generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

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

the metric entropy of endomorphisms we prove that for a c 2 non invertible but non degenerate map on a compact riemannian manifold without boundary an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents this generalizes ledrappier young s entropy formula 5 for negative lyapunov exponents of diffeomorphisms to the case of endomorphisms

extrema of young s modulus in hexagonal materials abstract the optimum values of young s modulus for hexagonal materials are obtained by using the representation of the elasticity tensor by ahmad and khan f ahmad r a khan eigenvectors of a rotation matrix q j mech appl math 62 2009 297 310 the expression of young s modulus e n for a hexagonal material is written in terms of n 3 only such that it reveals an axis of rotational symmetry in the direction x 3 which is perpendicular to the transverse isotropy plane that is x 1 x 2 plane indeed the components n 1 n 2 of the unit vector n have no influence on the value of young s modulus moreover young s modulus is expressed in terms of invariant quantities i e eigenvalues rather than components of the compliance tensor the problem is solved in a simple manner and is applied to some real materials

semi continuity of the entropy of sinai probability measures for partially hyperbolic diffeomorphisms abstract we establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

fourth rank tensors of the thirty two crystal classes multiplication tables the fourth rank tensors that embody the elastic or other properties of crystalline anisotropic substances can be partitioned into a number of sets in order that each shall acknowledge the symmetry of one or more of the crystal classes and moreover shall make up a closed linear associative algebra of hypercomplex numbers for the purpose of calculating the sums products and inverses of its constituent tensors to which end coordinate invariant expressions of the tensors are adopted the calculations are simplified immensely and ensuing physical analyses are well prepared for once the structure of every algebra is unravelled completely in terms of a number of separate subalgebras isomorphic to familiar algebras such as the binary one of the complex numbers the quaternary one of the 2x2 matrices and the octonary one of the complex quaternions the fourth rank tensors do not seem to have been submitted previously to the present algebraic point of view and nor do those of any other rank a parallel but less intricate development can be provided for the second rank ones

extrema of young s modulus for elastic solids with tetragonal symmetry abstract for a homogeneous and linearly elastic solid the general expression of young s modulus e n is given and a constrained extremum problem is formulated for the evaluation of the directions n corresponding to stationary values of the modulus the formulation follows that presented in international journal of solids and structures 40 2003 1713 1744 for the cubic and transversely isotropic elastic symmetries in this paper the tetragonal elastic symmetry class is considered and explicit solutions for the directions n associated to critical points of e n are analytically evaluated properties of these directions and of the corresponding values of the modulus are discussed in detail the results are presented in terms of three material parameters which are responsible of the degree of anisotropy for the tetragonal system the complete description of the directional dependence of young s modulus leads to the identification of 12 classes of behavior for each of these classes several examples of real materials are shown and suitable graphical representations of the function e n are given as well

mindlin second gradient elastic properties from dilute two phase cauchy elastic composites part ii higher order constitutive properties and application cases starting from a cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first order by a certain discrepancy tensor see part i of the present article it is shown that the equivalent second gradient mindlin elastic solid i is positive definite only when the discrepancy tensor is negative defined ii the non local material symmetries are the same of the discrepancy tensor and iii the nonlocal effective behaviour is affected by the shape of the rve which does not influence the first order homogenized response furthermore explicit derivations of non local parameters from heterogeneous cauchy elastic composites are obtained in the particular cases of a circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix b n polygonal cylindrical voids in an isotropic matrix and c circular cylindrical voids in an orthortropic matrix

yet another induction scheme for non uniformly expanding transformations abstract we introduce a new induction scheme for non uniformly expanding maps f of compact riemannian manifolds relying upon ideas of 33 and 10 we use this induction approach to prove that the existence of a gibbs markov young structure is a necessary condition for f to preserve an absolutely continuous probability with all its lyapunov exponents positive

general entropy like uncertainty relations in finite dimensions we revisit entropic formulations of the uncertainty principle up for an arbitrary pair of positive operator valued measures povm a and b acting on finite dimensional hilbert space salicr generalized entropies including r nyi and tsallis ones among others are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables we obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals which is valid for both pure and mixed states the bound depends on the overlap triplet with ca respectively cb being the overlap between the elements of the povm a respectively b and the overlap between the pair of povm our approach is inspired by that of de vicente and s nchez ruiz 2008 phys rev a 77 042110 and consists in a minimization of the entropy sum subject to the landau pollak inequality that links the maximum probabilities of both observables we solve the constrained optimization problem in a geometrical way and furthermore when dealing with r nyi or tsallis entropic formulations of the up we overcome the h lder conjugacy constraint imposed on the entropic indices by the riesz thorin theorem in the case of nondegenerate observables we show that for given the bound obtained is optimal and that for r nyi entropies our bound improves deutsch one but maassen uffink bound prevails when finally we illustrate by comparing our bound with known previous results in particular cases of r nyi and tsallis entropies

regularity of srb entropy for geometric lorenz attractors we consider the classical geometric lorenz attractors showing that the srb entropy admits gamma holder continuity for any 0 gamma 1

on the true extrema of young s modulus in hexagonal materials abstract in a paper which has been recently published in applied mathematics and computations khan and ahmad 2012 1 deal with the detection of the extrema of young s modulus e in hexagonal materials a few issues presented in that paper which deserve being outlined and thoroughly discussed are tackled moreover in the case of hexagonal materials a suitable classification is suggested an exhaustive panoramic view of the possible shape of the surface e n generated by young s modulus for all possible orientations n is illustrated and some meaningful numerical examples are proposed

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

the lyapunov exponents of generic volume preserving and symplectic maps we show that the integrated lyapunov exponents of c1 volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding oseledets splitting is trivial all lyapunov exponents are equal to zero or else dominated uniform hyperbolicity in the projective bundle almost everywhere we deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold almost every orbit either is projectively hyperbolic or has all lyapunov exponents equal to zero similarly for a residual subset of all c1 symplectic diffeomorphisms on any compact manifold either the diffeomorphism is anosov or almost every point has zero as a lyapunov exponent with multiplicity at least 2 finally given any set s gl d satisfying an accessibility condition for a residual subset of all continuous s valued cocycles over any measure preserving homeomorphism of a compact space the oseledets splitting is either dominated or trivial the condition on s is satisfied for most common matrix groups and also for matrices that arise from discrete schr nodinger operators

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

on herman s positive entropy conjecture abstract we show that any area preserving c r diffeomorphism of a two dimensional surface displaying an elliptic fixed point can be c r perturbed to one exhibiting a chaotic island whose metric entropy is positive for every 1 r this proves a conjecture of herman stating that the identity map of the disk can be c perturbed to a conservative diffeomorphism with positive metric entropy this implies also that the chirikov standard map for large and small parameter values can be c approximated by a conservative diffeomorphisms displaying a positive metric entropy a weak version of sinai s positive metric entropy conjecture finally this sheds light onto a herman s question on the density of c r conservative diffeomorphisms displaying a positive metric entropy we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable so conjecturally uniformly hyperbolic or display a chaotic island of positive metric entropy

random products of standard maps we develop a general geometric method to establish the existence of positive lyapunov exponents for a class of skew products the technique is applied to show non uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps notably for skew products whose fiber dynamics is given by a continuum of parameters in the froeschle family these types of coupled systems appear as some induced maps in models for the study of arnold diffusion consequently we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics the methods are also suitable for studying cocycles over shift spaces and do not demand any low dimensionality condition on the fiber

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

entropic heisenberg limits and uncertainty relations from the holevo information bound strong and general entropic and geometric heisenberg limits are obtained for estimates of multiparameter unitary displacements in quantum metrology such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions a key ingredient is the holevo bound on the shannon mutual information of a quantum communication channel this leads to a bayesian bound on performance in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group a geometric measure of performance related to entropy is proposed for general parameter estimation it is also shown how strong entropic uncertainty relations for mutually unbiased observables such as number and phase position and momentum energy and time and orthogonal spin 1 2 directions can be obtained from elementary applications of holevo s bound a geometric interpretation of results is emphasised in terms of the volumes of quantum and classical statistical ensembles

ruelle inequality of folding type for c 1 alpha maps for all c 1 alpha maps we prove an inequality conjectured by ruelle that the metric entropy is bounded from above by the folding entropy minus the sum of the negative lyapunov exponents

unstable entropy and unstable pressure for partially hyperbolic endomorphisms abstract in this paper unstable metric entropy unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated a version of shannon mcmillan breiman theorem is established and a variational principle is formulated which gives a relationship between unstable metric entropy and unstable pressure unstable topological entropy as an application of the variational principle some results on the u equilibrium states are given

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

on the continuity of the srb entropy for endomorphisms we consider classes of dynamical systems admitting markov induced maps under general assumptions which in particular guarantee the existence of srb measures we prove that the entropy of the srb measure varies continuously with the dynamics we apply our result to a vast class of non uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the srb measure in particular we show that the srb entropy of viana maps varies continuously with the map

uncertainty in quantum measurements the object of this letter is to show that except in the case of canonically conjugate observables the generalized heisenberg inequality does not properly express the quantum uncertainty principle it is in general too weak an inequality is obtained which does express the principle

mindlin second gradient elastic properties from dilute two phase cauchy elastic composites part i closed form expression for the effective higher order constitutive tensor abstract it is shown that second order homogenization of a cauchy elastic dilute suspension of randomly distributed inclusions yields an equivalent second gradient mindlin elastic material this result is valid for both plane and three dimensional problems and extends earlier findings by bigoni and drugan bigoni d drugan w j 2007 analytical derivation of cosserat moduli via homogenization of heterogeneous elastic materials j appl mech 74 741 753 from several points of view i the result holds for anisotropic phases with spherical or circular ellipsoid of inertia ii the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material iii a perfect energy match is found between heterogeneous and equivalent materials instead of an optimal bound the constitutive higher order tensor defining the equivalent mindlin solid is given in a surprisingly simple formula applications treatment of material symmetries and positive definiteness of the effective higher order constitutive tensor are deferred to part ii of the present article

asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals because of the uncertain irregular microgeometry a random polycrystalline aggregate the shape and crystalline orientations of the constituent anisotropic grains of which are uncorrelated may have elastic moduli scattered over some though small intervals while the conventional macroscopic homogeneity and isotropy hypotheses for it may be considered only as approximate with corresponding uncertainty our formal bounds are proposed to provide the estimates on those uncertainties in asymptotic sense explicit expressions for the aggregates of trigonal crystals classes 3m 32 3m are derived

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

measurement uncertainty relations for discrete observables relative entropy formulation we introduce a new information theoretic formulation of quantum measurement uncertainty relations based on the notion of relative entropy between measurement probabilities in the case of a finite dimensional system and for any approximate joint measurement of two target discrete observables we define the entropic divergence as the maximal total loss of information occurring in the approximation at hand for fixed target observables we study the joint measurements minimizing the entropic divergence and we prove the general properties of its minimum value such a minimum is our uncertainty lower bound the total information lost by replacing the target observables with their optimal approximations evaluated at the worst possible state the bound turns out to be also an entropic incompatibility degree that is a good information theoretic measure of incompatibility indeed it vanishes if and only if the target observables are compatible it is state independent and it enjoys all the invariance properties which are desirable for such a measure in this context we point out the difference between general approximate joint measurements and sequential approximate joint measurements to do this we introduce a separate index for the tradeoff between the error of the first measurement and the disturbance of the second one by exploiting the symmetry properties of the target observables exact values lower bounds and optimal approximations are evaluated in two different concrete examples 1 a couple of spin 1 2 components not necessarily orthogonal 2 two fourier conjugate mutually unbiased bases in prime power dimension finally the entropic incompatibility degree straightforwardly generalizes to the case of many observables still maintaining all its relevant properties we explicitly compute it for three orthogonal spin 1 2 components

generic continuity of metric entropy for volume preserving diffeomorphisms abstract let m be a compact manifold and diff m 1 m be the set of c 1 volume preserving diffeomorphisms of m we prove that there is a residual subset r diff m 1 m such that each f r is a continuity point of the map g h m g from diff m 1 m to r where h m g is the metric entropy of g with respect to volume measure m

variational estimates for the creep behaviour of polycrystals a variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high temperature creep the procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single crystal grains as such it constitutes an extension to locally anisotropic behaviour of the variational procedure devel oped by ponte castaneda 1991 for nonlinear heterogeneous media with locally isotropic behaviour by way of an example the procedure is applied to the de termination of bounds of the hashin shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals the bounds are computed for the special class of untextured fcc polycrystals with isotropic pure power law viscous behaviour first considered by hutchinson 1976 in the context of a calculation of the self consistent type the new bounds are found to be more restrictive than the corresponding classical taylor bishop hill bounds and also more re strictive if only slightly so than related bounds of the hashin shtrikman type by dendievel et al 1991 the new procedure has the advantage over the self consistent procedure of hutchinson 1976 that it may be applied without any essential complications to aggregates of crystals with slip systems exhibiting dif ferent creep rules with for example different power exponents and to general loading conditions however the distinctive feature of the new variational proce dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals

invariant measures satisfying an equality relating entropy folding entropy and negative lyapunov exponents in this paper we prove that for a c2 non invertible but non degenerate map on a compact manifold an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents if and under a condition on the jacobian of the map only if the measure has absolutely continuous conditional measures on the stable manifolds

density of near extreme events we provide a quantitative analysis of the phenomenon of crowding of near extreme events by computing exactly the density of states dos near the maximum of a set of independent and identically distributed random variables we show that the mean dos converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential faster than pure exponential or as a pure exponential function we argue that some of these results would remain valid even for certain correlated cases and verify it for power law correlated stationary gaussian sequences satisfactory agreement is found between the near maximum crowding in the summer temperature reconstruction data of western siberia and the theoretical prediction

srb measures and pesin s entropy formula for endomorphisms a celebrated result conjectured by ruelle and proved by ledrappier strelcyn and young in the smooth ergodic theory of diffeomorphisms is that an invariant measure of a c 2 diffeomorphism satisfies pesin s entropy formula if and only if it has absolutely continuous conditional measures on unstable manifolds 41 42 the later property is known as srb property of the invariant measure this result was also successfully generalized to random dynamical systems of diffeomorphisms 51

extended quantum conditional entropy and quantum uncertainty inequalities quantum states can be subjected to classical measurements whose incompatibility or uncertainty can be quantified by a comparison of certain entropies there is a long history of such entropy inequalities between position and momentum recently these inequalities have been generalized to the tensor product of several hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized our proofs utilize the technique of the original derivation of strong subadditivity of the von neumann entropy

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

ergodic properties for some non expanding non reversible systems abstract we study several properties of invariant measures obtained from preimages for non invertible maps on fractal sets which model non reversible dynamical systems we give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set we give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure this helps in estimating the unstable dimension a e with respect to this measure with the help of lyapunov exponents when there exist negative lyapunov exponents of this limit measure we study the conditional probabilities induced on the non uniform local stable manifolds by the limit measure and also its pointwise dimension on stable manifolds

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

on optimal zeroth order bounds with application to hashin shtrikman bounds and anisotropy parameters abstract a new result presented in this paper is the evaluation of the hashin shtrikman bounds for composites composed of arbitrarily anisotropic constituents to date evaluation of the hashin shtrikman bounds are limited to composites with isotropic constituents or to polycrystalline composites with specific crystal symmetries the generality of the exact result presented herein is achieved through a reinterpretation of kroner s j mech phys solids 25 1977 137 recursive relations for n th order bounds and the optimal zeroth order n 0 bound the definitions of optimal zeroth order bounds are extended to all even ordered tensors and procedures are presented to evaluate these bounds for all second and fourth order tensors while optimal zeroth order bounds are not new the ability to calculate them for fourth order tensors of arbitrary symmetry is new utilizing the zeroth order bounds material anisotropy parameters are defined which quantify the extent of anisotropy for even ordered tensors

extremal lyapunov exponents an invariance principle and applications we propose a new approach to analyzing dynamical systems that combine hyperbolic and non hyperbolic center behavior e g partially hyperbolic diffeomorphisms a number of applications illustrate its power

nonuniform hyperbolicity global dominated splittings and generic properties of volume preserving diffeomorphisms we study generic volume preserving diffeomorphisms on compact manifolds we show that the following property holds generically in the c1 topology either there is at least one zero lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component moreover if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold that is it has a positive measure intersection with any nonempty open set and there is a global dominated splitting for the proof we establish some new properties of independent interest that hold cr generically for any r 1 namely the continuity of the ergodic decomposition the persistence of invariant sets and the l1 continuity of lyapunov exponents

continuity properties of entropy on donne une borne superieure sur le defaut en semicontinuite superieure des entropies topologiques et metriques d une auto application c r d une variete c r r 1

extrema of young s modulus for cubic and transversely isotropic solids for a homogeneous anisotropic and linearly elastic solid the general expression of youngs modulus e n embracing all classes that characterize the anisotropy is given a constrained extremum problem is then formulated for the evaluation of those directions n at which e n attains stationary values cubic and transversely isotropic symmetry classes are dealt with and explicit solutions for such directions n are provided for each case relevant properties of these directions and corresponding values of the modulus are discussed as well results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy on the basis of such material parameters for cubic symmetry two classes of behavior can be distinguished and in the case of transversely isotropic solids the classes are found to be four for both symmetries and for each class of behavior some examples for real materials are shown and graphical representations of the dependence of youngs modulus on direction n are given as well

model order reduction of nonlinear homogenization problems using a hashin shtrikman type finite element method abstract this work presents a computational nonlinear homogenization approach the starting point of which is a model order reduction method based on data clustering to this end the micromechanical data from numerical experiments snapshots is analyzed in order to identify characteristic microstructural deformation patterns these describe how the macroscopic strain typically localizes within the microstructure the outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points within each cluster the strain is then approximated as being constant the mechanical problem is formulated in terms of a three field hashin shtrikman type variational formulation which is based on the introduction of a linear elastic reference medium after discretization most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns the resulting homogenization scheme includes as special cases the finite element method as well as hashin shtrikman and talbot willis type homogenization approaches with phase wise constant trial fields as well as related bounds the limit case finite element method allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material the method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy the results are shown to be in good agreement with full field fe simulations furthermore the method is used to compute a refined upper bound of the talbot willis type compared to phase wise constant trial fields which converges to the finite element solution with increasingly refined discretization

a note on the computation of the extrema of young s modulus for hexagonal materials a simple method to obtain the highest and lowest young s modulus for a material of the hexagonal class is presented it is based upon the use of tensor invariants of plane anisotropic elasticity in fact the cylindrical symmetry of the elastic tensor allows for transforming the 3d original problem into a planar one with a considerable simplification

on the ergodicity of partially hyperbolic systems pugh and shub have conjectured that essential accessibility implies ergodicity for a c partially hyperbolic volume preserving diffeomorphism we prove this conjecture under a mild center bunching assumption which is satisfied in particular by all partially hyperbolic systems with 1 dimensional center bundle we also obtain ergodicity results for c 1ci partially hyperbolic systems

entropy of physical measures for c infty c dynamical systems for a c infty map f on a compact manifold m we prove that for a lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the top lyapunov exponent of lambda df lambda tm circlearrowleft at x this contrasts with the well known ruelle inequality as a consequence we give some refinement of tsujii s work 23 relating physical and sinai ruelle bowen measures

uncertainty relation for smooth entropies uncertainty relations give upper bounds on the accuracy by which the outcomes of two incompatible measurements can be predicted while established uncertainty relations apply to cases where the predictions are based on purely classical data e g a description of the system s state before measurement an extended relation which remains valid in the presence of quantum information has been proposed recently berta et al nature phys 6 659 2010 here we generalize this uncertainty relation to one formulated in terms of smooth entropies since these entropies measure operational quantities such as extractable secret key length our uncertainty relation is of immediate practical use to illustrate this we show that it directly implies security of quantum key distribution protocols our security claim remains valid even if the implemented measurement devices deviate arbitrarily from the theoretical model

srb measures and homoclinic relation for endomorphisms in this paper we give an upper bound for the number of srb measures of saddle type of local diffeomorphisms of boundaryless manifolds in terms of maximal cardinality of set of periodic points without any homoclinic relation

partial hyperbolicity and robust transitivity throughout this paper m denotes a three dimensional boundaryless compact manifold and diff m the space of gl diffeomorphisms defined on m endowed with the usual cl topology a invariant set a is transitive if a w x for some xea here w x is the forward limit set of x the accumulation points of the positive orbit of x the maximal invariant set of in an open set u denoted by a u is the set of points whose whole orbit is contained in u i e a u i e z i u the set a u is robustly transitive if ar is transitive for every diffeomorphism r clclose to a diffeomorphism ediff m is transitive if m w x for some xem i e if a m m is transitive analogously is robustly transitive if every r glclose to also is transitive i e if a m m is robustly transitive in this paper we focus our attention on forms of hyperbolicity uniform partial and strong partial of a maximal invariant set a u derived from its robust transitivity observe that u can be equal to m and then is robustly transitive on one hand in dimension one there do not exist robustly transitive diffeomorphisms the diffeomorphisms with finitely many hyperbolic periodic points morse smale are open and dense in diff s1 on the other hand for two dimensional diffeomorphisms every robustly transitive set a u is a basic set i e a u is hyperbolic transitive and the periodic points of are dense in a u in particular every robustly transitive surface diffeomorphism is anosov and the unique surface which supports such diffeomorphisms is the torus t 2 these assertions follow from m3 and m4 in dimension bigger than or equal to three besides anosov hyperbolic diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type as far as we know three types of such diffeomorphisms have been constructed skew products

the smooth entropy formalism for von neumann algebras we discuss information theoretic concepts on infinite dimensional quantum systems in particular we lift the smooth entropy formalism as introduced by renner and collaborators for finite dimensional systems to von neumann algebras for the smooth conditional min and max entropy we recover similar characterizing properties and information theoretic operational interpretations as in the finite dimensional case we generalize the entropic uncertainty relation with quantum side information of tomamichel and renner and discuss applications to quantum cryptography in particular we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von neumann algebra

invariance principles for interval maps with an indifferent fixed point in this note we establish an almost sure invariance principle for a large class of interval maps with indifferent fixed points including the manneville pomeau map this implies a number of well known corollaries including the weak invariance principle and the law of the iterated logarithm

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

statistical stability and continuity of srb entropy for systems with gibbs markov structures we present conditions on families of diffeomorphisms that guarantee statistical stability and srb entropy continuity they rely on the existence of horseshoe like sets with infinitely many branches and variable return times as an application we consider the family of henon maps within the set of benedicks carleson parameters

on optimal zeroth order bounds of linear elastic properties of multiphase materials and application in materials design abstract zeroth order bounds of elastic properties have been discussed by kroner 1977 and by nadeau and ferrari 2001 these bounds enclose the effective linear elastic properties of multiphase materials constituted of materials with arbitrary symmetry and of an arbitrary number of phases by using solely the material constants of the single materials nadeau and ferrari showed that these bounds are isotropic tensors and presented an algorithm for the determination of the upper and the lower zeroth order bound it is shown in this paper that a problem arises for the lower bound since the algorithm presented in nadeau and ferrari 2001 results in a negative compression modulus and or shear modulus although the considered stiffness is positive definite a simple analytic example for this undesirable property is given together with a short mathematica code of the algorithm in the present work the definition of the lower bound by nadeau and ferrari is modified thereby assuring its positive definiteness the mathematica code of the corrected algorithm is also given furthermore new bounds for non diagonal components are derived which give information of in principle accessible values for non diagonal stiffness components using the zeroth order bounds of the present work the practical application of zeroth order bounds for local and online material data bases of stiffness tensors is presented in order to accelerate purposes in materials design through efficient materials screening

pesin s entropy formula for c 1 c 1 non uniformly expanding maps we prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials moreover we formulate a c 1 generalization of pesin s entropy formula all ergodic weak srb like measures satisfy pesin s entropy formula for c 1 non uniformly expanding maps we show that for weak expanding maps such that text leb a e x has positive frequency of hyperbolic times then all the necessarily existing ergodic weak srb like measures satisfy pesin s entropy formula and are equilibrium states for the potential psi log det df in particular this holds for any c 1 expanding map and in this case the set of invariant probability measures that satisfy pesin s entropy formula is the weak closed convex hull of the ergodic weak srb like measures

a gibbs energy barrier based computational micro sphere model for the simulation of martensitic phase transformations summary we introduce a material model for the simulation of polycrystalline materials undergoing solid to solid phase transformations as a basis we present a scalar valued phase transformation model where a helmholtz free energy function depending on volumetric and deviatoric strain measures is assigned to each phase the analysis of the related overall gibbs energy density allows for the calculation of energy barriers with these quantities at hand we use a statistical physics based approach to determine the resulting evolution of volume fractions though the model facilitates to take into account an arbitrary number of solid phases of the underlying material we restrict this work to the simulation of phase transformations between an austenitic parent phase and a martensitic tension and compression phase the scalar model is embedded into a computational micro sphere formulation in view of the simulation of three dimensional boundary value problems the final modelling approach necessary for macroscopic simulations is accomplished by a finite element formulation where the local material behaviour at each integration point is governed by the response of the micro sphere model copyright 2014 john wiley sons ltd

a c 1 generic dichotomy for diffeomorphisms weak forms of hyperbolicity or infinitely many sinks or sources we show that for every compact n dimensional manifold n 1 there is a residual subset of diff m of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities either it is contained in the closure of an infinite set of sinks or sources newhouse phenomenon or it presents some weak form of hyperbolicity called dominated splitting this is a generalization of a bidimensional result of mafine ma3 in particular we show that any cl robustly transitive diffeomorphism admits a dominated splitting

estimation of overall properties of random polycrystals with the use of invariant decompositions of hooke s tensor in the paper the theoretical analysis of bounds and self consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented in the study two invariant decompositions of hooke s tensors are used the applied method enables derivation of novel expressions for estimates of the bulk and shear moduli which depend on invariants of local stiffness tensor with use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained