dynamical behavior of combined detrended cross correlation analysis methods in random walks and levy flights abstract in this paper the utility and the correlation technique of the detrended cross correlation analysis and the combined detrended cross correlation analysis methods are treated for the stochastic properties of the random walk and the levy flight via nucleotide sequence data data are extracted definitely from twelve genomic dna sequences humhbb 1 6 amazonmolly bushbaby cow dolphin elephant and tarsier of the genbank databank we mainly simulate and analyze scaling exponents for three methods in 1d and 2d random walks and levy flights to compare each other between one gene and the other gene in the twelve genomic dna sequences the value of the detrended cross correlation coefficient d c c a in humhbb3 and dolphin genes is shown the largest value 0 214 in the 1d levy flight with 1 5 compared to other sets the detrended cross correlation coefficient d c c a in humhbb4 and humhbb6 genes is shown the smallest value 0 231 in the 1d levy flight with 1 5 we particularly find that the combined detrended cross correlation coefficient c d c c a in the two cases of humhbb3 bushbaby and humhbb6 dolphin has the largest value 0 331 in the 1d random walk compared to other sets while that in humhbb3 tarsier and humhbb4 elephant has the smallest value 0 295 in the 1d levy flight with 1 5

modeling record breaking stock prices abstract we study the statistics of record breaking events in daily stock prices of 366 stocks from the standard and poor s 500 stock index both the record events in the daily stock prices themselves and the records in the daily returns are discussed in both cases we try to describe the record statistics of the stock data with simple theoretical models the daily returns are compared to i i d rvs and the stock prices are modeled using a biased random walk for which the record statistics are known these models agree partly with the behavior of the stock data but we also identify several interesting deviations most importantly the number of records in the stocks appears to be systematically decreased in comparison with the random walk model considering the autoregressive ar 1 process we can predict the record statistics of the daily stock prices more accurately we also compare the stock data with simulations of the record statistics of the more complicated garch 1 1 model which in combination with the ar 1 model gives the best agreement with the observational data to better understand our findings we discuss the survival and first passage times of stock prices on certain intervals and analyze the correlations between the individual record events after recapitulating some recent results for the record statistics of ensembles of n stocks we also present some new observations for the weekly distributions of record events

record statistics for biased random walks with an application to financial data we consider the occurrence of record breaking events in random walks with asymmetric jump distributions the statistics of records in symmetric random walks was previously analyzed by majumdar and ziff phys rev lett 101 050601 2008 and is well understood unlike the case of symmetric jump distributions in the asymmetric case the statistics of records depends on the choice of the jump distribution we compute the record rate p n c defined as the probability for the nth value to be larger than all previous values for a gaussian jump distribution with standard deviation that is shifted by a constant drift c for small drift in the sense of c n 1 2 the correction to p n c grows proportional to arctan n and saturates at the value c 2 for large n the record rate approaches a constant which is approximately given by 1 2 c exp c 2 2 2 for c 1 these asymptotic results carry over to other continuous jump distributions with finite variance as an application we compare our analytical results to the record statistics of 366 daily stock prices from the standard poor s 500 index the biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices and after detrending the number of upper records is in good agreement with the symmetric random walk however the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified

correlations between record events in sequences of random variables with a linear trend the statistics of records in sequences of independent identically distributed random variables is a classic subject of study one of the earliest results concerns the stochastic independence of record events recently records statistics beyond the case of i i d random variables have received much attention but the question of independence of record events has not been addressed systematically in this paper we study this question in detail for the case of independent non identically distributed random variables specifically for random variables with a linearly moving mean we find a rich pattern of positive and negative correlations and show how their asymptotics is determined by the universality classes of extreme value statistics

record statistics and persistence for a random walk with a drift we study the statistics of records of a one dimensional random walk of n steps starting from the origin and in presence of a constant bias c at each time step the walker makes a random jump of length eta drawn from a continuous distribution f eta which is symmetric around a constant drift c we focus in particular on the case were f eta is a symmetric stable law with a levy index 0 mu leq 2 the record statistics depends crucially on the persistence probability which as we show here exhibits different behaviors depending on the sign of c and the value of the parameter mu hence in the limit of a large number of steps n the record statistics is sensitive to these parameters c and mu of the jump distribution we compute the asymptotic mean record number after n steps as well as its full distribution p r n we also compute the statistics of the ages of the longest and the shortest lasting record our exact computations show the existence of five distinct regions in the c 0 mu leq 2 strip where these quantities display qualitatively different behaviors we also present numerical simulation results that verify our analytical predictions

stability of slender bodies under compression and validity of the von karman theory since their formulation almost 100 years ago the von karman vk plate equations have been frequently used both by engineers and by analysts to study thin elastic bodies in particular their stability behaviour under applied loads at the same time the derivation of these equations met some harsh criticism and their precise mathematical status has been unclear until very recently following up on a recent variational derivation of the vk theory by friesecke james and muller from three dimensional nonlinear elasticity we study the predictions and the validity of the vk equation in the presence of in plane compressive forces the first main result is a stability alternative either the load leads to instability already in the nonlinear bending theory of plates kirchhoff love theory or it leads to an instability in a geometrically linear kl theory linearized instability or vk theory is valid the second main result states that under suitable conditions the critical loads for nonlinear stability and linearized instability coincide the third main result asserts this critical load also agrees with the load beyond which the infimum of the vk functional is the main ingredients are a sharp rigidity estimate for maps with low elastic energy and a study of the properties of isometric immersions from a set in mathbb r 2 to mathbb r 3 and their geometrically linear counterparts

a hierarchy of plate models derived from nonlinear elasticity by gamma convergence we derive a hierarchy of plate models from three dimensional nonlinear elasticity by convergence what distinguishes the different limit models is the scaling of the elastic energy per unit volume h where h is the thickness of the plate this is in turn related to the strength of the applied force h membrane theory derived earlier by le dret and raoult corresponds to 0 nonlinear bending theory to 2 von karman theory to 3 4 and linearized vk theory to 3 intermediate values of lead to certain theories with constraints a key ingredient in the proof is a generalization to higher derivatives of our rigidity result 29 which states that for maps v 0 1 3 3 the l2 distance of v from a single rotation is bounded by a multiple of the l2 distance from the set so 3 of all rotations

saint venant s problem and semi inverse solutions in nonlinear elasticity saint venant s problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross sectional resultants of force and moment using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite dimensional manifold in particular the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations moreover the set of semi inverse solutions can be analyzed locally

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

span observables when is a foraging rabbit no longer hungry be x t a random walk we study its span s i e the size of the domain visited up to time t we want to know the probability that s reaches 1 for the first time as well as the density of the span given t analytical results are presented and checked against numerical simulations we then generalize this to include drift and one or two reflecting boundaries we also derive the joint probability of the maximum and minimum of a process our results are based on the diffusion propagator with reflecting or absorbing boundaries for which a set of useful formulas is derived

statistics of superior records we study statistics of records in a sequence of random variables these identical and independently distributed variables are drawn from the parent distribution the running record equals the maximum of all elements in the sequence up to a given point we define a superior sequence as one where all running records are above the average record expected for the parent distribution we find that the fraction of superior sequences s n decays algebraically with sequence length n s n n in the limit n interestingly the decay exponent is nontrivial being the root of an integral equation for example when is a uniform distribution with compact support we find 0 450265 in general the tail of the parent distribution governs the exponent we also consider the dual problem of inferior sequences where all records are below average and find that the fraction of inferior sequences i n decays algebraically albeit with a different decay exponent i n n we use the above statistical measures to analyze earthquake data language en

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

a characterization of banach spaces containing it is proved that a banach space contains a subspace isomorphic to 11 if and only if it has a bounded sequence with no weak cauchy subsequence the proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or boolean independent a bounded sequence of elements fn in a banach space b is said to be equivalent to the usual 11 basis provided there is a a 0 so that for all n and choices of scalars c cn

rounding effects in record statistics we analyze record breaking events in time series of continuous random variables that are subsequently discretized by rounding to integer multiples of a discretization scale ensuremath delta g0 rounding leads to ties of an existing record thereby reducing the number of new records for an infinite number of random variables that are drawn from distributions with a finite upper limit the number of discrete records is finite while for distributions with a thinner than exponential upper tail fewer discrete records arise compared to continuous variables in the latter case the record sequence becomes highly regular at long times

time since maximum of brownian motion and asymmetric levy processes motivated by recent studies of record statistics in relation to strongly correlated time series we consider explicitly the drawdown time of a levy process which is defined as the time since it last achieved its running maximum when observed over a fixed time period we show that the density function of this drawdown time in the case of a completely asymmetric jump process may be factored as a function of t multiplied by a function of t t this extends a known result for the case of pure brownian motion we state the factors explicitly for the cases of exponential down jumps with drift and for the downward inverse gaussian levy process with drift

inverse statistics in economics the gain loss asymmetry abstract inverse statistics in economics is considered we argue that the natural candidate for such statistics is the investment horizons distribution this distribution of waiting times needed to achieve a predefined level of return is obtained from often detrended historic asset prices such a distribution typically goes through a maximum at a time called the optimal investment horizon since this defines the most likely waiting time for obtaining a given return by considering equal positive and negative levels of return we report on a quantitative gain loss asymmetry most pronounced for short horizons it is argued that this asymmetry reflects the market dynamics and we speculate over the origin of this asymmetry

universal statistics of longest lasting records of random walks and levy flights we study the record statistics of random walks after n steps x0 x1 xn with arbitrary symmetric and continuous distribution p of the jumps i xi xi 1 we consider the age of the records i e the time up to which a record survives depending on how the age of the current last record is defined we propose three distinct sequences of ages indexed by i ii iii associated to a given sequence of records we then focus on the longest lasting record which is the longest element among this sequence of ages to characterize the statistics of these longest lasting records we compute i the probability that the record of the longest age is broken at step n denoted by q n which we call the probability of record breaking and ii the duration of the longest lasting record we show that both q n and the full statistics of are universal i e independent of the jump distribution p we compute exactly the large n asymptotic behaviors of q n as well as when it exists and show that each case gives rise to a different universal constant associated to random walks including levy flights while two of them appeared before in the excursion theory of brownian motion for which we provide here a simpler derivation the third case gives rise to a non trivial new constant ciii 0 241 749 associated to the records of random walks other observables characterizing the ages of the records exhibiting an interesting universal behavior are also discussed

on a problem of h p rosenthal concerning operators on c 0 1 a problem of h p rosenthal asks whether every bounded linear operator which is an isomorphism on a closed linear infinite dimensional subspace x not containing any isomorph of c0 is actually an isomorphism on a subspace isomorphic to c 0 1 an affirmative answer to this problem is provided when t is a contraction whose restriction to x is an isometry

justification of the nonlinear kirchhoff love theory of plates as the application of a new singular inverse method in the framework of isotropic homogeneous nonlinear elasticity for a st venant kirchhoff material we consider a three dimensional plate of thickness and periodic in the two other directions using a new method that we call the singular inverse method we prove the existence of a solution rescaled uniformly in for small forces and at the same time we prove the rigorous convergence of this rescaled solution to the solution of the nonlinear kirchhoff love plate model we also state a 3d 2d error estimate

persistence of random walk records we study records generated by brownian particles in one dimension specifically we investigate an ordinary random walk and define the record as the maximal position of the walk we compare the record of an individual random walk with the mean record obtained as an average over infinitely many realizations we term the walk superior if the record is always above average and conversely the walk is said to be inferior if the record is always below average we find that the fraction of superior walks s decays algebraically with time s t in the limit t and that the persistence exponent is nontrivial 0 382 258 the fraction of inferior walks i also decays as a power law i t but the persistence exponent is smaller 0 241 608 both exponents are roots of transcendental equations involving the parabolic cylinder function to obtain these theoretical results we analyze the joint density of superior walks with a given record and position while for inferior walks it suffices to study the density as a function of position

convergence of equilibria of thin elastic plates under physical growth conditions for the energy density abstract the asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied as the thickness h of the plate goes to zero more precisely it is shown that critical points of the nonlinear elastic functional e h whose energies per unit thickness are bounded by ch 4 converge to critical points of the limit of h 4 e h this is proved under the physical assumption that the energy density w f blows up as det f 0

order statistics of 1 fα signals order statistics of periodic gaussian noise with 1 f power spectrum is investigated using simulations and phenomenological arguments we find three scaling regimes for the average gap d k x k x k 1 between the kth and k 1 st largest values of the signal the result d k k 1 known for independent identically distributed variables remains valid for 0 5 the spectra of average ordered values e k x 1 x k k is also examined the exponent is derived from the gap scaling as well as by relating e k to the density of near extreme states known results for the density of near extreme states combined with scaling suggest that 2 1 2 4 3 2 and 2 are exact values we also show that parallels can be drawn between e k and the quantum mechanical spectra of a particle in power law potentials

an extension of james s compactness theorem abstract let x and y be banach spaces and f b y endow y with the topology f of pointwise convergence on f assume that t x y is a bounded linear operator which is w f continuous assume further that every vector in the range of t attains its norm at some element of f that is for every x x there exists y f such that t x y t x then t is w w continuous the proof relies on rosenthal s l 1 theorem as a corollary to the above result one obtains an alternative proof of james s compactness theorem that a bounded subset k of a banach space e is relatively weakly compact provided that each functional in e attains its supremum on k

statistics of persistent events in the binomial random walk will the drunken sailor hit the sober man the statistics of persistent events recently introduced in the context of phase ordering dynamics is investigated in the case of the one dimensional lattice random walk in discrete time we determine the survival probability of the random walker in the presence of an obstacle moving ballistically with velocity v i e the probability that the random walker remains up to time n on the left of the obstacle three regimes are to be considered for the long time behavior of this probability according to the sign of the difference between v and the drift velocity v of the random walker in one of these regimes v v the survival probability has a nontrivial limit at long times which is discontinuous at all rational values of v an algebraic approach allows us to compute these discontinuities as well as several related quantities the mathematical structure underlying the solvability of this model combines elementary number theory algebraic functions and algebraic curves defined over the rationals

a justification of nonlinear properly invariant plate theories a single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame invariance properties of three dimensional finite elasticity by a successive scaling of the external loading on the three dimensional body the nonlinear membrane theory the nonlinear inextensional theory and the von karman equations are derived as the leading order terms in the asymptotic expansion of finite elasticity the governing equations of the nonlinear inextensional theory are of particular interest where 1 plane strain kinematics and plane stress constitutive equations are derived simultaneously from the asymptotic analysis 2 the theory can be phrased as a minimization problem over the space of isometric deformations of a surface and 3 the local equilibrium equations are identical to those arising in the one director cosserat shell model furthermore it can be concluded that with a regular single scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading order term of an asymptotic analysis

record statistics of a strongly correlated time series random walks and levy flights we review recent advances on the record statistics of strongly correlated time series whose entries denote the positions of a random walk or a l evy flight on a line after a brief survey of the theory of records for independent and identically distributed random variables we focus on random walks during the last few years it was indeed realized that random walks are a very useful laboratory to test the effects of correlations on the record statistics we start with the simple one dimensional random walk with symmetric jumps both continuous and discrete and discuss in detail the statistics of the number of records as well as of the ages of the records i e the lapses of time between two successive record breaking events then we review the results that were obtained for a wide variety of random walk models including random walks with a linear drift continuous time random walks constrained random walks like the random walk bridge and the case of multiple independent random walkers finally we discuss further observables related to records like the record increments as well as some questions raised by physical applications of record statistics like the effects of measurement error and noise

record statistics for multiple random walks we study the statistics of the number of records r n n for n identical and independent symmetric discrete time random walks of n steps in one dimension all starting at the origin at step 0 at each time step each walker jumps by a random length drawn independently from a symmetric and continuous distribution we consider two cases i when the variance ensuremath sigma 2 of the jump distribution is finite and ii when ensuremath sigma 2 is divergent as in the case of l evy flights with index 0l ensuremath mu l2 in both cases we find that the mean record number ensuremath langle r n n ensuremath rangle grows universally as ensuremath sim ensuremath alpha n sqrt n for large n but with a very different behavior of the amplitude ensuremath alpha n for ng1 in the two cases we find that for large n ensuremath alpha n ensuremath approx 2 sqrt mathrm ln n independently of ensuremath sigma 2 in case i in contrast in case ii the amplitude approaches to an n independent constant for large n ensuremath alpha n ensuremath approx 4 sqrt ensuremath pi independently of 0l ensuremath mu l2 for finite ensuremath sigma 2 we argue and this is confirmed by our numerical simulations that the full distribution of r n n sqrt n ensuremath 2 sqrt mathrm ln n sqrt mathrm ln n converges to a gumbel law as n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty in case ii our numerical simulations indicate that the distribution of r n n sqrt n converges for n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty to a universal nontrivial distribution independently of ensuremath mu we discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the standard poor s 500 index

optimal investment horizons for stocks and markets abstract the inverse statistics is the distribution of waiting times needed to achieve a predefined level of return obtained from detrended historic asset prices i simonsen m h jensen a johansen eur phys j 27 2002 583 m h jensen a johansen i simonsen physica a 234 2003 338 such a distribution typically goes through a maximum at a time coined the optimal investment horizon which defines the most likely waiting time for obtaining a given return by considering equal positive and negative levels of return we reported in m h jensen a johansen i simonsen physica a 234 2003 338 on a quantitative gain loss asymmetry most pronounced for short horizons in the present paper the inverse statistics for 2 3 of the individual stocks presently in the djia is investigated we show that this gain loss asymmetry established for the djia surprisingly is not present in the time series of the individual stocks nor their average this observation points towards some kind of collective movement of the stocks of the index synchronization