hyperasymptotics for nonlinear odes i a riccati equation we illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations the example is a riccati equation the main tools that we need are transseries expansions and the riemann sheet structure of the borel transform of the divergent asymptotic expansions hyperasymptotic expansions determine the solutions uniquely a numerical illustration is included

percolation under noise detecting explosive percolation using the second largest component we consider the problem of distinguishing between different rates of percolation under noise a statistical model of percolation is constructed allowing for the birth and death of edges as well as the presence of noise in the observations this graph valued stochastic process is composed of a latent and an observed nonstationary process where the observed graph process is corrupted by type i and type ii errors this produces a hidden markov graph model we show that for certain choices of parameters controlling the noise the classical erd s renyi percolation is visually indistinguishable from a more rapid form of percolation in this setting we compare two different criteria for discriminating between these two percolation models based on the interquartile range iqr of the first component s size and on the maximal size of the second largest component we show through data simulations that this second criterion outperforms the iqr of the first component s size in terms of discriminatory power the maximal size of the second component therefore provides a useful statistic for distinguishing between different rates of percolation under physically motivated conditions for the birth and death of edges and under noise the potential application of the proposed criteria for the detection of clinically relevant percolation in the context of applied neuroscience is also discussed

information transfer network of global market indices abstract we study the topological properties of the information transfer networks itn of the global financial market indices for six different periods itn is a directed weighted network in which the direction and weight are determined by the transfer entropy between market indices by applying the threshold method it is found that itn undergoes a crossover from the complete graph to a small world sw network sw regime of itn for a global crisis is found to be much more enhanced than that for ordinary periods furthermore when itn is in sw regime the average clustering coefficient is found to be synchronized with average volatility of markets we also compare the results with the topological properties of correlation networks

regularity of the hamiltonian along optimal trajectories this paper concerns state constrained optimal control problems in which the dynamic constraint takes the form of a differential inclusion if the differential inclusion does not depend on time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is independent of time if the differential inclusion is lipschitz continuous then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is lipschitz continuous these two well known results are examples of the following principle the hamiltonian evaluated along the optimal state trajectory and the co state trajectory inherits the regularity properties of the differential inclusion regarding its time dependence we show that this principle also applies to another kind of regularity if the differential inclusion has bounded variation w r t time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory has bounded variation two application

critical transitions and perturbation growth directions critical transitions occur in a variety of dynamical systems here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions as suitable indicator variables for critical transitions we consider changes in growth rates and directions of covariant lyapunov vectors studying critical transitions in several models of fast slow systems i e a network of coupled fitzhugh nagumo oscillators models for josephson junctions and the hindmarsh rose model we find that tangencies between covariant lyapunov vectors are a common and maybe generic feature during critical transitions we further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves in the presence of noise we find the alignment of covariant lyapunov vectors and changes in finite time lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as e g finite time estimates of the variance additionally we propose a new method for estimating approximations of covariant lyapunov vectors without knowledge of the future trajectory of the system we find that these approximated covariant lyapunov vectors can also be applied to predict critical transitions

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

scaling and memory in volatility return intervals in financial markets abstract for both stock and currency markets we study the return intervals between the daily volatilities of the price changes that are above a certain threshold q we find that the distribution function pq scales with the mean return interval as the scaling function f x is similar in form for all seven stocks and for all seven currency databases analyzed and f x is consistent with a power law form f x x with 2 we also quantify how the conditional distribution pq 0 depends on the previous return interval 0 and find that small or large return intervals are more likely to be followed by small or large return intervals this clustering of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long term correlations known to be present in the volatility econophysics fluctuations extreme values long term correlations long term memory

the semigroup property of value functions in lagrange problems the lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form the binary operation of inf composition applied to the value functions of a lagrange problem equates the principle of optimality with a semigroup property this paper finds the infinitesimal generator of the semigroup by differentiating at t 0 the type of limit is epigraphical convergence in a uniform sense moreover the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed

exponential asymptotics and boundary value problems keeping both sides happy at all orders we introduce templates for exponential asymptotic expansions that in contrast to matched asymptotic approaches enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter graphic 1 and have a single boundary layer at one end of the interval for linear equations the template is a transseries that takes the form of a sliding ladder of exponential scales for nonlinear equations the transseries template is a two dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed an exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of terms in the series beyond all orders in the nonlinear case we also demonstrate how these transseries can be resummed to generate multiple scales type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion 1 embed inline graphic 1 gif

coupled points in the calculus of variations and applications to periodic problems the aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions and to develop second order necessary conditions for optimality when one of the end points is fixed our necessary conditions reduce to the known ones involving conjugate points we also apply our results to the periodic problems of the calculus of variations

the classical problem of the calculus of variations in the autonomous case relaxation and lipschitzianity of solutions we consider the problem of minimizing b a l x t x t dt x a a x b b under the assumption that the lagrangian l is continuous and satisfies a growth assumption that does not imply superlinear growth we provide a result on the relaxation of the functional and show that a solution to the minimum problem is lipschitzian

path probability of stochastic motion a functional approach the path probability of a particle undergoing stochastic motion is studied by the use of functional technique and the general formula is derived for the path probability distribution functional the probability of finding paths inside a tube band the center of which is stipulated by a given path is analytically evaluated in a way analogous to continuous measurements in quantum mechanics then the formalism developed here is applied to the stochastic dynamics of stock price in finance

receiver operating characteristic curves for a simple stochastic process that carries a static signal the detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic roc curve in which the probability of correct detection is plotted against the probability for a false positive this kind of analysis is typically applied to the situation in which signal and noise are stochastic variables the detection problem emerges however also often in a context in which both signal and noise are stochastic processes and the correct or false detection is said to take place when the process crosses a threshold in a given time window here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process the well known ornstein uhlenbeck process we give exact but difficult to evaluate quadrature expressions for the detection rates for false positives and correct detections investigate systematically a simple sampling approximation suggested earlier compare to an approximation by stratonovich for the limit of high threshold and briefly explore the case of multiplicative signal all theoretical results are compared to extensive numerical simulations of the corresponding langevin equation our results demonstrate that the sampling approximation provides a reasonable description of the roc curve for this system and it clarifies limit cases for the roc curve

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

probabilities for large events in driven threshold systems many driven threshold systems display a spectrum of avalanche event sizes often characterized by power law scaling an important problem is to compute probabilities of the largest events black swans we develop a data driven approach to the problem by transforming to the event index frame and relating this to shannon information for earthquakes we find the 12 month probability for magnitude m 6 earthquakes in california increases from about 30 after the last event to 40 50 prior to the next one

moments of the first passage time under external driving a general theory is derived for the moments of the first passage time of a one dimensional markov process in the presence of a weak time dependent forcing the linear corrections to the moments can be expressed by quadratures of the potential and of the time dependent probability density of the unperturbed system or equivalently by its laplace transform if none of the latter functions is known the derived formulas may still be useful for specific cases including a slow driving or a driving with power at only small or large times in the second part of the paper explicit expressions for the mean and variance of the first passage time are derived for the cases of a linear or a parabolic potential and an exponentially decaying driving force the analytical results are found to be in excellent agreement with computer simulations of the respective first passage processes the particular examples furthermore demonstrate that already the effect of a simple exponential driving can be fairly involved implying a nontrivial nonmonotonic behavior of mean and variance as functions of the driving s time scale

relaxation of multiple integrals in the space bv ω rp a characterisation of the surface energy density for the relaxation in v r p of the functional is obtained a lemma of de giorgi is used to modify a sequence near the boundary without increasing its total energy

hyperasymptotics and hyperterminants exceptional cases a new method is introduced for the computation of hyperterminants it is based on recurrence relations and can also be used to compute the parameter derivatives of the hyperterminants these parameter derivatives are needed in hyperasymptotic expansions in exceptional cases numerical illustrations and an application are included

hyperasymptotics for nonlinear odes ii the first painleve equation and a second order riccati equation this paper is a sequel to olde daalhuis olde daalhuis 2005 proc r soc a 461 2503 2520 in which we constructed hyperasymptotic expansions for a simple first order riccati equation in this paper we illustrate that the method also works for more complicated nonlinear ordinary differential equations and that in those cases the riemann sheet structure of the so called borel transform is much more interesting the two examples are the first painleve equation and a second order riccati equation the main tools that we need are transseries expansions and stokes multipliers hyperasymptotic expansions determine the solutions uniquely some details are given about solutions that are real valued on the positive real axis

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

evaluation of stochastic resonance based detectors of weak harmonic signals in additive white gaussian noise a thorough evaluation of stochastic resonance in the framework of statistical detection theory is presented both as a nonlinear signal preprocessor and as a detector the pertinent receiver operating characteristics are compared with those of the known statistically optimum detector using extensive monte carlo simulations parameter optimization and computational budget aspects are discussed

neural population coding is optimized by discrete tuning curves the sigmoidal tuning curve that maximizes the mutual information for a poisson neuron or population of poisson neurons is obtained the optimal tuning curve is found to have a discrete structure that results in a quantization of the input signal the number of quantization levels undergoes a hierarchy of phase transitions as the length of the coding window is varied we postulate using the mammalian auditory system as an example that the presence of a subpopulation structure within a neural population is consistent with an optimal neural code

bolza problems with discontinuous lagrangians and lipschitz continuity of the value function we study the local lipschitz continuity of the value function v associated with a bolza problem in the presence of a lagrangian l x q convex and uniformly superlinear in q but only borel measurable in x under these assumptions the associated integral functional is not lower semicontinuous with respect to the suitable topology which ensures the existence of minimizers so all results known in the literature fail to apply yet the lipschitz regularity of v does not depend on the existence of minimizers in fact it is enough to control the derivatives of quasi minimal curves but the problem is nontrivial due to the general growth conditions assumed here on l x cdot we propose a new approach based on suitable reparameterization arguments to obtain suitable a priori estimates on the lipschitz constants of quasi minimizers as a consequence of our analysis we derive the lipschitz continuity of v and a compactness result for value functions associated with sequences of locally equibounded discontinuous lagrangians

another theorem of classical solvability in small for one dimensional variational problems in this paper we suggest a direct method for studying local minimizers of one dimensional variational problems which naturally complements the classical local theory this method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable the basis of these results is a regularity theory a priori estimates and compactness in c1 for solutions of obstacle problems with sufficiently close obstacles in these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory

relaxation dynamics of aftershocks after large volatility shocks in the ssec index abstract the relaxation dynamics of aftershocks after large volatility shocks are investigated based on two high frequency data sets of the shanghai stock exchange composite ssec index compared with previous relevant work we have defined main financial shocks based on large volatilities rather than large crashes we find that the occurrence rate of aftershocks with the magnitude exceeding a given threshold for both daily volatility constructed using 1 minute data and minutely volatility using intra minute data decays as a power law the power law relaxation exponent increases with the volatility threshold and is significantly greater than 1 taking financial volatility as the counterpart of seismic activity the power law relaxation in financial volatility deviates remarkably from the omori law in geophysics

relaxation and regularity in the calculus of variations in this work we prove that if l t u is a continuous function in t and u borel measurable in with bounded non convex pieces in then any absolutely continuous solution u to the variational problem min abl t u t u t dt u w01 1 a b is quasi regular in the sense of tonelli i e u is locally lipschitz on an open set of full measure of a b under the further assumption that either l is lipschitz continuous in u locally uniformly in but not necessarily in t or l is invariant under a group of c1 transformations as in the noether s theorem without one of those further assumptions the solution could be not regular as shown by a recent example in gratwick and preiss 2010 13 our result is then optimal in this sense moreover we improve the standard hypothesis used so far in buttazzo et al 1998 1 clarke and vinter 1985 5 6 csornyei et al 2008 7 tonelli 1915 15 which have been the lipschitz continuity of l in u locally uniform in and t and some growth condition in we also show that the relaxed and the original problem have the same solutions without assuming any of the two further assumptions above this extends a result in mariconda and treu 2004 14 to the non autonomous case

field theories in the modern calculus of variations two methods of construction of fields of extremals geodesic coverings in the generalized problem of bolza are given and as a conse quence sufficient conditions for optimality in a form similar to weierstrass are formulated the first field theory is an extension of young s field theory concourse of flights for our problem the other describes a nonclassical treat ment of field theory which allows one to reject the self multiplier restriction

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

holder estimates in space time for viscosity solutions of hamilton jacobi equations it is well known that solutions to the basic problem in the calculus of variations may fail to be lipschitz continuous when the lagrangian depends on t similarly for viscosity solutions to time dependent hamilton jacobi equations one cannot expect lipschitz bounds to hold uniformly with respect to the regularity of coefficients this phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm we will show that this is the case for a suitable holder norm obtaining uniform estimates in x t for solutions to first and second order hamilton jacobi equations our results apply to degenerate parabolic equations and require superlinear growth at infinity in the gradient variables of the hamiltonian proofs are based on comparison arguments and representation formulas for viscosity solutions as well as weak reverse holder inequalities 2010 wiley periodicals inc

wavelet and receiver operating characteristic analysis of heart rate variability multiresolution wavelet analysis has been used to study the heart rate variability in two classes of patients with different pathological conditions the scale dependent measure of thurner et al was found to be statistically significant in discriminating patients suffering from hypercardiomyopathy from a control set of normal subjects we have performed receiver operating characteristc roc analysis and found the roc area to be a useful measure by which to label the significance of the discrimination as well as to describe the severity of heart dysfunction

interpretation of lavrentiev phenomenon by relaxation the higher order case we consider integral functionals of the calculus of variations of the form f u fj f x u u u n dx defined for u e wn 0 1 and we show that the relaxed functional f with respect to the weak woc 1 0 1 convergence can be written as i f u j xu u n dx l u where the additional term l u the lavrentiev gap is explicitly identified in terms of f

interpretation of the lavrentiev phenomenon by relaxation we consider functionals of the calculus of variations of the form f u 01 f x u u dx defined for u w1 0 1 and we show that the relaxed functional f with respect to weak w1 1 0 1 convergence can be written as f u 01 f x u u dx l u where the additional term l u called the lavrentiev term is explicitly identified in terms of f

slow fluctuations in recurrent networks of spiking neurons networks of fast nonlinear elements may display slow fluctuations if interactions are strong we find a transition in the long term variability of a sparse recurrent network of perfect integrate and fire neurons at which the fano factor switches from zero to infinity and the correlation time is minimized this corresponds to a bifurcation in a linear map arising from the self consistency of temporal input and output statistics more realistic neural dynamics with a leak current and refractory period lead to smoothed transitions and modified critical couplings that can be theoretically predicted

the approximation of higher order integrals of the calculus of variations and the lavrentiev phenomenon we prove the following approximation theorem given a function x a b to mathbb r n in the sobolev space bf w nu 1 1 nu geq 1 and epsilon 0 there exists a function x epsilon in bf w nu 1 infty such that begin gather int a b sum i 1 m l i x epsilon nu x epsilon nu 1 psi i t x epsilon x epsilon dotsc x epsilon nu int a b sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu epsilon begin aligned x epsilon a x a quad x epsilon b x b x epsilon a x a quad x epsilon b x b vdots x epsilon nu a x nu a quad x epsilon nu b x nu b end aligned end gather provided that for every i in 1 dotsc m l i psi i is continuous in a neighborhood of x l i is convex in its second variable and psi i evaluated along x has positive sign we discuss the optimality of our assumptions comparing them with an example of sarychev j dynam control systems 3 1997 pp 565 588 as a consequence we obtain the nonoccurrence of the lavrentiev phenomenon in particular the integral functional int a b l x nu x nu 1 does not exhibit the lavrentiev phenomenon for any given boundary values x a a x b b x a a x b b dotsc x nu a a nu x nu b b nu furthermore we prove the following necessary condition an action functional with lagrangian of the form sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu with nu geq 0 exhibiting the lavrentiev phenomenon takes the value infty in any neighborhood of a minimizer

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

information transmission in parallel threshold arrays suprathreshold stochastic resonance the information transmitted through a parallel summing array of noisy threshold elements with a common threshold is considered in particular using theoretical and numerical analysis a recently reported n g stocks phys rev lett 84 2310 2000 form of stochastic resonance termed suprathreshold stochastic resonance ssr is studied in detail ssr is observed to occur in arrays with two or more elements and unlike stochastic resonance sr in a single element gives rise to noise induced information gains that occur independent of the setting of the threshold or the size of the signal the transmitted information is maximized when all thresholds are set to coincide with the signal mean in this situation and for large arrays the noise can enhance performance up to approximately half the theoretical noiseless channel capacity the theory is tested against digital simulation

scaling and memory of intraday volatility return intervals in stock markets we study the return interval tau between price volatilities that are above a certain threshold q for 31 intraday data sets including the standard and poor s 500 index and the 30 stocks that form the dow jones industrial index for different threshold q the probability density function pq tau scales with the mean interval tau as formula see text similar to that found in daily volatilities since the intraday records have significantly more data points compared to the daily records we could probe for much higher thresholds and still obtain good statistics we find that the scaling function f x is consistent for all 31 intraday data sets in various time resolutions and the function is well approximated by the stretched exponential f x similar to e ax gamma with gamma 0 38 0 05 and a 3 9 0 5 which indicates the existence of correlations we analyze the conditional probability distribution pq tau tau0 for tau following a certain interval tau0 and find pq tau tau0 depends on tau0 which demonstrates memory in intraday return intervals also we find that the mean conditional interval tau tau0 increases with tau0 consistent with the memory found for pq tau tau0 moreover we find that return interval records in addition to having short term correlations as demonstrated by pq tau tau0 have long term correlations with correlation exponents similar to that of volatility records

the riccati equation for optimal control problems with mixed state control constraints necessity and sufficiency the goal of this paper is to conduct a complete study of second order conditions for the optimal control problem with mixed state control constraints the conjugate point theory is presented and a necessary condition in terms of the corresponding riccati equation is obtained sufficiency criteria are developed in terms of strengthened necessary conditions including the riccati equation the results generalize the known ones for pure control constraints as well as for the mixed state control constraints

existence and regularity in the small in the calculus of variations abstract a local existence theorem is proved for the basic problem in the calculus of variations that of minimizing l t x x dt over a class of functions x assuming given boundary conditions the lagrangian l is only assumed to be locally lipschitz and strictly convex in its x variable

periodic solutions of hamilton s equations and local minima of the dual action the dual action is a functional whose extremals lead to solutions of hamilton s equations up to now extremals of the dual action have been obtained either through its global minimization or through application of crit ical point theory a new methodology is introduced in which local minima of the dual action are found to exist applications are then made to the existence of hamiltonian trajectories having prescribed period 1 introduction a classical and very interesting boundary value problem is concerned with periodic solutions of hamilton s equations 1 1 p i vxtf x i p i x t vph x t p t where the function h rn x rn r is called the hamiltonian any such hamil tonian trajectory x p is such that h x t p t is constant this constant is called the energy associated with x p if x 0 x t p 0 p t for some t 0 then x p is said to have period t one can ask for what energies and periods are their associated hamiltonian trajectories this and related issues have both a long history and an active present we refer to desolneux and moulis 16 mancini 20 and rabinowitz 23 for surveys and a more extensive bibliography than would be appropriate here in this article attention will be focused upon the case in which the hamiltonian is a convex function a situation which has drawn perhaps the most attention of late we begin by reviewing briefly the basis for much of this recent work a dual action principle introduced by clarke in 1978 6 let g denote the conjugate of h in the sense of convex analysis i e the function

evolution of moments and correlations in nonrenewal escape time processes the theoretical description of nonrenewal stochastic systems is a challenge analytical results are often not available or can be obtained only under strong conditions limiting their applicability also numerical results have mostly been obtained by ad hoc monte carlo simulations which are usually computationally expensive when a high degree of accuracy is needed to gain quantitative insight into these systems under general conditions we here introduce a numerical iterated first passage time approach based on solving the time dependent fokker planck equation fpe to describe the statistics of nonrenewal stochastic systems we illustrate the approach using spike triggered neuronal adaptation in the leaky and perfect integrate and fire model respectively the transition to stationarity of first passage time moments and their sequential correlations occur on a nontrivial time scale that depends on all system parameters surprisingly this is so for both single exponential and scale free power law adaptation the method works beyond the small noise and time scale separation approximations it shows excellent agreement with direct monte carlo simulations which allow for the computation of transient and stationary distributions we compare different methods to compute the evolution of the moments and serial correlation coefficients sccs and discuss the challenge of reliably computing the sccs which we find to be very sensitive to numerical inaccuracies for both the leaky and perfect integrate and fire models in conclusion our methods provide a general picture of nonrenewal dynamics in a wide range of stochastic systems exhibiting short and long range correlations

generalized gradients of lipschitz functionals abstract the purpose of this article is threefold i to present in a unified fashion the theory of generalized gradients whose elements are at present scattered in various sources ii to give an account of the ways in which the theory has been applied iii to prove new results concerning generalized gradients of summation functionals pointwise maxima and integral functionals on subspaces of l these last mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by rockafellar ioffe and levin valadier and others

multifunctions of bounded variation abstract consider control systems described by a differential equation with a control term or more generally by a differential inclusion with velocity set f t x certain properties of state trajectories can be derived when it is assumed that f t x is merely measurable w r t the time variable t but sometimes a refined analysis requires the imposition of stronger hypotheses regarding the time dependence stronger forms of necessary conditions for minimizing state trajectories can be derived for example when f t x is lipschitz continuous w r t time it has recently become apparent that significant addition properties of state trajectories can still be derived when the lipschitz continuity hypothesis is replaced by the weaker requirement that f t x has bounded variation w r t time this paper introduces a new concept of multifunctions f t x that have bounded variation w r t time near a given state trajectory of special relevance to control we provide an application to sensitivity analysis

stokes phenomena in discrete painleve i in this study we consider the asymptotic behaviour of the first discrete painleve equation in the limit as the independent variable becomes large using an asymptotic series expansion we identify two types of solutions which are pole free within some sector of the complex plane containing the positive real axis using exponential asymptotic techniques we determine stokes phenomena effects present within these solutions and hence the regions in which the asymptotic series expression is valid from a careful analysis of the switching behaviour across stokes lines we find that the first type of solution is uniquely defined while the second type contains two free parameters and that the region of validity may be extended for appropriate choice of these parameters

power law relaxation and cumulative information we associate the cumulative information of a system relaxing towards equilibrium with a divergent integral when a power law relaxation holds we discuss and illustrate numerically how this implies that a system that relaxes to equilibrium through a power law has a cumulative information content that progressively diverges from that of its equilibrium realization to which it is relaxing our findings shed light on some aspects of weak ergodicity breaking and suggest that power laws imply a form of complexity that does not require dissipation or built in disorder

dynamical structures of high frequency financial data abstract we study the dynamical behavior of high frequency data from the korean stock price index kospi using the movement of returns in korean financial markets the dynamical behavior of a binarized series of our models is not completely random in addition the conditional probability is numerically estimated from a return series of kospi tick data non trivial probability structures can be constituted from binary time series of autoregressive ar logit and probit models for which the akaike information criterion shows a minimum value at the 15th order from our results we find that the value of the correct match ratio for the ar model is slightly larger than that derived by other models

receiver operating characteristic analysis reveals superiority of scale dependent wavelet and spectral measures for assessing cardiac dysfunction receiver operating characteristic analysis was used to assess the suitability of various heart rate variability hrv measures for correctly classifying electrocardiogram records of varying lengths as normal or revealing the presence of heart failure scale dependent hrv measures were found to be substantially superior to scale independent measures scaling exponents for discriminating the two classes of data over a broad range of record lengths hours to minutes a jittered integrate and fire model built around a fractal gaussian noise kernel provides a realistic though not perfect simulation of heartbeat sequences s0031 9007 98 08087 9

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

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

modelling memory processes and internet response times weibull or power law abstract the weibull distribution is proposed as a model for response times theoretical support is offered by classical results for extreme value distributions fits of the weibull distribution to response time data in different contexts show that this distribution and the exponential distribution on small time scales perform better than the often suggested power law and logarithmic function this study suggests that the power law can be viewed as an approximation at neural level for the aggregate strength of superposed memory traces that have different decay rates in distinct parts of the brain as we predict this view does not find support at the level of induced response processes the distinction between underlying and induced processes might also be considered in other fields such as engineering biology and physics

maximally informative stimuli and tuning curves for sigmoidal rate coding neurons and populations a general method for deriving maximally informative sigmoidal tuning curves for neural systems with small normalized variability is presented the optimal tuning curve is a nonlinear function of the cumulative distribution function of the stimulus and depends on the mean variance relationship of the neural system the derivation is based on a known relationship between shannon s mutual information and fisher information and the optimality of jeffrey s prior it relies on the existence of closed form solutions to the converse problem of optimizing the stimulus distribution for a given tuning curve it is shown that maximum mutual information corresponds to constant fisher information only if the stimulus is uniformly distributed as an example the case of sub poisson binomial firing statistics is analyzed in detail

a new variational inequality in the calculus of variations and lipschitz regularity of minimizers abstract we consider a problem of the calculus of variations of the form p minimize i x a b t x t x t d t x a x b subject to x w 1 m a b r n x t c a e x t t a b where a b r n r n r is borel measurable c is a cone is a nonempty subset of r n and is an arbitrary extended valued function we prove that if t t x satisfies a nonsmooth version of cesari s condition s then any local minimizer of p in the ac norm is lipschitz whenever is coercive the proof is obtained via a new variational inequality formulated here that holds under the extended condition s just borel measurability if is autonomous for a e t in a b w t x t x t v v t x t x t p t v 1 v 0 where p t is an absolutely continuous arc the derivative of which belongs to a suitable subdifferential of x t x t a e on a b the proof of the directional weierstrass type condition w is based on clarke s nonsmooth recent versions of the maximum principle

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

the fine structure of volatility feedback i multi scale self reflexivity abstract we attempt to unveil the fine structure of volatility feedback effects in the context of general quadratic autoregressive qarch models which assume that today s volatility can be expressed as a general quadratic form of the past daily returns the standard arch or garch framework is recovered when the quadratic kernel is diagonal the calibration of these models on us stock returns reveals several unexpected features the off diagonal non arch coefficients of the quadratic kernel are found to be highly significant both in sample and out of sample although all these coefficients turn out to be one order of magnitude smaller than the diagonal elements this confirms that daily returns play a special role in the volatility feedback mechanism as postulated by arch models the feedback kernel exhibits a surprisingly complex structure incompatible with all models proposed so far in the literature its spectral properties suggest the existence of volatility neutral patterns of past returns the diagonal part of the quadratic kernel is found to decay as a power law of the lag in line with the long memory of volatility finally qarch models suggest some violations of time reversal symmetry in financial time series which are indeed observed empirically although of much smaller amplitude than predicted we speculate that a faithful volatility model should include both arch feedback effects and a stochastic component