generation and monitoring of discrete stable random processes using multiple immigration population models some properties of classical population processes that comprise births deaths and multiple immigrations are investigated the rates at which the immigrants arrive can be tailored to produce a population whose steady state fluctuations are described by a pre selected distribution attention is focused on the class of distributions with a discrete stable law which have power law tails and whose moments and autocorrelation function do not exist the separate problem of monitoring and characterizing the fluctuations is studied analysing the statistics of individuals that leave the population the fluctuations in the size of the population are transferred to the times between emigrants that form an intermittent time series of events the emigrants are counted with a detector of finite dynamic range and response time this is modelled through clipping the time series or saturating it at an arbitrary but finite level whereupon its moments and correlation properties become finite distributions for the time to the first counted event and for the time between events exhibit power law regimes that are characteristic of the fluctuations in population size the processes provide analytical models with which properties of complex discrete random phenomena can be explored and in addition provide generic means by which random time series encompassing a wide range of intermittent and other discrete random behaviour may be generated

associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

residence time distribution for a class of gaussian markov processes we study the distribution of residence time or equivalently that of mean magnetization for a family of gaussian markov processes indexed by a positive parameter alpha the persistence exponent for these processes is simply given by theta alpha but the residence time distribution is nontrivial the shape of this distribution undergoes a qualitative change as theta increases indicating a sharp change in the ergodic properties of the process we develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha for some special values of alpha we obtain closed form expressions of the distribution function s1063 651x 99 03306 1

the evolution of a batch immigration death process subject to counts a bivariate batch immigration death process is developed to study the degree to which the fundamental structure of a hidden stochastic process can be inferred purely from counts of escaping individuals this question is of immense importance in fields such as quantum optics where externally based radiation elucidates the nature of the underlying electromagnetic radiation process batches of i immigrants enter the population at rate qi and each individual dies independently at rate general expressions are developed for the population size cumulants and probabilities together with those for the associated counting process the strong link between these two structures is highlighted through two specific examples involving k batch immigration for i k and schoenberg batch immigration over i 2m m 0 1 2 and shows that high quality inferences on the hidden population process can be inferred purely from externally counted observations

power law relaxation in a complex system omori law after a financial market crash we study the relaxation dynamics of a financial market just after the occurrence of a crash by investigating the number of times the absolute value of an index return is exceeding a given threshold value we show that the empirical observation of a power law evolution of the number of events exceeding the selected threshold a behavior known as the omori law in geophysics is consistent with the simultaneous occurrence of i a return probability density function characterized by a power law asymptotic behavior and ii a power law relaxation decay of its typical scale our empirical observation cannot be explained within the framework of simple and widespread stochastic volatility models

distributivity equations in the class of semi t operators recently the distributivity equations have been discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this study we present the solutions to distributivity between semi t operators previous results related to the distributivity between nullnorms and between semi t operators and semi nullnorms can be obtained as simple corollaries

statistics of the occupation time for a random walk in the presence of a moving boundary we investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v for the continuous time problem brownian motion we provide a simple alternative proof of newman s recent result newman t j 2001 j phys a math gen 34 l89 using a method developed by kac we then discuss the same problem for the case of a random walk in discrete time with an arbitrary distribution of steps taking advantage of the general set of results of sparre andersen for the binomial random walk we analyse the corrections to the continuum limit on the example of the mean occupation time the case of cauchy distributed steps is also studied

statistical properties of functionals of the paths of a particle diffusing in a one dimensional random potential we present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one dimensional quenched random potential we demonstrate the implementation of the formalism in two specific examples 1 where the functional corresponds to the local time spent by the particle around the origin and 2 where the functional corresponds to the occupation time spent by the particle on the positive side of the origin within an observation time window of size t we compute the disorder average distributions of the local time the inverse local time the occupation time and the inverse occupation time and show that in many cases disorder modifies the behavior drastically

random walks with power law fluctuations in the number of steps in this paper we derive limiting distributions for the resultant of an anisotropic random walk where the individual steps comprising the walk are drawn from distributions of arbitrary but finite variance and where the number of steps fluctuate according to a discrete power law distribution the consideration of discrete distributions with power law tails is motivated by their recently discovered relevance to complex systems and networks when this random walk is unbiased a power law decay for large values of the amplitude of the resultant occurs for a small directional bias the power law tail persists only in the direction parallel to the bias exhibiting an exponential decay in all other directions we consider the relevance of the limiting distributions to non diffusive transport

the distributive equations for idempotent uninorms and nullnorms the distributive equations only involving idempotent uninorms and nullnorms are investigated the following two cases are proved i as for a nullnorm with an idempotent uninorm having distributivity the necessary and sufficient conditions are given ii as for an idempotent uninorm with a nullnorm having distributivity the necessary and sufficient conditions are given also

scaling and memory in the return intervals of realized volatility abstract we perform return interval analysis of 1 min realized volatility defined by the sum of absolute high frequency intraday returns for the shanghai stock exchange composite index ssec and 22 constituent stocks of ssec the scaling behavior and memory effect of the return intervals between successive realized volatilities above a certain threshold q are carefully investigated in comparison with the volatility defined by the closest tick prices to the minute marks the return interval distribution for the realized volatility shows a better scaling behavior since 20 stocks out of 22 stocks and the ssec pass the kolmogorov smirnov ks test and exhibit scaling behaviors among which the scaling function for 8 stocks could be approximated well by a stretched exponential distribution revealed by the ks goodness of fit test under the significance level of 5 the improved scaling behavior is further confirmed by the relation between the fitted exponent and the threshold q in addition the similarity of the return interval distributions for different stocks is also observed for the realized volatility the investigation of the conditional probability distribution and the detrended fluctuation analysis dfa show that both short term and long term memory exists in the return intervals of realized volatility

statistics of the occupation time for a class of gaussian markov processes we revisit the work of dhar and majumdar 1999 phys rev e 59 6413 on the limiting distribution of the temporal mean mt t 1 0tdu sign yu for a gaussian markovian process yt depending on a parameter which can be interpreted as brownian motion in the time scale t t2 this quantity the mean magnetization is simply related to the occupation time of the process that is the length of time spent on one side of the origin up to time t using the fact that the intervals between sign changes of the process form a renewal process on the time scale t we determine recursively the moments of the mean magnetization we also find an integral equation for the distribution of mt this allows a local analysis of this distribution in the persistence region mt 1 as well as its asymptotic analysis in the regime where is large finally we put the results thus found in perspective with those obtained by dhar and majumdar by another method based on a formalism due to kac

self adjointness for dirac operators via hardy dirac inequalities distinguished self adjoint extensions of dirac operators are constructed for a class of singular potentials including coulombic ones up to and including the critical case x 1 the method uses hardy dirac inequalities and quadratic form techniques

extreme value statistics in records with long term persistence many natural records exhibit long term correlations characterized by a power law decay of the autocorrelation function c s ensuremath sim s ensuremath ensuremath gamma with time lag s and correlation exponent 0l ensuremath gamma l1 we study how the presence of such correlations affects the statistics of the extreme events i e the maximum values of the signal within time segments of the fixed duration r we find numerically that i the integrated distribution function of the maxima converges to a gumbel distribution for large r similar to uncorrelated signals ii the deviations for finite r depend on the initial distribution of the records and on their correlation properties iii the maxima series exhibit long term correlations similar to those of the original data and most notably iv the maxima distribution as well as the mean maxima significantly depend on the history in particular on the previous maximum the last item implies that conditional mean maxima and conditional maxima distributions with the value of the previous maximum as condition should be considered for an improved extreme event prediction we provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long term correlated records

fluctuations in a coupled population model we investigate a discrete markov process in which the immigration of individuals into one population is controlled by the fluctuations in another we examine the effect of coupling back the second population to the first through a similar mechanism and derive exact solutions for the generating functions of the population statistics we show that a stationary state exists over a certain parameter range and obtain expressions for moments and correlation functions in this regime when more than two populations are coupled cyclically transient oscillations and periodic behaviour of correlation functions are predicted we demonstrate that if the initial distribution of either population is stable or more generally has a power law tail that falls off like n 1 0 1 then for certain parameter values there exists a stationary state that is also power law but not stable this stationary state cannot be accessed from a single multiple immigrant population model but arises solely from the nonlinear interaction of the coupled system

phase distribution and phase correlation of financial time series the scaling phase distribution and phase correlation of financial time series are investigated based on the dow jones industry average and nasdaq 10 min intraday data for a period from 1 aug 1997 to 31 dec 2003 the returns of the two indices are shown to have nice scaling behaviors and belong to stable distributions according to the criterion of l evy s ensuremath alpha stable distribution condition an approach catching characteristic features of financial time series based on the concept of instantaneous phase is further proposed to study the phase distribution and correlation analysis of the phase distribution concludes that return time series fall into a class which is different from other nonstationary time series the correlation between returns of the two indices probed by the distribution of phase difference indicates that there was a remarkable change of trading activities after the event of the 9 11 attack and this change persisted in later trading activities

zero crossing rates of functions of gaussian processes formulas for the expected zero crossing rates of random processes that are monotone transformations of gaussian processes can be obtained by using two different techniques the first technique involves derivation of the expected zero crossing rate for discrete time processes and extends the result of the continuous time case by using an appropriate limiting argument the second is a direct method that makes successive use of r price s 1958 theorem the chain rule for derivatives and s o rice s 1954 formula for the expected zero crossing rate of a gaussian process a constant which depends on the variance of the transformed process and a second moment of its derivative is derived multiplying rice s original expression by this constant yields the zero crossing formula for the transformed process the two methods can be used for the general level crossing problem of random processes that are monotone functions of a gaussian process

distinguishing population processes by external monitoring we investigate the statistical and correlation properties of two stochastic population models that give rise to identical first order probability densities we assume that the processes are monitored indirectly through measurement of the rate at which individuals emigrate from the population formulae characterizing the integrated statistics of these counting processes are derived and it is shown how they may be used to distinguish the population models

analytical results for generalized persistence properties of smooth processes we present a general scheme to calculate within the independent interval approximation generalized level dependent persistence properties for processes having a finite density of zero crossings our results are especially relevant for the diffusion equation evolving from random initial conditions one of the simplest coarsening systems exact results are obtained in certain limits and rely on a new method to deal with constrained multiplicative processes an excellent agreement of our analytical predictions with direct numerical simulations of the diffusion equation is found

residence times and other functionals of reflected brownian motion we study the residence and local times for a brownian particle confined by reflecting boundaries and propose a general solution to the problem of finding the related probability distribution its fourier transform characteristic function and laplace transform survival probability are obtained in a compact matrix form involving the laplace operator eigenbasis explicit combinatorial relations are derived for the moments and the probability distribution is shown to be nearly gaussian when the exploration time is long enough when the eigenbasis or a part of it is known the numerical computation of the residence time distributions is straightforward and accurate the present approach can also be applied to investigate other functionals of reflected brownian motion describing in particular restricted diffusion in an external field or potential e g nuclei diffusing in an inhomogeneous magnetic field theoretical results for the local times are confronted with monte carlo simulations on the unit interval disk and sphere

clustering of extreme and recurrent events in deterministic chaotic systems we study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems we find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events which leads to counterintuitive behavior in particular the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds in sharp contrast to what has been observed for stochastic processes with long range correlations as well as for independent and identically distributed random variables this has important implications for the time dependent hazard assessment of extreme events indicating that possible threshold dependencies should always be taken into account

the modularity condition for semi t operators and semi uninorms abstract the aim of this paper is mainly to solve the functional equations given by the modularity condition several years ago the modularity equations for t norms t conorms uninorms and t operators which are commutative and associative have been studied our investigations are motivated by modularity condition for generalizations of these operators by removing associativity or commutativity in this work the following main conclusions are proved 1 a continuous t norm with respect to a continuous semicopula is modular if and only if they are equal the case for a semicopula with respect to a strict t norm is also the same a semicopula with respect to a co semicopula is modular if and only if the semicopula is min and the co semicopula is max the modularity condition does not hold for a co semicopula with respect to a semicopula 2 necessary and sufficient conditions are given for a semi t operator with respect to a semi uninorm a pseudo uninorm with respect to a semi t operator to satisfy the modularity condition equation new solutions to the modularity condition equations of the case 1 are characterized

from doubly stochastic representations of k distributions to random walks and back again an optics tale a random walk model with a negative binomially fluctuating number of steps is considered in the case where the mean of the number fluctuations is finite the asymptotic behaviour of the resultant statistics in the large limit is derived and shown to give the k distribution the equivalence of this model to the hitherto unrelated doubly stochastic representation of the k distribution is also demonstrated the convergence to the k distribution of the probability density function generated by a random walk with a finite mean number of steps is examined along with the moments and the non gaussian statistics are shown to be a direct result of discreteness and bunching effects

preassociative aggregation functions the classical property of associativity is very often considered in aggregation function theory and fuzzy logic in this paper we provide axiomatizations of various classes of preassociative functions where preassociativity is a generalization of associativity recently introduced by the authors these axiomatizations are based on existing characterizations of some noteworthy classes of associative operations such as the class of aczelian semigroups and the class of t norms

generation and monitoring of a discrete stable random process a discrete stochastic process with stationary power law distribution is obtained from a death multiple immigration population model emigrations from the population form a random series of events which are monitored by a counting process with finite dynamic range and response time it is shown that the power law behaviour of the population is manifested in the intermittent behaviour of the series of events

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

lacunarity of the zero crossings of gaussian processes a lacunarity analysis of the zero crossings derived from gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process these bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a poisson random process both possessing the same mean interval size between crossings at very small and large scales compared with the correlation length of the random processes the lacunarity is similar to the poisson but exhibits significant departures from poisson behavior if there is a zero frequency component to the process s power spectrum a comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias

nonergodicity of a time series obeying levy statistics time averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied such a process called a levy walk describes dynamical behaviors of many physical systems fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example when the mean sojourn time diverges the process is non ergodic in that case the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function instead it remains random even in the limit of long measurement time several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges and favorably compared to monte carlo simulations nonergodicity of the power spectrum of the process is briefly discussed and a nonstationary wiener khintchine theorem relating the correlation functions and the power spectrum is presented the considered situation is in full contrast to the usual assumptions of ergodicity and stationarity

renormalization group and singular perturbations multiple scales boundary layers and reductive perturbation theory perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis with numerous examples we illustrate its application to ordinary differential equation problems involving multiple scales boundary layers with technically difficult asymptotic matching and wkb analysis in contrast to conventional methods the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally although the latter can be reproduced if desired by appropriate expansion of the renormalization group approximant we show that the renormalization group equation may be interpreted as an amplitude equation and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points deriving both amplitude equations and the center manifold s1063651x 96 00506 5

the modularity condition for semi t operators abstract the aim of this paper is mainly to solve the functional equations given by the modularity condition several years ago the modularity equations were discussed in families of certain commutative operations such as t norms t conorms uninorms 2 uninorms and t operators in this study we continue the investigation of this same topic by focusing on semi t operators which are generalization of t operators by omitting commutativity our results are not only the full characterization of this modularity condition for semi t operators with continuous underlying operators but also generalize the case for t operators under certain conditions

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

semi t operators on bounded lattices abstract we study semi t operators on bounded lattices and show the existence of semi t operators using the fact that there exist some pseudo t norms and pseudo t conorms on the arbitrary fixed bounded lattice l as a by product some constructive approaches to semi t operators are discussed using these approaches we obtain the least semi t operators and the greatest semi t operators on bounded lattices a partial order relation is also defined based on the semi t operator and some properties of this relation are examined

statistics of return intervals in long term correlated records we consider long term correlated data with several distribution densities gaussian exponential power law and log normal and various correlation exponents ensuremath gamma 0l ensuremath gamma l1 and study the statistics of the return intervals r j between events above some threshold q we show that irrespective of the distribution the return intervals are long term correlated in the same way as the original record but with additional uncorrelated noise due to this noise the correlations are difficult to observe by the detrended fluctuation analysis which exhibits a crossover behavior but show up very clearly in the autocorrelation function the distribution p q r of the return intervals is characterized at large scales by a stretched exponential with exponent ensuremath gamma and at short scales by a power law with exponent ensuremath gamma ensuremath 1 we discuss in detail the occurrence of finite size effects for large threshold values for all considered distributions we show that finite size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 10 6 data points finally in order to quantify the clustering of extreme events due to the long term correlations in the return intervals we study the conditional distribution function and the related moments we find that they show pronounced memory effects irrespective of the distribution of the original data

on the associativity functional equation abstract let a b be any bounded closed real interval the class of all continuous nondecreasing associative functions m a b 2 a b fulfilling the boundary conditions m a a a and m b b b is described

distributivity between semi t operators and semi nullnorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this paper we describe the solutions of distributivity between semi t operators and semi nullnorms previous results about distributivity between nullnorms can be obtained as simple corollaries

left and right distributivity equations for semi t operators and uninorms abstract recently the distributivity equation for different kind of operators has become a focus of research as the crucial tool in the numerous applications as utility and optimization theory or integration theory the main purpose of this paper is to study the distributivity equations between two special classes of aggregation operators namely semi t operators and uninorms without the commutativity assumption of semi t operators it is necessary to consider the left and the right distributivity conditions separately in both cases the obtained solutions are significantly different

characterization of structural reorganization in rice piles diagnostics applied to a rice pile cellular automaton reveal different mechanisms producing power law behaviors of statistical attributes of grains which are germane to self organised critical phenomena the probability distributions for these quantities can be derived from two distinct random walk models that account for correlated clustered behavior through incorporating fluctuations in the number of steps in the walk the first model describes the distribution for a spatial quantity the resultant flight length of grains this has a power law tail caused by grains moving through a discrete power law distributed number of random steps of finite length developing this model into a random walk obtains distributions for the resultant flight length with characteristics similar to l evy distributions the second random walk model is devised to explain a temporal quantity the distribution of trapping or residence times of grains at single locations in the pile diagnostics reveal that the trapping time can be constructed as a sum of subtrapping times which are described by a l evy distribution where the number of terms in the sum is a discrete random variable accurately described by a negative binomial distribution the infinitely divisible two parameter limit distribution for the resultant of such a random walk is discussed and describes a dual scale power law behavior if the number fluctuations are strongly clustered the form for the distribution of transit times of grains results as a corollary

statistical properties of volatility return intervals of chinese stocks abstract the statistical properties of the return intervals q between successive 1 min volatilities of 30 liquid chinese stocks exceeding a certain threshold q are carefully studied the kolmogorov smirnov ks test shows that 12 stocks exhibit scaling behaviors in the distributions of q for different thresholds q furthermore the ks test and weighted ks test show that the scaled return interval distributions of 6 stocks out of the 12 stocks can be nicely fitted by a stretched exponential function f e with 0 31 under the significance level of 5 where is the mean return interval the investigation of the conditional probability distribution p q 0 and the mean conditional return interval 0 demonstrates the existence of short term correlation between successive return interval intervals we further study the mean return interval 0 after a cluster of n intervals and the fluctuation f l using detrended fluctuation analysis and find that long term memory also exists in the volatility return intervals

return times for stochastic processes with power law scaling an analytical study of the return time distribution of extreme events for stochastic processes with power law correlation has been carried out the calculation is based on an expansion in the correlation exponent c t ensuremath mid t ensuremath mid ensuremath 1 the fixed point of the theory is associated with stretched exponential scaling of the distribution analytical expressions have been provided in the preasymptotic regime also the permanence time distribution appears to be characterized by stretched exponential scaling the conditions for application of the theory to non gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed

multifractal properties of return time statistics the global statistics of the return times of a dynamical system can be described by a new spectrum of generalized dimensions comparison with the usual multifractal analysis of measures is presented and the difference between the two corresponding sets of dimensions is established theoretical analysis and numerical examples of dynamical systems in the class of iterated functions are presented

on the conditional distributivity of continuous semi t operators over uninorms abstract the issue of conditional distributivity or how it is also called restricted distributivity which is a form of relaxed distributivity on the restricted domain is crucial for many different areas such as utility theory and integration theory the focus of this paper is on this specific form of distributivity for a continuous semi t operator with respect to a continuous t conorm and for a continuous semi t operator with respect to a uninorm of the form u min or u max with continuous underlying t norm and t conorm

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

a classical measure of evidence for general null hypotheses abstract in science the most widespread statistical quantities are perhaps p values a typical advice is to reject the null hypothesis h 0 if the corresponding p value is sufficiently small usually smaller than 0 05 many criticisms regarding p values have arisen in the scientific literature the main issue is that in general optimal p values based on likelihood ratio statistics are not measures of evidence over the parameter space here we propose an objective measure of evidence for very general null hypotheses that satisfies logical requirements i e operations on the subsets of that are not met by p values e g it is a possibility measure we study the proposed measure in the light of the abstract belief calculus formalism and we conclude that it can be used to establish objective states of belief on the subsets of based on its properties we strongly recommend this measure as an additional summary of significance tests at the end of the paper we give a short listing of possible open problems

dirac operators with a spherically symmetric δ shell interaction dirac operators with a contact interaction supported by a sphere are studied restricting attention to the operators that are rotationally and space reflection symmetric the partial wave operators are constructed using the self adjoint extension theory a particular attention being paid to those among them that can be interpreted as function shells of scalar and vector nature the class of interactions for which the sphere becomes impenetrable is specified and spectral properties of the obtained hamiltonians are discussed

distribution of first return times in correlated stationary signals we present an analytical expression for the first return time frt probability density function of a stationary correlated signal precisely we start by considering a stationary discrete time ornstein uhlenbeck ou process with exponential decaying correlation function the first return time distribution for this process is derived by adopting a well known formalism typically used in the study of the frt statistics for nonstationary diffusive processes then by a subordination approach we treat the case of a stationary process with power law tail correlation function and diverging correlation time we numerically test our findings obtaining in both cases a good agreement with the analytical results we notice that neither in the standard ou nor in the subordinated case a simple form of waiting time statistics like stretched exponential or similar can be obtained while it is apparent that long time transient may shadow the final asymptotic behavior

zero crossing statistics for non markovian time series in applications spanning from image analysis and speech recognition to energy dissipation in turbulence and time to failure of fatigued materials researchers and engineers want to calculate how often a stochastic observable crosses a specific level such as zero at first glance this problem looks simple but it is in fact theoretically very challenging and therefore few exact results exist one exception is the celebrated rice formula that gives the mean number of zero crossings in a fixed time interval of a zero mean gaussian stationary process in this study we use the so called independent interval approximation to go beyond rice s result and derive analytic expressions for all higher order zero crossing cumulants and moments our results agree well with simulations for the non markovian autoregressive model

left and right distributivity equations in the class of semi t operators abstract the issue of distributivity is crucial for many different theoretical areas such as the utility theory and the integration theory also this problem is of high interest for modelling some practical situations from e g fields of economics and social sciences thus recently the distributivity equations have been studied for different classes of aggregation operators by a significant number of researchers the focus of this paper in particular is on solving the left and the right distributivity equations between semi t operators

exact occupation time distribution in a non markovian sequence and its relation to spin glass models we compute exactly the distribution of the occupation time in a discrete non markovian toy sequence that appears in various physical contexts such as the diffusion processes and ising spin glass chains the non markovian property makes the results nontrivial even for this toy sequence the distribution is shown to have non gaussian tails characterized by a nontrivial large deviation function that is computed explicitly an exact mapping of this sequence to an ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model at a given temperature what is the distribution over disorder of the thermally averaged number of spins that are aligned to their local fields we show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the sherrington kirkpatrick model with gaussian disorder

power law distributed level crossings define fractal behavior a relationship is established between the autocorrelation function of continuous gaussian and non gaussian stochastic processes and the discrete process that describes their zero or level crossings random fractals occur when the distribution for the number of crossings is described by a class of markov processes whose singlefold statistics are the discrete analog of the levy stable continuous probability densities

relations between lagrangian models and synthetic random velocity fields the authors propose an alternative interpretation of markovian transport models based on the well mixed condition in terms of the properties of a random velocity field with second order structure functions scaling linearly in the space time increments this interpretation allows direct association of the drift and noise terms entering the model with the geometry of the turbulent fluctuations in particular the well known nonuniqueness problem in the well mixed approach is solved in terms of the antisymmetric part of the velocity correlations its relation with the presence of nonzero mean helicity and other geometrical properties of the flow is elucidated the well mixed condition appears to be a special case of the relation between conditional velocity increments of the random field and the one point eulerian velocity distribution allowing generalization of the approach to the transport of nontracer quantities application to solid particle transport leads to a model satisfying in the homogeneous isotropic turbulence case all the conditions on the behavior of the correlation times for the fluid velocity sampled by the particles in particular correlation times in the gravity and in the inertia dominated case respectively longer and shorter than in the passive tracer case in the gravity dominated case correlation times longer for velocity components along gravity than for the perpendicular ones the model produces in channel flow geometry particle deposition rates in agreement with experiments

a t norm embedding theorem for fuzzy sets it is well known that the class of upper semicontinuous normal convex fuzzy sets with compact supports can be embedded isometrically as a complete convex cone in a banach space we prove an analogous result for a subclass of fuzzy sets that is free from the normality limitation by exchanging the standard algebraic operations on fuzzy sets with operations based on strict t norms this allows us to investigate a new notion of fuzzy convexity that we call t convexity we show that the class of upper semicontinuous fuzzy t convex sets with nonempty compact supports can be embedded as a closed convex cone in a banach space this implies that fuzzy t convex sets satisfy the cancellation law we discuss a possible application of the embedding theorem in mathematical morphology

occupation times of the ornstein uhlenbeck process functional pca and evidence from electricity prices abstract we discuss the functional principal component analysis fpca of the occupation times of the ornstein uhlenbeck process for the eigenvalue problem of the covariance operator of the occupation times we derive the corresponding integral equation in the large time limit and we solve numerically for the principal components the formulation applies the path integral approach of feynman and kac the principal components are compared with those from empirical electricity price processes on energy markets the results indicate that fpca of the occupation times is a suitable tool in stochastic energy modeling to generate moderately sized scenario trees

aggregation functions means this two part state of the art overview on aggregation theory summarizes the essential information concerning aggregation issues an overview of aggregation properties is given including the basic classification on aggregation functions in this first part the stress is put on means i e averaging aggregation functions both with fixed arity n ary means and with multiple arities extended means

persistence exponents and the statistics of crossings and occupation times for gaussian stationary processes we consider the persistence probability the occupation time distribution and the distribution of the number of zero crossings for discrete or equivalently discretely sampled gaussian stationary processes gsps of zero mean we first consider the ornstein uhlenbeck process finding expressions for the mean and variance of the number of crossings and the partial survival probability we then elaborate on the correlator expansion developed in an earlier paper g c m a ehrhardt and a j bray phys rev lett 88 070602 2002 to calculate discretely sampled persistence exponents of gsps of known correlator by means of a series expansion in the correlator we apply this method to the processes d n x dt n ensuremath eta t with ng 3 incorporating an extrapolation of the series to the limit of continuous sampling we then extend the correlator method to calculate the occupation time and crossing number distributions as well as their partial survival distributions and the means and variances of the occupation time and number of crossings we apply these general methods to the d n x dt n ensuremath eta t processes for n 1 random walk n 2 random acceleration and larger n and to simple diffusion from random initial conditions in one to three dimensions the results for discrete sampling are extrapolated to the continuum limit where possible

on bisymmetric operators on a finite chain the bisymmetry property is studied for operators defined on a finite chain in particular a characterization of all bisymmetric commutative and smooth binary operators is given it is deduced from this characterization that bisymmetry and associativity are equivalent for finite chains under these additional conditions

conditional distributivity of continuous semi t operators over disjunctive uninorms with continuous underlying t norms and t conorms abstract the issue of conditional distributivity which is also known as restricted distributivity is crucial for many different areas such as utility theory and integration theory in the present study we focus on this specific form of distributivity for continuous semi t operators with respect to disjunctive uninorms with continuous underlying t norms and t conorms this study extends previous research by focusing on a much wider class of uninorms

dual multifractal spectra the multifractal formalism characterizes the scaling properties of a physical density rho as a function of the distance l to each singularity alpha of the field is attributed a fractal dimension for its support f alpha an alternative representation has been proposed by considering the distribution of distances associated to a fixed mass computing these spectra for a multifractal cantor set it is shown that these two approaches are dual to each other and that both spectra as well as the moment scaling exponents are simply related we apply the same inversion formalism to exponents obtained for turbulent statistics in the gledzer ohkitani yamada shell model and observe that the same duality relation holds here

fluctuations in the zeros of differentiable gaussian processes the stochastic point processes formed by the zero crossings or extremal points of differentiable stationary gaussian processes are studied as a function of their autocorrelation function the properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin the distribution for the number of zeros occurring in an asymptotically large interval are approximately negative binomial or binomial depending upon whether the relative variance or fano factor is greater or less than unity the correlation properties of the zeros are such that they are repelled from each other or are antibunched if the autocorrelation function of the gaussian process is characterized by a single scale size but occur in clusters if more than one characteristic scale size is present the intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves when bunching occurs the interval density becomes bimodal indicating the interval sizes within and between the clusters the interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the gaussian process at large delay times poisson distributed fluctuations of the zeros occur only exceptionally but never form a poisson process

recurrence duration statistics and time dependent intrinsic correlation analysis of trading volumes a study of chinese stock indices abstract the trading volume in stock markets is known as an important variable which reflects the liquidity of the financial markets and therefore is regarded to be greatly important for the measurement of market liquidity risk in this work a new concept called recurrence duration is introduced for study of daily trading volumes which is inspired by idea of the volatility duration that was proposed and studied in our previous work the recurrence duration is thought as the shortest passing time that the following days trading volume takes to exceed or go below the current trading volume which is time varying similar to the volatility duration distribution of the price returns the power law function could describe the empirical probability distribution of recurrence durations of trading volumes and their tail distributions can be fitted by two stretched exponential functions further the correlation relationships of trading volumes between chinese stock indices as well as the correlations of recurrence durations are investigated one approach employed is a recently proposed method time dependent intrinsic correlation tdic which is based on the empirical mode decomposition emd to decompose nonlinear and nonstationary signals into the intrinsic mode functions imfs the instantaneous periods of which are used then in determination of the sizes of sliding windows to compute the running correlation coefficients for the multiscale signals the empirical results reveal rich patterns of correlations for both trading volumes and recurrence durations at different scales for different modes another approach is the widely used dcca cross correlation coefficient by which the level of cross correlation is measured for both original series and imf modes of the stock indices

distributivity between semi t operators and mayor s aggregation operators the problem of distributivity was first posed over forty years ago and it has been investigated for families of certain operations such as t norms t conorms uninorms and nullnorms in this paper we investigate this topic further by focusing on mayor s aggregation operators and semi t operators which generalize the t operators by omitting commutativity the results that we obtain are complete and extend previous results concerning distributivity between mayor s aggregation operators and semi nullnorms

mimicking a turbulent signal sequential multiaffine processes an efficient method for the construction of a multiaffine process with prescribed scaling exponents is presented at variance with the previous proposals this method is sequential and therefore it is the natural candidate in numerical computations involving synthetic turbulence the application to the realization of a realistic turbulent like signal is discussed in detail the method represents a first step towards the realization of a realistic spatio temporal turbulent field

distinguished self adjoint extensions of dirac operators via hardy dirac inequalities we prove some hardy dirac inequalities with two different weights including measure valued and coulombic ones those inequalities are used to construct distinguished self adjoint extensions of dirac operators for a class of diagonal potentials related to the weights in the mentioned inequalities

residence time statistics for normal and fractional diffusion in a force field we investigate statistics of occupation times for an over damped brownian particle in an external force field using a backward fokker planck equation introduced by majumdar and comtet for an arbitrary potential field the distribution of occupation times is expressed in terms of solutions of the corresponding first passage time problem this general relationship between occupation times and first passage times is valid for normal markovian diffusion and for non markovian sub diffusion the latter modeled using the fractional fokker planck equation for binding potential fields we find in the long time limit ergodic behavior for normal diffusion while for the fractional framework weak ergodicity breaking is found in agreement with previous results of bel and barkai on the continuous time random walk on a lattice for non binding cases rich physical behaviors are obtained and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite our work establishes a link between fractional calculus and ergodicity breaking