the mathematical analysis of the structure of some songs abstract the mathematical structures of six songs were studied by treating them as complex systems the studies were based on i scattering diagram ii spiral structure iii graph theory and animal diagrams and iv entropy and organization in i the fractal dimension and the radius of gyration was determined for each melody in ii it was shown that the total number of notes used in a song display a spiral which can be expressed by a mathematical equation in iii the animal diagram was obtained by using the pathways between the consecutive notes of the song a ratio defined as the number of edges to the total number of unit edge for the animal diagram was used as a parameter to characterize a melody in iv the entropy the maximum entropy and the negentropy or the organization of the songs were determined in addition the rate of change of organization of the songs was also determined

noncommutative differential geometry and new models of gauge theory the noncommutative differential geometry of the algebra c v mn c of smooth mn c valued functions on a manifold v is investigated for n 2 the analog of maxwell s theory is constructed and interpreted as a field theory on v it describes a u n yang mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential the euclidean action which is positive vanishes on exactly two distinct gauge orbits which are interpreted as two vacua of the theory in one of the corresponding vacuum sectors the su n part of the yang mills field is massive for the case n 2 analogies with the standard model of electroweak theory are pointed out finally a brief description is provided of what happens if one starts from the analog of a general yang mills theory instead of maxwell s theory which is a particular case

bandt pompe approach to the classical quantum transition abstract by regarding the celebrated classical quantum transition problem as one pertaining to the domain of dynamic systems theory we present a discussion that exhibits the superiority of the bandt pompe approach to the extraction of a probability distribution from time series values

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

wavelet entropy of stochastic processes abstract we compare two different definitions for the wavelet entropy associated to stochastic processes the first one the normalized total wavelet entropy ntws family s blanco a figliola r q quiroga o a rosso e serrano time frequency analysis of electroencephalogram series iii wavelet packets and information cost function phys rev e 57 1998 932 940 o a rosso s blanco j yordanova v kolev a figliola m schurmann e basar wavelet entropy a new tool for analysis of short duration brain electrical signals j neurosci method 105 2001 65 75 and a second introduced by tavares and lucena physica a 357 1 2005 71 78 in order to understand their advantages and disadvantages exact results obtained for fractional gaussian noise 1 1 and fractional brownian motion 1 3 are assessed we find out that the ntws family performs better as a characterization method for these stochastic processes

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

the weierstrass mandelbrot process revisited we derive a functional central limit theorem for quasi gaussian processes in particular we prove that the limit of the mandelbrot weierstrass process is a complex fractional brownian motion

a study on improvisation in a musical performance using multifractal detrended cross correlation analysis abstract mfdfa the most rigorous technique to assess multifractality was performed on four hindustani music samples played on same raga sung by the same performer each music sample was divided into six parts and multifractal spectral width was determined for each part corresponding to the four samples the results obtained reveal that different parts of all the four sound signals possess spectral width of widely varying values this gives a cue of the so called musical improvisation in all music samples keeping in mind they belong to the bandish part of the same raga formal compositions in hindustani raga are juxtaposed with the improvised portions where an artist manoeuvers his her own creativity to bring out a mood that is specific for that particular performance which is known as improvisation further this observation hints at the association of different emotions even in the same bandish of the same raga performed by the same artist this interesting observation cannot be revealed unless rigorous non linear technique explores the nature of musical structure in the second part we applied mfdxa technique to explore more in depth about improvisation and association with emotion this technique is applied to find the degree of cross correlation x between the different parts of the samples pronounced correlation has been observed in the middle parts of the all the four samples evident from higher values of x whereas the other parts show weak correlation this gets further support from the values of spectral width from different parts of the sample width of those parts is significantly different from other parts this observation is extremely new both in respect of musical structure of so called improvisation and associated emotion the importance of this study in application area of cognitive music therapy is immense

optimal strategies for risk sensitive portfolio optimization problems for general factor models we consider constructing optimal strategies for risk sensitive portfolio optimization problems on an infinite time horizon for general factor models where the mean returns and the volatilities of individual securities or asset categories are explicitly affected by economic factors the factors are assumed to be general diffusion processes in studying the ergodic type bellman equations of the risk sensitive portfolio optimization problems we introduce some auxiliary classical stochastic control problems with the same bellman equations as the original ones we show that the optimal diffusion processes of the problem are ergodic and that under some condition related to integrability by the invariant measures of the diffusion processes we can construct optimal strategies for the original problems by using the solution of the bellman equations

comparison of two fractal interpolation methods abstract as a tool for studying complex shapes and structures in nature fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon numerous fractal interpolation methods have been proposed over the past few decades but they differ substantially in the form features and statistical properties in this study we simulated one and two dimensional fractal surfaces by using the midpoint displacement method and the weierstrass mandelbrot fractal function method and observed great differences between the two methods in the statistical characteristics and autocorrelation features from the aspect of form features the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases while the simulations of the weierstrass mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases from the aspect of statistical properties the peak heights from the weierstrass mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension and the variances are approximately two times larger when the fractal dimension equals to 1 2 1 4 1 6 and 1 8 the skewness is positive with the midpoint displacement method and the peaks are all convex but for the weierstrass mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero the kurtosis is less than one with the midpoint displacement method and generally less than that of the weierstrass mandelbrot fractal function method the autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness which is suitable for simulating aperiodic surface while the simulation of the weierstrass mandelbrot fractal function method has strong periodicity which is suitable for simulating periodic surface

the wishart autoregressive process of multivariate stochastic volatility abstract the wishart autoregressive war process is a dynamic model for time series of multivariate stochastic volatility the war naturally accommodates the positivity and symmetry of volatility matrices and provides closed form non linear forecasts the estimation of the war is straighforward as it relies on standard methods such as the method of moments and maximum likelihood for illustration the war is applied to a sequence of intraday realized volatility covolatility matrices from the toronto stock market tsx

characterization of gaussian self similar stochastic processes using wavelet based informational tools efficient tools to characterize stochastic processes are discussed quantifiers originally proposed within the framework of information theory like entropy and statistical complexity are translated into wavelet language which renders the above quantifiers into tools that exhibit the important localization advantages provided by wavelet theory two important and popular stochastic processes fractional brownian motion and fractional gaussian noise are studied using these wavelet based informational tools exact analytical expressions are obtained for the wavelet probability distribution finally numerical simulations are used to validate our analytical results

earthquake statistics and fractal faults we introduce a self affine asperity model sam for the seismicity that mimics the fault friction by means of two fractional brownian profiles that slide one over the other an earthquake occurs when there is an overlap of the two profiles representing the two fault faces and its energy is assumed proportional to the overlap surface the sam exhibits the gutenberg richter law with an exponent ensuremath beta related to the roughness index of the profiles apart from being analytically treatable the model exhibits a nontrivial clustering in the spatiotemporal distribution of epicenters that strongly resembles the experimentally observed one a generalized and more realistic version of the model exhibits the omori scaling for the distribution of the aftershocks the sam lies in a different perspective with respect to usual models for seismicity in this case in fact the critical behavior is not self organized but stems from the fractal geometry of the faults which in its turn is supposed to arise as a consequence of geological processes on very long time scales with respect to the seismic dynamics our approach is distinguished by the explicit introduction of the fault geometry as an active element of this complex phenomenology

on the spectra of riemann liouville fractional brownian motion we study the spectrum of the fractional brownian motion of riemann liouville type using two approaches namely the double frequency spectral density and the wigner ville spectrum the bifrequency representation gives a complex valued function which contains two distinctive terms these terms can be identified as the diagonal and off diagonal distribution of the spectrum in a frequency frequency plane the physical interpretation of these two terms is briefly discussed a calculation of wigner ville spectrum gives an alternative way of representing the spectrum of this nonstationary process in the time frequency plane asymptotic approximation of the wigner ville spectrum is obtained we show that the large time average spectrum of riemann liouville fractional brownian motion exhibits a power law in the high frequency range

recurrence and transience of optimal feedback processes associated with bellman equations of ergodic type the paper is concerned with ergodic type bellman equations arising typically in linear exponential quadratic gaussian control we are interested in giving recurrence transience criteria for associated optimal feedback diffusions in terms of qualitative properties of solutions to the bellman equation to establish such criteria we propose a new approach which is based on the lyapunov method it turns out that a certain convexity of the equation plays a key role in our arguments

self similarity of the 1 f noise called music abstract suggestions have been made that computer musicians should attempt to compose fractal music and questions have been raised whether there is such a thing as fractal music voss and clark observed that music is scaling or 1 f noise as analyzed on the basis of the amplitude loudness of the audio signals they failed to find a fractal distribution of acoustic frequencies music notes in music analyzing bach s and mozart s compositions we have shown that the incidence of the frequency intervals or of the changes of acoustic frequency has a fractal geometry fractal phenomena are characterized by scale independency the purpose of this investigation is to demonstrate the self similarity of music and to explore its implications

fractal landscapes in physics and biology the purpose of this talk is to describe some recent progress in three aspects of statistical physics that overlap biology the first of these represents a discovery about the real world for which we presently have no theoretical understanding 1 the second concerns the solution of a rich model of many body physics which should apply to a wide range of biological situations but presently applies primarily to mathematical ecology 2 the third concerns what on the one hand might be a mere numerical coincidence or on the other hand might represent evidence that diffusion limited aggregation describes the physical mechanism underlying the growth of nerve cells at least the quasi two dimensional nerve cells that grow in the retina 3

complexity entropy causality plane a useful approach for distinguishing songs nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies this is particularly true when dealing with music databases in this context it is essential to have techniques and tools able to discriminate properties from these massive sets in this work we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure building up the complexity entropy causality plane the results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs additionally we believe that the here reported method may be applied in practical situations since it is simple robust and has a fast numerical implementation

analysis of temporal fluctuations in bach s sinfonias abstract the correlation structures in 15 bach s sinfonias were analyzed each sinfonia is characterized by the superposition of three voices each voice is a sequence of pitches each voice was transformed in a time series in which the sampling time was given by the smallest pitch duration in that voice the scaling properties of the three voices of each sinfonia was quantified by means of the estimate of the scaling exponent performed using the power spectral density psd and the detrended fluctuation analysis dfa the results show that the voice time series are persistent the dfa was applied not only to any single voice time series but also to couples 2 dfa of voices and to the triple 3 dfa of voices it was found that the first voice of each sinfonia modulates the scaling behavior of the whole sinfonia

quantum deformation of brst algebra we investigate theq deformation of the brst algebra the algebra of the ghost matter and gauge fields on one spacetime point using the result of the bicovariant differential caculus there are two nilpotent operations in the algebra the brst transformation b and the derivatived we show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operations as well as the operation the antimultiplicativ e inner involution

viscoelastic behavior of stock indices abstract the scattering diagram of a stock index results in a complex network structure which can be used to analyze the viscoelastic properties of the index the change along x or y direction of the diagram corresponds to purely elastic or spring like movement whereas the diagonal change at an angle of 45 corresponds to purely viscous or dashpot like movement the viscous component pushes the price from its current value to any other value while the elastic component acts like a restoring force four indices namely dji s p 500 nasdaq 100 and nasdaq composite were studied for the period of 2001 2009 nasdaq composite displayed very high elasticity while nasdaq 100 displayed the highest fluidity in the time period considered the fluidity of dji and s p 500 came out to be close to each other and they are almost the same in the second half of the period

correlations in a mozart s music score k 73x with palindromic and upside down structure abstract in this work we study long range correlations in a scherzo duetto di mozart score k 73x for two violins this is a fascinating piece as the second violin part is upside down on the same sheet below the first violin and some parts are like a palindrome given such ingenious structure it is expected the existence of long range correlations in the score structure in order to quantify long range correlations we considered the music score as a sequence of integer numbers each of them corresponding to last common denominator units of note by using detrended fluctuation analysis dfa correlations are quantified by means of the scaling exponent that reflects the type of correlations for a given distance between neighbors note the following conclusions can be drawn from the analysis a for about 10 25 neighbor note distances correlations are similar to 1 f noise this is an interesting finding since it has been shown that pleasant sounds for humans display a behavior similar to 1 f noise b as the neighbor note distance increases the long range correlations decays continuously for some score sections the music score behaves like non correlated i e purely random noise summing up the results show that the studied mozart s score contains a certain degree of correlation for relatively small note distances and becomes close to non correlated behavior for long note distances we considered also the sequence constructed by considering the distance between the simultaneously played notes of the two violins interestingly for relatively small neighbor note distances a scaling behavior similar to that found for individual violins is also displayed in some sense this is an expression of the specific structure palindromes plus upside down construction used by mozart in the composition of this music score although we focused on a particular high art music score our results suggest that modern methods borrowed from statistical physics can be useful for the systematic study of music composition techniques

ordinal analysis of time series abstract in order to develop fast and robust methods for extracting qualitative information from non linear time series bandt and pompe have proposed to consider time series from the pure ordinal viewpoint on the basis of counting ordinal patterns which describe the up and down in a time series they have introduced the concept of permutation entropy for quantifying the complexity of a system behind a time series the permutation entropy only provides one detail of the ordinal structure of a time series here we present a method for extracting the whole ordinal information

b r s algebras analysis of the consistency equations in gauge theory we compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms by using the appropriate cohomological and algebraic methods we do it for all dimensions of spacetime and all structure groups with reductive lie algebras

holder estimates for degenerate elliptic equations with coercive hamiltonians we prove a priori estimates and regularity results for some quasi linear degenerate elliptic equations arising in optimal stochastic control problems our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally holder continuous and solutions to be locally lipschitz continuous we also give an existence result for the associated dirichlet problem

predictions for pulsed field gradient nmr experiments of diffusion in fractal spaces in this paper theoretical predictions are made for pulsed field gradient nmr experiments of diffusion in fractal spaces we obtain the general q space behaviour of the nmr signal which upon fourier transformation delivers the probability density for displacements we also show that there exists a power law behaviour in the limits of large q or small displacements

simple model for 1 f α noise we present a simple stochastic mechanism which generates pulse trains exhibiting a power law distribution of the pulse intervals and a 1 f ensuremath alpha power spectrum over several decades at low frequencies with ensuremath alpha close to 1 the essential ingredient of our model is a fluctuating threshold which performs a brownian motion whenever an increasing potential v t hits the threshold v t is reset to the origin and a pulse is emitted we show that if v t increases linearly in time the pulse intervals can be approximated by a random walk with multiplicative noise our model agrees with recent experiments in neurobiology and explains the high interpulse interval variability and the occurrence of 1 f ensuremath alpha noise observed in cortical neurons and earthquake data

analysis and perception of spectral 1 f characteristics of amplitude and period fluctuations in normal sustained vowels two kinds of fluctuations are always observed in the steady parts of normal sustained vowels one is amplitude fluctuation defined as the cyclic changes of maximum peak amplitudes the other is period fluctuation defined as the cyclic changes of pitch periods the primary purpose of this paper is to present quantitative descriptions of amplitude and period sequences obtained from normal sustained vowels these fluctuation sequences consisted of maximum peak amplitudes or pitch periods extracted successively from 512 consecutive pitch periods in the steady part results of the frequency analysis indicated that their frequency characteristics seemed to be subject to the spectral 1 f power law in order to investigate the possibility that the frequency characteristics of the fluctuation sequences influence the voice quality of sustained vowels psychoacoustic experiments were conducted amplitude and period sequences evaluated in the experiments were spectral 1 f0 white noise 1 f 1 f2 and 1 f3 sequences respectively the experimental results indicated that the subjective voice quality of synthesized sustained vowels could reflect the differences in the frequency characteristics of the fluctuation sequences

large time behavior of the heat kernel and the behavior of the green function near criticality for nonsymmetric elliptic operators abstract let p be a second order elliptic operator which is defined on a domain r n and let w x be a nonzero nonnegative function assume that p p w x admits a positive minimal green function g p x y for all lim 0 g p x y f x y where f 0 for x y unless p and p admit ground states x and x respectively with eigenvalue zero such that z w z z dz in this case f x y x y z w z z dx moreover if z z dz then the heat kernel k x y t of the parabolic equation lu u t pu 0 satisfies lim t k x y t x y z z dz

new method of structural functions for analysing fractal scaling properties of natural processes a method to reveal and estimate the fractal scaling properties of positive and negative increments in one valued functions that describe natural processes is proposed structural functions which were introduced by kolmogorov 1941 for analysing the scaling properties of small scale turbulence provide the basis of the method examples are given to illustrate the application of the proposed method for analysing the trace of one of the coordinates of brownian motion simulated asymmetric wave forms internal waves sand waves in one directional streams and river turbulence

brain electrical activity analysis using wavelet based informational tools ii tsallis non extensivity and complexity measures the processing of information by the brain is reflected in dynamical changes of the electrical activity in i time ii frequency and iii space therefore the concomitant studies require methods capable of describing the qualitative variation of the signal in both time and frequency the present work complements the efforts described in rosso et al physica a 313 2002 587 devoted to non extensive information measures a la tsallis here we show that the notion of complexity using 1 appropriate information theory tools adapted to a non extensive scenario and 2 scalp eeg data provides one with valuable insights into the dynamics of neural activity in particular using different complexity measures we detect the presence of states characterized by both order and maximal complexity

random fractals self affinity in noise music mountains and clouds abstract many of nature s seemingly complex shapes can be effectively characterized and modeled as random fractals based on generalizations of fractional brownian motion fbm as a function of one dimension t the trace vh t provides a model for the 1 noises extending fbm s to higher dimensions gives vh x y as landscapes and vh x y z as clouds although all such fbm s are statistically self affine as characterized by the parameter h or the spectral density exponent either zerosets or trails of independent fbm s are statistically self similar and may be represented by the fractal dimension d

discrete differential calculus graphs topologies and gauge theory differential calculus on discrete sets is developed in the spirit of noncommutative geometry any differential algebra on a discrete set can be regarded as a reduction of the universal differential algebra and this allows a systematic exploration of differential algebras on a given set associated with a differential algebra is a di graph where two vertices are connected by at most two antiparallel arrows the interpretation of such a graph as a hasse diagram determining a locally finite topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure it is shown that field theories and in particular gauge theories can be formulated on a discrete set in close analogy with the continuum case the framework presented generalizes ordinary lattice theory which is recovered from an oriented hypercubic lattice graph it also includes e g the two point space used by connes and lott and others in models of elementary particle physics the formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an internal discrete space a la connes et al to models of dimensionally reduced gauge fields furthermore a symmetric lattice is also studied which in a certain continuum limit turns out to be related to a noncommutative differential calculus on manifolds

bellman equations of risk sensitive control risk sensitive control problems are considered existence of a nonnegative solution to the bellman equation of risk sensitive control is shown the result is applied to prove that no breaking down occurs asymptotic behaviour of the nonnegative solution is studied in relation to ergodic control problems and the relationship between the asymptotics and the large deviation principle is noted

risk sensitive control on an infinite time horizon stochastic control problems on an infinite time horizon with exponential cost criteria are considered the donsker varadhan large deviation rate is used as a criterion to be optimized the optimum rate is characterized as the value of an associated stochastic differential game with an ergodic expected average cost per unit time cost criterion if we take a small noise limit a deterministic differential game with average cost per unit time cost criterion is obtained this differential game is related to robust control of nonlinear systems

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

dynamical complexity detection in pre seismic emissions using nonadditive tsallis entropy abstract we investigate within the framework of nonextensive tsallis entropy particular aspects of the relation between signal analysis and signals originating in a very special nonlinear system the focal area of an impending earthquake the tsallis like time dependent entropy is shown to be a rather powerful tool for providing a novel quantitative strategy for monitoring the focal area states just before the earthquake occurrence

on unbounded solutions of ergodic problems for non local hamilton jacobi equations abstract we study an ergodic problem associated to a non local hamilton jacobi equation defined on the whole space l u x d u x m f x and determine whether unbounded solutions exist or not we prove that there is a threshold growth of the function f that separates existence and non existence of solutions a phenomenon that does not appear in the local version of the problem moreover we show that there exists a critical ergodic constant such that the ergodic problem has solutions for and such that the only solution bounded from below which is unique up to an additive constant is the one associated to

on the structure of solutions of ergodic type bellman equation related to risk sensitive control bellman equations of ergodic type related to risk sensitive control are considered we treat the case that the nonlinear term is positive quadratic form on first order partial derivatives of solution which includes linear exponential quadratic gaussian control problem in this paper we prove that the equation in general has multiple solutions we shall specify the set of all the classical solutions and classify the solutions by a global behavior of the diffusion process associated with the given solution the solution associated with ergodic diffusion process plays particular role we shall also prove the uniqueness of such solution furthermore the solution which gives us ergodicity is stable under perturbation of coefficients finally we have a representation result for the solution corresponding to the ergodic diffusion

on the fractal dimension of self affine profiles one dimensional profiles f x can be characterized by a minkowski bouligand dimension d and by a scale dependent generalized roughness w f epsilon this roughness can be defined as the dispersion around a chosen fit to f x in an epsilon scale it is shown that d limepsilon to 0 2 in w f epsilon in epsilon holds for profiles nowhere differentiable this establishes a close connection between the roughness and the fractal dimension and proves that d 2 h for self affine profiles h is the roughness or hurst exponent two numerical algorithms based on the roughness one around the local average f x epsilon usual roughness and the other around the local rms straight line a generalized roughness are discussed the estimates of d for standard self affine profiles are reliable and robust especially for the last method

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

an investigation into the linear and nonlinear correlation of two music walk sequences abstract in order to investigate the features of a multiple part musical score which enhance its appeal to the listener s ear this study performs a robust analysis of the correlation between two musical sequences in the proposed approach a series of notes are extracted from seven well known classical pieces of music and are converted into one variable music walks the linear correlation between pairs of music walks is assessed using the conventional linear correlation coefficient while the nonlinear correlation is examined using the mutual information concept the results show that even though two musical time walks may exhibit virtually no linear correlation they invariably have a certain degree of nonlinear correlation in other words to truly understand the correlation between two musical sequences it is necessary to consider not only the linear correlation between them but also the nonlinear correlation in addition it is shown that the normalized mutual information coefficient between musical sequences has a relatively low value and varies significantly over the course of the musical score thus it can be inferred that the appeal of a musical score stems at least in part from significant variations in both the melody and the rhythm of the constituent parts such that the overall score has a rich and unpredictable property

intensive statistical complexity measure of pseudorandom number generators abstract a statistical complexity measure has been recently proposed to quantify the performance of chaotic pseudorandom number generators prng physica a 354 2005 281 here we revisit this quantifier and introduce two important improvements i consideration of an intensive statistical complexity physica a 334 2004 119 and ii following the prescription of brand and pompe phys rev lett 88 2002 174102 1 in evaluating the probability distribution associated with the prng the ensuing new measure is applied to a very well tested prng advanced by marsaglia

tsallis entropy and cortical dynamics the analysis of eeg signals abstract we undertake the study of eeg signals by recourse to a wavelet based multiresolution analysis as adapted to an information measure scenario different information measures are employed it is shown that non extensive ones seem to be of particular usefulness

generating two dimensional fractional brownian motion using the fractional gaussian process fgp algorithm abstract fractional brownian motion fbm is a random fractal that has been used to model many one two and multi dimensional natural phenomena the increments process of fbm has a gaussian distribution and a stationary correlation function the fractional gaussian process fgp algorithm is an exact algorithm to simulate gaussian processes that have stationary correlation functions the approximate second partial derivative of two dimensional fbm called 2d fractional gaussian noise is found to be a stationary isotropic gaussian process in this paper the expected correlation function for 2d fractional gaussian noise is derived the 2d fgp algorithm is used to simulate the approximate second partial derivative of 2d fbm fbm2 which is then numerically integrated to generate 2d fractional brownian motion fbm2 ensemble averages of surfaces simulated by the fgp2 algorithm show that the correlation function and power spectral density have the desired properties of 2d fractional brownian motion

ergodic bsdes and optimal ergodic control in banach spaces in this paper we introduce a new kind of backward stochastic differential equations called ergodic bsdes which arise naturally in the study of optimal ergodic control we study the existence uniqueness and regularity of solution to ergodic bsdes then we apply these results to the optimal ergodic control of a banach valued stochastic state equation we also establish the link between the ergodic bsdes and the associated hamilton jacobi bellman equation applications are given to the optimal ergodic control of stochastic partial differential equations

detrended fluctuation analysis for fractals and multifractals in higher dimensions one dimensional detrended fluctuation analysis dfa and multifractal detrended fluctuation analysis mfdfa are widely used in the scaling analysis of fractal and multifractal time series because they are accurate and easy to implement in this paper we generalize the one dimensional dfa and mfdfa to higher dimensional versions the generalization works well when tested with synthetic surfaces including fractional brownian surfaces and multifractal surfaces the two dimensional mfdfa is also adopted to analyze two images from nature and experiment and nice scaling laws are unraveled

gravity in non commutative geometry we study general relativity in the framework of non commutative differential geometry as a prerequisite we develop the basic notions of non commutative riemannian geometry including analogues of riemannian metric curvature and scalar curvature this enables us to introduce a generalized einstein hilbert action for non commutative riemannian spaces as an example we study a space time which is the product of a four dimensional manifold by a two point space using the tools of non commutative riemannian geometry and derive its generalized einstein hilbert action in the simplest situation where the riemannian metric is taken to be the same on the two copies of the manifold one obtains a model of a scalar field coupled to einstein gravity this field is geometrically interpreted as describing the distance between the two points in the internal space

multifractal analyses of music sequences multifractal analysis is applied to study the fractal property of music in this paper a method is proposed to transform both the melody and rhythm of a music piece into individual sets of distributed points along a one dimensional line the structure of the musical composition is thus manifested and characterized by the local clustering pattern of these sequences of points specifically the local holder exponent and the multifractal spectrum are calculated for the transformed music sequences according to the multifractal formalism the observed fluctuations of the holder exponent along the music sequences confirm the non uniformity feature in the structures of melodic and rhythmic motions of music our present result suggests that the shape and opening width of the multifractal spectrum plot can be used to distinguish different styles of music in addition a characteristic curve is constructed by mapping the point sequences converted from the melody and rhythm of a musical work into a two dimensional graph each different pieces of music has its own unique characteristic curve this characteristic curve which also exhibits a fractal trait unveils the intrinsic structure of music

thermodynamics of fractal signals based on wavelet analysis application to fully developed turbulence data and dna sequences we use the continuous wavelet transform to generalize the multifractal formalism to fractal functions we report the results of recent applications of the so called wavelet transform modulus maxima wtmm method to fully developed turbulence data and dna sequences we conclude by briey describing some works currently under progress which are likely to be the guidelines for future research c 1998 elsevier science b v all rights reserved

thermodynamics of relation based systems with applications in econophysics sociophysics and music abstract a methodology was developed to analyze relation based systems evolving in time by using the fundamental concepts of thermodynamics the behavior of such systems can be tracked from the scattering matrix which is actually a network of directed vectors or pathways connecting subsequent values which characterize an event such as the index values in stock markets a system behaves in a rigid elastic way to an external effect and resists permanent deformation or it behaves in a viscous or soft way and deforms in an irreversible way it was shown in the past that a formula derived using the slope of paths gives a measure about the extent of viscoelastic behavior of relation based systems gunduz 2009 5 gunduz and gunduz 2010 6 in this research the work associated with elastic component and heat associated with viscous component were discussed and elaborated in a simple two subsequent pathway system in a scattering diagram the first vector represents the cause and the second the effect by using work and heat energy relations that involve force and also storage and loss modulus terms respectively one can calculate the energy involved in relation based systems the modulus values can be found from the parallel and vertical components of the second vector with respect to the first vector once work like and heat like terms were determined the internal energy is also easily found from their summation the parallel and vertical components can also be used to calculate the magnitude of torque and torque energy in the system three cases i the behavior of the nasdaq 100 index ii a social revolt and iii the structure of a melody were analyzed for their work like heat like and torque like energies in the course of their evolution nasdaq 100 exhibits highly dissipative behavior and its work terms are very small but heat terms are of large magnitude its internal energy highly fluctuates in time in the social revolt studied work and heat terms are of comparable magnitude the melody depicts highly organized structure and usually has larger work terms than heat terms but at some intervals heat terms burst out and attain very large magnitudes torque terms reach high values when the system is recovering from a minimum value

revealing competitive behaviours in music by means of the multifractal detrended fluctuation analysis application to bach s sinfonias the one two and three dimensional multifractal detrended fluctuation analysis mfdfa was applied to bach s sinfonias which are characterized by the superposition of three different voices each voice represented as a time series can be considered as a component of a one two or three dimensional vector the one dimensional mf dfa was applied to any single voice while the two and three dimensional mf dfa was applied to the couples of voices and to the triple respectively each voice is characterized by a multifractal degree md indicated by the range of the generalized hurst exponents the higher the md the larger the amount of heterogeneity and irregularity competitive scaling multifractal behaviours in bach s sinfonias were revealed although one or two voices showed a relatively high md the other two voices or voice are characterized by a low md nevertheless the overall effect of the sinfonia measured by the md of the triple tends towards homogeneity or at least to an average between the different competitive scaling behaviour shown by the different voices

criticality and long range correlations in time series in classical and quantum systems we present arguments which indicate that a transitional state in between two different regimes implies the occurrence of 1 f time series and that this property is generic in both classical and quantum systems our study focuses on two particular examples the one dimensional module 1 logistic map and nuclear excitation spectra obtained with a schematic shell model hamiltonian we suggest that a transitional point is characterized by the long range correlations implied by 1 f time series we apply a fourier spectral analysis and the detrended fluctuation analysis method to study the fluctuations to each system

fractal analysis of surface roughness by using spatial data summary we develop fractal methodology for data taking the form of surfaces an advantage of fractal analysis is that it partitions roughness characteristics of a surface into a scale free component fractal dimension and properties that depend purely on scale particular emphasis is given to anisotropy where we show that for many surfaces the fractal dimension of line transects across a surface must either be constant in every direction or be constant in each direction except one this virtual direction invariance of fractal dimension provides another canonical feature of fractal analysis complementing its scale invariance properties and enhancing its attractiveness as a method for summarizing properties of roughness the dependence of roughness on direction may be explained in terms of scale rather than dimension and can vary with orientation scale may be described by a smooth periodic function and may be estimated nonparametrically our results and techniques are applied to analyse data on the surfaces of soil and plastic food wrapping for the soil data interest centres on the effect of surface roughness on retention of rain water and data are recorded as a series of digital images over time our analysis captures the way in which both the fractal dimension and the scale change with rainfall or equivalently with time the food wrapping data are on a much finer scale than the soil data and are particularly anisotropic the analysis allows us to determine the manufacturing process which produces the smoothest wrapping with least tendency for micro organisms to adhere

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

ergodicity of robust switching control and nonlinear system of quasi variational inequalities we analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities these equations are related to robust switching control problems introduced in 3 we prove that as time horizon goes to infinity resp discount factor goes to zero the long run average solution to the parabolic system resp the limiting discounted solution to the elliptic system is characterized by a solution of a nonlinear system of ergodic variational inequalities our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities

1 f noise structures in pollocks s drip paintings abstract the fractality and scaling properties of luminance patterns in pollock s drip paintings are studied in this paper the painting is represented as a matrix array where each entry represents the luminance at the corresponding canva position detrended fluctuation analysis dfa is used to characterize fractality and the results show that the whole luminance pattern is indeed fractal interestingly the fractality is associated to 1 f noise structures which can be inducing the aesthetic properties of the artistic object in fact for spatial scales below 30 cm pollock s drip painting are organized as 1 f noise structures which presumably are attuned for pleasant i e aesthetic perception

brs symmetry in connes non commutative geometry we extend the brs and anti brs symmetry to the two point space of connes non commutative model building scheme the constraint relations are derived and the quantum lagrangian constructed we find that the quantum lagrangian can be written as a functional of the curvature for symmetric gauges with the brs anti brs auxiliary field finding a geometrical interpretation as the extension of the higgs scalar

large time behavior of solutions to semilinear equations with quadratic growth in the gradient this paper studies the large time behavior of solutions to semilinear cauchy problems with quadratic nonlinearity in gradients the cauchy problem considered has a general state space and may degenerate on the boundary of the state space two types of large time behavior are obtained i pointwise convergence of the solution and its gradient and ii convergence of solutions to associated backward stochastic differential equations when the state space is mathbb r d or the space of positive definite matrices both types of convergence are obtained under growth conditions on coefficients these large time convergence results have direct applications in risk sensitive control and long term portfolio choice problems

fractional brownian motion fractional gaussian noise and tsallis permutation entropy abstract in this work we analyze two important stochastic processes the fractional brownian motion and fractional gaussian noise within the framework of the tsallis permutation entropy this entropic measure evaluated after using the bandt pompe method to extract the associated probability distribution is shown to be a powerful tool to characterize fractal stochastic processes it allows for a better discrimination of the processes than the shannon counterpart for appropriate ranges of values of the entropic index moreover we find the optimum value of this entropic index for the stochastic processes under study

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

extracting features of gaussian self similar stochastic processes via the bandt pompe approach by recourse to appropriate information theory quantifiers normalized shannon entropy and martin plastino rosso intensive statistical complexity measure we revisit the characterization of gaussian self similar stochastic processes from a bandt pompe viewpoint we show that the ensuing approach exhibits considerable advantages with respect to other treatments in particular clear quantifiers gaps are found in the transition between the continuous processes and their associated noises

wavelet packet computation of the hurst exponent wavelet packet analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis a new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power law to estimate the hurst exponent profile irregularity and long range correlations in self affine systems can be identified and indexed with the hurst exponent and synthetic one dimensional fractional brownian motion fbm type profiles are used to illustrate and test the proposed wavelet packet expansion we also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self affine trace or an enzyme walk the robustness of the wavelet approach applied to this stochastic system is presented and comparison is made between the wavelet packet method and the root mean square roughness and second moment approaches for both examples the wavelet packet method to estimate the global hurst exponent appears to have similar accuracy compared with other methods but its main advantage is the extensive choice of available analysing wavelet filter functions for characterizing periodic and oscillatory signals

fractal geometry of music abstract music critics have compared bach s music to the precision of mathematics what mathematics and what precision are the questions for a curious scientist the purpose of this short note is to suggest that the mathematics is at least in part mandelbrot s fractal geometry and the precision is the deviation from a log log linear plot

music walk fractal geometry in music abstract in this study sequences of musical notes from various pieces of music are converted into one variable random walks here termed music walks quantitative measurements of the properties of each musical composition are then performed by applying hurst exponent and fourier spectral analyses on these music walk sequences our results show that music shares the similar fractal properties of a fractional brownian motion fbm that is music displays an anti persistent trend in its tone changes melody over decades of musical notes and music sequence exhibits generally the 1 f type spectrum fractal property with apparently two different values in two different temporal scales