mrc performance for binary signals in nakagami fading with general branch correlation exact expressions are derived for the performance of predetection maximal ratio combiner diversity reception with l correlated branches in nakagami fading bit error rates are evaluated for both coherent and noncoherent binary phase shift keying and frequency shift keying signals starting from the l variate moment generating function of the random input power vector the new formulation presented for the bit error rate in which the covariance matrix of the fading at the l branches explicitly appears allows arbitrary branch correlation to be taken into account for any diversity order in the case of identical fading severity on the branches results are presented for evaluation of the outage probability for integer values of fading severity as well as for the effect of the presence of unbalanced channels with arbitrary correlation

canonical detection in spherically invariant noise the paper deals with the detection of signals with unknown parameters in impulsive noise modeled as a spherically symmetric random process the proposed model subsumes several interesting families of noise amplitude distributions generalized cauchy generalized laplace generalized gaussian contaminated normal it also allows handling of the case of correlated noise by a whitening approach the generalized maximum likelihood decision strategy is adopted resulting in a canonical detector which is independent of the amplitude distribution of the noise a general method for performance evaluation is outlined and a comprehensive performance analysis is carried out for the case of m ary equal energy orthogonal signals under several distributional assumptions for the noise the performance is contrasted with that of the maximum likelihood receiver for completely known signals so as to assess the loss due to the a priori uncertainty as to the signal parameters

numerical methods for high dimensional probability density function equations in this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables these types of equations arise naturally in many different areas of mathematical physics e g in particle systems liouville and boltzmann equations stochastic dynamical systems fokker planck and dostupov pugachev equations random wave theory malakhov saichev equations and coarse grained stochastic systems mori zwanzig equations we propose three different classes of new algorithms addressing high dimensionality the first one is based on separated series expansions resulting in a sequence of low dimensional problems that can be solved recursively and in parallel by using alternating direction methods the second class of algorithms relies on truncation of interaction in low orders that resembles the bogoliubov born green kirkwood yvon bbgky framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations the third class of algorithms is based on high dimensional model representations e g the anova method and probabilistic collocation methods a common feature of all these approaches is that they are reducible to the problem of computing the solution to high dimensional equations via a sequence of low dimensional problems the effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations with up to 120 variables

exact pdf equations and closure approximations for advective reactive transport mathematical models of advection reaction phenomena rely on advective flow velocity and bio chemical reaction rates that are notoriously random by using functional integral methods we derive exact evolution equations for the probability density function pdf of the state variables of the advection reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions these pdf equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly correlated random field our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large eddy diffusivity led closure approximation 1 we find that the solution to the led based pdf equation which is exact for uncorrelated reaction rates is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty

optimum detection of fading signals in impulsive noise this paper deals with the synthesis and the analysis of optimum receivers to detect one out of m equally likely equi energy fading signals in impulsive noise modelled as a compound gaussian possibly correlated process we show that the conventional coherent and incoherent detectors are still optimum independent of the noise as well as the fading probability density functions the performance analysis has been carried on with reference to the general case of arbitrarily distributed disturbance in order to simplify the analysis asymptotical expressions have been developed for high signal to noise ratios as well as high signal space dimensionality interestingly enough this allows separating the effect of the noise spikyness from that of the fading law results indicate that for deep fading the noise marginal distribution does not dramatically affect the error probability nor is it influential on the limit operating characteristics corresponding to infinite signal space dimensions for non fluctuating signals instead the noise distribution plays a primary role spiky noise usually produces performance impairment moreover the limit performance in impulsive disturbance may exhibit marked deviations from the well known stepwise shape which is typical of gaussian channels

intrinsic chaos and external noise in population dynamics abstract we address the problem of the relative importance of the intrinsic chaos and the external noise in determining the complexity of population dynamics we use a recently proposed method for studying the complexity of nonlinear random dynamical systems the new measure of complexity is defined in terms of the average number of bits per time unit necessary to specify the sequence generated by the system this measure coincides with the rate of divergence of nearby trajectories under two different realizations of the noise in particular we show that the complexity of a nonlinear time series model constructed from sheep populations comes completely from the environmental variations however in other situations intrinsic chaos can be the crucial factor this method can be applied to many other systems in biology and physics

not all possibly random sequences are created equal the need to assess the randomness of a single sequence especially a finite sequence is ubiquitous yet is unaddressed by axiomatic probability theory here we assess randomness via approximate entropy apen a computable measure of sequential irregularity applicable to single sequences of both even very short finite and infinite length we indicate the novelty and facility of the multidimensional viewpoint taken by apen in contrast to classical measures furthermore and notably for finite length finite state sequences one can identify maximally irregular sequences and then apply apen to quantify the extent to which given sequences differ from maximal irregularity via a set of deficit defm functions the utility of these defm functions which we show allows one to considerably refine the notions of probabilistic independence and normality is featured in several studies including i digits of e 2 and 3 both in base 2 and in base 10 and ii sequences given by fractional parts of multiples of irrationals we prove companion analytic results which also feature in a discussion of the role and validity of the almost sure properties from axiomatic probability theory insofar as they apply to specified sequences and sets of sequences in the physical world we conclude by relating the present results and perspective to both previous and subsequent studies

the modified cramer rao bound in vector parameter estimation in this paper we extend the scalar modified cramer rao bound mcrb to the estimation of a vector of nonrandom parameters in the presence of nuisance parameters the resulting bound is denoted with the acronym mcrvb where v stands for vector as with the scalar bound the mcrvb is generally looser than the conventional crvb but the two bounds are shown to coincide in some situations of practical interest the mcrvb is applied to the joint estimation of carrier frequency phase and symbol epoch of a linearly modulated waveform corrupted by correlated impulsive noise encompassing white gaussian noise as a particular case wherein data symbols and noise power are regarded as nuisance parameters in this situation calculation of the conventional crvb is infeasible while application of the mcrvb leads to simple useful expressions with moderate analytical effort when specialized to the case of white gaussian noise the mcrvb yields results already available in the literature in fragmentary form and simplified contexts

interference modeling in a direct sequence spread spectrum packet radio network a technique for characterizing multiuser interference and background noise in a direct sequence spread spectrum network is introduced and packet error probabilities are calculated the multiuser interference over a packet in the network is modeled as a compound gaussian multivariate random variable for moderate to large values of the processing gain the conditional variance is dependent on the number of users and their interference powers the method works for any interference with statistics of the block interference having a spherically symmetric distribution the best performance results in terms of the expected total interference power are obtained for the case of a large number of interferers with comparable interference powers as the number of interferers approaches infinity the performance is the same as that for gaussian noise for a small number of interferers the block error probability curve is broader than that for gaussian noise for small values of snr the probability of error is smaller than that corresponding to gaussian noise and for large values of snr it is larger modifications to the basic direct sequence scheme that improve performance in the case of one strong interferer are suggested

numerical approximation of high dimensional fokker planck equations with polynomial coefficients this paper is concerned with the numerical solution of high dimensional fokker planck equations related to multi dimensional diffusion with polynomial coefficients or pearson diffusions classification of multi dimensional pearson diffusion follows from the classification of one dimensional pearson diffusion there are six important classes of pearson diffusion three of them possess an infinite system of moments gaussian gamma beta while the other three possess a finite number of moments inverted gamma student and fisher snedecor numerical approximations to the solution of the fokker planck equation are generated using the spectral method the use of an adaptive reduced basis technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition svd the basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite dimensional subspace of the solution space this is achieved through basis enrichment and projection stages numerical results with different boundary conditions are presented to demonstrate the accuracy and efficiency of the numerical scheme

chaotic and stochastic functions abstract we study chaotic functions that are exact solutions to nonlinear maps a generalization of these functions which cannot be expressed as a recursive procedure anymore can produce truly random sequences even if the initial conditions are known exactly the next values are in principle unpredictable we present a mathematical formulation of an elementary stochastic process several applications are discussed

a survey of full length nonlinear shift register cycle algorithms shift registers have been used to generate sequences of 0 s and 1 s for over thirty years a wide variety of applications has been made of these sequences principally communications have made use of the sequences generated one particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences these sequences are periodic and of length 2 n and all 2 n different binary n tuples appear exactly one time in a periodic portion of the sequence in this paper we discuss various algorithms which have been suggested for generating these sequences

ambient noise statistics signal processing algorithms optimized for gaussian noise may degrade significantly in a non gaussian noise environment therefore it is important to characterize the noise accurately before including its structure in the formulation of algorithms a generic distribution suitable for modeling non gaussian ambient noise in the kurtosis range 1 8

spherically invariant vector random fields in space and time this paper is concerned with spherically invariant or elliptically contoured vector random fields in space and or time which are formulated as scale mixtures of vector gaussian random fields while a spherically invariant vector random field may or may not have second order moments a spherically invariant second order vector random field is determined by its mean and covariance matrix functions just like the gaussian one this paper explores basic properties of spherically invariant second order vector random fields and proposes an efficient approach to develop covariance matrix functions for such vector random fields

a zoo of computable binary normal sequences historically there has been a virtual absence of constructive methods to produce broad classes of certifiably random infinite sequences despite considerable interest in this endeavor previously we proved a theorem that yielded explicit algorithms to produce diverse sets of normal numbers reasonable candidates for random sequences given their limiting equidistribution of subblocks of all lengths herein we develop this algorithmic approach much further systematizing the normal number generation process in several ways we construct delineated distinct sets of normal numbers classified by the extent to which initial segments deviate from maximal irregularity with virtually any allowable specified rate of convergence to 0 of this deviation encompassing arbitrarily fast and slow rates and accommodating asymmetric behavior above or below a centered median as a corollary we provide an explicit construction of a normal number that satisfies the law of the iterated logarithm we also produce distinct families of biased normal numbers with virtually any specified rate of convergence of the bias to 0 this latter theory is in part motivated by the remarkable observation that the binary version of champernowne s number which is also normal is biased any initial segment has more 1s than 0s finally we construct an interesting normal sequence with arbitrarily fast convergence to equidistribution of singleton blocks yet arbitrarily slow convergence of pairs which has profound implications both for probability theory and for metrics to evaluate the near randomness of sequences

performance of ds ssma communications in impulsive channels part i linear correlation receivers the performance of digital linear correlation receivers is studied in a multiuser environment there are assumed to be two types of sources interfering with data transmission multiple access interference and additive channel noise which is attributed to impulsive noise sources in the environment the contribution of multiple access interference is examined by considering k asynchronous users transmitting simultaneously over a linear channel using the binary psk direct sequence spreadspectrum multiple access ds ssma technique alternatively the effects of the non gaussian impulsive channel in such a system are studied by modeling the samples of noise after front end filtering errorprobability performance under these conditions is compared to that for additive white gaussian noise awgn channels due to computational complexity exact analysis is limited here to systems utilizing short spreading sequences computationally simple methods are proposed for approximating the average error probability when the length of the signature sequences is large furthermore some asymptotic results are obtained for the case of infinitely long sequences in all cases performance variation is examined as the shape of the noise density varies with snr held constant the results of this analysis indicate that the presence of impulsive noise can cause significant performance degradation over that predicted from an awgn model even when the total noise power does not increase

non gaussian clutter modeling with generalized spherically invariant random vectors this paper describes the modeling of non gaussian clutter with a set of generalized spherically invariant random vectors sirv s the generalization extends the traditional model to account for dependence between successive sirv realizations significant properties of generalized sirv s are derived as well as a closed form expression for a family of generalized sirv density functions the density underlying recorded sonar reverberation is approximated with this function through appropriate choice of a shape parameter given this reverberation model the optimum detector is derived from the generalized sirv density likelihood ratio this paper concludes by showing how applying this optimum detector to non gaussian data leads to a reduction in the false alarm rate when compared to processing with a matched filter alone

a recipe for randomness despite many diverse theories that address closely related themes e g probability theory algorithmic complexity cryptoanalysis and pseudorandom number generation a near void remains in constructive methods certified to yield the desired random output herein we provide explicit techniques to produce broad sets of both highly irregular finite and normal infinite sequences based on constructions and properties derived from approximate entropy apen a computable formulation of sequential irregularity furthermore for infinite sequences we considerably refine normality by providing methods for constructing diverse classes of normal numbers classified by the extent to which initial segments deviate from maximal irregularity

detection of weak random signals in iid non gaussian noise this paper considers the detection of weak random signals in circularly symmetric independent identically distributed noise locally optimum detectors and ad hoc nonlinearities are considered with asymptotic expressions provided for evaluation of detection performance the analytical expressions are used to evaluate the robustness of detectors to mismatch in the noise models finite sample monte carlo simulation results indicate the reliability of these asymptotic measures in cases of practical interest the results show that as has been found for detection of weak known signals in non gaussian noise reasonably configured ad hoc nonlinearities are nearly optimum and robust to modest errors in the noise statistics

higher order dangers and precisely constructed taxa in models of randomness the certification construction and delineation of individual infinite length random sequences have been longstanding yet incompletely resolved problems we address this topic via the study of normal numbers which often have been viewed as reasonable proxies for randomness given their limiting equidistribution of subblocks of all lengths however limitations arise within this perspective first we explicitly construct a normal number that satisfies the law of the iterated logarithm yet exhibits pairwise bias toward repeated values rendering it inappropriate for any collection of random numbers accordingly we deduce that the evaluation of higher order block dynamics even beyond limiting equidistribution and fluctuational typicality is imperative in proper evaluation of sequential randomness second we develop several criteria motivated by classical theorems for symmetric random walks which lead to algorithms for generating normal numbers that satisfy a variety of attributes for the series of initial partial sums including rates of sign changes patterns of return times to 0 and the extent of fairness of the sequence such characteristics generally are unaddressed in most evaluations of randomness more broadly we can differentiate normal numbers both on the basis of multiple distinct qualitative attributes and quantitatively via a spectrum of rates within each attribute furthermore we exhibit a toolkit of techniques to construct normal sequences that realize diverse a priori specifications including profound biases overall we elucidate the vast diversity within the category of normal sequences

new evolution equations for the joint response excitation probability density function of stochastic solutions to first order nonlinear pdes by using functional integral methods we determine new evolution equations satisfied by the joint response excitation probability density function pdf associated with the stochastic solution to first order nonlinear partial differential equations pdes the theory is presented for both fully nonlinear and for quasilinear scalar pdes subject to random boundary conditions random initial conditions or random forcing terms particular applications are discussed for the classical linear and nonlinear advection equations and for the advection reaction equation by using a fourier galerkin spectral method we obtain numerical solutions of the proposed response excitation pdf equations these numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi element probabilistic collocation methods it is found that the response excitation approach yields accurate predictions of the statistical properties of the system in addition it allows to directly ascertain the tails of probabilistic distributions thus facilitating the assessment of rare events and associated risks the computational cost of the response excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the pde is subject to high dimensional random boundary or initial conditions the question of high dimensionality for evolution equations involving multidimensional joint response excitation pdfs is also addressed

performance of optimum and suboptimum receivers in the presence of impulsive noise modeled as an alpha stable process impulsive noise bursts in communication systems are traditionally handled by incorporating in the receiver a limiter which clips the received signal before integration an empirical justification for this procedure is that it generally causes the signal to noise ratio to increase recently very accurate models of impulsive noise were presented based on the theory of symmetric spl alpha stable probability density functions we examine the performance of optimum receivers designed to detect signals embedded in impulsive noise which is modeled as an infinite variance symmetric spl alpha stable process and compare it against the performance of several suboptimum receivers as a measure of receiver performance we compute an asymptotic expression for the probability of error for each receiver and compare it to the probability of error calculated by extensive monte carlo simulation

a fast solver for fokker planck equation applied to viscoelastic flows calculations 2d fene model in this paper we introduce a new method for solving efficiently the fokker planck equation arising in the simulation of dilute polymeric solutions for a two dimensional fene finitely extensible nonlinear elastic model the structure of the fokker planck equation is used to design a fast solver the resulting equations are discretized with a spectral spectral element method application of the method to flow of a fene fluid past a confined cylinder demonstrates readily the advantages of the proposed scheme over traditional stochastic simulations

optimum diversity detection over fading dispersive channels with non gaussian noise we consider the problem of m ary signal detection over a single input multiple output simo channel affected by frequency dispersive rayleigh distributed fading and corrupted by additive non gaussian noise modeled as a spherically invariant random process we derive both the optimum detection structure and a suboptimal reduced complexity receiver based on the low energy coherence approach interestingly both detection structures are canonical i e they are independent of the actual noise statistics we also carry out a performance analysis of both receivers with reference to the case that the channel is affected by a frequency selective fading and for a binary frequency shift keying signaling format the results obtained through both a chernoff bounding technique and monte carlo simulations reveal that the adoption of diversity also represents a suitable means to restore performance in the presence of dispersive fading and impulsive non gaussian noise interestingly it is also shown that the suboptimal receiver incurs a limited loss with respect to the optimum unrealizable receiving structure

non gaussian noise models in signal processing for telecommunications new methods an results for class a and class b noise models the subject here is generalized i e non gaussian noise models and specifically their first order probability density functions pdfs attention is focused primarily on the author s canonical statistical physical class a and class b models in particular class a noise describes the type of electromagnetic interference emi often encountered in telecommunication applications where this ambient noise is largely due to other intelligent telecommunication operations on the other hand ambient class b noise usually represents man made or natural nonintelligent i e nonmessage bearing noise and is highly impulsive class a noise is not an spl alpha stable process nor is it reducible to such except in the limiting gaussian cases of high density noise by the central limit theorem class b noise is also asymptotically normal before model approximation under rather broad conditions principally governed by the source propagation and distribution scenarios the pdf of class b noise alone no gaussian component can usually be approximated by 1 a symmetric gaussian spl alpha stable s spl alpha s model in the case of narrowband reception or when the pdf spl omega sub 1 spl alpha of the amplitude is symmetric and 2 a nonsymmetric spl alpha stable ns spl alpha s model no gaussian component can be constructed in broadband regimes new results here include i counting functional methods for constructing the general qth order characteristic functions cfs of class a and class b noise from which all moments and in principle the pdfs follow ii the first order cfs pdfs and cumulative probabilities apds of nonsymmetric broadband class b noise extended to include additive gauss noise agn iii proof of the existence of all moments in the basic class a and class b models iv the key physical role of agn and the fact that agn removes spl alpha stability v the explicit roles of the propagation and distribution scenarios and vi extension to noise fields although telecommunication applications are emphasized class a and class b noise models apply selectively but equally well to other physical regimes e g underwater acoustics and em radar optics etc supportive empirical data are included

fuzzy dynamical system approach for a dual parameter hybrid order robust control design abstract robust control for uncertain dynamical systems is considered the uncertainty is assumed to be bounded and the fuzzy set theory is adopted to describe the bound a new dual parameter hybrid order robust control design is then proposed in two steps first a class of robust controls is designed taking the advantage of the flexibility in choosing the order and a design parameter the controls are deterministic and are not if then fuzzy rules based deterministic performance including uniform boundedness and uniform ultimate boundedness for the controlled systems is guaranteed second a fuzzy set theoretic based performance index consisting of average and steady state system performance and control cost is proposed a constrained optimization problem is formulated with the intension of minimizing the performance index the objective of the optimization problem is to search for the optimal choice of dual parameters that minimizes the performance index we show that the solution to the optimization problem always exists as a result the resulting control design achieves two concurrent goals guaranteeing deterministic performance as well as minimizing the performance index the theory is illustrated by an unmanned aerial vehicle uav system control design

galerkin finite element approximation of symmetric space fractional partial differential equations abstract in this paper symmetric space fractional partial differential equations ssfpde with the riesz fractional operator are considered the ssfpde is obtained from the standard advection dispersion equation by replacing the first order and second order space derivatives with the riesz fractional derivatives of order 2 0 1 and 2 1 2 respectively we prove that the variational solution of the ssfpde exists and is unique using the galerkin finite element method and a backward difference technique a fully discrete approximating system is obtained which has a unique solution according to the lax milgram theorem the stability and convergence of the fully discrete schemes are derived finally some numerical experiments are given to confirm our theoretical analysis

high order approximation of pearson diffusion processes this paper focuses on pearson diffusions and the spectral high order approximation of their related fokker planck equations the pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient they are widely used in the physical and chemical sciences engineering rheology environmental sciences and financial mathematics in recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations analytical solutions have been derived for some of the pearson diffusions including the ornstein uhlenbeck cox ingersoll ross and jacobi processes however analytical investigations and computations for diffusions with so called heavy tailed ergodic distributions are more difficult to perform the novelty of this research is the development of an accurate and efficient numerical method to solve the fokker planck equations associated with pearson diffusions with different boundary conditions comparisons between the numerical predictions and available time dependent and equilibrium analytical solutions are made the solution of the fokker planck equation is approximated using a reduced basis spectral method the advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving fokker planck type equations

wavelet discretizations of parabolic integrodifferential equations we consider parabolic problems dot u au f in 0 t times omega t infty where omega subset mathbb r d is a bounded domain and a is a strongly elliptic classical pseudodifferential operator of order rho in 0 2 in tilde h rho 2 omega we use a theta scheme for time discretization and a galerkin method with n degrees of freedom for space discretization the full galerkin matrix for a can be replaced with a sparse matrix using a wavelet basis and the linear systems for each time step are solved approximatively with gmres we prove that the total cost of the algorithm for m time steps is bounded by o mn log n beta operations and o n log n beta memory we show that the algorithm gives optimal convergence rates up to logarithmic terms for the computed solution with respect to l2 in time and the energy norm in space

an efficient technique for simulations of viscoelastic flows derived from the brownian configuration field method in this paper we present a new approach for calculating viscoelastic flows this method is derived from the brownian configuration field method m a hulsen a p g van heel and b h a a van den brule j non newtonian fluid mech 70 1997 pp 79 101 for the simulation of flows of dilute polymeric solutions the method presented here has the same attractive features as those in a previous work c chauviere siam j sci comput 23 2002 pp 2123 2140 i e improved stability in comparison with traditional methods starting from closed form constitutive equations moreover the method developed here has the additional advantage of being significantly more cpu efficient results are presented by solving the benchmark problem of the flow of an oldroyd b fluid past a cylinder in a channel using a spectral element method to demonstrate the advantages of the proposed method

a numerical solution for multichannel detection a numerical approach to the vector signal detection problem is proposed by using an approximate series representation for vector complex valued random processes this technique allows us to provide computationally feasible suboptimum receivers for the problem of detecting an improper or proper complex signal in additive white noise its implementation requires the knowledge of the correlation matrix of the vector process involved the performance of the technique devised is assessed by some communication examples

high order numerical methods for the riesz space fractional advection dispersion equations abstract in this paper we propose high order numerical methods for the riesz space fractional advection dispersion equations rsfade on a finite domain the rsfade is obtained from the standard advection dispersion equation by replacing the first order and second order space derivative with the riesz fractional derivatives of order 0 1 and 1 2 respectively firstly we utilize the weighted and shifted grunwald difference operators to approximate the riesz fractional derivative and present the finite difference method for the rsfade specifically we discuss the crank nicolson scheme and solve it in matrix form secondly we prove that the scheme is unconditionally stable and convergent with the accuracy of o 2 h 2 thirdly we use the richardson extrapolation method rem to improve the convergence order which can be o 4 h 4 finally some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent with theoretical analysis

randomness and degrees of irregularity the fundamental question are sequential data random arises in myriad contexts often with severe data length constraints furthermore there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness e g to evaluate the efficacy of therapy in medicine we address both these issues from a computable framework via a quantification of regularity apen approximate entropy defining maximal randomness for sequences of arbitrary length indicating the applicability to sequences as short as n 5 points an infinite sequence formulation of randomness is introduced that retains the operational and computable features of the finite case in the infinite sequence setting we indicate how the foundational definition of independence in probability theory and the definition of normality in number theory reduce to limit theorems without rates of convergence from which we utilize apen to address rates of convergence of a deficit from maximal randomness refining the aforementioned concepts in a computationally essential manner representative applications among many are indicated to assess i random number generation output ii well shuffled arrangements and iii the quality of bootstrap replicates

optimum detection over rayleigh fading dispersive channels with non gaussian noise the paper discusses the problem of detecting one out of m gaussian correlated signals in impulsive noise modeled as a compound gaussian possibly correlated process we first show that for uncorrelated noise the detection problem admits one and the same optimum in the sense of attaining minimum error probability solution independent of the noise statistics the optimum detector in fact amounts to an estimation block aimed at measuring the short time noise power spectral density psd whose output is fed to a bank of estimators correlators each keyed to one of the m admissible waveforms and to the estimated noise psd we also give suggestions for realizing a suboptimum receiver with reduced complexity which again is canonical in its structure as for the performance analysis we focus on binary frequency shift keying bfsk signaling we provide numerical results for two particular channel correlations and for cauchy distributed noise these results indicate that like for the general gauss gauss case the performance depends on the energy contrast as well as on the time bandwidth product of the useful signal moreover noise spikiness seems to negatively affect the performance in the sense that heavier and heavier high amplitude tails of the noise marginal distribution give rise to higher and higher error probabilities for fixed energy contrast and time bandwidth signal product

on the solution of the fokker planck equation using a high order reduced basis approximation the numerical solution of the one dimensional fokker planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented the finitely extensible non linear elastic fene model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique this technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition svd the basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite dimensional subspace of the solution space this is achieved through basis enrichment and projection the reduced basis method is extended to the high dimensional fokker planck equation using the d dumbbell fene model by consideration of a high dimensional singular value decomposition some numerical results are presented to demonstrate the efficiency of the numerical scheme