stochastic flow cascades we introduce and explore a stochastic flow cascade sfc model a general statistical model for the unidirectional flow through a tandem array of heterogeneous filters examples include the flow of i liquid through heterogeneous porous layers ii shocks through tandem shot noise systems iii signals through tandem communication filters the sfc model combines together the langevin equation convolution filters and moving averages and poissonian randomizations a comprehensive analysis of the sfc model is carried out yielding closed form results levy laws are shown to universally emerge from the sfc model and characterize both heavy tailed retention times noah effect and long ranged correlations joseph effect

levy ornstein uhlenbeck and subordination spectral vs jump description unlike brownian motion which propagates diffusively and whose sample path trajectories are continuous non brownian levy motions propagate via jumps flights and their sample path trajectories are purely discontinuous when analyzing systems involving non brownian levy motions the common practice is to use either spectral or fractional calculus methods in this manuscript we suggest an alternative analytical approach using the poisson superposition jump structure of non brownian levy motions we demonstrate this approach in two exemplary topics i systems governed by l evy driven ornstein uhlenbeck dynamics and ii systems subject to temporal levy subordination we show that this approach yields answers and insights that are not attainable using spectral methods alone

power law shot noise the behavior of power law shot noise for which the associated impulse response functions assume a decaying power law form is explored expressions are obtained for the moments moment generating functions amplitude probability density functions autocorrelation functions and power spectral densities for a variety of parameters of the process for certain parameters the power spectral density exhibits 1 f type behavior over a substantial range of frequencies so that the process serves as a source of 1 f sup alpha shot noise for alpha in the range 0

shot noise systems with random relaxations we explore a model of a random relaxation shot noise system in which i the shot inflow is a general poisson point process with possibly infinite poissonian rates and ii the exponential relaxations governing the shot decay are randomized this system model is applicable to physical environments polluted by radioactive contamination of heterogeneous types the statistics of random relaxation shot noise systems are analyzed quantitatively and comprehensively stationary structure correlation structure process distribution fractality and asymptotic fractality and the display both separately and simultaneously of the noah and joseph effects results are obtained explicitly and in closed form and facilitate the design of tractable shot noise systems with unique features

a set probability technique for detecting relative time order across multiple neurons with the development of multielectrode recording techniques it is possible to measure the cell firing patterns of multiple neurons simultaneously generating a large quantity of data identification of the firing patterns within these large groups of cells is an important and a challenging problem in data analysis here we consider the problem of measuring the significance of a repeat in the cell firing sequence across arbitrary numbers of cells in particular we consider the question given a ranked order of cells numbered 1 to n what is the probability that another sequence of length n contains j consecutive increasing elements assuming each element of the sequence is drawn with replacement from the numbers 1 through n we derive a recursive formula for the probability of the sequence of length j or more for n 2j a closed form solution is derived for n 2j we obtain upper and lower bounds for these probabilities for various combinations of parameter values these can be computed very quickly for a typical case with small n 10 and large n 3000 sequences of 7 and 8 are statistically very unlikely a potential application of this technique is in the detection of repeats in hippocampal place cell order during sleep unlike most previous articles on increasing runs in random lists we use a probability approach based on sets of overlapping sequences

on representability of a finite local ring this article is devoted to the following problem let r be a finite local ring with identity can r be embedded into a matrix ring over a commutative ring the question is reduced to an explicitly described class of test cases

a langevin approach to the log gauss pareto composite statistical structure abstract the distribution of wealth in human populations displays a log gauss pareto composite statistical structure its density is log gauss in its central body and follows power law decay in its tails this composite statistical structure is further observed in other complex systems and on a logarithmic scale it displays a gauss exponential structure its density is gauss in its central body and follows exponential decay in its tails in this paper we establish an equilibrium langevin explanation for this statistical phenomenon and show that i the stationary distributions of langevin dynamics with sigmoidal force functions display a gauss exponential composite statistical structure ii the stationary distributions of geometric langevin dynamics with sigmoidal force functions display a log gauss pareto composite statistical structure this equilibrium langevin explanation is universal as it is invariant with respect to the specific details of the sigmoidal force functions applied and as it is invariant with respect to the specific statistics of the underlying noise

markov breaking and the emergence of long memory in ornstein uhlenbeck systems we consider a complex system composed of many non identical parts where i the dynamics of each part are ornstein uhlenbeck ii all parts are driven by a common external levy noise and iii the system s collective output is the averaged aggregate of the outputs of its parts whereas the dynamics on the microscopic parts level are markov the dynamics on the macroscopic system level are not markov and may display a long memory moreover the universal temporal scaling limit of the system s output in the presence of long memory is fractional brownian motion the model presented is analytically tractable and gives closed form quantitative characterizations of both the markov breaking phenomenon and the emergence of long memory

from ornstein uhlenbeck dynamics to long memory processes and fractional brownian motion this article establishes a natural physical path leading from regular ornstein uhlenbeck dynamics to anomalous long memory processes and thereafter to fractional brownian motion considering a system composed of n different parts each part conducting its own ornstein uhlenbeck dynamics and all parts being perturbed by a common external l evy noise we show that the collective system dynamics in the limit n ensuremath rightarrow ensuremath infty converges to a temporal moving average of the driving noise the limiting moving average process in turn can posses a long memory in which case when observed over large time scales further yields fractional brownian motion the temporal correlation structure of the limiting moving average process turns out to be determined by the structural statistical variability of the system s composing parts thus the emergence of a long memory is a consequence of the intrinsic quenched disorder present at the system s formation epoch rather than the consequence of the external annealed disorder carried in continuously by the driving noise

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

shot noise displaying simultaneously the noah and joseph effects shot noise processes displaying the noah effect are of infinite variance hence the measurement of the correlation structure of such processes indicating whether the joseph effect is displayed as well cannot be conducted via their autocovariance functions which are undefined to circumvent this problem a poisson based analysis applicable to shot noise processes with infinite variance as well as to shot noise processes with divergent noise levels is developed a poissonian autocovariance function which characterizes the process distribution of general shot noise processes is introduced in particular this function governs and quantifies the processes stationary structure amplitudal behavior and correlation structure temporal behavior the poissonian methodology developed enables a precise quantitative analysis of shot noise processes displaying simultaneously the noah and joseph effects

embedding rings in completed graded rings 3 algebras over general k abstract it is shown that any associative algebra r over a commutative ring k such that r n 1 0 can be embedded in a graded k algebra h i such that h i 0 only for 1 i 2 n 1 by the results of the first paper in this series r can therefore be embedded in 2 n 1 1 2 n 1 1 strictly upper triangular matrices over a commutative k algebra for k any integral domain not a field this is in fact a best result in contrast with the case of k a field where one can replace 2 n 1 by n see part 2 of this series more generally if r is a nonunital k algebra with a strictly positive integer valued filtration function then r can be mapped into a completed z graded k algebra s by a homomorphism f such that v a v f a 2 v a 1 where v denotes the given filtration function on r and the grading filtration on s if all values of the filtration function on r are powers of 2 one can even get v f a v a

embedding rings in completed graded rings 4 commutative algebras abstract if k is a commutative ring of finite characteristic and a a commutative associative k algebra with augmentation such that the intersection of the powers of the augmentation ideal is zero it is shown that a can be embedded in a completed graded commutative associative k algebra with degree 0 component just k the corresponding result is false in characteristic 0 though we do not know whether it can fail when k is a field the above finite characteristic result is also true with commutative associative algebra replaced by any variety of associative or nonassociative algebras defined by multilinear identities

embedding rings in completed graded rings 1 triangular embeddings abstract let k be a commutative ring with 1 r an associative k algebra without 1 and n a positive integer it is shown that the following conditions are equivalent a r can be embedded as a k algebra in the ring of strictly upper triangular n n matrices over an associative k algebra a b r has such an embedding with a further assumed commutative c r can be embedded as a k algebra in a graded associative k algebra c whose only nonzero components are c 1 c nn 1 the proof that the other conditions imply b is essentially a demonstration that a certain kind of k linear category always has a faithful functor into a commutative k algebra subsequent papers in this series determine sufficient conditions for a c above to hold in terms of nilpotence of r under various hypotheses

correlation cascades of levy driven random processes abstract we explore the correlation structure of a large class of random processes driven by non gaussian levy noise sources with possibly infinite variances examples of such processes include levy motions levy driven ornstein uhlenbeck motions levy driven moving average processes fractional levy motions and fractional levy noises based on the fact that non gaussian levy noises are continuum superpositions of poisson noises we unveil an underlying cascade of levy correlation functions which characterize the process distribution and the correlation structure of the processes under consideration in the case where the driving levy noise sources are fractal the resulting cascade admits a unique scale free form

the misconception of mean reversion the notion of random motion in a potential well is elemental in the physical sciences and beyond quantitatively this notion is described by reverting diffusions asymptotically stationary diffusion processes which are simultaneously i driven toward a reversion level by a deterministic force and ii perturbed off the reversion level by a random white noise the archetypal example of reverting diffusions is the ornstein uhlenbeck process which is mean reverting in this paper we analyze reverting diffusions and establish that i if the magnitude of the perturbing noise is constant then the diffusion s stationary density is unimodal and the diffusion is mode reverting ii if the magnitude of the perturbing noise is non constant then in general neither is the diffusion s stationary density unimodal nor is the diffusion mode reverting in the latter case we further establish a result asserting when unimodality and mode reversion do hold in particular we demonstrate that the notion of mean reversion which is fundamental in economics and finance is a misconception as mean reversion is an exception rather than the norm

bifurcation bimodality and finite variance in confined levy flights we investigate the statistical behavior of levy flights confined in a symmetric quartic potential well u x proportional variant x 4 at stationarity the probability density function features a distinct bimodal shape and decays with power law tails which are steep enough to give rise to a finite variance in contrast to free levy flights from a delta initial condition a bifurcation of the unimodal state is observed at t c 0 the nonlinear oscillator with potential u x ax 2 2 bx 4 4 a b 0 shows a crossover from unimodal to bimodal behavior at stationarity depending on the ratio a b

fractal levy correlation cascades the correlation structure of a wide class of random processes driven by stable non gaussian levy noise sources is explored since these noises are of infinite variance correlations cannot be measured via auto covariance functions exploiting the underlying poissonian structure of levy noises we present a cascade of poissonian correlation functions which characterize the correlation structure and the process distribution of the processes under consideration the theory developed is applied to various examples including motions ornstein uhlenbeck and moving average processes and fractional motions and noises all driven by stable non gaussian levy noises

embedding rings in completed graded rings 2 algebras over a field abstract it is shown that any z filtered associative algebra r over a field k can be embedded in the completion h of a z graded k algebra hi so that the filtration on r is induced by the grading of h a normal form for the universal such h is found a consequence is that if r is a k algebra such that rn 0 then r can be embedded in a graded k algebra hi of which all homogeneous components except h1 hn 1 are zero and so by the results of part 1 of this series can be embedded in n n strictly upper triangular matrices over a commutative k algebra this last result has also been proved by i v l vov

conditional probability based significance tests for sequential patterns in multineuronal spike trains we consider the problem of detecting statistically significant sequential patterns in multineuronal spike trains these patterns are characterized by ordered sequences of spikes from different neurons with specific delays between spikes we have previously proposed a data mining scheme to efficiently discover such patterns which occur often enough in the data here we propose a method to determine the statistical significance of such repeating patterns the novelty of our approach is that we use a compound null hypothesis that not only includes models of independent neurons but also models where neurons have weak dependencies the strength of interaction among the neurons is represented in terms of certain pair wise conditional probabilities we specify our null hypothesis by putting an upper bound on all such conditional probabilities we construct a probabilistic model that captures the counting process and use this to derive a test of significance for rejecting such a compound null hypothesis the structure of our null hypothesis also allows us to rank order different significant patterns we illustrate the effectiveness of our approach using spike trains generated with a simulator