generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

extended immersed boundary method using fem and rkpm abstract the proposed extended immersed boundary method eibm has several distinct features in comparison with the immersed boundary ib method in the ib method the interaction between the immersed elastic boundary and the surrounding viscous fluid is replaced with an equivalent body force distribution within the fluid domain the key ingredient of the ib method is the enforcement of the energy input from the equivalent body force into the fluid domain to be the same at all time as that of the elastic forces within the ib in eibm instead of the volumeless immersed elastic boundary we consider the submerged elastic solid which occupies a finite volume within the fluid domain in particular we replace the kinematic and dynamic matching of the fluid solid interface and the effect of the submerged solid with a judiciously chosen collection of equivalent nodal forces calculated in the context of finite element formulations the employment of the finite element representation enables a realistic stress analysis for the submerged solid subject to large deformations finally in eibm we also replace the discretized delta function commonly used in the ib method with the kernel functions in meshless reproducing kernel particle method rkpm with this improvement using the same finite support domain we could improve the discretized delta function from c 1 to c n continuity where n is chosen according to the required smoothness and resolution of such functions and more importantly enables the use of non uniform meshing in the fluid domain these extensions may eventually open doors to the modelling of complex biological fluid involving molecular cellular and flexible vessel flow interactions

new indicators of chaos an algorithm which allows to recognize whether the evolution of the system is periodic chaotic or hyperchaotic is proposed minimum length orbit is introduced its behavior depends on the kind of solution the calculations are presented for the henon map for the damped driven pendulum for the rossler system and for the hyperchaotic qi system

geometrical properties of local dynamics in hamiltonian systems the generalized alignment index gali method we investigate the detailed dynamics of multidimensional hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits the behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion the number of deviation vectors their linear in dependence and the spectrum of lyapunov exponents the different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on n dimensional tori more specifically we introduce the generalized alignment index of order k galik as the volume of a generalized parallelepiped whose edges are k initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step we show analytically and verify numerically on particular examples of n degree of freedom hamiltonian systems that for chaotic orbits galik tends exponentially to zero with exponents that involve the values of several lyapunov exponents in the case of regular orbits galik fluctuates around non zero values for 2 k n and goes to zero for n k 2n following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus t 2 k n m if 0 m k n and t k n if m k n the galik is a generalization of the smaller alignment index sali gali2 sali however galik provides significantly more detailed information on the local dynamics allows for a faster and clearer distinction between order and chaos than sali and works even in cases where the sali method is inconclusive

alignment indices a new simple method for determining the ordered or chaotic nature of orbits we introduce a new simple and efficient method for determining the ordered or chaotic nature of orbits in two dimensional 2d four dimensional 4d and six dimensional 6d symplectic maps the computation of the alignment indices for a given orbit we follow the evolution in time of two different initial deviation vectors computing the norms of the difference d parallel alignment index and the addition d antiparallel alignment index of the two vectors the time evolution of the smaller alignment index reflects the chaotic or ordered nature of the orbit in 2d maps the smaller alignment index tends to zero for both ordered and chaotic orbits but with completely different time rates which allows us to distinguish between the two cases in 4d and 6d maps the smaller alignment index tends to zero in the case of chaotic orbits while it tends to a positive non zero value in the case of ordered orbits the efficiency of the new method is also shown in a case of weak chaos and a comparison with other known methods that separate chaotic from regular orbits is presented

hyperchaotic states in the parameter space abstract in this paper we propose a numerical method to characterize hyperchaotic points in the parameter space of continuous time dynamical systems the method considers the second largest lyapunov exponent value as a measure of hyperchaotic motion to construct two dimensional parameter space color plots different levels of hyperchaos in these plots are represented by a continuously changing yellow red scale as an example a particular system modeled by a set of four nonlinear autonomous first order ordinary differential equations is considered practical applications of these plots include by instance walking in the parameter space of hyperchaotic systems along desirable paths

algebraic yuzvinski formula abstract topological entropy is very well understood for endomorphisms of compact abelian groups a fundamental result in this context is the so called yuzvinski formula which is the key step in finding the topological entropy of any compact group endomorphism the goal of this paper is to prove a perfect analog of the yuzvinski formula for the algebraic entropy namely the algebraic yuzvinski formula giving the value of the algebraic entropy of an endomorphism of a finite dimensional rational vector space as the mahler measure of its characteristic polynomial

entropy on abelian groups abstract we introduce the algebraic entropy for endomorphisms of arbitrary abelian groups appropriately modifying existing notions of entropy the basic properties of the algebraic entropy are given as well as various examples the main result of this paper is the addition theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups we give several applications of the addition theorem among them the uniqueness theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms furthermore we point out the delicate connection of the algebraic entropy with the mahler measure and lehmer problem in number theory

difficulties in detecting chaos in a complex system abstract the sequences given by a 7d map are analysed by means of the methods widely used to detect chaos in the real world in order to test their sensitivity to chaotic features of a non linear system determined by comparatively high number of parameters the same diagnostic approaches are applied to the 3d lorenz map for comparison the results show that for some of the sequences yielded from the 7d map the adopted methods are not able to give a straightforward answer to the question if the system is chaotic as in the 3d case since the sequences subject to the analysis are not contaminated by noise and are sufficiently long it could be assumed that such difficulties arise likely due to specific internal features of the more complex system it is found also that an increase from 0 01 to 0 5 of the sampling step determining the sequences obtained from the 7d map masks the chaos in some of them

algebraic entropy for abelian groups the theory of endomorphism rings of algebraic structures allows in a natural way a systematic approach based on the notion of entropy borrowed from dynamical systems here we study the algebraic entropy of the endomorphisms of abelian groups introduced in 1965 by adler konheim and mcandrew the so called addition theorem is proved this expresses the algebraic entropy of an endomorphism of a torsion group as the sum of the algebraic entropies of the restriction to a invariant subgroup and of the endomorphism induced on the quotient group particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy the significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy and we also investigate such groups a uniqueness theorem for the algebraic entropy of endomorphisms of torsion abelian groups is proved

endomorphisms of abelian groups with small algebraic entropy we study the endomorphisms of abelian groups g having a small algebraic entropy h where small usually means h log2 using essentially elementary tools from linear algebra we show that this study can be carried out in the group qd where an automorphism with h log2 must have all eigenvalues in the open circle of radius 2 centered at 0 and must leave invariant a lattice in qd i e be essentially an automorphism of zd in particular all eigenvalues of an automorphism with h 0 must be roots of unity this is a particular case of a more general fact known as algebraic yuzvinskii theorem we discuss other particular cases of this fact and we give some applications of our main results

a numerical method for suspension flow abstract peskin s j comput phys 25 220 1977 immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles as before the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations as a result one set of equations holds in the entire computational domain eliminating the need to apply boundary conditions on the surface of suspended objects the new method computes the force density by discretizing the stress strain constitutive equations for an elastic solid on a grid using data provided by clusters of lagrangian points this approach clearly specifies the material properties of the suspended objects a simple data structure for the lagrangian points makes it easy to model suspended solids with arbitrary shape and size the method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids with theory and analysis the capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems

algebraic entropy for amenable semigroup actions abstract we introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups s on discrete abelian groups a by endomorphisms they extend the classical algebraic entropy for endomorphisms of abelian groups corresponding to the case s n we investigate the fundamental properties of the algebraic entropy and compute it in several examples paying special attention to the case when s is an amenable group for actions of cancellative right amenable monoids on torsion abelian groups we prove the so called addition theorem in the same setting we see that a bridge theorem connects the algebraic entropy with the topological entropy of the dual action by means of pontryagin duality so that we derive an addition theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups

required criteria for recognizing new types of chaos application to the cord attractor after suggesting criteria to recognize a new system and a new attractor and to make a distinction between them the paper details the topological analysis of the cord attractor this attractor which resembles a cord between two leaves is produced by a three dimensional system that is obtained after a modification of the lorenz 84 model for the global atmospheric circulation l a aguirre and c letellier phys rev e 83 066209 2011 the nontrivial topology of the attractor is described in terms of a template that corresponds to a reverse horseshoe that is to a spiral rossler attractor with negative and positive global twists due to its particular structure and to the fact that such a system has two variables from which the dynamics is poorly observable this attractor qualifies as a challenging benchmark in nonlinear dynamics

detecting order and chaos in hamiltonian systems by the sali method we use the smaller alignment index sali to distinguish rapidly and with certainty between ordered and chaotic motion in hamiltonian flows this distinction is based on the different behaviour of the sali for the two cases the index fluctuates around non zero values for ordered orbits while it tends rapidly to zero for chaotic orbits we present a detailed study of sali s behaviour for chaotic orbits and show that in this case the sali exponentially converges to zero following a time rate depending on the difference of the two largest lyapunov exponents 1 2 i e sali e

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

milnor wolf theorem for group endomorphisms abstract we study the growth of group endomorphisms and we prove an analogue of chou s extension of milnor wolf theorem indeed if g is an elementary amenable group and g g is an endomorphism then has either polynomial or exponential growth this result follows by studying the growth of automorphisms of finitely generated groups where we prove some stronger results

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd