successive eigenvalue relaxation a new method for the generalized eigenvalue problem and convergence estimates we present a new subspace iteration method for the efficient computation of several smallest eigenvalues of the generalized eigenvalue problem au lambda bu for symmetric positive definite operators a and b we call this method successive eigenvalue relaxation or the ser method homoechon of the classical successive over relaxation or sor method for linear systems in particular there are two significant features of ser which render it computationally attractive i it can effectively deal with preconditioned large scale eigenvalue problems and ii its practical implementation does not require any information about the preconditioner used it can routinely accommodate sophisticated preconditioners designed to meet more exacting requirements e g three dimensional elasticity problems with small thickness parameters we endow ser with theoretical convergence estimates allowing for multiple and clusters of eigenvalues and illustrate their usefulness in a numerical example for a discretized partial differential equation exhibiting clusters of eigenvalues

on the hyperbolicity of the period doubling fixed point we give a new proof of the hyperbolicity of the fixed point for the period doubling renormalization operator using the local dynamics near a semi attractive fixed point in a banach space and the theory of holomorphic motions we also give a new proof of the exponential contraction of the feigenbaum renormalization operator in the hybrid class of the period doubling fixed point our proof uses the non existence of invariant line fields in the period doubling tower c mcmullen the topological convergence d sullivan and a new infinitesimal argument

application of time iterative schemes to incompressible flow the computation of steady incompressible flows by an euler implicit algorithm is studied using both the incompressible equations and the low mach number compressible equations the incompressible equations are handled by adding an artificial time derivative to the continuity equation this allows both the pressure and velocity to be obtained implicitly in one dimensional problems both systems converge rapidly even at low mach numbers where the eigenvalues are very stiff in two dimensions where approximate factorization is required the presence of stiff eigenvalues is highly detrimental stiffness can be avoided in the incompressible equations by selecting an appropriate pseudo mach number this insures reliable convergence and results in an efficient incompressible flow algorithm in the case of compressible equations the mach number cannot be chosen arbitrarily and the contamination introduced by approximate factorization must be removed a matrix preconditioning factor that accomplishes this is developed and demonstrated with this modification the convergence rate is the same as in the incompressible case and is independent of mach number rapid convergence is observed at mach numbers as low as 6 05

navier stokes equations for almost incompressible flow the isentropic navier stokes equations at low mach numbers describe almost incompressible flow by a variable transformation the equations take a symmetric form suitable for computation estimates of the solution and its derivatives are obtained for the linearized equations on domains with open boundaries this is a necessary basis for the application of any numerical method whenever some part of the computational boundary is not a solid wall a semi implicit difference approximation is applied to the nonlinear equations for computation of flow over a backward facing step

link wise artificial compressibility method the artificial compressibility method acm for the incompressible navier stokes equations is link wise reformulated referred to as lw acm by a finite set of discrete directions links on a regular cartesian mesh in analogy with the lattice boltzmann method lbm the main advantage is the possibility of exploiting well established technologies originally developed for lbm and classical computational fluid dynamics with special emphasis on finite differences at least in the present paper at the cost of minor changes for instance wall boundaries not aligned with the background cartesian mesh can be taken into account by tracing the intersections of each link with the wall analogously to lbm technology lw acm requires no high order moments beyond hydrodynamics often referred to as ghost moments and no kinetic expansion like finite difference schemes only standard taylor expansion is needed for analyzing consistency preliminary efforts towards optimal implementations have shown that lw acm is capable of similar computational speed as optimized bgk lbm in addition the memory demand is significantly smaller than bgk lbm importantly with an efficient implementation this algorithm may be among the few which are compute bound and not memory bound two and three dimensional benchmarks are investigated and an extensive comparative study between the present approach and state of the art methods from the literature is carried out numerical evidences suggest that lw acm represents an excellent alternative in terms of simplicity stability and accuracy

renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion we establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result namely we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same size of a break

global hyperbolicity of renormalization for c r unimodal mappings in this paper we extend m lyubichi s recent results on the global hyperbolicity of renormalization of quadratic like germs to the space of cr unimodal maps with quadratic critical point we show that in this space the boundedtype limit sets of the renormalization operator have an invariant hyperbolic structure provided r i 2 a with a close to one as an intermediate step between lyubichi s results and ours we prove that the renormalization operator is hyperbolic in a banach space of real analytic maps we construct the local stable manifolds and prove that they form a continuous lamination whose leaves are c1 codimension one banach submanifolds of the ambient space and whose holonomy is c1 for some 0 we also prove that the global stable sets are c1 immersed codimension one submanifolds as well provided r i 3 a with a close to one as a corollary we deduce that in generic oneparameter families of cr unimodal maps the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a cantor set with hausdorff dimension less than one

application of the entropically damped artificial compressibility model to direct numerical simulation of turbulent channel flow abstract the feasibility of the entropically damped artificial compressibility edac model for simulation of wall bounded turbulence is examined a favourable feature of edac is the purely parabolic type of governing equations resulting from the introduction of a damping term to the pressure evolution equation this significantly reduces noise in the velocity divergence field when central schemes are applied for the spatial discretisation the method of lines is used for the solution of the edac equations a conservative finite difference method of purely 2nd and mixed 4th 2nd order is applied the explicit 4th order six stage low storage runge kutta method is used for time advancement as expected the mixed scheme is superior to the 2nd order one in terms of both accuracy and computational efficiency for the first time the edac model is assessed for the direct numerical simulation of wall bounded turbulent flow requiring a non uniform mesh as a particular test case we have chosen the channel flow at the friction reynolds numbers r e 180 and 395 a very good agreement with the reference data is obtained as documented by the chosen one point velocity statistics the mean and r m s profiles and the budgets of the reynolds stresses the use of the explicit time discretisation and local rather than compact spatial schemes results in easily and efficiently parallelisable solution algorithm we achieved very high computational performance on a desktop computer when solving the edac equations using a code dedicated for the graphics processing unit gpu the aspects of the gpu implementation are discussed

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy

multifractality of the feigenbaum attractor and fractional derivatives it is shown that fractional derivatives of the integrated invariant measure of the feigenbaum map at the onset of chaos have power law tails in their cumulative distributions whose exponents can be related to the spectrum of singularities f this is a new way of characterizing multifractality in dynamical systems so far applied only to multifractal random functions frisch and matsumoto j stat phys 108 1181 2002 the relation between the thermodynamic approach vul sinai and khanin russian math surveys 39 1 1984 and that based on singularities of the invariant measures is also examined the theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations

comparison of the lattice boltzmann method and the artificial compressibility method for navier stokes equations this paper compares in detail the lattice boltzmann method and an isothermal navier stokes method it is found that these two methods are closely related to each other both methods satisfy similar macroscopic governing equations in their continuous forms but they differ from each other in their discrete forms besides the obvious differences in stencils for spatial discretization these two methods also differ from each other in temporal discretization numerical tests show that these differences have little impact on the simulation of velocity fields but do generate noticeable differences in the pressure fields both methods are capable of simulating transient flows and exhibit oscillatory behavior due to the propagation of pressure waves the lattice boltzmann method may be more accurate for capturing the pressure waves

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

knotted periodic orbits in suspensions of smale s horseshoe period multiplying and cabled knots abstract following earlier work on knotted periodic orbits in a suspension of smale s 1 horseshoe diffeomorphism holmes and williams 2 we define a notion of iterated horseshoe knots we show that approximately half the horseshoe knots are cabled and that among these cablings are infinitely many non isotopic iterated torus knots we obtain uniqueness results for certain families of resonant torus knots related to cascades of period doubling and period multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the henon maps we give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots

low mach number limit of the full navier stokes equations the low mach number limit for classical solutions of the full navier stokes equations is here studied the combined effects of large temperature variations and thermal conduction are taken into account in particular we consider general initial data the equations lead to a singular problem depending on a small scaling parameter whose linearized system is not uniformly well posed yet it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the mach number ma 0 1 the reynolds number re 1 and the peclet number pe 1 based on uniform estimates in sobolev spaces and using a theorem of g metivier s schochet 30 we next prove that the penalized terms converge strongly to zero this allows us to rigorously justify at least in the whole space case the well known computations given in the introduction of p l lions book 26

conservative high order finite difference schemes for low mach number flows three finite difference algorithms are proposed to solve a low mach number approximation for the navier stokes equations these algorithms exhibit fourth order spatial and second order temporal accuracy they are dissipation free and thus well suited for dns and les of turbulent flows the key ingredient common to each of the methods presented is a poisson equation with variable coefficient that is solved for the hydrodynamic pressure this feature ensures that the velocity field is constrained correctly it is shown that this approach is needed to avoid violation of the conservation of kinetic energy in the inviscid limit which would otherwise arise through the pressure term in the momentum equation an existing set of finite difference formulae for incompressible flow is generalized to handle arbitrary large density fluctuations with no violation of conservation through the non linear convective terms an algorithm which conserves mass momentum and kinetic energy fully is obtained when an approximate equation of state is used instead of the exact one results from a model problem are used to show both spatial and temporal convergence rates and several test cases are presented to illustrate the performance of the algorithms

analysis of artificial pressure equations in numerical simulations of a turbulent channel flow abstract recently several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation avoiding the resolution of a poisson equation these methods can be seen as various levels of approximation of the compressible navier stokes equation in the low mach number limit we study the simulation of incompressible wall bounded flows using several artificial pressure equations in order to determine the most relevant approximations the simulations are stable using a finite difference method in a staggered grid system even without diffusive term and converge to the incompressible solution both in direct numerical simulations and for coarser meshes to be used in large eddy simulations a pressure equation with a convective and a diffusive term produces a more accurate solution than a compressible solver or methods involving more approximations this suggests that it is near to an optimal level of approximation the presence of a convective term in the pressure evolution equation is in particular crucial for the accuracy of the method the rate of convergence of the solution in terms of artificial mach number is studied numerically and validates the theoretical quadratic convergence rate we demonstrate that this property can be used to accelerate the rate of convergence using an extrapolation in terms of artificial mach number since the approach is based on an explicit and local system of equations the numerical procedure is massively parallelisable and has low memory requirements

artificial compressibility method revisited asymptotic numerical method for incompressible navier stokes equations the artificial compressibility method for the incompressible navier stokes equations is revived as a high order accurate numerical method fourth order in space and second order in time similar to the lattice boltzmann method the mesh spacing is linked to the mach number an accuracy higher than that of the lattice boltzmann method is achieved by exploiting the asymptotic behavior of the solution of the artificial compressibility equations for small mach numbers and the simple lattice structure an easy method for accelerating the decay of acoustic waves which deteriorate the quality of the numerical solution and a simple cure for the checkerboard instability are proposed the high performance of the scheme is demonstrated not only for the periodic boundary condition but also for the dirichlet type boundary condition

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

a parallel algorithm for the nonsymmetric eigenvalue problem this paper describes a parallel algorithm for computing the eigenvalues and eigenvectors of a nonsymmetric matrix the algorithm is based on a divide and conquer procedure and uses an iterative refinement technique

fuzzy rule based classification with featureselector and modified threshold accepting abstract this paper highlights the need to reduce the dimension of the feature space in classification problems of high dimensions without sacrificing the classification power considerably we propose a methodology for classification tasks which comprises three phases i feature selection ii automatic generation of fuzzy if then rules and iii reduction of the rule base while retaining its high classification power the first phase is executed by using featureselector a software developed solely for feature extraction in pattern recognition and classification problems this is for the first time that the featureselector is used as a preprocessor for rule based classification systems in the second phase a standard fuzzy rule based classification system is modified and invoked with the most important features extracted by the featureselector as the new set of features in the third phase a modified threshold accepting algorithm mta proposed elsewhere by the authors ravi et al fuzzy sets and systems forthcoming is used for minimizing the number of rules in the classification system while guaranteeing high classification power the number of rules used and the classification power are taken as the objectives for this multi objective combinatorial global optimization problem the methodology proposed here has been successfully demonstrated for two well known problems i the wine classification problem which includes 13 feature variables in its original form and ii the wisconsin breast cancer determination problem which has 9 feature variables in conclusion the results are encouraging as there is no remarkable reduction in the classification power in both the problems despite the fact that some features have been deleted from the study by resorting to feature selection also the mta outperformed the original threshold accepting algorithm for the test problems considered here the authors suggest that classification problems having higher feature dimensions can be solved successfully within the framework of the methodology presented here the high classification powers obtained for both the problems when working with less feature variables than the original number is the significant achievement of this study

mining axiomatic fuzzy set association rules for classification problems abstract in this paper we propose a novel method to mine association rules for classification problems namely afsrc afs association rules for classification realized in the framework of the axiomatic fuzzy set afs theory this model provides a simple and efficient rule generation mechanism it can also retain meaningful rules for imbalanced classes by fuzzifying the concept of the class support of a rule in addition afsrc can handle different data types occurring simultaneously furthermore the new model can produce membership functions automatically by processing available data an extensive suite of experiments are reported which offer a comprehensive comparison of the performance of the method with the performance of some other methods available in the literature the experimental result shows that afsrc outperforms most of other methods when being quantified in terms of accuracy and interpretability afsrc forms a classifier with high accuracy and more interpretable rule base of smaller size while retaining a sound balance between these two characteristics

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

a shorter proof of the existence of the feigenbaum fixed point we use the leray schauder fixed point theorem to prove the existence of an analytic fixed point for the period doubling accumulation renormalization operator our argument does not however show that the linearization of the renormalization operator at this fixed point is hyperbolic

a three dimensional incompressible navier stokes flow solver using primitive variables an implicit finite difference computer code has been developed to solve the incompressible navier stokes equations in a three dimensional curvilinear coordinate system the pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation the solution procedure employs an implicit approximate factorization scheme the reynolds stresses which are uncoupled from the implicit scheme are lagged by one time step to facilitate implementing various levels of the turbulence model test problems for external and internal flows are computer and the results are compared with existing experimental data the application of this technique for general three dimensional problems is then demonstrated

one dimensional dynamics for a discontinuous map abstract a discontinuous two parameter family of one dimensional maps is shown to posses many of the features of the quadratic logistic map for instance a unique twice differentiable maximum however paths may be chosen in the two dimensional parameter space which give rise to a wealth of possible routes to chaos in particular paths may be found for the feigenbaum route to chaos period doubling cascade of bifurcations with a resulting single parameter in the map converging geometrically with a constant n directly to chaotic motion

computation of low speed flow with heat addition a perturbation expansion is used to obtain a system of conservation laws for compressible flows that is valid at arbitrarily low mach numbers these equations are rendered hyperbolic by adding an artificial time derivative to the energy equation thus introducing pseudoacoustic waves with a speed the same order as the particle velocity traditional time iterative schemes are shown to be effective in solving this system numerically stability calculations of the complete vector system indicate unconditional stability at all mach numbers in the absence of gravity but the source term introduced by buoyancy becomes destabilizing at froude numbers below about 1 this instability is amplified by approximate factorization thus precluding solutions with gravity below this mach number level computations of strong heat addition in low mach number flow both with and without gravity confirm the stability predictions convergent solutions are maintained and are used to show the effect of gravity on such flowfields

a numerical method for incompressible and compressible flow problems with smooth solutions abstract a semi implicit difference method of second order in space is introduced for the numerical solution of the euler equations if the mach number e is small the solutions are second order accurate also in time in particular the solutions converge to an approximate solution of the incompressible equations as e tends to zero numerical experiments are presented for channel flow and the theoretical results given for the linearized equations are shown to be valid also for the real nonlinear problem

period doubling in two parameter families abstract we study two aspects of the cross over from dissipative to area preserving behaviour for period doubling cascades first we consider the smoothness of the limiting bifurcation curve at the crossover point we make a conjecture and give supporting numerical evidence second we study the spectrum of the linearised doubling operator acting on arbitrary mappings and show that it can be expressed in terms of the spectrum of its restriction to area preserving mappings

fractal boundary of domain of analyticity of the feigenbaum function and relation to the mandelbrot set the universal map for the period doubling transition to chaos is studied numerically in the complex plane the boundary of the domain of analyticity of this function is obtained graphically and is shown to be a fractal with self similar properties obtained by rescaling with the universal constants and in the complex parameter plane this domain is shown asymptotically to be similar to part of the mandelbrot set

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

kinetically reduced local navier stokes equations for simulation of incompressible viscous flows recently another approach to study incompressible fluid flow was suggested s ansumali i karlin and h ottinger phys rev lett 94 080602 2005 the kinetically reduced local navier stokes krlns equations we consider a simplified two dimensional krlns system and compare it with chorin s artificial compressibility method a comparison of the two methods for steady state computation of the flow in a lid driven cavity at various reynolds numbers shows that the results from both methods are in good agreement with each other however in the transient flow it is demonstrated that the krlns equations correctly describe the time evolution of the velocity and of the pressure unlike the artificial compressibility method

fuzzy pattern recognition for automatic detection of different teeth substances abstract a method for characterizing the surface of a test object on the basis of its oscillatory behaviour is presented the basic principle of the method is the excitation of oscillations in a small surface element on the elastic solid under investigation and the appropriate measurement of the reaction to this excitation after calculation of the fourier transform a total of 400 magnitude values of the oscillation are available as characteristic features for the classification of an unkown surface for testing the measuring system experiments were conducted on various test objects and the results of measurements performed on different tooth surface are presented the automatic classification of tooth surface is important for the detection of the cementoenamel junction in parodontological practice and the soft removal of dental calculus because in this application the spectra are subject to severe fluctuations by variations in the test object a fuzzy classifier with weighted distance function is employed for the automatic classification for solving the essential problem of selecting from a total of 400 spectral features those which appear to be suited for the respective classification problem a new feature selection algorithm is presented the highly successful result of these experiments indicates possibilities of applying new diagnostic and therapeutic instruments in the field of periodontology

artificial compressibility method and lattice boltzmann method similarities and differences the artificial compressibility method and the lattice boltzmann method yield the solutions of the incompressible navier stokes equations in the limit of the vanishing mach number the inclusion of the bulk viscosity is considered to be one of the reasons for the success of the lattice boltzmann method at least in the 2d case in the present paper the robustness of the artificial compressibility method is enhanced by introducing a new dissipation term which makes high cell reynolds number computation possible the increase of the stability is also confirmed in the linear stability analysis the magnitude of the eigenvalues are drastically reduced for low resolution comparisons are made with the lattice boltzmann method it is confirmed that the fortified acm is more robust as well as more accurate than the lattice boltzmann method

a precise calculation of the feigenbaum constants the feigenbaum constants arise in the theory of iteration of real functions we calculate here to high precision the constants ca and 5 associated with period doubling bifurcations for maps with a single maximum of order z for 2 0 that is the sequence 2 x i 1 z xi i 1 2 x0 0 in 1979 feigenbaum 8 observed that there exist bifurcations in the set of limit points of 2 that is in the set of all points which are the limit of some infinite subsequence as the parameter u is increased for fixed z roughly speaking if the sequence 2 is asymptotically periodic with period p for a particular parameter value j that is there exists a stable p cycle then as 1u is increased the period will be observed to double so that a stable 2p cycle appears we denote the critical u value at which the 2j cycle first appears by hii feigenbaum also conjectured that there exist certain universal scaling constants associated with these bifurcations specifically 3 a lim yi itz i 0 yo j 1 yj exists and 2 is about 4 669 similarly if d is the value of the nearest cycle element to 0 in the 2j cycle then di 4 a lim j exists and a2 is about 2 503 received november 22 1989 revised september 10 1990 1980 mathematics subject classification 1985 revision primary lly60 26a18 39a10 65q05 1991 american mathematical society 0025 5718 91 1 00 25 per page

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

period doubling forc2 mappings the rigorous treatment of period doubling cascades developed by lanford for analytic families is extended to coverc2 families for any 0 this requires spectral analysis of the linearised doubling operator on spaces of differentiable mappings and a version of stable manifold theory which takes account of the non differentiability of the doubling operator on these spaces

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

asymptotic single and multiple scale expansions in the low mach number limit an asymptotic analysis of the euler equations in the limit of vanishing mach number is presented which can be employed to extend the validity of numerical methods from the compressible to the low mach number regime r klein j comput phys 121 1995 pp 213 237 l hoffmann and a meister in proceedings of the fourth eccomas computational fluid dynamics conference vol 1 c hirsch et al eds athens 1998 pp 567 572 the work is based on the analysis recently proposed by klein whereby the paper is devoted to a rigorous mathematical justification of this asymptotic investigation in order to give reliable statements concerning the assumptions under which the results hold in contrast to klein a single scale as well as multiple scale expansion is used depending on the spatial domain under consideration

kinetically reduced local navier stokes equations an alternative approach to hydrodynamics an alternative approach the kinetically reduced local navier stokes krlns equations for the grand potential and the momentum is proposed for the simulation of low mach number flows the taylor green vortex flow is considered in the krlns framework and compared to the results of the direct numerical simulation of the incompressible navier stokes equations the excellent agreement between the krlns equations and the incompressible nonlocal navier stokes equations for this nontrivial time dependent flow indicates that the former is a viable alternative for computational fluid dynamics at low mach numbers

a unified approach to compressible and incompressible flows abstract we present a finite element formulation for solving the compressible navier stokes equations which accommodates the use of any set of variables if primitive variables p u t or entropy variables are used the incompressible limit is well behaved therefore one formulation can be used to solve both compressible and incompressible flows

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

periodicity versus chaos in one dimensional dynamics we survey recent results in one dimensional dynamics and as an application we derive rigorous basic dynamical facts for two standard models in population dynamics the ricker and the hassell families we also informally discuss the concept of chaos in the context of one dimensional discrete time models first we use the model case of the quadratic family for an informal exposition we then review precise generic results before turning to the population models our focus is on typical asymptotic behavior seen for most initial conditions and for large sets of maps parameter sets corresponding to different types of attractors are described in particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models natural measures asymptotic distributions and their stability properties are considered

rigorous estimates for a computer assisted kam theory nonautonomous hamiltonian systems of one degree of freedom close to integrable ones are considered let e be a positive parameter measuring the strength of the perturbation and denote by e c the critical value at which a given kam kolmogorov arnold moser torus breaks down a computer assisted method that allows one to give rigorous lower bounds for e c is presented this method has been applied in celletti falcolini porzio to be published in ann inst h poincare to the escande and doveil pendulum yielding a bound which is within a factor 40 2 of the value indicated by numerical experiments

an inverse kinetic theory for the incompressible navier stokes equations abstract an inverse kinetic theory applying specifically to incompressible newtonian fluids which permits us to avoid the n 2 algorithmic complexity of the poisson equation for the fluid pressure is presented the theory is based on the construction of a suitable kinetic equation in phase space which permits us to determine exactly the fluid equations by means of the velocity moments of the kinetic distribution function it is found that the fluid pressure can also be determined as a moment of the distribution function without solving the poisson equation as is usually required in direct solution methods for the incompressible fluid equations finally the dynamical system underlying the incompressible navier stokes equations and advancing in time the fluid fields has been also identified and proven to produce an unique set of fluid equations