singular hyperbolic attractors are chaotic we prove that a singular hyperbolic attractor of a 3 dimensional flow is chaotic in two different strong senses first the flow is expansive if two points remain close at all times possibly with time reparametrization then their orbits coincide second there exists a physical or sinai ruelle bowen measure supported on the attractor whose ergodic basin covers a full lebesgue volume measure subset of the topological basin of attraction moreover this measure has absolutely continuous conditional measures along the center unstable direction is a u gibbs state and is an equilibrium state for the logarithm of the jacobian of the time one map of the flow along the strong unstable direction this extends to the class of singular hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic or axiom a attractors for flows in particular these results can be applied i to the flow defined by the lorenz equations ii to the geometric lorenz flows iii to the attractors appearing in the unfolding of certain resonant double homoclinic loops iv in the unfolding of certain singular cycles and v in some geometrical models which are singular hyperbolic but of a different topological type from the geometric lorenz models in all these cases the results show that these attractors are expansive and have physical measures which are u gibbs states

multifractal analysis of conformal axiom a flows we develop the multifractal analysis of conformal axiom a flows this includes the study of the hausdorff dimension of basic sets of the flow the description of the dimension spectra for pointwise dimension and for lyapunov exponents and the multifractal decomposition associated with these spectra the main tool of study is the thermodynamic formalism for hyperbolic flows by bowen and ruelle examples include suspensions over axiom a conformal diffeomorphisms anosov flows and in particular geodesic flows on compact smooth surfaces of negative curvature

non linear strain displacement equations exactly representing large rigid body motions part ii enhanced finite element technique abstract the precise representation of arbitrarily large rigid body motions in the displacement patterns of curved timoshenko mindlin type tm shell elements has been considered in part i of the present work this consideration required the development of strain displacement equations of the finite deformation tm shell theory written in local curvilinear coordinates with regard to their consistency with the large rigid body motions for this purpose the displacement vectors of the face surfaces are introduced and resolved in the reference surface frame in this paper economical schemes of evaluating the stiffness matrix by means of the analytical integration inside the element and an advanced approach for solving incremental equilibrium equations are discussed in detail the developed approach may allow the use of load increments that are much larger than possible with the approach proposed in part i the numerical results are presented to demonstrate the high accuracy and effectiveness of the developed four node curved shell elements and to compare their performance with non linear solid shell elements extracted from the literature

a hybrid mixed four node quadrilateral plate element based on sampling surfaces method for 3d stress analysis summary the hybrid mixed assumed natural strain four node quadrilateral element using the sampling surfaces sas technique is developed the sas formulation is based on choosing inside the plate body n not equally spaced sas parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables such choice of unknowns with the consequent use of lagrange polynomials of degree n 1 in the thickness direction permits the presentation of the plate formulation in a very compact form the sas are located at chebyshev polynomial nodes that allow one to minimize uniformly the error due to the lagrange interpolation to avoid shear locking and have no spurious zero energy modes the assumed natural strain concept is employed the developed hybrid mixed four node quadrilateral plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations it can be useful for the 3d stress analysis of thin and thick plates because the sas formulation gives the possibility to obtain solutions with a prescribed accuracy which asymptotically approach the 3d exact solutions of elasticity as the number of sas tends to infinity copyright 2016 john wiley sons ltd

finite deformation higher order shell models and rigid body motions this paper focuses on the developments of higher order shell models by employing the new concept of interpolation surfaces i surfaces inside the shell body we introduce n n p 3 i surfaces and choose the values of displacements with correspondence to these surfaces as fundamental shell unknowns such choice allows first to develop various higher order shell models in a very compact form and second to derive non linear strain displacement relationships which are completely free for arbitrarily large rigid body motions the general 3n parameter shell model is proposed in the framework of the lagrangian description the special 9 12 and 15 parameters cases corresponding to n 3 4 and 5 have been dealt in detail the proposed shell models account for thickness stretching and the complete 3d constitutive equations are utilized the displacement vectors of equally located i surfaces for each model are represented in a convected curvilinear coordinate system allowing one to develop directly finite deformation geometrically exact shell finite elements which could be discussed in future works 2008 elsevier ltd all rights reserved

the lyapunov spectrum for conformal expanding maps and axiom a surface diffeomorphisms we provide a detailed description of the decomposition of a conformal repeller by the level sets of the lyapunov exponent along with a similar result for axiom a surface diffeomorphisms

non linear exact geometry 12 node solid shell element with three translational degrees of freedom per node this paper presents the finite rotation exact geometry eg 12 node solid shell element with 36 displacement degrees of freedom the term eg reflects the fact that coefficients of the first and second fundamental forms of the reference surface and christoffel symbols are taken exactly at each element node the finite element formulation developed is based on the 9 parameter shell model by employing a new concept of sampling surfaces s surfaces inside the shell body we introduce three s surfaces namely bottom middle and top and choose nine displacements of these surfaces as fundamental shell unknowns such choice allows one to represent the finite rotation higher order eg solid shell element formulation in a very compact form and to derive the strain displacement relationships which are objective that is invariant under arbitrarily large rigid body shell motions in convected curvilinear coordinates the tangent stiffness matrix is evaluated by using 3d analytical integration and the explicit presentation of this matrix is given the latter is unusual for the non linear eg shell element formulation copyright 2011 john wiley sons ltd

assumed strain stabilization procedure for the 9 node lagrange shell element an assumed strain strain interpolation method is used to construct a stabilization matrix for the 9 node shell element the stabilization procedure can be justified based on the hellinger reissner variational method it involves a projection vector which is orthogonal to both linear and quadratic fields in the local co ordinate system of each quadrature point all terms in the development involve 2 2 quadrature in the 9 node element example problems show good accuracy and an almost optimal rate of convergence

multifractal analysis of hyperbolic flows we establish the multifractal analysis of hyperbolic flows and of suspension flows over subshifts of finite type a non trivial consequence of our results is that for every holder continuous function non cohomologous to a constant the set of points without birkhoff average has full topological entropy

thermodynamic formalism for contracting lorenz flows we study the expansion properties of the contracting lorenz flow introduced by rovella via thermodynamic formalism specifically we prove the existence of an equilibrium state for the natural potential hat varphi t x y z t log j x y z cu for the contracting lorenz flow and for t in an interval containing 0 1 we also analyse the lyapunov spectrum of the flow in terms of the pressure

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

a stabilization procedure for the quadrilateral plate element with one point quadrature a stabilization procedure is developed for controlling the kinematic modes of the four node bilinear quadrilateral element when single point quadrature is used these kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness in this stabilization procedure additional generalized strains are defined which are activated by the kinematic modes these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral by using a scaling law developed in an earlier paper the stabilization parameters are defined so they do not adversely affect the element s performance several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems

a multifractal formalism let x be a metric space and a borel probability measure on x for q t r and e x write formula then hq t and pq t are borel measures hq t is a multifractal generalization of the centered hausdorff measure and pq t is a multifractal generalization of the packing measure the measures hq t and pq t define for a fixed q in the usual way a generalized hausdorff dimension dimq e and a generalized packing dimension dimq e of subsets e of x we study the functions b q dimq supp b q dimq supp and their relation to the so called multifractal spectra functions of formula we prove among other things that f f is bounded from above by the legendre transform of b b and that equality holds for graph directed self similar measures and cookie cutter measures finally we discuss the connection with generalized renyi dimensions

a family of ans four node exact geometry shell elements in general convected curvilinear coordinates the non conventional exact geometry shell elements based on the timoshenko mindlin kinematics with five displacement degrees of freedom are proposed the term exact geometry exg reflects the fact that coefficients of the first and second fundamental forms of the reference surface and christoffel symbols are taken exactly at every gauss integration point the choice of only displacements as fundamental shell unknowns gives an opportunity to derive strain displacement relationships which are invariant under rigid body shell motions in a convected curvilinear coordinate system this paper presents a newly developed family consisting of three hybrid and one displacement based four node exg shell elements to avoid shear and membrane locking and have no spurious zero energy modes the ans concept is employed the ans interpolations satisfy exactly the plate compatibility equation for in plane strains as a result all exg shell elements developed pass membrane and bending plate patch tests and exhibit a superior performance in the case of distorted coarse mesh configurations copyright 2010 john wiley sons ltd

tensor based finite element formulation for geometrically nonlinear analysis of shell structures in the present study a finite element computational model for the nonlinear analysis of shell structures is presented a tensor based finite element formulation is presented to describe the mathematical model of a shell in a natural and simple way by using curvilinear coordinates in addition a family of high order elements with lagrangian interpolations is used to avoid membrane and shear locking and no mixed interpolations are employed a first order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the lagrangian description this approach takes into account thickness stretching and therefore 3d constitutive equations are utilized numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites as well as functionally graded shells are found to be excellent and show the validity of the developed finite element model moreover the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation of shells

a stabilized one point integrated quadrilateral reissner mindlin plate element a new quadrilateral reissner mindlin plate element with 12 element degrees of freedom is presented for linear isotropic elasticity a hellinger reissner functional with independent displacements rotations and stress resultants is used within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces the hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically this leads to a part obtained by one point integration and a stabilization matrix the element possesses a correct rank does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates the bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements copyright 2004 john wiley sons ltd

a continuum based shell theory for non linear applications the paper introduces a non linear shell theory which provides a complete three dimensional state of stress since the theory is derived from simple three dimensional continuum mechanics it is very easy to understand as an example the theory is applied to quadrilateral shell elements which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface it is proposed to eliminate this degree of freedom on element level so that the elements have the same layout as the equivalent brick elements but have a better behaviour in bending have stress resultants and are cheaper with respect to computational effort the advantages with respect to implementation in a finite element program as well as in special applications are obvious however well known conditioning problems in thin shell applications must be expected therefore emphasis is put on this issue in the example problems it is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non linear applications which will be considered in a forthcoming paper

refined shear deformation models for composite laminates with finite rotations abstract for arbitrary multilayered shell structures made particularly of composite material layers a refined finite rotation theory with seven independent displacement variables is developed approximating the displacement field by a cubic series expansion of thickness coordinates this model allows a quadratic shear deformation distribution across the thickness procedures are given permitting a unique determination of the first order displacement term in the case of finite rotations kinematic relations are formulated in two alternative forms suitable for both classical and isoparametric finite element formulations the constitutive relations presented model orthotropic material properties varying arbitrarily across the thickness this third order single layer theory is then transformed by introducing further constraints into three simplified models a third order theory with five independent displacement variables a mindlin reissner type theory and a kirchhoff love type theory these four models differ however from each other essentially in the constraints imposed on the first and third order displacement variables a significant advantage for a unified finite element development finally the mindlin reissner type theory is generalized to a layer wise model being the most predictive one in dealing with local interlaminar effects the theoretical models are transformed into adequate finite shell elements and then compared by means of appropriate examples concerning their prediction capability also examples are given demonstrating their applicability to finite rotation phenomena

high order corrections to the time dependent born oppenheimer approximation ii coulomb systems we study the dynamics of molecular systems with coulomb forces we prove that if the nuclear masses are proportional to e 4 then certain solutions to the time dependent schrodinger equation have asymptotic expansions to arbitrarily high order in powers of e as e 0

multifractal analysis of lyapunov exponent for continued fraction and manneville pomeau transformations and applications to diophantine approximation we extend some of the theory of multifractal analysis for conformal expanding systems to two new cases the non uniformly hyperbolic example of the manneville pomeau equation and the continued fraction transformation a common point in the analysis is the use of thermodynamic formalism for transformations with infinitely many branches

on the lyapunov spectrum for rational maps we study the dimension spectrum for lyapunov exponents for rational maps on the riemann sphere

hybrid mixed ans finite elements for stress analysis of laminated composite structures sampling surfaces plate formulation abstract a hybrid mixed ans four node quadrilateral plate element through the use of the sampling surfaces sas technique is developed the sas formulation is based on choosing inside the n th layer i n not equally spaced sas parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables such choice of unknowns with utilizing the lagrange polynomials of degree i n 1 in the thickness direction for each layer leads to a very compact form of the governing equations the sas are located inside each layer at chebyshev polynomial nodes that allows one to minimize uniformly the error due to the lagrange interpolation to avoid shear locking and have no spurious zero energy modes the hybrid mixed method and the ans concept are employed the developed four node quadrilateral laminated plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations it can be useful for the 3d stress analysis of thick and thin laminated composite plates because the sas formulation gives the possibility to obtain the numerical solutions with a prescribed accuracy which asymptotically approach the exact solutions of elasticity as the number of sas tends to infinity

natural equilibrium states for multimodal maps this paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps this class contains but it is larger than collet eckmann for a map in this class we prove existence and uniqueness of equilibrium states for the geometric potentials t log df for the largest possible interval of parameters t we also study the regularity and convexity properties of the pressure function completely characterising the first order phase transitions results concerning the existence of absolutely continuous invariant measures with respect to the lebesgue measure are also obtained

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

suspension flows over countable markov shifts we introduce the notion of topological pressure for suspension flows over countable markov shifts and we develop the associated thermodynamic formalism in particular we establish a variational principle for the topological pressure and an approximation property in terms of the pressure on compact invariant sets as an application we present a multifractal analysis for the entropy spectrum of birkhoff averages for suspension flows over countable markov shifts the domain of the spectrum may be unbounded and the spectrum may not be analytic we provide explicit examples where this happens we also discuss the existence of full measures on the level sets of the multifractal decomposition

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects