hurst entropy a method to determine predictability in a binary series based on a fractal related process shannon s concept of information is related to predictability in a binary series the value of information relies on the frequency of 0 s and 1 s or how it is expected to occur however information entropy does not consider the bias in randomness related to autocorrelation in fact it is possible for a binary temporal series to carry both short and long term memories related to the sequential distribution of 0 s and 1 s although the hurst exponent measures the range of autocorrelation there is a lack of mathematical connection between information entropy and autocorrelation present in the series to fill this important gap we combined numerical simulations and an analytical approach to determine how information entropy changes according to the frequency of 0 s and 1 s and the hurst exponent indeed we were able to determine how predictability depends on both parameters our findings are certainly useful to several fields when binary times series are applied such as neuroscience to econophysics

applications of atiyah hirzebruch spectral sequences for motivic cobordism in this paper we study applications of atiyah hirzebruch spectral sequences for motivic cobordism recently found by hopkins and morel for example we compute the mod 2 motivic cohomology of a reduced cech complex of a splitting variety according to orlov vishik and voevodsky the cobordism ring mgl 2 spec mathbb r is computed and compared with the results of the real cobordism theory of hu and kriz moreover we study algebraic cobordism of the classifying spaces bg for algebraic groups over mathbb c for example the chow ring ch bg 2 2 for the exceptional lie group g 2 is determined by using motivic cobordism and motivic cohomology

examples of non algebraic classes in the brown peterson tower for every n ge 0 we construct classes in the brown peterson cohomology bp langle n rangle of smooth projective complex algebraic varieties which are not in the image of the cycle map from the corresponding motivic brown peterson cohomology this generalizes the examples of atiyah and hirzebruch to all finite levels in the brown peterson tower

persistent random motion with maximally correlated fluctuations how often should a random walker change its direction of motion in order to maximize correlation in velocity fluctuations over a finite time interval we address this optimal diffusion problem in the context of the one dimensional persistent random walk where we evaluate the correlation and mutual information in velocity trajectories as a function of the persistence level and the observation time we find the optimal persistence level corresponds to the average number of direction reversals asymptotically scaling as the square root of the observation time this square root scaling law makes the relative growth between the average number of direction reversals and the persistence length invariant with respect to changes in the overall time duration of the random walk

integrated approach to the assessment of long range correlation in time series data to assess whether a given time series can be modeled by a stochastic process possessing long range correlation one usually applies one of two types of analysis methods the spectral method and the random walk analysis the first objective of this work is to show that each one of these methods used alone can be susceptible to producing false results we thus advocate an integrated approach which requires the use of both methods in a consistent fashion we provide the theoretical foundation of this approach and illustrate the main ideas using examples the second objective relates to the observation of long range anticorrelation hurst exponent hl1 2 in real world time series data the very peculiar nature of such processes is emphasized in light of the stringent condition under which such processes can occur using examples we discuss the possible factors that could contribute to the false claim of long range anticorrelations and demonstrate the particular importance of the integrated approach in this case

information theoretical analysis of resting state eeg microstate sequences non markovianity non stationarity and periodicities we present an information theoretical analysis of temporal dependencies in eeg microstate sequences during wakeful rest we interpret microstate sequences as discrete stochastic processes where each state corresponds to a representative scalp potential topography testing low order markovianity of these discrete sequences directly we find that none of the recordings fulfils the markov property of order 0 1 or 2 further analyses show that the microstate transition matrix is non stationary over time in 80 window size 10 s 60 window size 20 s and 44 window size 40 s of the subjects and that transition matrices are asymmetric in 14 20 70 subjects to assess temporal dependencies globally the time lagged mutual information function autoinformation function of each sequence is compared to the first order markov model defined by the classical transition matrix approach the autoinformation function for the markovian case is derived analytically and numerically for experimental data we find non markovian behaviour in the range of the main eeg frequency bands where distinct periodicities related to the subject s eeg frequency spectrum appear in particular the microstate clustering algorithm induces frequency doubling with respect to the eeg power spectral density while the tail of the autoinformation function asymptotically reaches the first order markov confidence interval for time lags above 1000 ms in summary our results show that resting state microstate sequences are non markovian processes which inherit periodicities from the underlying eeg dynamics our results interpolate between two diverging models of microstate dynamics memoryless markov models on one side and long range correlated models on the other microstate sequences display more complex temporal dependencies than captured by the transition matrix approach in the range of the main eeg frequency bands but show finite memory content in the long run

selmer companion curves we say that two elliptic curves e1 e2 over a number field k are n selmer companions for a positive integer n if for every quadratic character of k there is an isomorphism seln e 1 k seln e 2 k between the nselmer groups of the quadratic twists e 1 e 2 we give sufficient conditions for two elliptic curves to be n selmer companions and give a number of examples of non isogenous pairs of companions

torsion cohomology classes and algebraic cycles on complex projective manifolds abstract atiyah and hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold which are not poincare dual to an algebraic cycle we notice that the order of these classes must be small compared to the dimension of the manifold however building upon a construction of kollar one can provide such examples with arbitrary high prime order the dimension being fixed this method also provides examples of torsion algebraic cycles which are non trivial in the griffiths group and lie in a arbitrary high level of the h saito filtration on chow groups

scaling laws and persistence in human brain activity abstract we study the temporal variability of human brain activity in timeseries of functional magnetic resonance imaging fmri data we find that these timeseries show scaling behavior which is quantified by computing various scaling exponents we demonstrate that mentally active zones are one to one related to large exponents or equivalently to highly temporally correlated processes while mentally inactive zones are well described by a simple random walk model activation maps of scaling exponents are presented and compared to standard results in fmri in contrast to standard model based activation analyses no prior knowledge of the experimental stimulation paradigm has to be assumed for extracting activation patterns from fmri timeseries we mention consequences of persistence in fmri timeseries for standard statistical analysis

modular diagonal quotient surfaces the main objective of this paper is to analyze the geometry of the modular diagonal quotient surface zn e e x n x n which classifies pairs of elliptic curves e1 and e2 together with an isomorphism of determinant e between their associated modular representations mod n in particular we calculate some of the numerical invariants of its minimal desingularization zn e such as its betti and chern numbers and determine its place in the enriques kodaira classification table

analytical and empirical fluctuation functions of the eeg microstate random walk short range vs long range correlations we analyze temporal autocorrelations and the scaling behaviour of eeg microstate sequences during wakeful rest we use the recently introduced random walk approach and compute its fluctuation function analytically under the null hypothesis of a short range correlated first order markov process the empirical fluctuation function and the hurst parameter h as a surrogate parameter of long range correlations are computed from 32 resting state eeg recordings and for a set of first order markov surrogate data sets with equilibrium distribution and transition matrices identical to the empirical data in order to distinguish short range correlations h 0 5 from previously reported long range correlations h 0 5 statistically confidence intervals for h and the fluctuation functions are constructed under the null hypothesis comparing three different estimation methods for h we find that only one data set consistently shows h 0 5 compatible with long range correlations whereas the majority of experimental data sets cannot be consistently distinguished from markovian scaling behaviour our analysis suggests that the scaling behaviour of resting state eeg microstate sequences though markedly different from uncorrelated zero order markov processes can often not be distinguished from a short range correlated first order markov process our results do not prove the microstate process to be markovian but challenge the approach to parametrize resting state eeg by single parameter models

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

mutual information identifies spurious hurst phenomena in resting state eeg and fmri data long range memory in time series is often quantified by the hurst exponent h a measure of the signal s variance across several time scales we analyze neurophysiological time series from electroencephalography eeg and functional magnetic resonance imaging fmri resting state experiments with two standard hurst exponent estimators and with the time lagged mutual information function applied to discretized versions of the signals a confidence interval for the mutual information function is obtained from surrogate markov processes with equilibrium distribution and transition matrix identical to the underlying signal for eeg signals we construct an additional mutual information confidence interval from a short range correlated tenth order autoregressive model we reproduce the previously described hurst phenomenon h 0 5 in the analytical amplitude of alpha frequency band oscillations in eeg microstate sequences and in fmri signals but we show that the hurst phenomenon occurs without long range memory in the information theoretical sense we find that the mutual information function of neurophysiological data behaves differently from fractional gaussian noise fgn for which the hurst phenomenon is a sufficient condition to prove long range memory two other well characterized short range correlated stochastic processes ornstein uhlenbeck cox ingersoll ross also yield h 0 5 whereas their mutual information functions lie within the markovian confidence intervals similar to neural signals in these processes which do not have long range memory by construction a spurious hurst phenomenon occurs due to slow relaxation times and heteroscedasticity time varying conditional variance in summary we find that mutual information correctly distinguishes long range from short range dependence in the theoretical and experimental cases discussed our results also suggest that the stationary fgn process is not sufficient to describe neural data which seem to belong to a more general class of stochastic processes in which multiscale variance effects produce hurst phenomena without long range dependence in our experimental data the hurst phenomenon and long range memory appear as different system properties that should be estimated and interpreted independently

cohomologie non ramifiee et conjecture de hodge entiere en nous appuyant sur la conjecture de bloch kato en k theorie de milnor nous etablissons un lien general entre le defaut de la conjecture de hodge entiere pour la cohomologie de degre 4 et le troisieme groupe de cohomologie non ramifiee a coefficients q z ceci permet de montrer que sur un solide en anglais threefold uniregle le troisieme groupe de cohomologie non ramifiee a coefficients q z s annule ce que la k theorie algebrique ne permet d obtenir que dans certains cas ceci permet a l inverse de deduire d exemples ayant leur source en k theorie que la conjecture de hodge entiere pour la cohomologie de degre 4 peut etre en defaut pour les varietes rationnellement connexes pour certaines familles a un parametre de surfaces on etablit un lien entre la conjecture de hodge entiere et l indice de la fibre generique

torsion algebraic cycles and etale cobordism we prove that the classical integral cycle class map from algebraic cycles to etale cohomology factors through a quotient of l adic etale cobordism over an algebraically closed field of positive characteristic this shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of atiyah hirzebruch and totaro also work in positive characteristic

examples for the mod p motivic cohomology of classifying spaces let bg be the classifying space of a compact lie group g some examples of computations of the motivic cohomology h eg z p are given by comparing with h bg z p ch eg and bp bg

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

random maximal isotropic subspaces and selmer groups under suitable hypotheses we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over f p a random subspace chosen with respect to this measure is discrete with probability 1 and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative integer valued random variable we then prove that the p selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over f p by modeling the first subspace as being random we can explain the known phenomena regarding distribution of selmer ranks such as the theorems of heath brown swinnerton dyer and kane for 2 selmer groups in certain families of quadratic twists and the average size of 2 and 3 selmer groups as computed by bhargava and shankar our model is compatible with delaunay s heuristics for p torsion in shafarevich tate groups and predicts that the average rank of elliptic curves over a fixed number field is at most 1 2 many of our results generalize to abelian varieties over global fields

etale realization on the a1 homotopy theory of schemes abstract we compare friedlander s definition of the etale topological type for simplicial schemes to another definition involving realizations of pro simplicial sets this can be expressed as a notion of hypercover descent for etale homotopy we use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over s to the category of pro spaces after completing away from the characteristics of the residue fields of s we get a functor from the morel voevodsky a 1 homotopy category of schemes to the homotopy category of pro spaces

exact numerical simulation of the ornstein uhlenbeck process and its integral a numerical simulation algorithm that is exact for any time step ensuremath delta tg0 is derived for the ornstein uhlenbeck process x t and its time integral y t the algorithm allows one to make efficient unapproximated simulations of for instance the velocity and position components of a particle undergoing brownian motion and the electric current and transported charge in a simple r l circuit provided appropriate values are assigned to the ornstein uhlenbeck relaxation time ensuremath tau and diffusion constant c a simple taylor expansion in ensuremath delta t of the exact simulation formulas shows how the first order simulation formulas which are implicit in the langevin equation for x t and the defining equation for y t are modified in second order the exact simulation algorithm is used here to illustrate the zero ensuremath tau limit theorem textcopyright 1996 the american physical society

ranks of twists of elliptic curves and hilbert s tenth problem in this paper we investigate the 2 selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields we give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2 selmer rank and we give lower bounds for the number of twists with bounded conductor that have a given 2 selmer rank as a consequence under appropriate hypotheses we can find many twists with trivial mordell weil group and assuming the shafarevich tate conjecture many others with infinite cyclic mordell weil group using work of poonen and shlapentokh it follows from our results that if the shafarevich tate conjecture holds then hilbert s tenth problem has a negative answer over the ring of integers of every number field

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

dynamics of electroencephalogram entropy and pitfalls of scaling detection in recent studies a number of research groups have determined that human electroencephalograms eeg have scaling properties in particular a crossover between two regions with different scaling exponents has been reported herein we study the time evolution of diffusion entropy to elucidate the scaling of eeg time series for a cohort of 20 awake healthy volunteers with closed eyes we find that the diffusion entropy of eeg increments obtained from eeg waveforms by differencing exhibits three features short time growth an alpha wave related oscillation whose amplitude gradually decays in time and asymptotic saturation which is achieved after approximately 1 s this analysis suggests a linear stochastic ornstein uhlenbeck langevin equation with a quasiperiodic forcing whose frequency and or amplitude may vary in time as the model for the underlying dynamics this model captures the salient properties of eeg dynamics in particular both the experimental and simulated eeg time series exhibit short time scaling which is broken by a strong periodic component such as alpha waves the saturation of eeg diffusion entropy precludes the existence of asymptotic scaling we find that the crossover between two scaling regions seen in detrended fluctuation analysis dfa of eeg increments does not originate from the underlying dynamics but is merely an artifact of the algorithm this artifact is rooted in the failure of the trend plus signal paradigm of dfa

stable etale realization and etale cobordism abstract we show that there is a stable homotopy theory of profinite spaces and use it for two main applications on the one hand we construct an etale topological realization of the stable a 1 homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero on the other hand we get a natural setting for etale cohomology theories in particular we define and discuss an etale topological cobordism theory for schemes it is equipped with an atiyah hirzebruch spectral sequence starting from etale cohomology finally we construct maps from algebraic to etale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a bott element

markov processes hurst exponents and nonlinear diffusion equations with application to finance abstract we show by explicit closed form calculations that a hurst exponent h 1 2 does not necessarily imply long time correlations like those found in fractional brownian motion fbm we construct a large set of scaling solutions of fokker planck partial differential equations pdes where h 1 2 thus markov processes which by construction have no long time correlations can have h 1 2 if a markov process scales with hurst exponent h 1 2 then it simply means that the process has nonstationary increments for the scaling solutions we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient d x t is specified as an example we generate a class of student t like densities from the class of quadratic diffusion coefficients notably the tsallis density is one member of that large class the tsallis density is usually thought to result from a nonlinear diffusion equation but instead we explicitly show that it follows from a markov process generated by a linear fokker planck equation and therefore from a corresponding langevin equation having a tsallis density with h 1 2 therefore does not imply dynamics with correlated signals e g like those of fbm a short review of the requirements for fbm is given for clarity and we explain why the usual simple argument that h 1 2 implies correlations fails for markov processes with scaling solutions finally we discuss the question of scaling of the full green function g x t x t of the fokker planck pdes

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

arithmetic of elliptic curves upon quadratic extension this paper is a study of variations in the rank of the mordell weil group of an elliptic curve e defined over a number field f as one passes to quadratic extensions k of f let s k be the selmer group for multiplication by 2 on e k in analogy with genus theory we describe s k in terms of various objects defined over f and the local norm indices dimf2 ft norm ah for each completion fv of f in particular we show that dim s k dim e k 2 has the same parity as zi we compute i when e has good or multiplicative reduction modulo v assuming that the 2 primary component of the tate shafare vitch group u1 k is finite as conjectured we obtain the parity of rank e k for semistable elliptic curves defined over q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of birch and swinnerton dyer 1 introduction let e be an elliptic curve defined over a number field f our motivating question is this what can be said about variations in the rank of the mordell weil group e k over quadratic extensions k f dl 2 1 let e d denote the twist of e which becomes isomorphic to e over k but not over f concretely if we choose for e a model over f of the form y2 f x then a model for d is given by dy2 f x if a denotes the generator of gal k f then e f can be identified with the l eigenspace and e d f with the l eigenspace of a acting on e k it follows that rank e k rank e f rank e d f an equiv alent question therefore is to describe changes in the rank of e d f as d varies for certain specific curves defined over q this question has been discussed for

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

brown peterson and ordinary cohomology theories of classifying spaces for compact lie groups the steenrod algebra structures of h bg z p for compact lie groups are studied using these brown peterson cohomology and morava k theory are computed for many concrete cases all these cases have properties similar as torsion free lie groups or finite groups e g bp odd bg 0

hurst exponents markov processes and fractional brownian motion there is much confusion in the literature over hurst exponents recently we took a step in the direction of eliminating some of the confusion one purpose of this

assessment of long range correlation in time series how to avoid pitfalls due to the ubiquity of time series with long range correlation in many areas of science and engineering analysis and modeling of such data is an important problem while the field seems to be mature three major issues have not been satisfactorily resolved i many methods have been proposed to assess long range correlation in time series under what circumstances do they yield consistent results ii the mathematical theory of long range correlation concerns the behavior of the correlation of the time series for very large times a measured time series is finite however how can we relate the fractal scaling break at a specific time scale to important parameters of the data iii an important technique in assessing long range correlation in a time series is to construct a random walk process from the data under the assumption that the data are like a stationary noise process due to the difficulty in determining whether a time series is stationary or not however one cannot be 100 sure whether the data should be treated as a noise or a random walk process is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process and vice versa in this paper we seek to gain important insights into these issues by examining three model systems the autoregressive process of order 1 on off intermittency and l evy motions and considering an important engineering problem target detection within sea clutter radar returns we also provide a few rules of thumb to safeguard against misinterpretations of long range correlation in a time series and discuss relevance of this study to pattern recognition

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method