nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

distributions of time averages for weakly chaotic systems the role of infinite invariant density distributions of time averaged observables are investigated using deterministic maps with n indifferent fixed points and n state continuous time random walk processes associated with them in a weakly chaotic phase namely when separation of trajectories is subexponential maps are characterized by an infinite invariant density we find that the infinite density can be used to calculate the distribution of time averages of integrable observables with a formula recently obtained by rebenshtok and barkai as an example we calculate distributions of the average position of the particle and average occupation fractions our work provides the distributional limit theorem for time averages for a wide class of nonintegrable observables with respect to the infinite invariant density in other words it deals with the situation where the darling kac aaronson theorem does not hold

sporadic randomness the transition from the stationary to the nonstationary condition we address the study of sporadic randomness by means of the manneville map we point out that the manneville map is the generator of fluctuations yielding the levy processes and that these processes are currently regarded by some authors as statistical manifestations of a nonextensive form of thermodynamics for this reason we study the sensitivity to initial conditions with the help of a nonextensive form of the lyapunov coefficient the purpose of this research is twofold the former is to assess whether a finite diffusion coefficient might emerge from the nonextensive approach this property at first sight seems to be plausible in the nonstationary case where conventional kolmogorov sinai analysis predicts a vanishing lyapunov coefficient the latter purpose is to confirm or reject conjectures about the nonextensive nature of levy processes we find that the adoption of a nonextensive approach does not serve any predictive purpose it does not even signal a transition from a stationary to a nonstationary regime these conclusions are reached by means of both numerical and analytical calculations that shed light on why the levy processes do not imply any need to depart from the adoption of traditional complexity measures

ergodic theory for markov fibred systems and parabolic rational maps a paraboiic rational map of the riemann sphere admits a non atomic h conformal measure on its julia set where h the hausdorff dimension of the julia set and satisfies 1 2 h 2 with respect to this measure the rational map is conservative exact and there is an equivalent a finite invariant measure finiteness of the measure is characterised central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one a theory of markov fibred systems is developed and parabolic rational maps are considered within this framework

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

aging arcsine law in brownian motion and its generalization classical arcsine law states that the fraction of occupation time on the positive or the negative side in brownian motion does not converge to a constant but converges in distribution to the arcsine distribution here we consider how a preparation of the system affects the arcsine law i e aging of the arcsine law we derive an aging distributional theorem for occupation time statistics in brownian motion where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution furthermore we show that this result can be generalized as an aging distributional limit theorem in renewal processes

power associative nilalgebras of dimension 9 abstract in 19 authors described some properties about commutative power associative nilalgebras of nilindex 5 here we will get new results about the structure of this class of algebras those results will allow us to prove that every commutative power associative algebra of dimension 9 and nilindex 5 over a field of characteristic different from 2 3 and 5 is solvable consequently the famous albert s conjecture 1 problem 1 1 is setted for dimension 9 and characteristic 0 since the case of dimension 9 and nilindex 5 have already been examined in 27 11 14 and 20

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

from h to p efficiently implementing finite and spectral hp element methods to achieve optimal performance for low and high order discretisations the spectral hp element method can be considered as bridging the gap between the traditionally low order finite element method on one side and spectral methods on the other side consequently a major challenge which arises in implementing the spectral hp element methods is to design algorithms that perform efficiently for both low and high order spectral hp discretisations as well as discretisations in the intermediate regime in this paper we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders furthermore based upon this efficient implementation we analyse which spectral hp discretisation which specific combination of mesh size h and polynomial order p minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy we investigate this question for a set of both smooth and non smooth problems

insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures this is the peer reviewed version of the following article ruiz girones e oliver a socorro g cascon j escobar j m montenegro r sarrate j insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures international journal for numerical methods in engineering 2018 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5706 pdf this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

separation of trajectories and its relation to entropy for intermittent systems with a zero lyapunov exponent i t where is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points the mean of is determined by the infinite invariant density using semianalytical arguments we calculate the infinite invariant density for the pomeaumanneville map and with it we obtain excellent agreement between numerical simulation and theory we show that is equal to krengel s entropy and to the complexity calculated by the lempel ziv compression algorithm this generalized pesin s identity shows that and krengel s entropy are the natural generalizations of usual lyapunov exponent and entropy for these systems

simultaneous untangling and smoothing of tetrahedral meshes abstract the quality improvement in mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function the objective function is derived from some quality measure of the local submesh that is the set of tetrahedra connected to the adjustable or free node although these objective functions are suitable to improve the quality of a mesh in which there are non inverted elements they are not when the mesh is tangled this is due to the fact that usual objective functions are not defined on all r 3 and they present several discontinuities and local minima that prevent the use of conventional optimisation procedures otherwise when the mesh is tangled there are local submeshes for which the free node is out of the feasible region or this does not exist in this paper we propose the substitution of objective functions having barriers by modified versions that are defined and regular on all r 3 with these modifications the optimisation process is also directly applicable to meshes with inverted elements making a previous untangling procedure unnecessary this simultaneous procedure allows to reduce the number of iterations for reaching a prescribed quality to illustrate the effectiveness of our approach we present several applications where it can be seen that our results clearly improve those obtained by other authors

robust untangling of curvilinear meshes this paper presents a technique that allows to untangle high order curvilinear meshes the technique makes use of unconstrained optimization where element jacobians are constrained to lie in a prescribed range through moving log barriers the untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded jacobians bounds on jacobians are computed using the results of johnen et al 2012 2013 1 2 the technique is applicable to any kind of polynomial element for surface volume hybrid or boundary layer meshes a series of examples demonstrate both the robustness and the efficiency of the technique the final example involving a time explicit computation shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure

computation of true chaotic orbits using cubic irrationals abstract we introduce a method that enables us to generate long true orbits of discrete time dynamical systems defined by one dimensional piecewise linear fractional maps with integer coefficients the method uses cubic irrationals to represent numbers and involves only integer arithmetic to compute true orbits by applying the method to the bernoulli map and a modified bernoulli map we show that it successfully generates true chaotic and intermittent orbits respectively in contrast with conventional simulation methods we demonstrate through simulations concerning invariant measures and the power spectrum that the statistical properties of the true orbits generated agree well with those of typical orbits of the two maps

role of infinite invariant measure in deterministic subdiffusion statistical properties of the transport coefficient for deterministic subdiffusion are investigated from the viewpoint of infinite ergodic theory we find that the averaged diffusion coefficient is characterized by the infinite invariant measure of the reduced map we also show that when the time difference is much smaller than the total observation time the time averaged mean square displacement depends linearly on the time difference furthermore the diffusion coefficient becomes a random variable and its limit distribution is characterized by the universal law called the mittag leffler distribution

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

a distortion measure to validate and generate curved high order meshes on cad surfaces with independence of parameterization the first author was sponsored by cur from diue of the generalitat de catalunya and the european social fund grants fi dgr 11 and be dgr 12 and by the ferran sunyer i balaguer foundation grant fsb 13 the work of the corresponding author was partially supported by the boeing co us air force office of scientific research european commission through the boeing mit alliance the computational math program marie sklodowska curie actions hipermegaflows individual fellowship grant

geometry representation issues associated with p version finite element computations this paper addresses issues related to accurate geometry representation for p version finite elements on curved three dimensional domains specific options to account for domain geometry information during element level computation are identified accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second order elliptic boundary value problems are given an element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations or used to construct element level geometric approximations of required accuracy smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of poisson s equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence

infinite invariant density in a semi markov process with continuous state variables we report on a fundamental role of a non normalized formal steady state i e an infinite invariant density in a semi markov process where the state is determined by the interevent time of successive renewals the state describes certain observables found in models of anomalous diffusion e g the velocity in the generalized levy walk model and the energy of a particle in the trap model in our model the interevent time distribution follows a fat tailed distribution which makes the state value more likely to be zero because long interevent times imply small state values we find two scaling laws describing the density for the state value which accumulates in the vicinity of zero in the long time limit these laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density moreover we provide two distributional limit theorems for time averaged observables in these nonstationary processes we show that the infinite invariant density plays an important role in determining the distribution of time averages

generalized lyapunov exponent as a unified characterization of dynamical instabilities the lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits a positive lyapunov exponent exponential dynamical instability is a manifestation of chaos here we propose the lyapunov pair which is based on the generalized lyapunov exponent as a unified characterization of nonexponential and exponential dynamical instabilities in one dimensional maps chaos is classified into three different types i e superexponential exponential and subexponential chaos using one dimensional maps we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the lyapunov pair in subexponential chaos we show superweak chaos which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth the scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process which is related to infinite ergodic theory

thin shell analysis from scattered points with maximum entropy approximants this is the accepted version of the following article millan d rosolen a and arroyo m 2011 thin shell analysis from scattered points with maximum entropy approximants int j numer meth engng 85 723 751 doi 10 1002 nme 2992 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 2992 abstract

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

mesh generation in curvilinear domains using high order elements the ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high order algorithms in very complex geometries compact high order algorithms such as spectral element or p type finite element techniques offer the potential of high accuracy if the solution is smooth and a well behaved mapping exists between the local sub domains and a standard region in this paper we address the problem of coupling high order algorithms with unstructured mesh generation and cad representation techniques we present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings the first strategy investigates the influence of an anisometric parametric surface representation on the quality of high order unstructured meshes we show that a straightforward splitting of a standard linear mesh for use in a high order method can lead to the generation of distorted possibly invalid meshes even for simple computational domains such as cubes the second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries in the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher order information about the surface into the mesh generation using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high order discretization of the computational domain copyright 2001 john wiley sons ltd

mid node admissible spaces for quadratic triangular arbitrarily curved 2d finite elements this paper presents the extension of the mid node admissible space mas concept for two dimensional quadratic triangular element 2dqte to the case when more than one edge is curved the mathematical background for the mas for 2dqte with more than one edge curved is developed a metric based on the determination of the jacobian distribution whose evaluation and implementation is aided through the use of the mas is developed the evaluation of the quality of an element based on this metric and procedures to repair an element to produce better results are also presented copyright 2001 john wiley sons ltd

distributional behavior of time averages of non l 1 observables in one dimensional intermittent maps with infinite invariant measures in infinite ergodic theory two distributional limit theorems are well known one is characterized by the mittag leffler distribution for time averages of l 1 m functions i e integrable functions with respect to an infinite invariant measure the other is characterized by the generalized arc sine distribution for time averages of non l 1 m functions here we provide another distributional behavior of time averages of non l 1 m functions in one dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure observation functions considered here are non l 1 m functions which vanish at the indifferent fixed point we call this class of observation functions weak non l 1 m function our main result represents a first step toward a third distributional limit theorem i e a distributional limit theorem for this class of observables in infinite ergodic theory to prove our proposition we propose a stochastic process induced by a renewal process to mimic a birkoff sum of a weak non l 1 m function in the one dimensional intermittent maps

nilpotence of a class of commutative power associative nilalgebras abstract let a be a commutative algebra over a field f of characteristic 2 3 in m gerstenhaber on nilalgebras and linear varieties of nilpotent matrices ii duke math j 27 1960 21 31 m gerstenhaber proved that if a is a nilalgebra of bounded index t and the characteristic of f is zero or greater than 2 t 3 then the right multiplication r x is nilpotent and r x 2 t 3 0 for all x a in this work we prove that this result is also valid for commutative power associative algebras of characteristic t in section 3 we prove that when a is a power associative nilalgebra of dimension 6 then a is nilpotent or a 2 2 0 in section 4 we prove that every power associative nilalgebra a of dimension n and nilindex t n 1 is either nilpotent of index t or isomorphic to the suttles example

algebraic mesh quality metrics quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics the theory based on the jacobian and related matrices provides a means of constructing classifying and evaluating mesh quality metrics the jacobian matrix is factored into geometrically meaningful parts a nodally invariant jacobian matrix can be defined for simplicial elements using a weight matrix derived from the jacobian matrix of an ideal reference element scale and orientation invariant algebraic mesh quality metrics are defined the singular value decomposition is used to study relationships between metrics equivalence of the element condition number and mean ratio metrics is proved the condition number is shown to measure the distance of an element to the set of degenerate elements algebraic measures for skew length ratio shape volume and orientation are defined abstractly with specific examples given two combined metrics shape volume and shape volume orientation are algebraically defined and examples of such metrics are given algebraic mesh quality metrics are extended to nonsimplicial elements a series of numerical tests verifies the theoretical properties of the metrics defined

algorithms for determining invertible two and three dimensional quadratic isoparametric finite element transformations this paper presents an algorithm which determines the invertibility of any planar triangular quadratic isoparametric finite element transformation extensions of the algorithm to three dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10 node tetrahedra and 8 node bricks the mathematical basis for the algorithm focuses on the jacobian as a continuous function defined over a compact set where the jacobian attains a maximum and a minimum value the algorithm then determines whether these values are of opposite sign

generalized arcsine law and stable law in an infinite measure dynamical system limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied it is known that intermittent phenomena such as the rayleigh benard convection and belousov zhabotinsky reaction are described by infinite measure dynamical systems we show that the time average of the observation function which is not the l 1 m function whose average with respect to the invariant measure m is finite converges to the generalized arcsine distribution this result leads to the novel view that the correlation function is intrinsically random and does not decay moreover it is also numerically shown that the time average of the observation function converges to the stable distribution when the observation function has the infinite mean

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

optimization of a regularized distortion measure to generate curved high order unstructured tetrahedral meshes we present a robust method for generating high order nodal tetrahedral curved meshes the approach consists of modifying an initial linear mesh by first introducing high order nodes second displacing the boundary nodes to ensure that they are on the cad surface and third smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high order mesh the smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh this means that whenever possible the resulting mesh preserves the geometrical features of the initial linear mesh such as shape stretching and size we present several examples to illustrate the performance of the proposed algorithm furthermore the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements we consider cases with polynomial approximations up to degree ten large deformations of the curved boundaries concave boundaries and highly stretched boundary layer elements the meshes obtained are suitable for high order finite element analyses

aging generates regular motions in weakly chaotic systems using intermittent maps with infinite invariant measures we investigate the universality of time averaged observables under aging conditions according to aaronson s darling kac theorem in non aged dynamical systems with infinite invariant measures the distribution of the normalized time averages of integrable functions converges to the mittag leffler distribution this well known theorem holds when the start of observations coincides with the start of the dynamical processes introducing a concept of the aging limit where the aging time t a and the total measurement time t goes to infinity while the aging ratio t a t is kept constant we obtain a novel distributional limit theorem of time averaged observables integrable with respect to the infinite invariant density applying the theorem to the lyapunov exponent in intermittent maps we find that regular motions and a weakly chaotic behavior coexist in the aging limit this mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in hamiltonian systems the probability of finding regular motions in non aged processes is zero while in the aging regime it is finite and it increases when system ages

efficiency of high order elements for continuous and discontinuous galerkin methods summary to evaluate the computational performance of high order elements a comparison based on operation count is proposed instead of runtime comparisons more specifically linear versus high order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution continuous and discontinuous galerkin methods are considered in two dimensional and three dimensional domains for simplices and parallelotopes moreover both element wise and global operations arising from different galerkin approaches are studied the operation count estimates show that for implicit solvers high order methods are more efficient than linear ones copyright 2013 john wiley sons ltd

distributional response to biases in deterministic superdiffusion we report on a novel response to biases in deterministic superdiffusion for its reduced map we show using infinite ergodic theory that the time averaged velocity tav is intrinsically random and its distribution obeys the generalized arcsine distribution a distributional limit theorem indicates that the tav response to a bias appears in the distribution which is an example of what we term a distributional response induced by a bias although this response in single trajectories is intrinsically random the ensemble averaged tav response is linear

geometrical validity of curvilinear finite elements in this paper we describe a way to compute accurate bounds on jacobian determinants of curvilinear polynomial finite elements our condition enables to guarantee that an element is geometrically valid i e that its jacobian determinant is strictly positive everywhere in its reference domain it also provides an efficient way to measure the distortion of curvilinear elements the key feature of the method is to expand the jacobian determinant using a polynomial basis built using bezier functions that has both properties of boundedness and positivity numerical results show the sharpness of our estimates

inverse mapping and distortion measures for quadrilaterals with curved boundaries by using the theory of geodesics in differential geometry inverse relations of the mapping for 8 node quadrilaterals with curved boundaries are derived and expressed in terms of the element geodesic co ordinates defined at the local origin the parameters in the expressions of the geodesic co ordinates in terms of the isoparametric co ordinates are suggested to be the distortion measures of the element the mathematical meanings of these parameters can be discussed and explained by using the coefficient vectors in the equations defining the isoparametric co ordinate transformation this method of defining the element distortion measures is consistent and general and can be applied to any other two or three dimensional isoparametric finite elements