coevolutionary dynamics on scale free networks we investigate bak sneppen coevolution models on scale free networks with various degree exponents ensuremath gamma including random networks for ensuremath gamma g3 the critical fitness value f c approaches a nonzero finite value in the limit n ensuremath rightarrow ensuremath infty whereas f c approaches zero as 2l ensuremath gamma ensuremath leqslant 3 these results are explained by showing analytically f c n ensuremath simeq a k 1 2 n on the networks with size n the avalanche size distribution p s shows the normal power law behavior for ensuremath gamma g3 in contrast p s for 2l ensuremath gamma ensuremath leqslant 3 has two power law regimes one is a short regime for small s with a large exponent ensuremath tau 1 and the other is a long regime for large s with a small exponent ensuremath tau 2 phantom rule 0 3em 0ex ensuremath tau 1 g ensuremath tau 2 the origin of the two power regimes is explained by the dynamics on an artificially made star linked network

sur les ideaux induits dans les algebres enveloppantes soient mathfrak g une algebre de lie de dimension finie et mathfrak h une sousalgebre de lie de mathfrak g je montre que l induction qui envoie les ideaux de l algebre enveloppante de mathfrak h dans les ideaux de l algebre enveloppante de mathfrak g commute a une torsion pres a l anti automorphisme principal la resolution libre standard de la representation triviale d une algebre de lie joue un role important dans la demonstration

analysis of the block error probability of concatenated polar code ensembles in this paper we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check crc codes separated by an interleaver in the short and moderate block length regimes the analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding the analysis is carried out by introducing the concatenated polar code cpc ensembles whose distance properties can be analyzed for sufficiently short block lengths by means of the uniform interleaver approach at moderate block lengths we resort to the monte carlo simulations results show that if the inner polar code possesses a low minimum distance and the outer crc code has a sufficiently large amount of redundancy then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting cpc

an inverse image functor for lie algebroids abstract extending material from the theory of d modules to lie algebroids see also s chemla a duality property for complex lie algebroids math z 1999 367 388 we introduce an inverse image functor and show that it preserves coherence under appropriate circumstances as in the case of d modules a suitable non characteristicity notion enables us to give a sufficient condition in order for the duality functor and the inverse image functor to commute this generalizes a result for d modules due to kashiwara kawai and sato but even in the case of d modules our proof is different from theirs in particular we obtain a new duality formula for complexes of modules over an ordinary lie algebra and as a special case we get a new adjunction formula for modules over lie algebras moreover our result will shed some light on the behaviour of poisson cohomology a notion introduced by lichnerowitz under a poisson map in the analytic case

rank driven markov processes we study a class of markovian systems of n elements taking values in 0 1 that evolve in discrete time t via randomized replacement rules based on the ranks of the elements these rank driven processes are inspired by variants of the bak sneppen model of evolution in which the system represents an evolutionary fitness landscape and which is famous as a simple model displaying self organized criticality our main results are concerned with long time large n asymptotics for the general model in which at each time step k randomly chosen elements are discarded and replaced by independent u 0 1 variables where the ranks of the elements to be replaced are chosen independently at each time step according to a distribution n on 1 2 n k our main results are that under appropriate conditions on n the system exhibits threshold behavior at s 0 1 where s is a function of n and the marginal distribution of a randomly selected element converges to u s 1 as t and n of this class of models results in the literature have previously been given for special cases only namely the mean field or random neighbor bak sneppen model our proofs avoid the heuristic arguments of some of the previous work and use foster lyapunov ideas our results extend existing results and establish their natural more general context we derive some more specialized results for the particular case where k 2 one of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic markov chains on increasing state spaces which may be of independent interest

a duality approach to the symmetry of bernstein sato polynomials of free divisors abstract in this paper we prove that the bernstein sato polynomial of any free divisor for which the d s module d s h s admits a spencer logarithmic resolution satisfies the symmetry property b s 2 b s this applies in particular to locally quasi homogeneous free divisors for instance to free hyperplane arrangements or more generally to free divisors of linear jacobian type we also prove that the bernstein sato polynomial of an integrable logarithmic connection e and of its dual e with respect to a free divisor of linear jacobian type are related by the equality b e s b e s 2 our results are based on the behaviour of the modules d s h s and d s e s h s under duality

projectively simple rings abstract an infinite dimensional n graded k algebra a is called projectively simple if dim k a i for every nonzero two sided ideal i a we show that if a projectively simple ring a is strongly noetherian is generated in degree 1 and has a point module then a is equal in large degree to a twisted homogeneous coordinate ring b b x l here x is a smooth projective variety is an automorphism of x with no proper invariant subvariety we call such automorphisms wild and l is a ample line bundle we conjecture that if x admits a wild automorphism then every irreducible component of x is an abelian variety we prove several results in support of this conjecture in particular we show that the conjecture is true if dim x 2 in the case where x is an abelian variety we describe all wild automorphisms of x finally we show that if a is projectively simple and admits a balanced dualizing complex then proj a is cohen macaulay and gorenstein

interleaved polar i polar codes by inserting interleavers between intermediate stages of the polar encoder a new class of polar codes termed interleaved polar i polar codes is proposed by the uniform interleaver assumption we derive the weight enumerating function wef and the input output weight enumerating function iowef averaged over the ensemble of i polar codes the average wef can be used to calculate the upper bound on the average block error rate bler of a code selected at random from the ensemble of i polar codes also we propose a concatenated coding scheme that employs p high rate codes as the outer code and q i polar codes as the inner code with an interleaver in between the average wef of the concatenated code is derived based on the uniform interleaver assumption simulation results show that bler upper bounds can well predict bler performance levels of the concatenated codes the results show that the performance of the proposed concatenated code with p q 2 is better than that of the crc aided i polar code with p q 1 of the same length and code rate at high signal to noise ratios snrs moreover the proposed concatenated code allows multiple decoders to operate in parallel which can reduce the decoding latency and hence is suitable for ultra reliable low latency communications urllc

non triviality of a discrete bak sneppen evolution model consider the following evolution model proposed in ref 1 by bak and sneppen put n vertices on a circle spaced evenly each vertex represents a certain species we associate with each vertex a random variable representing the state or fitness of the species with values in 0 1 the dynamics proceeds as follows every discrete time step we choose the vertex with minimal fitness and assign to this vertex and to its two neighbours three new independent fitnesses with a uniform distribution on 0 1 a conjecture of physicists based on simulations is that in the stationary regime the one dimensional marginal distributions of the fitnesses converges when n to a uniform distribution on f 1 for some threshold f 1 in this paper we consider a discrete version of this model proposed in ref 2 in this discrete version the fitness of a vertex can be either 0 or 1 the system evolves according to the following rules each discrete time step we choose an arbitrary vertex with fitness 0 if all the vertices have fitness 1 then we choose an arbitrary vertex with fitness 1 then we update the fitnesses of this vertex and of its two neighbours by three new independent fitnesses taking value 0 with probability 0 q 1 and 1 with probability p 1 q we show that if q is close enough to one then the mean average fitness in the stationary regime is bounded away from 1 uniformly in the number of vertices this is a small step in the direction of the conjecture mentioned above and also settles a conjecture mentioned in ref 2 our proof is based on a reduction to a continuous time particle system

fluctuations and correlations in an individual based model of biological coevolution we extend our study of a simple model of biological coevolution to its statistical properties starting with a complete description in terms of a master equation we provide its relation to the deterministic evolution equations used in previous investigations the stationary states of the mutationless model are generally well approximated by gaussian distributions so that the fluctuations and correlations of the populations can be computed analytically several specific cases are studied by monte carlo simulations and there is an excellent agreement between the data and the theoretical predictions

network growth with preferential attachment for high indegree and low outdegree abstract we study the growth of a directed transportation network such as a food web in which links carry resources we propose a growth process in which new nodes or species preferentially attach to existing nodes with high indegree in food web language number of prey and low outdegree or number of predators this scheme which we call inverse preferential attachment is intended to maximize the amount of resources available to each new node we show that the outdegree predator distribution decays at least exponentially fast for large outdegree and is continuously tunable between an exponential distribution and a delta function the indegree prey distribution is poissonian in the large network limit

stochastic model for the species abundance problem in an ecological community we propose a model based on coupled multiplicative stochastic processes to understand the dynamics of competing species in an ecosystem this process can be conveniently described by a fokker planck equation we provide an analytical expression for the marginalized stationary distribution our solution is found in excellent agreement with numerical simulations and compares rather well with observational data from tropical forests

rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators abstract in the first part of this article we compute the rigid dualizing complex of a quantum enveloping algebra we consider the generic case and the case of a specialization at a non root of unity this answers a question of yekutieli j pure appl algebra 150 2000 85 in bull soc math france 122 1994 371 and math z 232 1999 367 we generalized d module theory to lie algebroids using these results we compute explicitly the rigid dualizing complex of the algebra of differential operators defined by an affine lie algebroid this generalizes results of yekutieli j pure appl algebra 150 2000 85

construction of polar codes for arbitrary discrete memoryless channels it is known that polar codes can be efficiently constructed for binary input channels at the same time existing algorithms for general input alphabets are less practical because of high complexity we address the construction problem for the general case and analyze an algorithm that is based on successive reduction of the output alphabet size of the subchannels in each recursion step for this procedure we estimate the approximation error as o mu 1 q 1 where q is the input alphabet size and mu is the quantization parameter i e the maximum size of the subchannel output alphabet allowed by the algorithm the complexity of the code construction scales as o n mu 2 log mu where n is the length of the code we also show that if the polarizing operation relies on modulo q addition it is possible to merge subsets of output symbols without any loss in subchannel capacity performing this procedure before each approximation step results in a further speed up of the code construction and the resulting codes have smaller gap to capacity we also show that a similar acceleration can be attained for polar codes over finite field alphabets experimentation shows that the suggested construction algorithms can be used to construct long polar codes for alphabets of size q 16 and more with acceptable loss of the code rate for a variety of polarizing transforms

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

weighted scale free networks in euclidean space using local selection rule received 23 july 2005 revised manuscript received 21 july 2006 published 22 september 2006a spatial scale free network is introduced and studied whose motivation originated in the growing internetas well as airport networks we argue that in these real world networks a new node necessarily selects one ofits neighboring local nodes for connection and is not controlled by preferential attachment as in the barabasi albert ba model this observation is mimicked in our model where the nodes pop up at randomly locatedpositions in the euclidean space and are connected to one end of the nearest link in spite of this crucialdifference it is observed that the leading behavior of our network is like that of the ba model de ning the linkweight as an algebraic power of its euclidean length the weight distribution and the nonlinear dependence ofthe nodal strength on the degree are analytically calculated it is claimed that a power law decay of the linkweights with time ensures such nonlinear behavior switching off the euclidean space from the same modelyields a much simpler de nition of the ba model where numerical effort grows linearly with

relaxed polar codes polar codes are the latest breakthrough in coding theory as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary input discrete memoryless channels polar encoding and successive cancellation decoding have the complexities of n log n for code length n although the complexity bound of n log n is asymptotically favorable we report in this work methods to further reduce the encoding and decoding complexities of polar coding the crux is to relax the polarization of certain bit channels without performance degradation we consider schemes for relaxing the polarization of both very good and very bad bit channels in the process of channel polarization relaxed polar codes are proved to preserve the capacity achieving property of polar codes analytical bounds on the asymptotic and finite length complexity reduction attainable by relaxed polarization are derived for binary erasure channels we show that the computation complexity can be reduced by a factor of six while preserving the rate and error performance we also show that relaxed polar codes can be decoded with significantly reduced latency for additive white gaussian noise channels with medium code lengths we show that relaxed polar codes can have lower error probabilities than conventional polar codes while having reduced encoding and decoding computation complexities

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

performance limits and practical decoding of interleaved reed solomon polar concatenated codes a scheme for concatenating the recently invented polar codes with non binary mds codes as reed solomon codes is considered by concatenating binary polar codes with interleaved reed solomon codes we prove that the proposed concatenation scheme captures the capacity achieving property of polar codes while having a significantly better error decay rate we show that for any e 0 and total frame length n the parameters of the scheme can be set such that the frame error probability is less than 2 n 1 e while the scheme is still capacity achieving this improves upon 2 n 0 5 e the frame error probability of arikan s polar codes the proposed concatenated polar codes and arikan s polar codes are also compared for transmission over channels with erasure bursts we provide a sufficient condition on the length of erasure burst which guarantees failure of the polar decoder on the other hand it is shown that the parameters of the concatenated polar code can be set in such a way that the capacity achieving properties of polar codes are preserved we also propose decoding algorithms for concatenated polar codes which significantly improve the error rate performance at finite block lengths while preserving the low decoding complexity

multilevel codes theoretical concepts and practical design rules this paper deals with 2 sup l ary transmission using multilevel coding mlc and multistage decoding msd the known result that mlc and msd suffice to approach capacity if the rates at each level are appropriately chosen is reviewed using multiuser information theory it is shown that there is a large space of rate combinations such that mlc and full maximum likelihood decoding mld can approach capacity it is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of mld the capacity rule the balanced distances rules and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies moreover topics relevant in practical applications such as signal set labeling dimensionality of the constituent constellation and hard decision decoding are emphasized bit interleaved coded modulation proposed by caire et al see ibid vol 44 p 927 46 1998 is reviewed in the context of mlc finally the combination of signal shaping and coding is discussed significant shaping gains are achievable in practice only if these design rules are taken into account

randomized polar subcodes with optimized error coefficient a method for construction of polar subcodes with reduced error coefficient is presented the proposed approach relies on explicit enumeration of low weight non zero codewords in a polar code and construction of dynamic freezing constraints which define a subcode not containing most of these codewords the obtained codes provide a large performance gain in the high snr region compared to non optimized polar subcodes and polar codes with crc

probabilistic approach to the bak sneppen model we study here the bak sneppen model a prototype model for the study of self organized criticality in this model several species interact and undergo extinction with a power law distribution of activity bursts species are defined through their fitness whose distribution in the system is uniform above a certain threshold run time statistics is introduced for the analysis of the dynamics in order to explain the peculiar properties of the model this approach based on conditional probability theory takes into account the correlations due to memory effects in this way we may compute analytically the value of the fitness threshold with the desired precision this represents a substantial improvement with respect to the traditional mean field approach

on a local version of the bak sneppen model a major difficulty in studying the bak sneppen model is in effectively comparing it with well understood models this stems from the use of two geometries complete graph geometry to locate the global fitness minimizer and graph geometry to replace the species in the neighborhood of the minimizer over the years a number of models inspired by bak sneppen were studied usually by introducing different or new features e g discretizing fitness randomized neighbors or population size we present a variant that only uses features present in bak sneppen and whose difference from the bak sneppen is that only the graph geometry is used for the evolution this allows to obtain the stationary distribution through random walk dynamics while preserving the geometric nature of the model we use this to show that for constant degree graphs the stationary fitness distribution converges to an iid law as the number of vertices tends to infinity we also discuss exponential ergodicity through coupling and avalanches for the model

shortened polar codes an optimization algorithm for finding a shortening pattern and a set of frozen symbols for polar codes is proposed the structure of polar codes is exploited to eliminate many equivalent shortening patterns thus reducing the search space a reduced complexity suboptimal algorithm is proposed for finding shortening patterns for long polar codes shortened codes obtained with the proposed method are shown to outperform low density parity check ldpc codes

weight distributions for successive cancellation decoding of polar codes in this paper we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes though we do not compute the distance spectrum of polar codes the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels also we establish the minimal distance between two cosets associated with two paths that differ in two positions this can be regarded as a first step toward analyzing the weight distributions for successive cancellation list decoding

on distance properties of convolutional polar codes a lower bound on the minimum distance of convolutional polar codes is provided the bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix moreover a construction of convolutional polar subcodes is proposed which provides improved performance under successive cancellation list decoding for sufficiently large list size the decoding complexity of convolutional polar subcodes appears to be lower compared with arikan polar subcodes with the same performance the error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of arikan polar subcodes with the same list size

extremal dynamics on complex networks analytic solutions the bak sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks by using the rate equation and the random walk approaches we obtain the analytic solution of the fitness threshold x c to be 1 k f 1 where k f k 2 k k in the quenched annealed updating case where k n is the n text th moment of the degree distribution thus the threshold is zero finite for the degree exponent ensuremath gamma l3 ensuremath gamma g3 for the quenched case in the thermodynamic limit the theoretical value x c fits well to the numerical simulation data in the annealed case only avalanche size defined as the duration of successive mutations below the threshold exhibits a critical behavior as its distribution follows a power law p a s ensuremath sim s ensuremath 3 2

how to construct polar codes a method for efficiently constructing polar codes is presented and analyzed although polar codes are explicitly defined straightforward construction is intractable since the resulting polar bit channels have an output alphabet that grows exponentially with the code length thus the core problem that needs to be solved is that of faithfully approximating a bit channel with an intractably large alphabet by another channel having a manageable alphabet size we devise two approximation methods which sandwich the original bit channel between a degraded and an upgraded version thereof both approximations can be efficiently computed and turn out to be extremely close in practice we also provide theoretical analysis of our construction algorithms proving that for any fixed e 0 and all sufficiently large code lengths n polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n

on rank driven dynamical systems we investigate a class of models related to the bak sneppen bs model initially proposed to study evolution the bs model is extremely simple and yet captures some forms of complex behavior such as self organized criticality that is often observed in physical and biological systems in this model random fitnesses in 0 1 are associated to agents located at the vertices of a graph g their fitnesses are ranked from worst 0 to best 1 at every time step the agent with the worst fitness and some others with a priori given rank probabilities are replaced by new agents with random fitnesses we consider two cases the exogenous case where the new fitnesses are taken from an a priori fixed distribution and the endogenous case where the new fitnesses are taken from the current distribution as it evolves we approximate the dynamics by making a simplifying independence assumption we use order statistics and dynamical systems to define a rank driven dynamical system that approximates the evolution of the distribution of the fitnesses in these rank driven models as well as in the bs model for this simplified model we can find the limiting marginal distribution as a function of the initial conditions agreement with experimental results of the bs model is excellent

individual based predator prey model for biological coevolution fluctuations stability and community structure we study an individual based predator prey model of biological coevolution using linear stability analysis and large scale kinetic monte carlo simulations the model exhibits approximate 1 f noise in diversity and population size fluctuations and it generates a sequence of quasisteady communities in the form of simple food webs these communities are quite resilient toward the loss of one or a few species which is reflected in different power law exponents for the durations of communities and the lifetimes of species the exponent for the former is near ensuremath 1 while the latter is close to ensuremath 2 statistical characteristics of the evolving communities including degree predator and prey distributions and proportions of basal intermediate and top species compare reasonably with data for real food webs

shape of ecological networks we study the statistics of ecosystems with a variable number of coevolving species the species interact in two ways by prey predator relationships and by direct competition with similar kinds the interaction coefficients change slowly through successful adaptations and speciations they are treated as quenched random variables these interactions determine long term topological features of the species network which are found to agree with those of biological systems

generalized partial orders for polar code bit channels we study partial orders pos for the synthesized bit channels of polar codes first we give an alternative proof of an existing po for bit channels with the same hamming weight and use the underlying idea to extend the bit channel ordering to some additional cases in particular the bit channel ordering for a given code block length is used to generate additional bit channel ordering relationships for larger block lengths generalizing previously known pos next we consider pos especially for the binary erasure channel bec we identify a symmetry property of the bhattacharyya parameters of complementary bit channel pairs on the bec and provide a condition for the alignment of polarized sets of bit channels for the bec and general binary input memoryless symmetric bms channels numerical examples and further properties about the pos for the bit channels with different hamming weights are provided to illustrate the new pos the bit channels with universal ordering positions which are independent of the channel erasure probability are verified for all of the code block lengths finally we show the threshold behavior of the bhattacharyya parameters of some bit channels by approximating the threshold values the corresponding value for a bit channel can be used to determine whether it is good or bad when the underlying channel is known

dynamic scflip decoding of polar codes this paper proposes a generalization of the recently introduced successive cancellation flip scflip decoding of polar codes characterized by a number of extra decoding attempts where one or several positions are flipped from the standard sc decoding to make such an approach effective we first introduce the concept of higher order bit flips and propose a new metric to determine the bit flips that are more likely to correct the trajectory of the sc decoding we then propose a generalized scflip decoding algorithm referred to as dynamic scflip d scflip which dynamically builds a list of candidate bit flips while guaranteeing that the next attempt has the highest probability of success among the remaining ones simulation results show that d scflip is an effective alternative to sc list decoding of polar codes by providing very good error correcting performance with an average computation complexity close to the one of the sc decoder

efficient design and decoding of polar codes polar codes are shown to be instances of both generalized concatenated codes and multilevel codes it is shown that the performance of a polar code can be improved by representing it as a multilevel code and applying the multistage decoding algorithm with maximum likelihood decoding of outer codes additional performance improvement is obtained by replacing polar outer codes with other ones with better error correction performance in some cases this also results in complexity reduction it is shown that gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them

asymmetric dynamics and critical behavior in the bak sneppen model abstract we investigate using mean field theory and simulation the effect of asymmetry on the critical behavior and probability density of bak sneppen models two kinds of anisotropy are investigated i different numbers of sites to the left and right of the central minimum site are updated and ii sites to the left and right of the central site are renewed in different ways of particular interest is the crossover from symmetric to asymmetric scaling for weakly asymmetric dynamics and the collapse of data with different numbers of updated sites but the same degree of asymmetry all non symmetric rules studied fall independent of the degree of asymmetry in the same universality class conversely symmetric variants reproduce the exponents of the original model our results confirm the existence of two symmetry based universality classes for extremal dynamics

effects of preference for attachment to low degree nodes on the degree distributions of a growing directed network and a simple food web model we study the growth of a directed network in which the growth is constrained by the cost of adding links to the existing nodes we propose a preferential attachment scheme in which a new node attaches to an existing node i with probability ensuremath pi k i ensuremath propto k i ensuremath 1 where k i is the number of outgoing links at i we calculate the degree distribution for the outgoing links in the asymptotic regime t ensuremath rightarrow ensuremath infty n k both analytically and by monte carlo simulations the distribution decays like k ensuremath mu k ensuremath gamma k for large k where ensuremath mu is a constant we investigate the effect of this preferential attachment scheme by comparing the results to an equivalent growth model with a degree independent probability of attachment which gives an exponential outdegree distribution also we relate this mechanism to simple food web models by implementing it in the cascade model we show that the low degree preferential attachment mechanism breaks the symmetry between in and outdegree distributions in the cascade model it also causes a faster decay in the tails of the outdegree distributions for both our network growth model and the cascade model

numerical analysis of the master equation applied to the master equation the usual numerical integration methods such as runge kutta method become inefficient when the rates associated with various transitions differ by several orders of magnitude we introduce an integration scheme that remains stable with much larger time increments than can be used in standard methods when only the stationary distribution is required a direct iteration method is even more rapid this method may be extended to construct the quasistationary distribution of a process with an absorbing state applications to birth and death processes reveal gains in efficiency of two or more orders of magnitude

some homological invariants of local pi algebras during the last several years artin jorgensen stafford yekutieli and others have generalized many homological results from the commutative case to the noncommutative connected graded case az sz1 jo1 jo2 ye in this paper we generalize some of those to the noncommutative local case our main interest is on noetherian local pi polynomial identity algebras though some results hold in a more general setting throughout a is an algebra over a base field k i is its jacobson radical and a0 a i if a0 is artinian a is called semilocal if a0 is a simple artinian algebra respectively a division algebra then a is called local respectively scalar local unless otherwise stated we are working with left modules and sometimes we use the term aop module for a right a module where aop is the opposite ring of a a finite a module means finitely generated over a we use noetherian respectively artinian for two sided noetherian respectively two sided artinian

performance bounds of concatenated polar coding schemes a concatenated coding scheme over binary memoryless symmetric bms channels using a polarization transformation followed by outer sub codes is analyzed achievable error exponents and upper bounds on the error rate are derived the first bound is obtained using outer codes which are typical random linear codes as a byproduct of this bound it determines the rates of the outer codes a lower bound on the error exponent that holds for all bms channels with a given capacity is then derived improved bounds and approximations for finite blocklength codes using channel dispersions normal approximation as well as converse and approximate converse results are also obtained the bounds are compared with actual simulation results from the literature for the cases considered when transmitting over the binary input additive white gaussian noise channel there was only a small gap between the channel dispersion based approximation and the actual error rate of concatenated bch polar codes

on singular probability densities generated by extremal dynamics abstract extremal dynamics is the mechanism that drives the bak sneppen model into a self organized critical state marked by a singular stationary probability density p x with the aim of understanding this phenomenon we study the bs model and several variants via mean field theory and simulation in all cases we find that p x is singular at one or more points as a consequence of extremal dynamics furthermore we show that the extremal barrier x i always belongs to the prohibited interval in which p x 0 our simulations indicate that the bak sneppen universality class is robust with regard to changes in the updating rule we find the same value for the exponent for all variants mean field theory which furnishes an exact description for the model on a complete graph reproduces the character of the probability distribution found in simulations for the modified processes mean field theory takes the form of a functional equation for p x

on the thresholds probability densities and critical exponents of bak sneppen like models abstract we report a simple method to accurately determine the threshold and the exponent of the bak sneppen bs model and also investigate the bs universality class for the random neighbor version of the bs model we find the threshold x 0 33332 3 in agreement with the exact result x 1 3 given by mean field theory for the one dimensional original model we find x 0 6672 2 in good agreement with the results reported in the literature for the anisotropic bs model we obtain x 0 7240 1 we study the finite size effect x l x l l observed in a system with l sites and find 1 00 1 for the random neighbor version 1 40 1 for the original model and 1 58 1 for the anisotropic case finally we discuss the effect of defining the extremal site as the one which minimizes a general function f x instead of simply f x x as in the original updating rule we emphasize that models with extremal dynamics have singular stationary probability distributions p x our simulations indicate the existence of two symmetry based universality classes

a homological reformulation of the link condition abstract we prove an equivalent condition for the existence of a link between prime ideals in terms of the structure of a certain cohomology module we use this formulation to answer an open question regarding the nature of module extensions over one sided noetherian rings we apply the techniques developed in this paper to the local link structure of prime ideals of small homological height and examine when certain noetherian rings satisfy the density condition

branching process in a stochastic extremal model we considered a stochastic version of the bak sneppen model sbsm of ecological evolution where the number m of sites mutated in a mutation event is restricted to only two here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion the critical behavior of the sbsm is found to be the same as the bs model in dimensions d 1 and 2 however on the scale free graphs the critical fitness value is nonzero even in the thermodynamic limit but the critical behavior is mean field like finally m has been made even smaller than two by probabilistically updating the mutation zone which also shows the original bs model behavior we conjecture that a sbsm on any arbitrary graph with any small branching factor greater than unity will lead to a self organized critical state

soft decision decoding of reed muller codes as generalized multiple concatenated codes constructs reed muller codes by generalized multiple concatenation of binary block codes of length 2 as a consequence of this construction a new decoding procedure is derived that uses soft decision information the algorithm is designed for low decoding complexity and is applicable to all reed muller codes it gives better decoding performance than soft decision bounded distance decoding its decoding complexity is much lower than that of maximum likelihood trellis decoding of reed muller codes especially for long codes

a duality property for complex lie algebroids interpreting lie algebroid theory in terms of cal d modules we define a duality functor for a lie algebroid as well as a direct image functor for a morphism of lie algebroids generalizing the work of schneiders see also the work of schapira schneiders and making assumptions analog to his we show that the duality functor and the direct image functor commute as an application we extend to lie algebroids some duality properties already known for lie algebras

construction of polar codes with sublinear complexity consider the problem of constructing a polar code of block length n for a given transmission channel w previous approaches require one to compute the reliability of the n synthetic channels and then use only those that are sufficiently reliable however we know from two independent works by schurch and by bardet et al that the synthetic channels are partially ordered with respect to degradation hence it is natural to ask whether the partial order can be exploited to reduce the computational burden of the construction problem we show that if we take advantage of the partial order we can construct a polar code by computing the reliability of roughly a fraction 1 log 3 2 n of the synthetic channels in particular we prove that n log 3 2 n is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor log log n this set of roughly n log 3 2 n synthetic channels is universal in the sense that it allows one to construct polar codes for any w and it can be identified by solving a maximum matching problem on a bipartite graph our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set as such this method is general and it can be used to further improve the complexity of the construction problem in case a refined partial order on the synthetic channels of polar codes is discovered

analytic results for scaling function and moments for a different type of avalanche in the bak sneppen evolution model starting from the master equation for the hierarchical structure of avalanches of a different kind within the frame of the bak sneppen evolution model we derive the exact formula of the scaling function describing the probability distribution of avalanches the scaling function displays features required by the scaling ansatz and verified by simulations using the scaling function we investigate the avalanche moment denoted by s k ensuremath delta overline f it is found that for any non negative integer k s k ensuremath delta overline f diverges as ensuremath delta f ensuremath k which gives an infinite group of exact critical exponents simulation outcomes of avalanche moments with k 1 2 3 are found to be consistent with the corresponding analytical results

criticality in simple models of evolution we consider two apparently similar models of biological evolution that have been claimed to exhibit self organized critical behavior a careful reanalysis of these models including several analytic results for one of them suggests that they are qualitatively different we demonstrate the limitations of the mean field description of these systems we argue that a more precise definition of self organized criticality is desirable and establish several criteria in this connection

effects of correlated interactions in a biological coevolution model with individual based dynamics models of biological coevolution have recently been proposed and studied in which a species is defined by a genome in the form of a finite bitstring and the interactions between species i and j are given by a fixed matrix with independent randomly distributed elements mij a consequence of the stochastic independence is that species whose genotypes differ even by a single bit may have completely different phenotypes as defined by their interactions with the other species this is clearly unrealistic as closely related species should be similar in their interactions with the rest of the ecosystem here we therefore study a model in which the mij are correlated to a controllable degree by means of a local averaging scheme we calculate both analytically and numerically the correlation function for matrix elements mij and mkl versus the hamming distance between the bitstrings representing the species the agreement between the analytical and numerical calculations is excellent for correlations of limited range but explainable differences arise for correlation ranges that are a significant fraction of the length of the bitstring we compare long kinetic monte carlo simulations of coevolution models with uncorrelated and correlated interactions respectively in particular we consider the probability density for the lifetimes of individual species the species lifetime distribution is close to a power law with an exponent near 2 over eight decades in time in both uncorrelated and correlated cases the durations of quasi steady states and power spectral densities for the diversity indices display noticeable differences however some qualitative features such as 1 f behaviour in power spectral densities for the diversity indices are not affected by the correlations in the interaction matrix

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

absorbing state phase transitions with extremal dynamics extremal dynamics represents a path to self organized criticality in which the order parameter is tuned to a value of zero the order parameter is associated with a phase transition to an absorbing state given a process that exhibits a phase transition to an absorbing state we define an extremal absorbing process providing the link to the associated extremal nonabsorbing process stationary properties of the latter correspond to those at the absorbing state phase transition in the former studying the absorbing version of an extremal dynamics model allows to determine certain critical exponents that are not otherwise accessible in the case of the bak sneppen bs model the absorbing version is closely related to the f avalanche introduced by paczuski maslov and bak phys rev e 53 414 1996 or in spreading simulations to the bs branching process also studied by these authors the corresponding nonextremal process belongs to the directed percolation universality class we revisit the absorbing bs model obtaining refined estimates for the threshold and critical exponents in one dimension we also study an extremal version of the usual contact process using mean field theory and simulation the extremal condition slows the spread of activity and modifies the critical behavior radically defining an extremal directed percolation universality class of absorbing state phase transitions asymmetric updating is a relevant perturbation for this class even though it is irrelevant for the corresponding nonextremal class

bak sneppen type models and rank driven processes the bak sneppen model is a well known stochastic model of evolution that exhibits self organized criticality only a few analytical results have been established for it so far we report a surprising connection between bak sneppen type models and more tractable markov processes that evolve without any reference to an underlying topology specifically we show that in the case of a large number of species the long time behavior of the fitness profile in the bak sneppen model can be replicated by a model with a purely rank based update rule whose asymptotics can be studied rigorously

cohomology of schematic algebras letrbe a positively graded noetheriank algebra and let proj r r grbe the quotient category of the category of gradedr modulesr grmodulo those of finite length ifris schematic cf f van oystaeyen and l willaert j pure appl algebra104 1995 109 122 and has finite global dimension we show that the cohomology groups ofr as defined in m artin and j j zhang adv in math 109 1994 228 287 are finite dimensional furthermore we introduce cech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups these generalized cech cohomology groups are still manageable we illustrate this with an example

polar codes bounds on bhattacharyya parameters and their applications this paper proposes a novel lower bound on the bhattacharyya parameter of synthesized channels in polar codes the proposed bound is a result of a newly defined merge operation that reduces the cardinality of output alphabets of a symmetric binary input discrete memoryless channel by repeated applications of the proposed merge operation we establish a universal bound on the bhattacharyya parameters of synthesized channels that holds for any symmetric binary input discrete memoryless transmission channel we then discuss some applications of the new bound for example using the derived universal lower bound along with the existing upper bound we find a partial order among the bhattacharyya parameters of those synthesized channels that cannot be ordered through existing results this ordering can help efficient design of polar codes in addition some extremal behaviors of the binary symmetric channel are also proved

bak sneppen model near zero dimension we consider the bak sneppen model near zero dimension where the avalanche exponent tau is close to 1 and the exponents m and sigma are close to 0 we demonstrate that tau 1 m sigma exp m 1 gamma small middle dot small middle dot small middle dot in this limit where gamma is euler s constant the avalanche hierarchy equation is rewritten in a form that makes it possible to find the relation between the critical exponents sigma and m with high accuracy we obtain precise values of the critical exponents for the one and two dimensional bak sneppen model and for the one dimensional anisotropic bak sneppen model

homomorphisms between generalized verma modules let g be a finite dimensional complex semisimple lie algebra and p a parabolic subalgebra the first result is a necessary and sufficient condition in the spirit of the bernstein gelfand gelfand theorem on verma modules for lepowsky s standard map between two generalized verma modules for g to be zero the main result gives a complete description of all homomorphisms between the generalized verma modules induced from one dimensional p modules in the hermitian symmetric situation

analytical solution of a model for complex food webs we investigate numerically and analytically a recently proposed model for food webs nature 404 180 2000 in the limit of large web sizes and sparse interaction matrices we obtain analytical expressions for several quantities with ecological interest in particular the probability distributions for the number of prey and the number of predators we find that these distributions have fast decaying exponential and gaussian tails respectively we also find that our analytical expressions are robust to changes in the details of the model

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor

stochastic model for tumor growth with immunization we analyze a stochastic model for tumor cell growth with both multiplicative and additive colored noises as well as nonzero cross correlations in between whereas the death rate within the logistic model is altered by a deterministic term characterizing immunization the birth rate is assumed to be stochastically changed due to biological motivated growth processes leading to a multiplicative internal noise moreover the system is subjected to an external additive noise which mimics the influence of the environment of the tumor the stationary probability distribution p s is derived depending on the finite correlation time the immunization rate and the strength of the cross correlation p s offers a maximum which becomes more pronounced for increasing immunization rate the mean first passage time is also calculated in order to find out under which conditions the tumor can suffer extinction its characteristics are again controlled by the degree of immunization and the strength of the cross correlation the behavior observed can be interpreted in terms of a biological model of tumor evolution

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

convolutional polar kernels a family of polarizing kernels is presented together with polynomial complexity algorithm for computing scaling exponent the proposed convolutional polar kernels are based on convolutional polar codes also known as b mera codes for these kernels a polynomial complexity algorithm is proposed to find weight spectrum of unrecoverable erasure patterns needed for computing scaling exponent as a result we obtain scaling exponent and polarization rate for convolutional polar kernels of size up to 1024