measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

distribution of periodic torus orbits and duke s theorem for cubic fields we study periodic torus orbits on spaces of lattices using the action of the group of adelic points of the underlying tori we define a natural equivalence relation on these orbits and show that the equivalence classes become uniformly distributed this is a cubic analogue of duke s theorem about the distribution of closed geodesics on the modular surface suitably interpreted the ideal classes of a cubic totally real field are equidistributed in the modular 5 fold sl3 z sl3 r so3 in particular this proves a stronger form of the folklore conjecture that the collection of maximal compact flats in sl3 z sl3 r so3 of volume infinity

upper and lower bounds at s 1 for certain dirichlet series with euler product estimates of the form l j s mathscr a ll epsilon j mathscr d a mathscr r epsilon mathscr a in the range s 1 ll 1 log mathscr r a for general l functions where mathscr r a is a parameter related to the functional equation of l s mathscr a can be quite easily obtained if the ramanujan hypothesis is assumed we prove the same estimates when the l functions have euler product of polynomial type and the ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions as a consequence we obtain an upper bound of this type for every l s pi where pi is an automorphic cusp form on rm gl mathbf d mathbb a k we employ these results to obtain siegel type lower bounds for twists by dirichlet characters of the third symmetric power of a maass form

multifractal detrended cross correlation analysis for two nonstationary signals we propose a method called multifractal detrended cross correlation analysis to investigate the multifractal behaviors in the power law cross correlations between two time series or higher dimensional quantities recorded simultaneously which can be applied to diverse complex systems such as turbulence finance ecology physiology geophysics and so on the method is validated with cross correlated one and two dimensional binomial measures and multifractal random walks as an example we illustrate the method by analyzing two financial time series

equidistribution in shrinking sets and l 4 norm bounds for automorphic forms we study two closely related problems stemming from the random wave conjecture for maas forms the first problem is bounding the l 4 norm of a maas form in the large eigenvalue limit we complete the work of spinu to show that the l 4 norm of an eisenstein series e z 1 2 it g restricted to compact sets is bounded by sqrt log t g the second problem is quantum unique ergodicity in shrinking sets we show that by averaging over the centre of hyperbolic balls in gamma backslash mathbb h quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the planck scale this result is conditional on the generalised lindelof hypothesis for hecke maas eigenforms but is unconditional for eisenstein series we also show that equidistribution for hecke maas eigenforms need not hold at or below the planck scale finally we prove similar equidistribution results in shrinking sets for heegner points and closed geodesics associated to ideal classes of quadratic fields

detrending moving average algorithm for multifractals the detrending moving average dma algorithm is a widely used technique to quantify the long term correlations of nonstationary time series and the long range correlations of fractal surfaces which contains a parameter ensuremath theta determining the position of the detrending window we develop multifractal detrending moving average mfdma algorithms for the analysis of one dimensional multifractal measures and higher dimensional multifractals which is a generalization of the dma method the performance of the one dimensional and two dimensional mfdma methods is investigated using synthetic multifractal measures with analytical solutions for backward ensuremath theta 0 centered ensuremath theta 0 5 and forward ensuremath theta 1 detrending windows we find that the estimated multifractal scaling exponent ensuremath tau q and the singularity spectrum f ensuremath alpha are in good agreement with the theoretical values in addition the backward mfdma method has the best performance which provides the most accurate estimates of the scaling exponents with lowest error bars while the centered mfdma method has the worse performance it is found that the backward mfdma algorithm also outperforms the multifractal detrended fluctuation analysis the one dimensional backward mfdma method is applied to analyzing the time series of shanghai stock exchange composite index and its multifractal nature is confirmed

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

on the hausdorff dimension of fractal attractors we consider such mappingsx n 1 f xn of an interval into itself for which the attractor is a cantor set for the same class of mappings for which the feigenbaum scaling laws hold we show that the hausdorff dimension of the attractor is universally equal tod 0 538

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

multifractal fourier detrended cross correlation analysis of traffic signals abstract multifractal detrended cross correlation analysis mf dxa has been developed to detect the long range power law cross correlation of considered signals in the presence of non stationarity however crossovers arising from extrinsic periodic trends make the scaling behavior difficult to analyze we introduce a fourier filtering method to eliminate the trend effects and systematically investigate the multifractal cross correlation of simulated and real traffic signals the crossover locations are found approximately corresponding to the periods of underlying trend traffic velocity on one road and flows on adjacent roads show strong cross correlation they also present weak multifractality after periodic trends are removed the traffic velocity and flow are cross correlated in opposite directions which is accordant to their actual evolution

bounds for eigenforms on arithmetic hyperbolic 3 manifolds on a family of arithmetic hyperbolic 3 manifolds of square free level we prove an upper bound for the sup norm of hecke maas cusp forms with a power saving over the local geometric bound simultaneously in the laplacian eigenvalue and the volume by a novel combination of diophantine and geometric arguments in a noncommutative setting we obtain bounds as strong as the best corresponding results on arithmetic surfaces

bounds for rankin selberg integrals and quantum unique ergodicity for powerful levels reference epfl article 202508doi 10 1090 s0894 0347 2013 00779 1view record in web of science record created on 2014 10 23 modified on 2017 12 10

equidistribution of cusp forms in the level aspect let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing square free level q infinity we prove that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the poincare measure

quantum unique ergodicity for half integral weight automorphic forms we investigate the analogue of the quantum unique ergodicity que conjecture for half integral weight automorphic forms assuming the generalized riemann hypothesis grh we establish que for both half integral weight holomorphic hecke cusp forms for 0 4 lying in kohnen s plus subspace and for half integral weight hecke maa cusp forms for 0 4 lying in kohnen s plus subspace by combining the former result along with an argument of rudnick it follows that under grh the zeros of these holomorphic hecke cusp equidistribute with respect to hyperbolic measure on 0 4 h as the weight tends to infinity

singularity power spectrum distribution abstract fractal and multifractal signal processing are hot topics in current studies at present the research mainly concentrates in fractal dimension multifractal spectrum and time singularity spectrum distribution which are based on differentiability of fractal subsets however fail to reflect the energy and power measurement of fractal singular subset therefore singularity power spectrum distribution spsd is proposed in this paper based on the traditional power spectrum density function and multifractal spectrum in theory basic concept of spsd is put forward based on the related singularity measure the theoretical expression of spsd is deduced and the compatibility between spsd and conventional power spectrum analysis is proved in algorithm the discrete approximation algorithm of spsd for discrete fractal time series is introduced simulation based on the multifractal brownian motion mfbm and actual sea clutter indicates that spsd can reveal effectively the singular power distribution of sea clutter and mfbm and identify fractal signals even for those with approximate multifractal spectra or approximate power spectra the spsd will provide application prospect of multifractal signal processing detection and distinguishing of multifractal time series

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

implementation of a constrained ritz series modeling technique for acoustic cavity structural systems a prior study ginsberg j h 2010b j acoust soc am 127 2749 2758 used ritz series in conjunction with hamilton s principle to derive general equations describing the time domain response of an acoustic cavity bounded by an elastic structure the equations of motion are supplemented by constraint equations that explicitly enforce velocity continuity at the cavity s surface these constraints are imposed by the surface traction which is represented by unknown coefficients of ritz type series the resulting set of equations are differential algebraic type three methods are presented to convert the governing equations to forms that are familiar to structural acoustics including one that transforms them from differential algebraic type to the standard ordinary differential equations associated with linear multi degree of freedom vibratory systems in cases where only the structure is excited the formulation offers options as to how displacement velocity boundary conditions on the nonstructural b

subconvex equidistribution of cusp forms reduction to eisenstein observables let traverse a sequence of cuspidal automorphic representations of gl2 with large prime level unramified central character and bounded infinity type for g gl1 pgl2 let h g denote the assertion that subconvexity holds for g twists of the adjoint l function of with polynomial dependence upon the conductor of the twist we show that h gl1 implies h pgl2 in geometric terms h pgl2 corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term h gl1 to the special case in which the relevant sequence of measures is tested against an eisenstein series

roughness exponents to calculate multi affine fractal exponents abstract we propose a new method to determine the multi affine fractal exponents based on a generalization of the roughness concept for synthetic multi affine fractal profiles with exactly known exponents we show that the results obtained in this method are more accurate than the results obtained with the height height correlation function method also in the present method scaling is observed using profiles with a number of points more than one order mangnitude smaller

trilinear forms and the central values of triple product l functions we give an explicit formula for certain global trilinear forms that appear in jacquet s conjecture in terms of local trilinear forms and the central values of triple product l functions

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

weak subconvexity for central values of l functions we describe a general method to obtain weak subconvexity bounds for many classes of l functions we give several examples of our bound and our work has applications to a conjecture of rudnick and sarnak for the mass equidistribution of hecke eigenforms

sieving for mass equidistribution we approach the holomorphic analogue to the quantum unique ergodicity conjecture through an application of the large sieve we deal with shifted convolution sums as in ho109 with various simplifications in our analysis due to the knowledge of the ramanujan petersson conjecture in this holomorphic case

multifractal moving average analysis and test of multifractal model with tuned correlations we address two common major problems in the study of time series characterizing fluctuations in complex systems multifractal analysis and multifractal modeling specifically we introduce a multi fractal centered moving average mf cma analysis which is computationally easier but equivalently performing compared with the well established multi fractal detrended fluctuation analysis mf dfa with linear detrending in addition we study in detail a generalized binomial multi fractal model gb mfm to conveniently and reliably generate multifractal surrogate data with arbitrary singularity strengths and arbitrary long term persistence we use the data generated by this model as well as realistic by construction monofractal data series with crossovers and trends to test and compare the multifractal analysis methods and discuss finite size effects as well as limitations due to spurious multifractality

mass equidistribution for hecke eigenforms in this work we study the mass equidistribution for holomorphic hecke eigenforms and establish by employing incomplete poincare series and the petersson formula sharp equidistribution results when the average is performed over intervals much shorter than before a key feature is the analysis of the off diagonal terms that result from this shortening of intervals 2003 wiley periodicals inc

the fractal energy measurement and the singularity energy spectrum analysis abstract the singularity exponent se is the characteristic parameter of fractal and multifractal signals based on se the fractal dimension reflecting the global self similar character the instantaneous se reflecting the local self similar character the multifractal spectrum mfs reflecting the distribution of se and the time varying mfs reflecting pointwise multifractal spectrum were proposed however all the studies were based on the depiction of spatial or differentiability characters of fractal signals taking the se as the independent dimension this paper investigates the fractal energy measurement fem and the singularity energy spectrum ses theory firstly we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain propose the conception of fem and ses of multifractal signals and investigate the hausdorff measure and the local direction angle of the fractal energy element then we prove the compatibility between fem and traditional energy and point out that ses can be measured in the fractal space finally we study the algorithm of ses under the condition of a continuous signal and a discrete signal and give the approximation algorithm of the latter and the estimations of fem and ses of the gaussian white noise fractal brownian motion and the multifractal brownian motion show the theoretical significance and application value of fem and ses

a comparative study of two dimensional multifractal detrended fluctuation analysis and two dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum multifractal detrended fluctuation analysis mfdfa and multifractal detrended moving average mfdma algorithm have been established as two important methods to estimate the multifractal spectrum of the one dimensional random fractal signal they have been generalized to deal with two dimensional and higher dimensional fractal signals this paper gives a brief introduction of the two dimensional multifractal detrended fluctuation analysis 2d mfdfa and two dimensional multifractal detrended moving average 2d mfdma algorithm and a detailed description of the application of the two dimensional fractal signal processing by using the two methods by applying the 2d mfdfa and 2d mfdma to the series generated from the two dimensional multiplicative cascading process we systematically do the comparative analysis to get the advantages disadvantages and the applicabilities of the two algorithms for the first time from six aspects such as the similarities and differences of the algorithm models the statistical accuracy the sensitivities of the sample size the selection of scaling range the choice of the q orders and the calculation amount the results provide a valuable reference on how to choose the algorithm from 2d mfdfa and 2d mfdma and how to make the schemes of the parameter settings of the two algorithms when dealing with specific signals in practical applications

triple product formula and the subconvexity bound of triple product l function in level aspect in this paper we derive a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two as an application we generalize venkatesh and woodbury s work on the subconvexity bound of triple product l function in level aspect allowing joint ramifications higher ramifications general unitary central characters and general special values of local epsilon factors

detrending fluctuation analysis based on moving average filtering abstract detrended fluctuation analysis dfa is a scaling method commonly used for detecting long range correlations in nonstationary time series applications range from financial time series to physiological data however as the removal of trends in dfa is based on discontinuous polynomial fitting oscillations in the fluctuation function and significant errors in crossover locations can be introduced to reduce the problems induced by discontinuous fitting moving average ma methods have been proposed previously by alesio et al eur j phys b 27 2002 197 in this work a variant of such ma methods is studied specifically the performance and characteristics of a ma method based on central differences is studied some important properties of this ma method are analyzed and illustrated with several artificial and real time series

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

mass equidistribution for hecke eigenforms we prove a conjecture of rudnick and sarnak on the mass equidistribution of hecke eigenforms this builds upon independent work of the authors