complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

a time scales version of a wirtinger type inequality and applications in this paper we derive a time scales version of a wirtinger type inequality the result is applied to derive nonoscillation of a certain second order euler type dynamic equation on time scales

quadratic maps and bockstein closed group extensions let be a central extension of the form where and are elementary abelian groups associated to there is a quadratic map given by the power map which uniquely determines the extension this quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of we say that is bockstein closed if is an ideal closed under the bockstein operator we find a direct condition on the quadratic map that characterizes when the extension is bockstein closed using this characterization we show for example that quadratic maps induced from the fundamental quadratic map given by yield bockstein closed extensions on the other hand it is well known that an extension is bockstein closed if and only if it lifts to an extension for some lattice in this situation one may write for a binding matrix with entries in we find a direct way to calculate the module structure of in terms of using this we study extensions where the lattice is diagonalizable triangulable and find interesting equivalent conditions to these properties

long time dynamics close to a percolation threshold abstract the response function for random walks on a bond disordered lattice is calculated in the effective medium approximation ema the basis for this approximation is reviewed and several exact limits preserved by this model are noted for low density bond disorder the velocity autocorrelation function is calculated from the ema and compared with exact expansions to order c2 more generally the long time dynamics is shown to have a universal form over a wide range of densities however at densities close to the percolation threshold this form is modified in particular the amplitudes of the long time tails are found to be divergent at the threshold indicating a crossover to a qualitatively different dynamics this crossover phenomenon is investigated analytically and numerically for the ema

the n n disconjugacy of a 2nth order linear difference equation abstract we give a formulation of generalized zeros and n n disconjugacy for even order formally self adjoint scalar difference equations positive definiteness of an associated quadratic functional is shown to imply n n disconjugacy this study is motivated by the open question of existence of a reid roundabout theorem for singular discrete linear hamiltonian systems

the algebra of grand unified theories the standard model is the best tested and most widely accepted theory of elementary particles we have today it may seem complicated and arbitrary but it has hidden patterns that are revealed by the relationship between three grand unied theories theories that unify forces and particles by extending the standard model symmetry group u 1 su 2 su 3 to a larger group these three are georgi and glashow s su 5 theory georgi s theory based on the group spin 10 and the pati salam model based on the group su 2 su 2 su 4 in this expository account for mathematicians we explain only the portion of these theories that involves nite dimensional group representations this allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve

a new finite size scaling approach to random walks we present a new finite size scaling method for the random walks rw superseding a previously widely used renormalization group approach which is shown here to be inconsistent the method is valid in any dimension and is based on the exact solution for the two point correlation function and on finite size scaling as an example the phase diagram is derived for the random walk in two dimensions with a surface bulk interaction where the system has either a surface or a defect line we also discuss an initial calculation of the corresponding phase diagram for the case of a critically diluted lattice

anomalous diffusion localization aging and subaging effects in trap models at very low temperature we study in detail the dynamics of the one dimensional symmetric trap model via a real space renormalization procedure which becomes exact in the limit of zero temperature in this limit the diffusion front in each sample consists of two delta peaks which are completely out of equilibrium with each other the statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit t 0 we first compute disorder averages of one time observables such as the diffusion front the thermal width the localization parameters the two particle correlation function and the generating function of thermal cumulants of the position we then study aging and subaging effects our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two time correlations we also extend the real space renormalization group method to include systematic corrections to the previous zero temperature procedure via a series expansion in t we then consider the generalized trap model with parameter alpha in 0 1 and obtain that the large scale effective model at low temperature does not depend on alpha in any dimension so that the only observables sensitive to alpha are those that measure the local persistence such as the probability to remain exactly in the same trap during a time interval finally we extend our approach at a scaling level for the trap model in d 2 and obtain the two relevant time scales for aging properties

lie algebras and higher torsion in p groups abstract the primary aim of this paper is to study exceptional torsion in the integral cohomology of a family of p groups associated to p adic lie algebras a spectral sequence e r g is defined for any lie algebra g which models the bockstein spectral sequence of the corresponding group in characteristic p this spectral sequence is then studied for complex semisimple lie algebras like sl n c and the results there are transferred to the corresponding p group via the intermediary arithmetic lie algebra defined over z the results obtained this way for a fixed lie algebra scheme like sl n hold in a range in the corresponding bockstein spectral sequence for all but finitely many primes depending on the chosen range over c it is shown that e 1 g h g u g h bg where u g is the filtered dual of the universal enveloping algebra of g equipped with the dual adjoint action and bg is the free loop space of the classifying space of an associated compact connected real form lie group g to g when passing to characteristic p in the corresponding bockstein spectral sequence a char 0 to char p phase transition is observed for example it is shown that the algebra e 1 sl 2 f p requires at least 17 generators unlike its characteristic zero counterpart which only requires two

group actions and group extensions in this paper we study finite group extensions represented by special cohomology classes as an application we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces in particular we prove that if z p acts freely on s then r k

on a conjecture of alley and alder for fluids and lorentz models we discuss a conjecture of alley and alder predicting a relation between the four point and the two point velocity autocorrelation functions for fluids and lorentz models at sufficiently long times if the conjecture is correct a modified burnett coefficient can be defined which has a finite value contrary to the ordinary burnett coefficient which is divergent the conjecture is tested for four classes of models with different methods for three dimensional fluids mode coupling theory yields a negative result the conjecture is confirmed for thed dimensional deterministic lorentz gas d 2 and for a class ofd dimensional stochastic lorentz models d 1 by low density kinetic theory as well as by rigorous results available for one dimension for yet another class of one dimensional stochastic lorentz models which are exactly solvable in one dimension the result is negative again all four classes of models show long time tails in the velocity autocorrelation function and have a finite diffusion coefficient

transport properties of the one dimensional stochastic lorentz model i velocity autocorrelation function point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables stationary renewal process for a fixed configuration of scatterers a particle performs the following random walk the particle starts at the pointx with velocity 1 in between scatterers the particle moves freely at a scatterer the particle is either transmitted or reflected both with probability 1 2 for given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations if this variable is averaged over the distribution of scatterers it decays not faster thant 3 2

long time tails in a lorentz gas to investigate the approach to the asymptotic long time behaviour in a d dimensional lorentz gas we have calculated the t 2 d2 correction to the long time tail t 1 d2 in the velocity autocorrelation function using low density kinetic theory the results are compared with existing computer simulations

nonoscillation of second order linear difference systems with varying coefficients abstract this paper deals with nonoscillation problem about the non autonomous linear difference system x n a n x n 1 n 1 2 where a n is a 2 2 variable matrix that is nonsingular for n n in the special case that a is a constant matrix it is well known that all non trivial solutions are nonoscillatory if and only if all eigenvalues of a are positive real numbers namely det a 0 tr a 0 and det a tr a 2 1 4 the well known result can be said to be an analogy of ordinary differential equations the results obtained in this paper extend this analogy result in other words this paper clarifies the distinction between difference equations and ordinary differential equations our results are explained with some specific examples in addition figures are attached to facilitate understanding of those examples

log periodic oscillations for diffusion on self similar finitely ramified structures under certain circumstances the time behavior of a random walk is modulated by logarithmic periodic oscillations using heuristic arguments we give a simple explanation of the origin of this modulation for diffusion on a substrate with two properties self similarity and finite ramification order on these media the time dependence of the mean square displacement shows log periodic modulations around a leading power law which can be understood on the basis of a hierarchical set of diffusion constants both the random walk exponent and the period of oscillations are analytically obtained for a pair of examples one is fractal and the other is nonfractal and confirmed by monte carlo simulations the last example shows that the anomalous diffusion can arise from substrates without holes of all sizes

intermittency in fibonacci chains the devil s staircase spectrum of a fibonacci chain is obtained using a simple non linear map this map exhibits intermittency for all fibonacci number iterates of the function whenever the frequency lies in a gap of the spectrum the self similar nature of the eigenfunctions is shown to be a consequence of the intermittency behaviour certain simple patterns that emerge in the gap labelling theorem are discussed

renormalization group approach to random walks on disordered lattices one dimensional random walks with static disorder are analyzed using a real space renormalization group procedure the presence of disorder leads to a non markovian description of the macroscopic behavior of the random walk we consider random walks with nearest neighbor hopping described by a master equation with both on site and site to site disorder in the transition matrix site to site disorder leads to a generalized diffusion coefficient with at 3 2 long time tail whereas on site disorder leads to a generalized burnett coefficient with at 1 2 long time tail

diffusion on a quasiperiodic chain an exact solution to the problem of diffusion on a quasiperiodic chain is provided by a decimation method involving scaling by an irrational length factor

intrinsic randomness of transport coefficient in subdiffusion with static disorder fluctuations in the time averaged mean square displacement for random walks on hypercubic lattices with static disorder are investigated it is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking for two and higher dimensional systems the distribution function of the diffusion coefficient is found to be the mittag leffler distribution which is the same as for the continuous time random walk whereas for one dimensional systems a different distribution a modified mittag leffler distribution arises we also present a comparison of these two distributions in terms of an ergodicity breaking parameter and show that the modified mittag leffler distribution has a larger deviation from ergodicity some remarks on similarities between these results and observations in biological experiments are presented

bockstein closed 2 group extensions and cohomology of quadratic maps abstract a central extension of the form e 0 v g w 0 where v and w are elementary abelian 2 groups is called bockstein closed if the components q i h w f 2 of the extension class of e generate an ideal which is closed under the bockstein operator in this paper we study the cohomology ring of g when e is a bockstein closed 2 power exact extension the mod 2 cohomology ring of g has a simple form and it is easy to calculate the main result of the paper is the calculation of the bocksteins of the generators of the mod 2 cohomology ring using an eilenberg moore spectral sequence we also find an interpretation of the second page of the bockstein spectral sequence in terms of a new cohomology theory that we define for bockstein closed quadratic maps q w v associated to the extensions e of the above form

asymptotic and oscillatory behavior of solutions of certain second order nonlinear difference equations mingshu peng department of applied mathematics beijing institute of technology beijing 100081 p r china qianli xu department of mathematics yiyang teachers college yiyang 413049 p r china lihong huang department of applied mathematics hunan university changsha 410082 p r china weigao ge

long time tails in stationary random media ii applications in a previous paper we developed a mode coupling theory to describe the long time properties of diffusion in stationary statistically homogeneous random media here the general theory is applied to deterministic and stochastic lorentz models and several hopping models the mode coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse grained diffusion coefficient and a coarse grained free volume for one dimensional models these amplitudes can be evaluated and the mode coupling theory is shown to agree with exact solutions obtained for these models for higher dimensional lorentz models the formal theory yields expressions which are difficult to evaluate for these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

random walks on site disordered lattices random walk models with disordered waiting times site disorder are analysed using a real space renormalisation group method anomalous diffusion is found in all dimensions whenever the first moment of the distribution of waiting times diverges above two dimensions the fixed points are described by continuous time random walks with non analytic waiting time functions below two dimensions the fixed points are themselves disordered random walks where the disorder is characterised by a stable probability distribution a fractal dimension is defined for site disordered lattices and below two dimensions a hyperscaling law is found which relates the fractal dimension the euclidean dimension and the exponent describing the displacement of the walker

on a non linear fluctuation theorem for the ageing dynamics of disordered trap models we consider the dynamics of the disordered trap model which is known to be completely out of equilibrium and to present strong localization effects in its ageing phase we are interested in the influence of an external force when it is applied from the very beginning at t 0 or only after a waiting time tw we obtain a non linear fluctuation theorem for the corresponding one time and two time diffusion fronts in any given sample which implies the following consequences i for fixed times there exists a linear response regime where the fluctuation dissipation relation or einstein relation is valid even in the ageing time sector in contrast with other ageing disordered systems ii for a fixed waiting time and fixed external field the validity of the linear response regime is limited in time by a characteristic time depending on the external force iii in the non linear response regime there exists a very simple relation for the asymmetry in the position for the one time and two times disorder averaged diffusion fronts in the presence of an external force in contrast to other models of random walks in random media the present non linear fluctuation theorem is a consequence of a special dynamical symmetry of the trap model

cohomology of uniformly powerful p groups in this paper we will study the cohomology of a family of p groups associated to fp lie algebras more precisely we study a category bgrp of p groups which will be equivalent to the category of fp bracket algebras lie algebras minus the jacobi identity we then show that for a group g in this category its fp cohomology is that of an elementary abelian p group if and only if it is associated to a lie algebra we then proceed to study the exponent of h g z in the case that g is as sociated to a lie algebra l to do this we use the bockstein spectral sequence and derive a formula that gives b 2 in terms of the lie algebra cohomologies of l we then expand some of these results to a wider category of p groups in particular we calculate the cohomology of the p groups n k which are defined to be the kernel of the mod p reduction gln z p k 1 z mod gln fp

diffusion in a one dimensional lattice with random asymmetric transition rates the authors study one dimensional continuous time random walks for which the pairs wn wn 1 of nearest neighbour transition rates are assumed to be independent equally distributed random variables the long time asymptotic behaviour of the mean displacement x t is determined exactly for a specific model system in which diodes u 0 and two way bonds lambda v v occur with probabilities p and 1 p respectively for lambda 1 p we find that x t approximately tnu f beta 1ln t where nu ln 1 p ln lambda and beta ln lambda and where f is a periodic function with period 1 the mean displacement thus not only increases slower than linearly in time but exhibits superimposed non decaying oscillations

oscillations in the tsallis income distribution abstract oscillations in the complementary cumulative distribution function ccdf of individual income data have been found in the data of various countries studied by different authors at different time periods but the dynamical origins of this behavior are currently unknown although these datasets can be fitted by different functions at different income ranges the tsallis distribution has recently been found capable of fitting the whole distribution by means of only two parameters this procedure showed clearly such oscillatory feature in the entire income range feature but made it particularly visible at the tail of the distribution although log periodic functions fitted to the data are capable of describing this behavior a different approach to naturally disclose such oscillatory characteristics is to allow the tsallis q parameter to become complex in this paper we use this idea in order to describe the behavior of the ccdf of the brazilian personal income recently studied empirically by soares et al 2016 typical elements of periodic motion such as amplitude and angular frequency coupled to this income analysis were obtained by means of this approach a highly non linear function for the ccdf was obtained through this methodology and a numerical test showed it capable of recovering the main oscillatory feature of the original ccdf of the personal income data of brazil

cohomology of metacyclic groups group n by a finite cyclic group k using homological perturbation theory we introduce the beginning of a free resolution of the integers z over the group ring zg of g in such a way that the resolution reflects the structure of g as an extension of n by k and we use this resolution to compute the additive structure of the integral cohomology of g in many cases we proceed by first establishing a number of special cases thereafter constructing suitable cohomology classes thereby obtaining a lower bound then computing characteristic classes introduced in an earlier paper and finally exploiting these classes obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension e the calculation is then completed by comparing the two bounds

space and time dimensions of algebras with application to lorentzian noncommutative geometry and quantum electrodynamics an analogy with real clifford algebras on even dimensional vector spaces suggests assigning an ordered pair s t of space and time dimensions or equivalently an ordered pair m n of metric and ko dimensions modulo 8 to any algebraic structure that we call cpt corepresentation represented over a hilbert space by two self adjoint involutions and an anti unitary operator having specific commutation relations it is shown that this assignment is compatible with the tensor product the space and time dimensions of the tensor product of two cpt corepresentations are the sums of the space and time dimensions of its factors and the same holds for the metric and ko dimensions this could provide an interpretation of the presence of such algebras in pt symmetric hamiltonians or the description of topological matter this construction is used to build an indefinite i e pseudo riemannian version of the spectral triple of noncommutative geometry defined over a krein space and classified by the pair m n instead of the ko dimension only within this framework we can express the lagrangian both bosonic and fermionic of a lorentzian almost commutative spectral triple we exhibit a space of physical states that solves the fermion doubling problem the example of quantum electrodynamics is described an analogy with real clifford algebras on even dimensional vector spaces suggests assigning an ordered pair s t of space and time dimensions or equivalently an ordered pair m n of metric and ko dimensions modulo 8 to any algebraic structure that we call cpt corepresentation represented over a hilbert space by two self adjoint involutions and an anti unitary operator having specific commutation relations it is shown that this assignment is compatible with the tensor product the space and time dimensions of the tensor product of two cpt corepresentations are the sums of the space and time dimensions of its factors and the same holds for the metric and ko dimensions this could provide an interpretation of the presence of such algebras in pt symmetric hamiltonians or the description of topological matter this construction is used to build an indefinite i e pseudo riemannian version of the spectral triple of noncommutative geometry defined over a krein space and classified by the pair m n

random walk in dynamically disordered chains poisson white noise disorder exact solutions are given for a variety of models of random walks in a chain with time dependent disorder dynamic disorder is modeled by white poisson noise models with site independent global and site dependent local disorder are considered results are described in terms of an affective random walk in a nondisordered medium in the cases of global disorder the effective random walk contains multistep transitions so that the continuous limit is not a diffusion process in the cases of local disorder the effective process is equivalent to usual random walk in the absence of disorder but with slower diffusion difficulties associated with the continuous limit representation of random walk in a disordered chain are discussed in particular we consider explicit cases in which taking the continuous limit and averaging over disorder sources do not commute

on birth and death processes in symmetric random environment we prove a limit theorem for a process in a random one dimensional medium which has been considered before as a model for hopping conduction in a disordered medium to the edge between the two integersj and j 1 a rate j 0 is attached these j j integral are taken as independent identically distributed random variables and represent the medium for given values j x t is a markov chain in continuous time which jumps fromj to j 1 and from j 1 toj at the same rate j we show that in many cases there exists normalizing constants y t which tend to oo witht such that the distribution of x t t or more generally of the whole processx st t s 0 converges to a limit as t the limit process is continuous and self similar

linear and quadratic ranges in representation stability abstract we prove two general results concerning spectral sequences of fi modules these results can be used to significantly improve stable ranges in a large portion of the stability theorems for fi modules currently in the literature we work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range previously the best stable ranges known in these examples were exponential up to an additive constant our work on congruence subgroups verifies a conjecture of djament

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

a class of sturm liouville difference equations non oscillation constants and property bd abstract in this paper we study oscillatory properties of the higher order sturm liouville difference equation 1 n n k n yk q k y k n r in particular we derive sufficient conditions for non oscillation of and compute explicitly its non oscillation constants the obtained results are used to derive necessary and sufficient conditions for discreteness and boundedness below the spectrum of associated difference operators we discuss also non oscillation of the euler type difference equation

diffusion in random one dimensional systems diffusion on the one dimensional lattice is described by a master equation with nearest neighbor transfer rates symmetric or asymmetric the transfer rates associated with bonds are assumed to be independent equally distributed random variables under various conditions on their common distribution the large time behavior of averaged site probabilities and or related quantities is exhibited

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

diffusion in layered random flows polymers electrons in random potentials and spin depolarization in random fields several related models are studied in a common framework we first reconsider the model of matheron and de marsilly for anomalous tracer dispersion in a stratified porous medium in each horizontal layer the flow velocity is constant parallel to the layer and depends randomly on the vertical coordinate z this model is mapped onto ad 1 localization problem in a random potential and equivalently onto ad 1 polymer at larget theaveraged distribution of horizontal displacementsx takes the scaling form p x t z 0 at 5 4q bxt 3 4 whereq y is independent of the details of the model q y a andb are obtained exactly for a large class of models from the lifschitz tails of the localization problem we find in the regionx t3 4 i e y thatq y y exp c y 4 3 we also obtain exactly ind 1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field by mapping onto a polymer problem as well as the average local concentration for diffusion in the presence of random sources and sinks these mappings are then used to study higher dimensional extensions of these models

oscillation theorems for second order neutral difference equations abstract oscillation criteria for the second order difference equations of the form 2xn qnf xn r 0 and the neutral difference equation 2 xn pxn k qnf xn r 0 are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn is a sequence of nonnegtive real numbers and f r r is continuous

two sided bounds on minimum error quantum measurement on the reversibility of quantum dynamics and on maximum overlap using directional iterates in a unified framework we estimate the following quantities of interest in quantum information theory 1 the minimum error distinguishability of arbitrary ensembles of mixed quantum states 2 the approximate reversibility of quantum dynamics in terms of entanglement fidelity this is referred to as channel adapted quantum error recovery when applied to the composition of an encoding operation and a noise channel 3 the maximum overlap between a bipartite pure quantum state and a bipartite mixed state that may be achieved by applying a local quantum operation to one part of the mixed state and 4 the conditional min entropy of bipartite quantum states a refined version of the author s techniques j tyson j math phys 50 032016 2009 for bounding the first quantity is employed to give two sided estimates of the remaining three quantities we obtain a closed form approximate reversal channel using a state dependent kraus decomposition our reversal may be interpreted as a quadratically

corrections to scaling for diffusion in disordered media the authors study the diffusion of a particle in a d dimensional lattice where disorder arises from a random distribution of waiting times associated with each site of the lattice using scaling arguments they derive in addition to the leading asymptotic behaviour the correction to scaling terms for the mean square displacement they also perform detailed monte carlo simulations for one two and three dimensions which give results in substantial agreement with the scaling argument predictions

monte carlo approach to dendritic growth the connection between diffusion limited aggregation and the equations of dendritic growth is critically examined a different type of monte carlo simulation is proposed and used to construct two dimensional dendrite like patterns the wavelength occurring for short times is in good agreement with the linear stability analysis the time dependence of the characteristic wavelengths is also determined

discrete spectra criteria for certain classes of singular differential and difference operators in this paper we consider singular differential and difference operators of the form m y 1 n w t r t y n n and b y k n 1 n w k delta n r k delta n y k as well as their full term analogies we establish criteria for discreteness and boudnedness bellow of their spectrum property bd difference operators are studied as the discrete counterparts of the continuous ones

real space renormalisation group method for the dilute gaussian model and its relation to the random walk problems near the percolation threshold an isomorphism between the gaussian model in the magnetic field and the time dependent conduction diffusion problem permits the author to develop a real space renormalisation group method for the diluted version of this model by which all single cluster results of the phenomenological scaling theory of gefen aharony and alexander 1983 are correctly reproduced an extension of this model to the case of strong bias then provides some restrictions on the validity of their results the result for the density of states which the author reproduces in the form predicted by alexander and orbach 1982 is subject to the same kind of restrictions

subdiffusion and localization in the one dimensional trap model we study a one dimensional generalization of the exponential trap model using both numerical simulations and analytical approximations we obtain the asymptotic shape of the average diffusion front in the subdiffusive phase our central result concerns the localization properties we find the dynamical participation ratios to be finite but different from their equilibrium counterparts therefore the idea of a partial equilibrium within the limited region of space explored by the walk is not exact even for long times where each site is visited a very large number of times we discuss the physical origin of this discrepancy and characterize the full distribution of dynamical weights we also study two different two time correlation functions which exhibit different aging properties one is sub aging whereas the other one shows full aging therefore two diverging time scales appear in this model we give intuitive arguments and simple analytical approximations that account for these differences and obtain new predictions for the asymptotic short time and long time behavior of the scaling functions finally we discuss the issue of multiple time scalings in this model

dynamical properties of quasi crystals fibonacci chain and penrose lattice the discrete scale invariance of quasi periodic systems leads to a scaling relationship omega l zf l between a characteristic dynamic variable omega and length l where f l is a periodic function of ln l the dynamic exponent z can be calculated for diffusion spin wave and phonon dynamics where omega is respectively gamma omega omega 2 where gamma and omega are characteristic rates or frequencies respectively by exploiting a crossover argument which results in z df t where df is the fractal dimension and t is the length scaling exponent for the conductance the latter exponents and hence z are calculated for a fibonacci chain model df t z 1 1 2 and via an exact bond moving technique for the penrose lattice df t z 2 0 2 the technique also provides the thermal exponent of the heisenberg spin model on the two quasi crystals nu 1 infinity respectively

systematic density expansion of transport properties of random lattices abstract using kinetic theory methods we develop systematic expansions in powers of the concentration of impurities scatterers super non conducting bonds sites and apply them to calculate explicity to linear and quadratic order in the concentration moments of displacement velodity autocorrelation functions etc of random walks on square and simple cubic lattices with site or bond disorder lattice lorentz gas ant and termite models site and bond percolation special attention is given to the velocity autocorrelation function cage effect long time tail and its comparison with results of computer simulations

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

anomalous diffusion in stochastic systems with nonhomogeneously distributed traps the stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed the particle is subjected to a stochastic stimulation obeying a general levy stable statistics and experiences long rests due to nonhomogeneously distributed traps the memory is taken into account by subordination of that process to a random time then the subordination equation is position dependent the problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power law position dependence of the memory in the case of the gaussian statistics the density distribution and moments are derived depending on geometry and memory parameters the system may reveal both the subdiffusion and enhanced diffusion the similar analysis is performed for the levy flights where the finiteness of the variance follows from a variable noise intensity near a boundary two diffusion regimes are found in the bulk and near the surface the anomalous diffusion exponent as a function of the system parameters is derived

equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings the two main techniques for the generation of quasiperiodic tilings de bruijn s grid method 1981 and the projection formalism are generalised a very broad class of quasiperiodic tilings is obtained in this way the two generalised methods are shown to be equivalent the standard calculation of fourier spectra is extended to the whole general class of tilings various applications are discussed

crossover dynamics in bond disordered lattices the asymptotic dynamics of the percolation model for a bond disordered lattice is studied the velocity autocorrelation function vacf is investigated for arbitrary concentration of disorder in two and three dimensions using an effective medium approximation ema corrections to the long time tails away from the percolation threshold and to the percolation tails at the threshold are calculated a characteristic time scale for the long time tails is identified and found to diverge at the threshold sufficiently close to the threshold the two types of asymptotic dynamics can be identified clearly for times greater than and less than this characteristic time respectively an approximate scaling of the ema equation is obtained near the threshold for investigation of the crossover region more generally the ema equation is solved numerically for arbitrary concentration in two dimensions to exhibit the complete time dependence of the vacf in all domains near and far from the threshold

a simple derivation of quasi crystalline spectra a quasi crystalline solid is an aperiodic array of atoms which gives rise to bragg like diffraction peaks using the projection method for generating such structures the authors provide a simple derivation of this paradoxical behaviour

unbounded positive solutions of higher order difference equations with singular nonlinear term in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form mxn 1 f n xn 0 nen0 where m 2 1 or 1 and f n0 r r is a real valued continuous function such that f n x 0 for all n x n0 0 where n0 0 1

scaling treatment of anisotropic diffusion exact scaling treatments are given of anisotropic diffusion in pure and diluted chains and in a fractal a dynamic decimation method is used to obtain detailed dynamic scaling descriptions the results for chains show crossover to drift or localised behaviour induced by anisotropy or dilution the fractal results include bias induced crossover with exponent unity from anomalous diffusion behaviour exponent log2 5 to drift behaviour and scaling corrections exponent log2 5 3 arising from rotational anisotropy

strongly disordered chain of impedances theoretical analysis and numerical results the authors study the input impedance of a chain of random resistors chosen according to a broad distribution i e decaying as a power law for large arguments as a function of frequency the modulus of the input impedance varies as a power law while its phase has a well defined value a simple theoretical analysis based on estimations of sums of random variables yields the value of the exponents which are in perfect agreement with numerical simulations even when logarithmic corrections come into play the analysis can be extended further to investigate the sample to sample fluctuations of the impedance again agreement with simulations is very good

anomalous diffusion on fractal lattices with site disorder like random walks on euclidean lattices random walks on fractal lattices are modified by a waiting time site disorder when the first moment of the waiting time distribution diverges it is shown that for lattices which support recurrent walks spectral dimension smaller than two the inverse of the diffusion exponent in the presence of disorder is increased by the difference between the waiting time fractal dimension and the usual fractal dimension this hyperscaling relation is derived for sierpinski gaskets in arbitrary dimension with scale dependent waiting times this provides qualitative insight into this problem and the exponent relation derived should also hold for statistically self similar structure such as percolation clusters for lattices whose usual spectral dimension is larger than two a mean field result holds

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

primitives and central detection numbers in group cohomology abstract fix a prime p given a finite group g let h g denote its mod p cohomology in the early 1990s henn lannes and schwartz introduced two invariants d 0 g and d 1 g of h g viewed as a module over the mod p steenrod algebra they showed that in a precise sense h g is respectively detected and determined by h d c g v for d d 0 g and d d 1 g with v running through the elementary abelian p subgroups of g the main goal of this paper is to study how to calculate these invariants we find that a critical role is played by the image of the restriction of h g to h c where c is the maximal central elementary abelian p subgroup of g a measure of this is the top degree e g of the finite dimensional hopf algebra h c h g f p a number that tends to be quite easy to calculate our results are complete when g has a p sylow subgroup p in which every element of order p is central using the benson carlson duality we show that in this case d 0 g d 0 p e p and a similar exact formula holds for d 1 as a bonus we learn that h e g p contains nontrivial essential cohomology reproving and sharpening a theorem of adem and karagueuzian in general we are able to show that d 0 g max e c g v v g if certain cases of benson s regularity conjecture hold in particular this inequality holds for all groups such that the difference between the p rank of g and the depth of h g is at most 2 when we look at examples with p 2 we learn that d 0 g 14 for all groups with 2 sylow subgroup of order up to 64 with equality realized when g su 3 4 en route we study two objects of independent interest if c is any central elementary abelian p subgroup of g then h g is an h c comodule and we prove that the subalgebra of h c primitives is always noetherian of krull dimension equal to the p rank of g minus the p rank of c if the depth of h g equals the rank of z g we show that the depth essential cohomology of g is nonzero reproving and extending a theorem of green and cohen macauley in a certain sense and prove related structural results

nonoscillation criteria for half linear second order difference equations abstract nonoscillation criteria for the half linear second order difference equation are established these criteria are derived using the riccati and variational technique