class properties of the extension of gl n q generated by the inverse transpose involution abstract letting denote the inverse transpose automorphism of gl n q a formula is obtained for the number of g in gl n q so that gg is equal to a given element h this generalizes a result of gow and macdonald for the special case that h is the identity we conclude that for g random gg behaves like a hybrid of symplectic and orthogonal groups it is shown that our formula works well with both cycle index generating functions and asymptotics and is related to the theory of random partitions the derivation makes use of models of representation theory of gl n q and of symmetric function theory including a new identity for hall littlewood polynomials we obtain information about random elements of finite symplectic groups in even characteristic and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of gl n q generated by the inverse transpose automorphism we give a second approach to these results using the theory of bilinear forms over a field the results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups and we give a few examples in this direction

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

a stochastic analysis of macroscopic dispersion in fractured media a stochastic modeling technique has been developed to investigate mass transport in a network of discrete fractures the model is based on the repetitive generation of realizations of a fracture network from probability distributions describing the fracture geometry and on a solution for mass transport within each network using a particle tracking technique the system we work with consists of two orthogonal fracture sets of finite length oriented at various angles with respect to the direction of the mean hydraulic gradient emphasis is placed on describing the character of dispersion which develops as a consequence of fracture interconnectivity and on testing the validity of the conventional diffusion based model of dispersion in describing transport in fractured media results show that mass distributions have a complex form marked longitudinal dispersion can develop even a short distance from a source the distribution of mass in the direction of flow has a consistent negative skew this pattern of dispersion arises from the limited number of pathways for mass to migrate through the network controlling factors in the transport process are the orientation of the fracture sets with respect to the mean hydraulic gradient the difference in the mean flow velocity in the two fracture sets and the standard deviation in velocity for fracture set 1 transport patterns can change greatly as the orientation of the hydraulic gradient changes with respect to the two fracture sets a conventional diffusion based model of dispersion cannot characterize transport in these fracture networks a skewed spatial distribution of mass is observed much more frequently than a gaussian distribution when the mean velocities in the two fracture sets are not equal the form of mass spreading is described by a more general skewed distribution that accounts for the bias in the probability of mass moving along one fracture set over another there is a tendency for mass to form a more symmetric distribution as the orientation of the two fracture sets is rotated toward a 45 angle with respect to the direction of the mean hydraulic gradient furthermore constant dispersivity values or simple dispersivity functions are not definable because of the sensitivity of transport to the local velocity field in the fracture network

continuous time random walks for analyzing the transport of a passive tracer in a single fissure 1 the selection of the correct model for describing contaminant transport in fractured media remains a challenging problem in subsurface hydrology in this work the experimental data corresponding to nonsorbing tracer runs in natural fissures of moreno et al 1985 are analyzed the advection dispersion equation ade model yields a poor fit in the late time regions of the experimental breakthrough curves denoting the existence of anomalous or non fickian transport continuous time random walks ctrw are proposed for describing this kind of transport using two of the most studied forms for the transition rate probability the asymptotic and truncated power law models the ctrw approach produces better fits to the experimental data than the ade model in all the simulations a moderately dispersive transport was found which is interpreted as being as a result of the effects of the small and intermediate scale heterogeneities

connectivity of random fault networks following a power law fault length distribution we present a theoretical and numerical study of the connectivity of fault networks following power law fault length distributions n l l a as expected for natural fault networks different regimes of connectivity are identified depending on a for a 3 faults smaller than the system size rule the network connectivity and classical laws of percolation theory apply on the opposite for a 1 the connectivity is ruled by the largest fault in the system for 1 a 3 both small and large faults control the connectivity in a ratio which depends on a the geometrical properties of the fault network and of its connected parts density scaling properties are established at the percolation threshold finally implications are discussed in the case of fault networks with constant density in particular we predict the existence of a critical scale at which fault networks are always connected whatever a smaller than 3 and whatever their fault density

casimir operators of semidirect sum lie algebras a new method for constructing the casimir operators of a class of non semisimple lie algebras is presented this class includes the semidirect sums of the unitary orthogonal or symplectic algebras with a heisenberg weyl algebra w n the construction procedure is illustrated with the simple example of the wu n n u n algebras its power is then demonstrated on the more difficult case of the wsp 2n r w n sp 2n r algebras detailed results are given for the eigenvalues of the wsp 6 r casimir operators relevant to a microscopic theory of nuclear collective motions

abelian varieties isogenous to a jacobian we dene a notion of weyl cm points in the moduli space ag 1 of g dimensional principally polarized abelian varieties and show that the andr e oort conjecture or the grh implies the following statement for any closed subvariety x ag 1 over q a there exists a weyl special point b 2 ag 1 q a such that b is not isogenous to the abelian variety a underlying any point a 2 x the title refers to the case when g 4 and x is the torelli locus in this case tsimerman has proved the statement unconditionally the notion of weyl special points is generalized to the context of shimura varieties and we prove a corresponding conditional statement with the ambient spaceag 1 replaced by a general shimura variety

the probability that the number of points on the jacobian of a genus 2 curve is prime in 2000 galbraith and mckee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points we show how their heuristics can be generalized to jacobians of curves of higher genus we then elaborate this in genus g 2 and study various related issues such as the probability of cyclicity and the probability of primality of the number of points on the curve itself finally we discuss the asymptotic behavior for g

monte carlo simulation of superdiffusion and subdiffusion in macroscopically heterogeneous media 1 monte carlo simulations are developed to approximate one dimensional superdiffusion and subdiffusion in macroscopically heterogeneous media with discontinuous or continuous transport parameters for superdiffusion characterized by a space fractional order derivative model one empirical reflection scheme is built to track particle trajectory across an interface with discontinuous dispersion coefficient d where the reflection probability depends on both and the ratio of d different from the superdiffusive case anomalous diffusion described by a time fractional derivative model can be decomposed into a motion component and a hitting time process where the discontinuity affects only the motion process implying an efficient monte carlo simulation of decoupled continuous time random walks the discontinuity of effective porosity n is also discussed and results show the influence of the ratio of n on solute particle dynamics in addition for anomalous superdiffusion and subdiffusion in heterogeneous media with spatially continuous d and n langevin analysis reveals that the corresponding particle dynamics contain three independent stable levy noises scaled by d the gradient of d and the gradient of ln n a new implicit eulerian finite difference method is also developed to solve the spatiotemporal fractional derivative models and then extensively cross verify the lagrangian solutions further testing against one field example of mixed superdiffusion and subdiffusion reveals the applicability and flexibility of the novel monte carlo approach in simulating realistic plumes in macroscopically heterogeneous media with locally variable transport parameters

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