geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

shape elongation from optimal encasing rectangles let s be a shape with a polygonal boundary we show that the boundary of the maximally elongated rectangle r s which encases the shape s contains at least one edge of the convex hull of s such a nice property enables a computationally efficient construction of r s in addition we define the elongation of a given shape s as the ratio of the length of r s determined by the longer edge of r s and the width of r s determined by the shorter edge of r s and show that a so defined shape elongation measure has several desirable properties several examples are given in order to illustrate the behavior of the new elongation measure as a by product of the method developed here we obtain a new method for the computation of the shape orientation where the orientation of a given shape s is defined by the direction of the longer edge of r s

equations of motion in a non integer dimensional space equations of motion are derived for a fractional dimensional system of n spatial coordinates to be used as an effective description of anisotropic and confined systems an existing measure theoretic approach is extended to multiple variables and different degrees of confinement in orthogonal directions and comparisons are made with the analytic continuation of gaussian integrals this is applied to the variational principle and equations of motion for a field described by a lagrange density are found a specific example is looked at in schrodinger wave mechanics particularly in three coordinate systems

discrete time linear quadratic control with arbitrary correlated noise in this technical note a control problem for a discrete time linear stochastic system with a general correlated noise process and a cost functional that is quadratic in the system state and the control is solved an optimal control is given explicitly as the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and the prediction of the response of a system to the future noise process the optimal cost is also given explicitly

computation of the euler number using the contact perimeter we present the computation of the euler number of shapes using the contact perimeter the contact perimeter was initially defined in e bribiesca measuring 2d shape compactness using the contact perimeter comput math appl 33 1997 1 9 in this reference the contact perimeter was used to define a measure of compactness for 2d shapes now in this paper we use the contact perimeter to compute the euler number of unit width objects composed of different side connected cells and face connected polyhedrons in two and three dimensions respectively finally we present some applications of this computation in knot and graph theory

chaos in one dimensional collision complexes the classical scattering of a particle by an oscillating potential well is investigated it is shown that the singularities of the scattering map s form a cantor set for this system this result is obtained from an analysis of the particle s return map which contains a smale s horseshoe and consequently a hyperbolic invariant set with chaotic dynamics the cantor set structure of the s singularities involves arbitrarily complex scattering behaviour

linear quadratic control for stochastic equations in a hilbert space with fractional brownian motions a linear quadratic control problem with a finite time horizon for some infinite dimensional controlled stochastic differential equations driven by a fractional gaussian noise is formulated and solved the feedback form of the optimal control and the optimal cost are given explicitly the optimal control is the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise the covariance of the noise as well as the control operator in the system equation can in general be unbounded so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain some examples of controlled stochastic partial differential equations are given

persistency of two dimensional self avoiding walks the authors investigate the persistency of self avoiding walks on the square lattice by studying the distribution of endpoint displacements projected in the direction of the first step exact enumeration and the constant fugacity monte carlo data indicate that the mean displacement after n steps xn increases logarithmically with n while the higher odd moments x2k 1n vary as n2kv in n where v is the correlation length exponent this unusual behaviour for the moments is found to arise from a simple scaling behaviour of the asymmetric component of the displacement distribution which in the constant fugacity ensemble can be written in the form a x approximately n 2v in n exp bx nv where n is the average number of steps in the walk at a fixed value of the fugacity and b is a coefficient of order unity

bose like oscillator in fractional dimensional space the system of a fractional dimensional bose like oscillator of one degree of freedom whose canonical variables satisfy the general wigner commutation relations is investigated momentum position uncertainty relations are obtained for states without definite parity the well known one dimensional heisenberg uncertainty principle is retained but for odd and even states momentum position uncertainty inequalities depending on the dimension of the space are obtained the motions of both the free particle and the harmonic oscillator in a fractional dimensional space are studied through the probability density function the existence of compression spread of the probability density for dimensions d 1 is shown fractional dimensional bose like operators are also deduced they together with the reflection operator form an r deformed heisenberg algebra with a deformation parameter depending on the dimension of the space

universal principal axes an easy to construct tool useful in defining shape orientations for almost every kind of shape abstract a convenient tool for defining shape orientations called universal principal axes is introduced these axes are formed of half lines starting from the shape centroid with their directional angles expressed as functions of the polar angle of the first non zero complex number encountered in the sequence x iy 1 d a with 1 2 3 4 provided that the integration domain is taken to be the given two dimensional 2d shape and the origin is assumed to be the shape centroid universal principal axes are shown to be independent of the translation scaling and rotation of the coordinate system used and they are therefore qualified in defining shape orientations a major benefit of using the proposed universal principal axes is that they exist for almost every kind of shape another benefit is that there is no need to judge in advance whether a given shape is mirror symmetric rotationally symmetric irregular etc moreover in the case of rotationally symmetric shapes the universal principal axes make the preprocessing procedure for detecting the number of folds contained in the given shape unnecessary although such procedure is essential for many existing tools designed to define orientations of rotationally symmetric shapes defining shape orientations by universal principal axes is therefore quite convenient an algorithm describing the construction of universal principal axes and several examples showing the detected universal principal axes for some shapes of distinct types are given the relationship between universal and conventional principal axes is also discussed

upper and lower bounds on the renyi dimensions and the uniformity of multifractals abstract we prove that for arbitrary probability measures the generalized renyi dimensions d q satisfy the inequality d q q 1 q d q q 1 q d q q 1 q q q q q 0 1 sign q sign q i n particular 1 2 d 2 d d 2 and 1 2 d d 1 d the upper bound corresponds to uniform measures whereas the lower bound describes non uniform measures such as measures with a power law singularity in order to characterize intermediate cases we introduce a quantity that measures the uniformity of multifractals we discuss the relation to the spectrum and apply the formalism to several examples such as power law measures the feigenbaum attractor and the attractor and mode locking structure of the critical circle map

anisotropic fractal media by vector calculus in non integer dimensional space a review of different approaches to describe anisotropic fractal media is proposed in this paper differentiation and integration non integer dimensional and multi fractional spaces are considered as tools to describe anisotropic fractal materials and media we suggest a generalization of vector calculus for non integer dimensional space by using a product measure method the product of fractional and non integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models the integration over non integer dimensional spaces is considered in this paper differential operators of first and second orders for fractional space and non integer dimensional space are suggested the differential operators are defined as inverse operations to integration in spaces with non integer dimensions non integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media the poisson s equation for fractal medium the euler bernoulli fractal beam and the timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media

linearity measure for curve segments in this paper we consider linearity measure for a bounded length curves first we define a new linearity measure for open curve segments and then extend method to closed curves contours the derived measures for both open curve segments and closed curves are invariant with respect to similarity transformations the linearity measure for open curve segments picks the value 1 if and only if the measured open line segment is a perfect straight line segment while the established linearity measures for closed curves never reach 1 as preferred

linear quadratic fractional gaussian control in this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean square integrable stochastic process with continuous sample paths and a cost functional that is quadratic in the system state and the control is solved an optimal control is given explicitly as the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and the prediction of the response of a system to the future noise process the optimal cost is also given the special case of a noise process that is an arbitrary standard fractional brownian motion is noted explicitly with an explicit expression for the prediction of the future response of a system to the noise process that is used the optimal control

is the dimension of chaotic attractors invariant under coordinate changes several different dimensionlike quantities which have been suggested as being relevant to the study of chaotic attractors are examined in particular we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points it is found that some are and some are not it is suggested that the word dimension be reversed only for those quantities have this invariance property

linear quadratic mean field type games jump diffusion process with regime switching in this paper we study a linear quadratic mean field type game problem under jump diffusion regime switching state dynamics that include mean field terms in the states and in the cost function in the cost functional we consider conditional variance of both the states and control actions mean of both states and control actions and covariance terms the underlying problem is solved in a semiexplicit way by using the direct method

a scaling law how an attractor s volume depends on noise level abstract we investigate the meaning of the dimension of a strange attractor for systems with noise more specifically we investigate the effect of adding noise of magnitude e to a deterministic system with d degrees of freedom if the attractor has dimension d and d d then its volume is zero the addition of noise may be an important physical probe for experimental situations useful for determining how much of the observed phenomena in a system is due to noise already present when the noise is added the attractor a e has positive volume we conjecture that the generalized volume of a e is proportional to e d d for e near 0 and show this relationship is valid in several cases for chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly

phase transitions in thermodynamics of a local lyapunov exponent for fully developed chaotic systems fluctuations in the divergence of nearby orbits are studied at a crisis point of chaos a statistical thermodynamic method for the description of the fluctuations is developed by using symbolic dynamics which can explicitly write a relation between a fluctuation and reference orbit the thermodynamics the free energy and entropy is exactly analyzed on a nonhyperbolic attractor of maps conjugate to the map u u a for 0 u a andu 1 u 1 a fora u 1 te free energy has discontinuities in its slope the entropy is directly calculated from the partition function then it becomes clear that the collision of a chaotic attractor with a particular fixed point yields a singular local structure in the distribution of fluctuations the existence of first order phase transitions depends on the asymmetry of a map it is shown that each of the coexisting states at the phase transition points is realized with the same probability in the thermodynamic limit

computing a shape s moments from its boundary abstract moments have been among the most commonly used tools in the analysis of 2d shapes important information about a shape such as its size center location and orientation are all moment based attributes there is also a number of very useful shape features such as maximum and minimum moments of inertia moment invariants shape spreadness and shape elongation which are also derived from moments traditionally moments of a shape are computed from the occupancy array representation of the shape that is all the shape s pixels are involved in the computation in this paper we propose a method which computes the same set of moments using only the corner pixels along a shape s boundary the basic approach is to construct a set of triangles using the shape s corners and the origin of the coordinate system the moments of these triangles are computed first the moments of the shape are then derived from the triangles moments we have found the suggested method to be much more efficient than the traditional occupancy array based methods

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

on the organisation of transient chaos application to irregular scattering it is shown how the local structure of chaotic repellers being responsible for transient chaotic behaviour is deduced from the properties of hyperbolic periodic orbits relations between static and dynamical multifractal spectra with respect to the natural invariant measure on the repeller are derived for invertible maps of the plane the results obtained for maps with unit jacobian apply to hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

a comment on the solutions of the equation a ka it is pointed out here that solenoidal solutions of the equation del a ka exist in magnetoelectric media in which case a must be identified as a linear combination of the electric and the magnetic fields

fractal dimensions and homeomorphic conjugacies we investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes when the homeomorphic conjugacy is not lipschitz continuous we discuss how its singularities can affect the whole set of generalized fractal dimensions we give applications to homeomorphisms that conjugate critical circle maps with irrational golden mean winding numbers we present numerical studies corroborating the theoretical predictions

a simple proof of indefinite linear quadratic stochastic optimal control with random coefficients we consider the indefinite linear quadratic optimal control problem for stochastic systems with random coefficients where the corresponding cost parameters need not be definite matrices although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic hamilton jacobi bellman hjb equation our approach is simple and direct specifically we develop a direct approach also known as the completion of squares method to characterize the explicit optimal solution and the optimal cost the corresponding optimal solution is linear characterized in terms of the stochastic riccati differential equation srde and the linear backward stochastic differential equation bsde in our approach the completion of squares method together with the srde and the bsde allows us to construct an equivalent cost functional that is quadratic in the control u provided that an additional positive definiteness condition in terms of the srde holds then the optimal control and the associated cost can be obtained by eliminating the quadratic term of u in the equivalent cost functional the additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term we verify the optimal solution by solving the stochastic hjb equation

measure of predictability many techniques have been developed to measure the difficulty of forecasting data from an observed time series this paper introduces a measure which we call the forecast entropy designed to measure the predictability of a time series we use attractors reconstructed from the time series and the distributions in the regular and tangent spaces of the data which comprise the attractor we then consider these distributions on different scales we present a formula for calculating the forecast entropy to provide a standard of predictability we define an idealized random system whose forecast entropy will be maximal we then use this measure to rescale the forecast entropy to lie in the range 0 1 the time series obtained from several chaotic systems as well as from a pseudorandom system are studied using this measure we present evidence that the forecast entropy can be used as a tool for determining optimal delays and embedding dimensions used for reconstructing better attractors we also show that the forecast entropy of a random system has completely different characteristics from that of a deterministic one

cantor set structures in the singularities of classical potential scattering a classical mechanical system is analysed which exhibits complicated scattering behaviour in the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular though in the complement of this cantor set the deflection function is regular one can choose impact parameter intervals leading to arbitrarily complicated trajectories the authors show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow this invariant set contains infinitely many unstable localised orbits periodic and aperiodic ones the stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function

fractal properties of scattering singularities we discuss properties of the set of scattering singularities in regions of irregular scattering we show how a symbolic organisation of the set can be used to determine the fractal dimension and the scaling function this yields information on the distribution of lyapunov exponents of bounded orbits the specific model studied is the motion of a particle in a plane elastically reflected by three circular discs centred on the corners of an equilateral triangle

extraction of dynamical equations from chaotic data abstract a method is described for extracting from a chaotic time series a system of equations whose solution reproduces the general features of the original data even when these are contaminated with noise the equations facilitate calculation of fractal dimension lyapunov exponents and short term predictions the method is applied to data derived from numerical solutions of the logistic equation the henon equations with added noise the lorenz equations and the rossler equations

nonconcave entropies in multifractals and the thermodynamic formalism we discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as legendre fenchel transforms of functions analogous to free energy functions we show that the legendre fenchel transform of a free energy function yields the correct multifractal spectrum only when the latter is wholly concave if the spectrum has no definite concavity then the transform yields the concave envelope of the spectrum rather than the spectrum itself some mathematical and physical examples are given to illustrate this result which lies at the root of the nonequivalence of the microcanonical and canonical ensembles on a more positive note we also show that the impossibility of expressing nonconcave multifractal spectra through legendre fenchel transforms of free energies can be circumvented with the help of a generalized free energy function which relates to a recently introduced generalized canonical ensemble analogies with the calculation of rate functions in large deviation theory are finally discussed

dimensional reduction via dimensional shadowing dimensional shadowing is introduced as a formal method to reduce extra dimensions in configuration space by considering a fractal subset the hausdorff dimension of the fractal is then perceived as the physical dimension of configuration space

complete poincare sections and tangent sets trying to extend a local definition of a surface of a section and the corresponding poincare map to a global one one can encounter severe difficulties we show that global transverse sections often do not exist for hamiltonian systems with two degrees of freedom as a consequence we present a method to generate the so called w section which by construction will be intersected by almost all orbits depending on the type of tangent set in the surface of the section we distinguish five types of w sections the method is illustrated by a number of examples most notably the quartic potential and the double pendulum w sections can also be applied to higher dimensional hamiltonian systems and to dissipative systems

exact series studies of self avoiding walks on two dimensional critical percolation clusters the author presents results of exact series studies of self avoiding walks on percolation clusters on the square lattice performed very close to the percolation threshold and estimates the critical exponents nu and gamma defined by the disorder averages of the mean square radius of gyration or the end to end distance and the number of self avoiding walks respectively the self avoiding walk configurations are enumerated exactly up to 35 steps the result for nu indicates a large increase compared with the self avoiding walks on fully occupied lattice the result for gamma however indicates a behaviour similar to that on a fully occupied lattice the result for nu is completely different from that of lee nakanishi and kim 1989

on solutions of the equation a ka the vector differential equation del a ka with k constant appears in models of force free magnetic fields in plasma physics and in astrophysics and also in a model for force free electromagnetic waves solutions of this equation which are identified with physical fields exhibit several inconsistencies this letter identifies the source of these inconsistencies as the lack of covariance of this equation with respect to transformations

partial information linear quadratic control for jump diffusions we study a stochastic control problem in which the state process is described by a stochastic differential equation sde that is driven by a brownian motion and a poisson random measure and is affine in both the state and the control the performance functional is quadratic in the state and the control all the coefficients are allowed to be random and non markovian moreover we may allow the control to be predictable to a given subfiltration of the filtration of the brownian motion and the random measure partial information control

the last remake of the mixmaster universe model we review the existing evidence on the non integrability of the mixmaster universe model we show how a local painleve analysis can be used to study the possible existence of essential singularities in agreement with recent studies we find that the mixmaster model possesses critical essential singularities the multivalued character of which is incompatible with integrability our analysis is complemented by a numerical study of the spectrum of the stretching numbers of the system we show that the zero energy case is characterized by a vanishing maximal lyapunov characteristic number

semiexplicit solutions to some nonlinear nonquadratic mean field type games a direct method this article examines mean field type game problems by means of a direct method we provide various solvable examples beyond the classical linear quadratic game problems these include quadratic quadratic games and games with power logarithmic sine square hyperbolic sine square payoffs nonlinear state dynamics such as log state control dependent regime switching quadratic state cotangent state and hyperbolic cotangent state are considered we identify equilibrium strategies and equilibrium payoffs in state and conditional mean field type feedback form it is shown that a simple direct method can be used to solve broader classes of nonquadratic mean field type games under jump diffusion regime switching gauss volterra processes which include fractional brownian motions and multifractional brownian motions we provide semiexplicit solutions to the fully cooperative noncooperative nonzero sum and adversarial game problems

multifractal analysis of nonhyperbolic coupled map lattices application to genomic sequences symbolic sequences generated by coupled map lattices cmls can be used to model the chaotic like structure of genomic sequences in this study it is shown that diffusively coupled chebyshev maps of order 4 corresponding to a shift of four symbols very closely reproduce the multifractal spectrum d q of human genomic sequences for coupling constant 0 35 0 01 if q 0 the presence of rare configurations causes deviations for q 0 which disappear if the rare event statistics of the cml is modified such rare configurations are known to play specific functional roles in genomic sequences serving as promoters or regulatory elements

rooted spiral trees in dimensions 2 3 and 4 the authors have studied rooted spiral trees in dimensions 2 3 and 4 by exact enumeration and by monte carlo methods numerical estimates of critical exponents thus obtained support the hypothesis that a dimensional reduction by four occurs in this problem a possible mechanism of the origin of this dimensional reduction is proposed

an easy measure of compactness for 2d and 3d shapes an easy measure of compactness for 2d two dimensional and 3d three dimensional shapes composed of pixels and voxels respectively is presented the work proposed here is based on the two previous works of the measure of discrete compactness e bribiesca measuring 2 d shape compactness using the contact perimeter comput math appl 33 1997 1 9 e bribiesca a measure of compactness for 3d shapes comput math appl 40 2000 1275 1284 the measure of compactness proposed here improves and simplifies the previous measure of discrete compactness now using this proposed measure of compactness it is possible to compute measures for any kind of object including porous and fragmented objects also the computation of the measures is very simple by means of the use of only one equation the measure of compactness proposed here depends in large part on the sum of the contact perimeters of the side connected pixels for 2d shapes or on the sum of the contact surface areas of the face connected voxels for 3d shapes relations between the perimeter and the contact perimeter for 2d shapes and between the area of the surface enclosing the volume and the contact surface area are presented the measure presented here of compactness is invariant under translation rotation and scaling in this work the term of compactness does not refer to point set topology but is related to intrinsic properties of objects finally in order to prove our measure of compactness we calculate the measures of discrete compactness of different objects also we present an important application for brain structures quantification by means of the use of the new proposed measure of discrete compactness

stochastic control for linear systems driven by fractional noises this paper is concerned with optimal control of stochastic linear systems involving fractional brownian motion fbm first as a prerequisite for studying the underlying control problems some new results on stochastic integrals and stochastic differential equations associated with fbm are established then three control models are formulated and studied in the first two models the state is scalar valued and the control is taken as markovian either the problems are completely solved based on a riccati equation for model 1 where the cost is a quadratic functional on state and control variables or optimality is characterized for model 2 where the cost is a power functional the last control model under investigation is a general one where the system involves the stratonovich integral with respect to fbm the state is multidimensional and the admissible controls are not limited to being markovian a new riccati type equation which is a backward stochastic differential equation involving both fbm and normal brownian motion is introduced optimal control and optimal value of the model are explicitly obtained based on the solution to this riccati type equation

the equation curl b kb and magnetic fields in two recent letters salingaros has put forward physical arguments to claim that solutions of curl b kb cannot represent real magnetic fields however the first part of his assertions is only a restatement of the well known result that this equation cannot be satisfied everywhere over a non vanishing magnetic field however it does not rule out the use of the equation locally over a restricted region of the field the second part suffers from an incorrect assumption which invalidates his conclusion

alternative area perimeter ratios for measurement of 2d shape compactness of habitats area perimeter ratios are often used to quantify 2d shape compactness shape compactness is of main importance to evaluate the effect of external disturbance on natural habitats reference values a m i n p a m a x p p m i n a p m a x a for area a and perimeter p are defined and proven the calculation of these reference values is based on the characteristics of the object studied and on pixel geometry the cases for 4 and 8 connectivity are discussed using these references alternative area perimeter ratios are composed examples are elaborated to illustrate the use of the indices and their performance

discrete time lqg controls with control dependent noise this paper presents some studies on partially observed linear quadratic gaussian lqg models where the stochastic disturbances depend on both the states and the controls and the measurements are bilinear in the noise and the states controls while the separation theorem of standard lqg design does not apply suboptimal linear state estimate feedback controllers are derived based on certain linearizations the controllers are useful for nonlinear stochastic systems where the linearized models include terms bilinear in the noise and states controls and are significantly more accurate than if the bilinear terms are set to zero the controllers are calculated by solving a generalized discrete time riccati equation which in turn has properties relating to well posedness of the associated lqg problem

biased diffusion in percolation systems indication of multifractal behaviour the authors study diffusion in percolation systems at criticality in the presence of a constant bias field e using the exact enumeration method they show that the mean displacement of a random walker varies as r t approximately log t a e where a e in 1 e 1 e for small e more generally diffusion on a given configuration is characterised by the probability p r t that the random walker is on site r at time t they find that the corresponding configurational average shows simple scaling behaviour and is described by a single exponent in contrast their numerical results indicate that the averaged moments pq t sigma rpq r t are described by an infinite hierarchy of exponents for zero bias field however all moments are determined by a single gap exponent

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

a measure of compactness for 3d shapes abstract a measure of compactness for 3d three dimensional shapes composed of voxels is presented the work proposed here improves and extends to the measure of discrete compactness 1 from 2d two dimensional domain to 3d the measure of discrete compactness proposed here corresponds to the sum of the contact surface areas of the face connected voxels of 3d shapes a relation between the area of the surface enclosing the volume and the contact surface area is presented the concept of contact surfaces is extended to 3d shapes composed of different polyhedrons which divide space generating different 3d lattices the measure proposed here of discrete compactness is invariant under translation rotation and scaling in this work the term of compactness does not refer to point set topology but is related to intrinsic properties of objects finally in order to prove our measure of compactness we calculate the measures of discrete compactness of different volcanos which are compared with their classical measures from the valley of mexico using digital elevation model dem data

construction of phenomenological models from numerical scalar time series abstract the problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed techniques are exemplified for rossler and lorenz chaotic attractors with emphasis on improved algorithms with respect to those previously published the steps which are still required in order to investigate experimental noisy data are discussed

a possible scheme for classifying phase transitions in the generalized entropy spectrum of a nonhyperbolic system abstract in this contribution the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system the system is used to discuss the occurence of combination of phase transitions among different entropy like scaling functions for the discussion the associated generalized entropy function is calculated from an approximation of finite level and for the asymptotic case from this unifying description of the system the information contained in the particular spectra as well as the relation among the different spectra can be derived and made visible in an easy way it is shown how the phase transition like behavior leads to the manifestation of a critical line in the surface described by the generalized entropy function which in turn determines the occurence of phase transitions for the particular spectra furthermore it is indicated how apparent phase transitions are encountered in approximations of a finite level which do not persist in the asymptotic limit

fractal dimension of spatially extended systems properties of the invariant measure are numerically investigated in 1d chains of diffusively coupled maps the coarse grained fractal dimension is carefully computed in various embedding spaces observing an extremely slow convergence towards the asymptotic value this is in contrast with previous simulations where the analysis of an insufficient number of points led the authors to underestimate the increase of fractal dimension with increasing the dimension of the embedding space orthogonal decomposition is also performed confirming that the slow convergence is intrinsically related to local nonlinear properties of the invariant measure finally the kaplan yorke conjecture is tested for short chains showing that despite the noninvertibility of the dynamical system a good agreement is found between lyapunov dimension and information dimension

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

linear exponential quadratic stochastic differential games in this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved the optimal strategies form a nash equilibrium the value of the game is also determined explicitly the solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

on solutions of the equation a ka ii the magnetic force free model for pt i see ibid vol 19 l101 4 1986 solutions of the vector differential equation del a ka k constant not 0 have been shown not to correspond to physical magnetic fields this is true even though they are solenoidal del a 0 which is a necessary but not sufficient condition for a vector field to be a magnetic field here the magnetic force free model is very carefully analysed and the inapplicability of the above equation is demonstrated using physical arguments

bounds for the fractal dimension of space the authors discuss the possibility that space has to be described as being of non integer dimension they elaborate on some of the problems associated with this idea and investigate the chance to observe effects due to such a fractional dimension two experiments show rather stringent bounds for the deviation of the space dimension from the value three these are the perihelion shift of planetary motion and the lamb shift in hydrogen

statistical thermodynamic approach to a chaotic dynamical system exactly solvable examples the static and dynamic properties of a chaotic attractor of a two dimensional map are studied which belongs to a particular class of piecewise continuous invertible maps coverings of a natural size to cover the attractor are introduced so that the microscopic information of the attractor is written on each box composing the cover the statistical thermodynamics of the scaling indices and the size indices of the boxes is formulated analytic forms of the free energy functions of the scaling indices and the size indices of the boxes are obtained for examples of a hyperbolic and a nonhyperbolic chaotic attractor the statistical thermodynamics of local lyapunov exponents is also studied and a relation between the thermodynamics of scaling indices and of local lyapunov exponents is invetigated for the nonhyperbolic example the free energy and entropy functions of local lyapunov exponents are obtained in analytic forms these results display the existence of phase transitions a phase transition is seen in the thermodynamics of scaling indices also

measuring 2 d shape compactness using the contact perimeter a new perimeter for shapes composed of cells is defined this perimeter is called the contact perimeter which corresponds to the sum of the boundaries of neighboring cells of the shape also a relation between the perimeter of the shape and the contact perimeter is presented the contact perimeter corresponds to the measure of compactness proposed here called discrete compactness in this case the term compactness does not refer to point set topology but is related to intrinsic properties of objects

discrete time linear quadratic mean field type repeated games perfect incomplete and imperfect information in this paper we study linear quadratic mean field type repeated games with arbitrary number of interacting decision makers different information structures are examined i perfect knowledge of the model ii imperfect knowledge of the model under perfect state observation we provide semi explicit solutions for both non cooperative and cooperative cases the state dynamics is given by a stochastic difference equation that includes mean field terms i e the expected value for both the state and control inputs in addition the cost function incorporates not only the expected value of the states but also the variance of both system state and control inputs the effect of the information is quantified using price of anarchy and price of imperfect information finally some numerical examples are presented

on the geometric structure of non hyperbolic attractors the authors discuss the f alpha spectrum of non hyperbolic attractors of the henon type they elucidate the origin of the phase transition found in a previous paper and give a lower bound to the spectrum in the non hyperbolic phase where kaplan yorke type formulae no longer hold their results disagree with other recent attempts numerical simulations for the henon map agree with analytical estimates

quantum mechanical models in fractional dimensions we formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators p q satisfy the r deformed heisenberg relations which depend on an operator we find representations of p q in which the dimension d and angular momentum l appear as parameters related to the eigenvalues of we analyse the domain of p and find conditions which ensure that p is hermitian we investigate plane wave solutions and also free particle wavefunctions in fractional dimensions and show that as a consequence of wavefunction continuity l is quantized the representations of p q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d both algebraically and analytically we demonstrate that the formalism extends also to time dependent hamiltonians by solving the time dependent harmonic oscillator in any dimension d 0 using the method of lewis and riesenfeld

quantum field theory on fractal spacetime a new regularisation method spacetime is modelled as a fractal subset of rn analysis on homogeneous sets with non integer hausdorff dimensions is applied to the low order perturbative renormalisation of quantum electrodynamics this new regularisation method implements the dirac matrices and tensors in r4 without difficulties is gauge invariant covariant and differs from dimensional regularisation in some aspects

spatial aggregation of two dimensional objects in raster data structures raster data structures are characterized by pixel geometry and resolution a method is proposed to reduce resolution to a minimum using spatial aggregation without information loss the method is based on a matrix approach and developed for a two dimensional spatial object perimeter measures and fractal dimension are shown not to be invariant to spatial aggregation

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

an algorithm for enumerating rigid clusters a more efficient algorithm for enumerating rigid clusters is presented using this algorithm the author has extended the series of site rigid animals enumerated by prunet and blanc 1986 by five more terms and has also calculated for the first time the exponent governing the growth of the radius of gyration for rigid animals

on solutions to the equation a ka in a recent letter salingaros ibid vol 19 l101 4 1986 claimed that the equation curl a ka cannot be used to describe a magnetic field b a he bases this on the assertion that the magnetic field b caused by a current density j must be orthogonal to j this assertion is shown to be incorrect

hopping conduction in insulator conductor superconductor mixtures the coherent medium approximation is applied to study the frequency dependent diffusion constant in two and three dimensional ant termite like mixtures where three types of jump rates normal metallic superconducting and insulating are distributed randomly the dc diffusion constant vanishes when p r pc where p and r are the probabilities that a given bond is metallic and superconducting respectively and z is the coordination number of the lattice pc rc identical to 2 z in the low frequency limit the real and imaginary parts of the ac diffusion constant go as a z d p r f omega and b z d p r g omega respectively a behaviour similar to the pure ant limit the behaviour of f omega and g omega is examined below at and above the percolation thresholds in both two and three dimensions the coefficients a and b of these leading terms diverge as r to rc superconducting region percolates indicating enhancement in the dielectric constant