holder continuity of the rotation number for quasi periodic co cycles in sl 2 mathbb r we prove two results on the rotation number of the skew product system omega a theta y in mathbb t d times mathbb r 2 mapsto theta omega a theta y in mathbb t d times mathbb r 2 where is diophantine and a theta in sl 2 mathbb r is homotopic to the identity on the one hand we prove that this function has the behavior of a frac 1 2 holder function on the other we show that the length of the gaps has a sub exponential estimate which depends on its label given by the gap labeling theorem we give also an estimate of the complement of the spectrum these results are obtained by studying the reducibility of the quasi periodic co cycle a

wavelet based fractal analysis of dna sequences abstract the fractal scaling properties of dna sequences are analyzed using the wavelet transform mapping nucleotide sequences onto a dna walk produces fractal landscapes that can be studied quantitatively by applying the so called wavelet transform modulus maxima method this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized oscillating boxes from the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima this method allows us to determine the singularity spectrum of the considered signal and thereby to achieve a complete multifractal analysis moreover by considering analyzing wavelets that make the wavelet transform microscope blind to patches of different nucleotide composition that are observed in genomic sequences we demonstrate and quantify the existence of long range correlations in the noncoding regions although the fluctuations in the patchy landscape of the dna walks reconstructed from both noncoding and protein coding regions are found homogeneous with gaussian statistics our wavelet based analysis allows us to discriminate unambiguously between the fluctuations of the former which behave like fractional brownian motions from those of the latter which cannot be distinguished from uncorrelated random brownian walks we discuss the robustness of these results with respect to various legitimate codings of the dna sequences finally we comment about the possible understanding of the origin of the observed long range correlations in noncoding dna sequences in terms of the nonequilibrium dynamical processes that produce the isochore structre of the genome

almost reducibility and non perturbative reducibility of quasi periodic linear systems in this paper we prove that a quasi periodic linear differential equation in sl 2 with two frequencies 1 is almost reducible provided that the coefficients are analytic and close to a constant in the case that is diophantine we get the non perturbative reducibility we also obtain the reducibility and the rotations reducibility for an arbitrary irrational under some assumption on the rotation number and give some applications for schrodinger operators our proof is a generalized kam type iteration adapted to all irrational

molecular evolution modeled as a fractal statistical process abstract modeling the rate of nucleotide substitutions in dna as a dichotomous stochastic process with an inverse power law correlation function describes evolution by a fractal stochastic process fsp this fsp model agrees with recent findings on the relationship between the variance and mean number of synonymous and nonsynonymous substitutions in 49 different genes in mammals that being a power law increase in the ratio of the variance to the mean the index of dispersion with the number of substitutions in a protein the probability of a given number of substitutions occuring in a time t is determined by a fractional diffusion equation whose solution is a truncated levy distribution implying that evolution is a levy process in time and yields the same functional behavior for the variance in the number of substitutions as does the fsp model in addition to obtaining these relationships the fsp model implies lognormal statistics for the index of dispersion as a function of the mean number of substitutions in a protein which is confirmed in the regression of the fsp model to data lognormal statistics suggest that molecular evolution can be viewed as a multiplicative stochastic process rather than the linear additive processes of darwinian selection and drift

power law shot noise the behavior of power law shot noise for which the associated impulse response functions assume a decaying power law form is explored expressions are obtained for the moments moment generating functions amplitude probability density functions autocorrelation functions and power spectral densities for a variety of parameters of the process for certain parameters the power spectral density exhibits 1 f type behavior over a substantial range of frequencies so that the process serves as a source of 1 f sup alpha shot noise for alpha in the range 0

application of computational mechanics to the analysis of natural data an example in geomagnetism we discuss how the ideal formalism of computational mechanics can be adapted to apply to a noninfinite series of corrupted and correlated data that is typical of most observed natural time series specifically a simple filter that removes the corruption that creates rare unphysical causal states is demonstrated and the concept of effective soficity is introduced we believe that computational mechanics cannot be applied to a noisy and finite data series without invoking an argument based upon effective soficity a related distinction between noise and unresolved structure is also defined noise can only be eliminated by increasing the length of the time series whereas the resolution of previously unresolved structure only requires the finite memory of the analysis to be increased the benefits of these concepts are demonstrated in a simulated times series by a the effective elimination of white noise corruption from a periodic signal using the expletive filter and b the appearance of an effectively sofic region in the statistical complexity of a biased poisson switch time series that is insensitive to changes in the word length memory used in the analysis the new algorithm is then applied to an analysis of a real geomagnetic time series measured at halley antarctica two principal components in the structure are detected that are interpreted as the diurnal variation due to the rotation of the earth based station under an electrical current pattern that is fixed with respect to the sun earth axis and the random occurrence of a signature likely to be that of the magnetic substorm in conclusion some useful terminology for the discussion of model construction in general is introduced

quantitative inequality for the eigenvalue of a schrodinger operator in the ball abstract the aim of this article is to prove a quantitative inequality for the first eigenvalue of a schrodinger operator in the ball more precisely we optimize the first eigenvalue v of the operator l v v with dirichlet boundary conditions with respect to the potential v under l 1 and l constraints on v the solution has been known to be the characteristic function of a centered ball but this article aims at proving a sharp growth rate of the following form if v is a minimizer then v v c v v l 1 2 for some c 0 the proof relies on two notions of derivatives for shape optimization parametric derivatives and shape derivatives we use parametric derivatives to handle radial competitors and shape derivatives to deal with normal deformation of the ball a dichotomy is then established to extend the result to all other potentials we develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball finally we add some remarks regarding the coercivity norm of the second order shape derivative in this context

on the first and second variations of a nonlocal isoperimetric problem we consider a nonlocal perturbation of an isoperimetric variational prob lem the problem may be viewed as a mathematical paradigm for the ubiquitous phenom enon of energy driven pattern formation associated with competing short and long range interactions in particular it arises as a g limit of a model for microphase separation of diblock copolymers in this article we establish precise conditions for criticality and stability i e we explicitly compute the first and second variations we also present some applications

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

network amplification of local fluctuations causes high spike rate variability fractal firing patterns and oscillatory local field potentials we investigate a model for neural activity in a two dimensional sheet of leaky integrate and fire neurons with feedback connectivity consisting of local excitation and surround inhibition each neuron receives stochastic input from an external source independent in space and time as recently suggested by softky and koch 1992 1993 independent stochastic input alone cannot explain the high interspike interval variability exhibited by cortical neurons in behaving monkeys we show that high variability can be obtained due to the amplification of correlated fluctuations in a recurrent network furthermore the cross correlation functions have a dual structure with a sharp peak on top of a much broader hill this is due to the inhibitory and excitatory feedback connections which cause hotspots of neural activity to form within the network these localized patterns of excitation appear as clusters or stripes that coalesce disintegrate or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters the synaptic current impinging upon a single neuron shows large fluctuations at many time scales leading to a large coefficient of variation cv for the interspike interval statistics the power spectrum associated with single units shows a 1 f decay for small frequencies and is flat at higher frequencies while the power spectrum of the spiking activity averaged over many cells equivalent to the local field potential shows no 1 f decay but a prominent peak around 40 hz in agreement with data recorded from cat and monkey cortex gray et al 1990 eckhorn et al 1993 firing rates exhibit self similarity between 20 and 800 msec resulting in 1 f like noise consistent with the fractal nature of neural spike trains teich 1992

noisy levy walk analog of two dimensional dna walks for chromosomes of s cerevisiae there has been interest in the physics community in studying dna sequences from a physical or mathematical point of view the findings of some recent works suggest that the sequence of base pairs or nucleotides in dna displays power law correlations 1 2 and several controversial points have been discussed in recent years most of these works are based on a one dimensional mapping of the sequence here we study a two dimensional mapping that will be described immediately below in particular in this work we analyze the two dimensional mapping of dna sequences of the organism saccharomyces cerevisiae s c baker s yeast the first eukaryote whose complete genome has been sequenced 3 this provides us with several very large sequences corresponding to the same organism such sequences consist of a succession of four symbols a adenine g guanine t thymine and c cytosine typically the frequency of a and of t is around 0 30 and that of g and c is about 0 20 the frequency of both a1g purines and c1t pyrimidines is very close to 0 5 we will envisage these sequences as realizations of a stochastic process for its analysis we introduce a mapping into a two dimensional walk with each c g symbol we associate one step in the positive negative direction along the vertical y axis with each t a symbol we associate one step to the right left along the horizontal x axis in this way we obtain a roughly unbiased walk a typical resulting pattern is displayed by chromosome ii see fig 1 among the many quantities useful to characterize a walk the mean square displacement msd is one of the most important as it is closely related to the correlations 4 for a standard random walk rw the msd r 2 s is a linear function of the number of steps s r 2 s 52dds the proportionality constant 2 dd where d is the dimension of the space defines the diffusion coefficientd this purely linear behavior is an immediate consequence of the fact that a rw is a sum of identically distributed independent random variables the total displacement after s steps r s is the sum of

extreme value theory for constrained physical systems we investigate extreme value theory for physical systems with a global conservation law which describes renewal processes mass transport models and long range interacting spin models as shown previously a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support we expose exact relationships between constrained extreme value theory and well known quantities of the underlying stochastic dynamics all valid beyond the midpoint in general i e even far from the thermodynamic limit for example for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events in the thermodynamic limit we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory for example for the renewal process we unravel dual scaling of the extreme value distribution pointing out two types of limiting laws a normalizable scaling function for the typical statistics and a non normalized state describing the rare events

dynamical model for dna sequences we address the problem of dna sequences developing a dynamical method based on the assumption that the statistical properties of dna paths are determined by the joint action of two processes one deterministic with long range correlations and the other random and ensuremath delta function correlated the generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos that are responsible for the birth of anomalous diffusion it is assumed that the deterministic process corresponds to unknown biological rules that determine the dna path whereas the noise mimics the influence of an infinite dimensional environment on the biological process under study we prove that the resulting diffusion process if the effect of the random process is neglected is an ensuremath alpha stable l evy process with 12 we also show that if the diffusion process is determined by the joint action of the deterministic and the random process the correlation effects of the deterministic dynamics are cancelled on the short range scale but show up in the long range one we denote our prescription to generate statistical sequences as the copying mistake map cmm we carry out our analysis of several dna sequences and their cmm realizations with a variety of techniques and we especially focus on a method of regression to equilibrium which we call the onsager analysis with these techniques we establish the statistical equivalence of the real dna sequences with their cmm realizations we show that long range correlations are present in exons as well as in introns but are difficult to detect since the exon dynamics is shown to be determined by the entanglement of three distinct and independent cmm s

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

recognition of an organism from fragments of its complete genome this paper considers the problem of matching a fragment to an organism using its complete genome our method is based on the probability measure representation of a genome we first demonstrate that these probability measures can be modeled as recurrent iterated function systems rifs consisting of four contractive similarities our hypothesis is that the multifractal characteristics of the probability measure of a complete genome as captured by the rifs is preserved in its reasonably long fragments we compute the rifs of fragments of various lengths and random starting points and compare with that of the original sequence for recognition using the euclidean distance a demonstration on five randomly selected organisms supports the above hypothesis

power spectrum of mass and activity fluctuations in a sandpile we consider a directed abelian sandpile on a strip of size 2 ifmmode times else texttimes fi n driven by adding a grain randomly at the left boundary after every t timesteps we establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary base representation of the position of a random walker on a ring of size 3 n we find that while the fluctuations of mass have a power spectrum that varies as 1 f for frequencies in the range 3 ensuremath 2n ensuremath ll f ensuremath ll 1 t the activity fluctuations in the same frequency range have a power spectrum that is linear in f

faber krahn inequalities in sharp quantitative form the classical faber krahn inequality asserts that balls uniquely minimize the first eigenvalue of the dirichlet laplacian among sets with given volume in this paper we prove a sharp quantitative enhancement of this result thus confirming a conjecture by nadirashvili and bhattacharya weitsman more generally the result applies to every optimal poincare sobolev constant for the embeddings w 1 2 0 omega hookrightarrow l q omega

including long range dependence in integrate and fire models of the high interspike interval variability of cortical neurons many different types of integrate and fire models have been designed in order to explain how it is possible for a cortical neuron to integrate over many independent inputs while still producing highly variable spike trains within this context the variability of spike trains has been almost exclusively measured using the coefficient of variation of interspike intervals however another important statistical property that has been found in cortical spike trains and is closely associated with their high firing variability is long range dependence we investigate the conditions if any under which such models produce output spike trains with both interspike interval variability and long range dependence similar to those that have previously been measured from actual cortical neurons we first show analytically that a large class of high variability integrate and fire models is incapable of producing such outputs based on the fact that their output spike trains are always mathematically equivalent to renewal processes this class of models subsumes a majority of previously published models including those that use excitation inhibition balance correlated inputs partial reset or nonlinear leakage to produce outputs with high variability next we study integrate and fire models that have nonpoissonian renewal point process inputs instead of the poisson point process inputs used in the preceding class of models the confluence of our analytical and simulation results implies that the renewal input model is capable of producing high variability and long range dependence comparable to that seen in spike trains recorded from cortical neurons but only if the interspike intervals of the inputs have infinite variance a physiologically unrealistic condition finally we suggest a new integrate and fire model that does not suffer any of the previously mentioned shortcomings by analyzing simulation results for this model we show that it is capable of producing output spike trains with interspike interval variability and long range dependence that match empirical data from cortical spike trains this model is similar to the other models in this study except that its inputs are fractional gaussian noise driven poisson processes rather than renewal point processes in addition to this model s success in producing realistic output spike trains its inputs have long range dependence similar to that found in most subcortical neurons in sensory pathways including the inputs to cortex analysis of output spike trains from simulations of this model also shows that a tight balance between the amounts of excitation and inhibition at the inputs to cortical neurons is not necessary for high interspike interval variability at their outputs furthermore in our analysis of this model we show that the superposition of many fractional gaussian noise driven poisson processes does not approximate a poisson process which challenges the common assumption that the total effect of a large number of inputs on a neuron is well represented by a poisson process

fractal renewal processes two relatively simple renewal processes whose power spectral densities vary as 1 f sup d are constructed 1 a standard renewal point process with 0

non gaussian statistics of anomalous diffusion the dna sequences of prokaryotes this article discusses a non gaussian indicator to measure the deviation from gaussian statistics of a diffusion process generated by dichotomous fluctuations with infinite memory

an alternative description of power law correlations in dna sequences abstract we analyze the coding sequence for the homo sapiens via a model which naturally embraces power law correlations plc among the bases in dna sequences of living organisms this model is based on a principle of universal optimization which is the core of all statistical arguments being associated with the power law distribution function of the length of dna measured in base pairs bp this distribution provides a plc parameter introduced through a nonadditive framework in which such parameter measures the plc in the dna sequence the results show that the short range correlations src always present in coding dna sequences are appropriately captured through the power law distribution adequately describing the cumulative length distribution of dna bases in contrast with the case of the traditional exponential statistical model we use an empirical cumulative distribution function and the database of proteins compiled by the ensembl project to show that the power law distribution provides the best description of the data a bayesian analysis of the data further confirms this result

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

existence and stability for a non local isoperimetric model of charged liquid drops we consider a variational problem related to the shape of charged liquid drops at equilibrium we show that this problem never admits local minimizers with respect to l1 perturbations preserving the volume however we prove that the ball is stable under small c1 1 perturbations when the charge is small enough

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

on periodic critical points and local minimizers of the ohta kawasaki functional abstract in this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem arising in the modeling of diblock copolymers in the main result by means of a purely variational procedure we show that it is possible to construct locally minimizing periodic critical points whose shape resemble that of any given strictly stable constant mean curvature periodic hypersurface along the way we establish several auxiliary results of independent interest

examples of discontinuity of lyapunov exponent in smooth quasiperiodic cocycles we study the regularity of the lyapunov exponent for quasiperiodic cocycles t a where t is an irrational rotation x x 2 on s1 and a cl s1 sl 2 r 0 l for any fixed l 0 1 2 and any fixed of bounded type we construct dl cl s1 sl 2 r such that the lyapunov exponent is not continuous at dl in cl topology we also construct such examples in a smaller schrodinger class

scaling of swimming sequences in copepod behavior data analysis and simulation abstract we consider symbolic sequences of copepod behavior classified as break and slow swimming states successive residence times for each state have been recorded for 327 symbolic times series obtained from 52 different trajectories we performed a symbolic dynamics analysis of the succession of states and showed that the probability densities of the duration of each state obey power laws for large times we analyzed the transition between states and showed as expected the non markovian properties of this dynamics we further simulate a copepod trajectory using an alternate renewal process for the generation of the sequence time series and using a brownian motion for the copepod motion during swimming periods

measuring correlations in symbol sequences abstract the paper is devoted to relations between correlation functions and mutual information it is shown that in sequences over an alphabet of symbols statistical dependences are measured by 1 2 independent parameters however not all of them can be determined by autocorrelation functions appropriate sets of correlation functions including crosscorrelations are introduced which allow the detection of all dependences the results are exemplified for binary ternary and quaternary symbol sequences as an application it is discussed that a nonuniform codon usage in protein coding dna sequences introduces periodic correlations even at distances in the order of 1000 base pairs

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

isoperimetry and stability properties of balls with respect to nonlocal energies we obtain a sharp quantitative isoperimetric inequality for nonlocal s perimeters uniform with respect to s bounded away from 0 this allows us to address local and global minimality properties of balls with respect to the volume constrained minimization of a free energy consisting of a nonlocal s perimeter plus a non local repulsive interaction term in the particular case s 1 the s perimeter coincides with the classical perimeter and our results improve the ones of knuepfer and muratov comm pure appl math 66 7 1129 1162 2013 comm pure appl math 2014 concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term more precisely their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

absolutely continuous spectrum of multifrequency quasiperiodic schrodinger operator abstract in this paper we prove that for any d frequency analytic quasiperiodic schrodinger operator if the frequency is weak liouvillean and the potential is small enough then the corresponding operator has absolutely continuous spectrum moreover in the case d 2 we even establish the existence of ac spectrum under small potential and some super liouvillean frequency and this result is optimal due to a recent counterexample of avila and jitomirskaya 6

hierarchical multifractal representation of symbolic sequences and application to human chromosomes the two dimensional density correlation matrix is constructed for symbolic sequences using contiguous segments of arbitrary size the multifractal spectrum obtained from this matrix motif is shown to characterize the correlations in the symbolic sequences this method is applied to entire human chromosomes shuffled human chromosomes reconstructed human genomic sequences and to artificial random sequences it is shown that all human chromosomes have common characteristics in their multifractal spectrum and deviate substantially from random and uncorrelated sequences of the same size small deviations are observed between the longer and the shorter chromosomes especially for the higher in absolute values statistical moments the correlations are crucial for the form of the multifractal spectrum surrogate shuffled chromosomes present randomlike spectrum distinctly different from the actual chromosomes analytical approaches based on hierarchical superposition of tensor products show that retaining pair correlations in the sequences leads to a closer representation of the genomic multifractal spectra especially in the region of negative exponents due to the underrepresentation of various functional units such as the cytosine guanine cg combination and its complementary gc complex retaining higher order correlations in the construction of the tensor products is a way to approach closer the structure of the multifractal spectra of the actual genomic sequences this hierarchical approach is generic and is applicable to other correlated symbolic sequences

stability in shape optimization with second variation we are interested in the question of stability in the eld of shape optimization precisely we prove that under structural assumptions on the hessian of the considered shape functions the necessary second order minimality conditions imply that the shape hessian is coercive for a given norm moreover under an additional continuity condition for the second order derivatives we derive precise optimality results in the class of regular perturbations of a domain these conditions are quite general and are satised for the volume the perimeter the torsional rigidity and the rst dirichlet eigenvalue of the laplace operator as an application we provide non trivial examples of inequalities obtained in this way

fractal renewal processes generate 1 f noise 1 mathit f mathit d noise occurs in an impressive variety of physical systems and numerous complex theories have been proposed to explain it we construct two relatively simple renewal processes whose power spectral densities vary as 1 mathit f mathit d i a standard renewal point process with 0d1 and ii a finite valued alternating renewal process with 0d2 the resulting event number statistics coincidence rates minimal coverings and autocorrelation functions are shown also to follow power law forms these fractal characteristics derive from interevent time probability density functions which themselves decay in a power law fashion a number of applications are considered trapping in amorphous semiconductors electronic burst noise movement in systems with fractal boundaries the digital generation of 1 mathit f mathit d noise and ionic currents in cell membranes

statistical properties of one dimensional binary sequences with power law power spectrum abstract by the fourier filtering method we generate one dimensional binary sequences from coarse grained continuous sequences with preset exponents 0 using the spectrum analysis we find that the corresponding binary sequences have pure 1 f power spectrum and spectrum exponents 0 0 2 0 where f is the frequency we evaluate numerically the relation between and 0 using the autocorrelation function analysis the detrended fluctuation analysis the duration time analysis and the entropy analysis we investigate extensively the statistical properties of such binary sequences we find that the statistical properties are basically different for 1 and 1 and binary sequences become more and more ordered as increases

floquet solutions for the 1 dimensional quasi periodic schrodinger equation we show that the 1 dimensional schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a floquet representation for almost every energye in the upper part of the spectrum we prove that the upper part of the spectrum is purely absolutely continuous and that for a generic potential it is a cantor set we also show that for a small potential these results extend to the whole spectrum

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

universal generation of 1 f noises this paper establishes a universal mechanism for the generation of 1 f noises the mechanism is based on a signal superposition model which superimposes signals transmitted from independent sources all sources transmit a statistically common yet arbitrary stochastic signal pattern each source has its own random transmission parameters amplitude frequency and initiation epoch we explore randomizations of the transmission parameters which render the power spectrum of the super imposed signal invariant with respect of the stochastic signal pattern transmitted analysis shows that the aforementioned randomizations yield superimposed signals which are 1 f noises the result obtained is in effect a randomized central limit theorem for 1 f noises

on second order shape optimization methods for electrical impedance tomography this paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion omega inside a body omega from boundary measurement this inverse problem known as electrical impedance tomography has many important practical applications and hence has been the focus of much attention during the past few years however to the best of our knowledge no work has yet considered a second order approach for this problem this paper aims to fill that void we investigate the existence of second order derivative of the state u with respect to perturbations of the shape of the interface partial omega then we choose a cost function in order to recover the geometry of partial omega and derive the expression of the derivatives needed to implement the corresponding newton method we then investigate the stability of the process and explain why this inverse problem is severely ill posed by proving the compactness of the hessian at the global minimizer

a non extensive approach to the entropy of symbolic sequences symbolic sequences with long range correlations are expected to result in a slow regression to a steady state of entropy increase however we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained provided that the extensive entropy of tsallis with entropic index q is adopted thereby resulting in a new form of entropy that we shall refer to as kolmogorov sinai tsallis kst entropy we assume that the same symbols either 1 or 1 are repeated in strings of length l with the probability distribution p l 1 l the numerical evaluation of the kst entropy suggests that at the value 2 a sort of abrupt transition might occur for the values of in the range 1 3 it is argued that this phase transition like property signals the onset of the thermodynamical regime at 2

dry friction avalanches experiment and theory experimental evidence and theoretical models are presented supporting the conjecture that dry friction stick slip is described by self organized criticality we use the data obtained with a pin on disk tribometer set to measure lateral force to examine the variation of the friction force as a function of time we study nominally flat surfaces of matching aluminum and steel the probability distribution of force drops follows a negative power law with exponents ensuremath mu in the range 3 2 3 5 the frequency power spectrum follows a 1 f ensuremath alpha pattern with ensuremath alpha in the range 1 1 8 we first compare these experimental results with the well known robin hood model of self organized criticality we find good agreement between theory and experiment for the force drop distribution but not for the power spectrum we explain this on a physical basis and propose a model which takes explicitly into account the stiffness and inertia of the tribometer specifically we numerically solve the equation of motion of a block on a friction surface pulled by a spring and show that for certain spring constants the motion is characterized by the same power law spectrum as in experiments we propose a physical picture relating the fluctuations of the force drops to the microscopic geometry of the surface

poincare recurrences and multifractal properties of genomic sequences we propose the return times spectra as a tool to analyze the genomic sequences the spectra discriminate between regular chaotic and mixed dynamical systems since generally their decay follows respectively an exponential law a power law or is a linear combination of them the analysis of words of n bases from a dna sequence exhibits the same exponential behavior for the coding and noncoding component the multifractal analysis shows a denite dierence which suggests a less uniform structure for the noncoding part

sharp phase transitions for the almost mathieu operator it is known that the spectral type of the almost mathieu operator amo depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency we study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place which solves jitomirskaya s conjecture together with a previous work by avila this gives the sharp description of phase transitions for the amo for the a e phase

analytically solvable autocorrelation function for weakly correlated interevent times long term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions such temporal correlations can be understood not only by heterogeneous interevent times iets but also by correlations between iets in contrast to the role of heterogeneous iets on the autocorrelation function little is known about the effects due to the correlations between iets to rigorously study these effects we derive an analytical form of the autocorrelation function for the arbitrary iet distribution in the case with weakly correlated iets where the farlie gumbel morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive iets our analytical results are confirmed by numerical simulations for exponential and power law iet distributions for the power law case we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between iets our analytical approach enables us to better understand long term temporal correlations induced by the correlations between iets

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

informational and causal architecture of continuous time renewal processes we introduce the minimal maximally predictive models epsilon text machines of processes generated by certain hidden semi markov models their causal states are either discrete mixed or continuous random variables and causal state transitions are described by partial differential equations as an application we present a complete analysis of the epsilon text machines of continuous time renewal processes this leads to closed form expressions for their entropy rate statistical complexity excess entropy and differential information anatomy rates

fast exact synthesis of gaussian and nongaussian long range dependent processes 1 f noise and statistically self similar random processes such as fractional brownian motion fbm and fractional gaussian noise fgn are fundamental models for a host of real world phenomena from network traffic to dna to the stock market synthesis algorithms play a key role by providing the feedstock of data necessary for running complex simulations and accurately evaluating analysis techniques unfortunately current algorithms to correctly synthesize these long range dependent lrd processes are either abstruse or prohibitively costly which has spurred the wide use of inexact approximations to fill the gap we develop a simple fast o n log n operations for a length n signal framework for exactly synthesizing a range of gaussian and nongaussian lrd processes as a bonus we introduce and study a new bi scaling fbm process featuring a kinked correlation function that exhibits distinct scaling laws at coarse and fine scales

scale free duplication dynamics a model for ultraduplication empirical studies of the genome wide length distribution of duplicated sequences have revealed an algebraic tail common to nearly all clades the decay of the tail is often well approximated by a single exponent that takes values within a limited range we propose and study here scale free duplication dynamics a class of model for genome sequence evolution that generates the observed shapes of this distribution a transition between self similar and non self similar regimes is exhibited our model accounts plausibly for the observed form of the algebraic tail which is not produced by standard models for generating long range sequence correlations

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

what can we learn with wavelets about dna sequences we use the wavelet transform to explore the complexity of dna sequences long range correlations are clearly identified and shown to be related to the sequence gc content the significance of this observation to gene evolution is discussed

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ for convex bodies d in rn the deviation d from spherical shape is estimated from above in terms of the dimensionless isoperimetric deficiency a of d as follows d f a for a sufficiently small here f is an explicit elementary function vanishing continuously at 0 the estimate is sharp as regards the order of magnitude of f the dimensions n 2 and 3 present anomalies as to the form of f in the planar case n 2 the result is contained in an inequality due to t bonnesen a qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies d in rn in the sense that as d varies d p 0 for a a 0 the proof of the estimate d f a is based on a related estimate in the case of domains not necessarily convex that are supposed a priori to be nearly spherical in a certain sense

uniform positivity and continuity of lyapunov exponents for a class of c 2 quasiperiodic schrodinger cocycles abstract we show that for a class of c 2 quasiperiodic potentials and for any fixed diophantine frequency the lyapunov exponent of the corresponding schrodinger cocycles as a function of energies are uniformly positive and weakly holder continuous as a corollary we obtain that the corresponding integrated density of states is weakly holder continuous as well our approach is of purely dynamical systems which depends on a detailed analysis of asymptotic stable and unstable directions we also apply it to more general sl 2 r cocycles which in turn can be applied to get uniform positivity and continuity of lyapunov exponents around unique nondegenerate extremal points of any smooth potential and to a certain class of c 2 szeg cocycles

theory of non markovian stochastic resonance we consider a two state model of non markovian stochastic resonance sr within the framework of the theory of renewal processes residence time intervals are assumed to be mutually independent and characterized by some arbitrary nonexponential residence time distributions which are modulated in time by an externally applied signal making use of a stochastic path integral approach we obtain general integral equations governing the evolution of conditional probabilities in the presence of an input signal these equations generalize earlier integral renewal equations by cox and others to the case of driving induced nonstationarity on the basis of these equations a response theory of two state renewal processes is formulated beyond the linear response approximation moreover a general expression for the linear response function is derived the connection of the developed approach with the phenomenological theory of linear response for manifest non markovian sr put forward i goychuk and p hanggi phys rev lett 91 070601 2003 is clarified and its range of validity is scrutinized the theory is then applied to sr in symmetric non markovian systems and to the class of single ion channels possessing a fractal kinetics

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

spectral 1 f noise derived from extremized physical information we consider the evolution of a temporal signal x t that is an intrinsic random field of order zero in the sense of a certain measurement estimation experiment the state of disorder of x t should increase toward an equilibrium state the disorder of x t is measured by its physical information scri and the equilibrium state is determined by the condition that scri be an extremum the equilibrium state is shown to have a power spectrum s ensuremath omega of the form mathrm ensuremath omega mathrm ensuremath mathrm ensuremath alpha 1 ensuremath le ensuremath alpha ensuremath le 2 that of 1 f noise

symbolic strings and spatial 1 f spectra abstract some string dynamics which are expected to show non trivial correlations are proposed and studied it is because we were motivated by the observation that some dna sequences show spatial 1 f correlation our model is based on a rule by which one string is split into two parts in the middle and another string is inserted between them when the strings are made longer the basic model shows long range correlations with a structure characteristic of cyclic patterns on the log log plot of the spectrum the effects of string cross over and point mutation are also studied the cross over destroys the structures appearing in the basic model and makes 1 f like correlation clear the point mutation makes the power spectrum flat as is expected from the white noise nature the analogy of the cross over model with real dna evolution dynamics is suggested

estimating the intermittency of point processes with applications to human activity and viral dna abstract the intermittency of a point process is the extent to which the number of events in a time window has pronounced departures from typical values combining point process and multifractal formalisms indicates that the correlation codimension can be used to quantify intermittency the correlation codimension is easily estimated and is simply related to other second order scaling exponents such as those of the fano factor and spectral density the correlation codimensions are derived for various uncorrelated fractal and fractal rate point processes in addition the estimation of intermittency as the correlation codimension of experimental events is illustrated with applications to experimental data human activity during bed rest is highly intermittent while other human activity and viral dna composition are non intermittent

holder continuity of absolutely continuous spectral measure for multi frequency schrodinger operators abstract we establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi frequency schrodinger operators with diphantine frequencies and small analytic potentials

waveform mapping and time frequency processing of dna and protein sequences current state of the art approaches for biological sequence querying and alignment require preprocessing and lack robustness to repetitions in the sequence in addition these approaches do not provide much support for efficiently querying subsequences a process that is essential for tracking localized database matches we propose a query based alignment method for biological sequences that first maps sequences to time domain waveforms before processing the waveforms for alignment in the time frequency plane the mapping uses waveforms such as gaussian functions with unique sequence representations in the time frequency plane the proposed alignment method employs a robust querying algorithm that utilizes a time frequency signal expansion whose basis function is matched to the basic waveform in the mapped sequences the resulting wavequery approach was demonstrated for both deoxyribonucleic acid dna and protein sequences using the matching pursuit decomposition as the signal basis expansion we specifically evaluated the alignment localization of wavequery over repetitive database segments and we demonstrated its operation in real time without preprocessing we also demonstrated that wavequery significantly outperformed the biological sequence alignment method blast for queries with repetitive segments for dna sequences a generalized version of the wavequery approach with the metaplectic transform is also described for protein sequence structure prediction

embedding of analytic quasi periodic cocycles into analytic quasi periodic linear systems and its applications in this paper we prove that any analytic quasi periodic cocycle close to constant is the poincare map of an analytic quasi periodic linear system close to constant which bridges both methods and results in quasi periodic linear systems and cocycles we also show that the almost reducibility of an analytic quasi periodic linear system is equivalent to the almost reducibility of its corresponding poincare cocycle by the local embedding theorem and the equivalence we transfer the recent local almost reducibility results of quasi periodic linear systems hou and you in invent math 190 209 260 2012 to quasi periodic cocycles and the global reducibility results of quasi periodic cocycles avila in almost reducibility and absolute continuity 2010 avila et al in geom funct anal 21 1001 1019 2011 to quasi periodic linear systems finally we give a positive answer to a question of avila et al geom funct anal 21 1001 1019 2011 and use it to study point spectrum of long range quasi periodic operator with liouvillean frequency the embedding also holds for some nonlinear systems