computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

equivariant iwasawa theory and non abelian stark type conjectures we discuss three di erent formulations of the equivariant iwasawa main conjecture attached to an extension k k of totally real elds with galois group g where k is a number eld and g is a p adic lie group of dimension 1 for an odd prime p all these formulations are equivalent and hold if iwasawa s invariant vanishes under mild hypotheses we use this to prove non abelian generalizations of brumer s conjecture the brumer stark conjecture and a strong version of the coates sinnott conjecture provided that 0

the attractor basin portrait of a cellular automaton local space time structures such as domains and the intervening dislocations dominate a wide class of cellular automaton ca behavior for such spatially extended dynamics regular domains vicinities and attractors are introduced as organizing principles to identify the discretized analogs of attractors basins and separatrices structures used in classifying dissipative continuous state dynamical systems we describe the attractor basin portrait of nonlinear elementary ca rule 18 whose global dynamics is largely determined by a single regular attracting domain the latter s basin is analyzed in terms of subbasin and portal structures associated with particle annihilation the conclusion is that the computational complexity of such ca is more apparent than real transducer machines are constructed that automatically identify domain and dislocation structures in space time count the number of dislocations in a spatial pattern and implement an isomorphism between rule 18 and rule 90 we use a transducer to trace dislocation trajectories and confirm that in rule 18 isolated dislocation trajectories as well as a dislocation gas agree extremely well with the classical model of annihilating diffusive particles the ca efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space time

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

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

congruences between derivatives of abelian l functions at s 0 let k k be a finite abelian extension of global fields we prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s 0 of derivatives of the dirichlet l functions associated to k k we also show that these congruences provide a universal approach to the refined abelian stark conjectures formulated by inter alia stark gross rubin popescu and tate we thereby obtain the first proofs of amongst other things the rubin stark conjecture and the refined class number formulas of both gross and tate for all extensions k k in which k is either an abelian extension of or is a function field

turbulent pattern bases for cellular automata abstract unpredictable patterns generated by cellular automata ca can be decomposed with respect to a turbulent positive entropy rate pattern basis the resulting filtered patterns uncover significant structural organization in a ca s dynamics and information processing capabilities we illustrate the decompostion technique by analyzing a binary range 2 cellular automaton having two invariant chaotic domains of different complexities and entropies once identified the domains are seen to organize the ca s state space and to dominate its evolution starting from the domains structures we show how to construct a finite state transducer that performs nonlinear spatial filtering such that the resulting space time patterns reveal the domains and the intervening walls and dislocations to show the statistical consequences of domain detection we compare the entropy and complexity densities of each domain with the globally averaged quantities a more graphical comparison uses difference patterns and difference plumes which trace the space time influence of a single site perturbation we also investigate the diversity of walls and particles emanating from the interface between two adjacent domains

chaos and diffusion in deterministic cellular automata abstract it is shown that some of the deterministic one dimensional cellular automata studied recently by wolfram exhibit a kind of spontaneous symmetry breaking the associated kinks bloch walls perform annihilating diffusive walks for infinite lattices and large times in addition close analogies between these automata and strange attractors are exhibited by mapping the automata onto dynamical systems defined by iterations of the unit square

the atomic strain tensor abstract a definition of the local atomic strain increments in three dimensions and an algorithm for computing them is presented an arbitrary arrangement of atoms is tessellated into delaunay tetrahedra identifying interstices and voronoi polyhedra identifying atomic domains the deformation gradient increment tensor for interstitial space is obtained from the displacement increments of the corner atoms of delaunay tetrahedra the atomic site strain increment tensor is then obtained by finding the intersection of the delaunay tetrahedra with the voronoi polyhedra accumulating the individual deformation gradient contributions of the intersected delaunay tetrahedra into the voronoi polyhedra an example application is discussed showing how the atomic strain clarifies the relative local atomic movement for a polymeric glass treated at the atomic level

multivariate granger causality and generalized variance granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables a shortcoming of the standard framework for granger causality is that it only allows for examination of interactions between single univariate variables within a system perhaps conditioned on other variables however interactions do not necessarily take place between single variables but may occur among groups or ensembles of variables in this study we establish a principled framework for granger causality in the context of causal interactions among two or more multivariate sets of variables building on geweke s seminal 1982 work we offer additional justifications for one particular form of multivariate granger causality based on the generalized variances of residual errors taken together our results support a comprehensive and theoretically consistent extension of granger causality to the multivariate case treated individually they highlight several specific advantages of the generalized variance measure which we illustrate using applications in neuroscience as an example we further show how the measure can be used to define partial granger causality in the multivariate context and we also motivate reformulations of causal density and granger autonomy our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems neural and otherwise

tamagawa numbers for motives with noncommutative coefficients ii we provide corroborative evidence for the equivariant tamagawa number conjecture which was formulated in the first part of this article

cohomology of units and l values at zero let klk be a finite galois extension of number fields with galois group g and let s be a finite g stable set of primes of k containing all archimedean primes we denote the g module of s units of k by e es and let as be the kernel of the augmentation map zs z which sends each basis element p c s to 1 we are concerned with invariants of k k which are associated to a g homomorphism p as e inducingq0 as cq0e these invariants were defined by tate ta2 and chinburg chl when s is large i e when s contains all ramified primes of k k and the s class group cl cls of k is trivial there are two of them and each is a function of the complex characters x of g the first aw has a x equal to the tate regulator at x divided by the leading coefficient c x of the taylor expansion at s 0 of the artin l function l s x with the euler factors at primes of s omitted according to stark s conjecture st in tate s form a takes algebraic values which satisfy a o x5 a x

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

on the equivariant tamagawa number conjecture for tate motives let l be a finite abelian extension of and let k be any subfield of l for each integer r with r 0 we prove the equivariant tamagawa number conjecture for the pair h0 spec l r gal l k

multiresolution molecular mechanics atomistic continuum coupling methods combine accurate atomistic methods and efficient continuum methods to simulate the behavior of highly ordered crystalline systems coupled methods utilize the advantages of both approaches to simulate systems at a lower computational cost while retaining the accuracy associated with atomistic methods many concurrent atomistic continuum coupling methods have been proposed in the past however their true computational efficiency has not been demonstrated the present work presents an efficient implementation of a concurrent coupling method called the multiresolution molecular mechanics mmm for serial parallel and adaptive analysis first we present the features of the software implemented along with the associated technologies the scalability of the software implementation is demonstrated and the competing effects of multiscale modeling and parallelization are discussed then the algorithms contributing to the efficiency of the software are presented these include algorithms for eliminating latent ghost atoms from calculations and measurement based dynamic balancing of parallel workload the efficiency improvements made by these algorithms are demonstrated by benchmark tests the efficiency of the software is found to be on par with lammps a state of the art molecular dynamics md simulation code when performing full atomistic simulations speed up of the mmm method is shown to be directly proportional to the reduction of the number of the atoms visited in force computation finally an adaptive mmm analysis on a nanoindentation problem containing over a million atoms is performed yielding an improvement of 6 3 8 5 times in efficiency over the full atomistic md method for the first time the efficiency of a concurrent atomistic continuum coupling method is comprehensively investigated and demonstrated

on galois structure invariants associated to tate motives we establish the equivalence of two definitions of invariants measuring the galois module structure of k groups of rings of integers in number fields due to chinburg et al on the one hand and the authors on the other we also make some remarks concerning the possibility of yet another such definition via lichtenbaum complexes

generalized bridging domain method for coupling finite elements with discrete elements abstract the concurrent coupling of finite elements and discrete elements is an effective dof reduction methodology for reproducing the granular flow phenomenon as discrete element method does in this paper we present a novel coupling strategy named the generalized bridging domain method this method introduces independent functions to weight the material properties of the continuum and those of the discrete element model and then the equilibrium of each new model is guaranteed by imposing compensation forces the compensation force of continuum is determined by force and displacement compatibility requirements of continuum with respect to discrete elements and vice versa utilizing different weighting functions four typical coupling methods the bridging domain method edge to edge coupling separate domain coupling and separate edge coupling are obtained additionally a new integration algorithm with multiple time steps is developed for the separate edge coupling the numerical performance of the separate domain coupling where displacement compatibility condition of continuum and that of discrete elements are individually enforced by the lagrange multiplier method has been investigated in detail results of several numerical examples show that the separate domain coupling outperforms other methods in avoiding spurious reflection and the separate edge coupling is effective enough for coupling finite elements with discrete elements due to the truncation of high frequency waves there exists energy loss but at an acceptable level if the waves can be resolved by the macroscopic finite element model

a bridging domain and strain computation method for coupled atomistic continuum modelling of solids we present a multiscale method that couples atomistic models with continuum mechanics the method is based on an overlapping domain decomposition scheme constraints are imposed by a lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap an efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material an iterative scheme is utilized to optimize the coupled configuration conditions for the regularity of the constrained matrices are determined a method for computing strain in atomistic models and handshake domains is formulated based on a moving least square approximation which includes both extensional and angle bending terms it is shown that this method exactly computes the linear strain field applications to the fracture of defected single layer atomic sheets and nanotubes are given copyright 2006 john wiley sons ltd

an embedded statistical method for coupling molecular dynamics and finite element analyses the coupling of molecular dynamics md simulations with finite element methods fem yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales the theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks various specialized methods have been developed to solve particular classes of problems many of these methods link the kinematics of individual md atoms with finite element fe nodes at their common interface necessarily requiring that the fe mesh be refined to atomic resolution some of these coupling approaches also require simulations to be carried out at 0 k and restrict modelling to two dimensional material domains due to difficulties in simulating full three dimensional material processes in the present work a new approach to md fem coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain the method replaces a direct linkage of individual md atoms and fe nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each fe node in an interface region the fem and md computational systems are effectively independent and communicate only through an iterative update of their boundary conditions thus the method lends itself for use with any fem or md code with the use of statistical averages of the atomistic quantities to couple the two computational schemes the developed approach is referred to as an embedded statistical coupling method escm escm provides an enhanced coupling methodology that is inherently applicable to three dimensional domains avoids discretization of the continuum model to atomic scale resolution and permits finite temperature states to be applied published in 2009 by john wiley sons ltd

enrichment and coupling of the finite element and meshless methods a mixed hierarchical approximation based on finite elements and meshless methods is presented two cases are considered the first one couples regions where finite elements or meshless methods are used to interpolate continuity and consistency is preserved the second one enriches a finite element mesh with particles thus there is no need to remesh in adaptive refinement processes in both cases the same formulation is used convergence is studied and examples are shown copyright 2000 john wiley sons ltd

inferring statistical complexity in the dripping faucet experiment abstract we have used topological e machine reconstruction to calculate statistical complexity associated to period five orbits in the dripping faucet experiment basic structures related to the periodic movement and to peripheral tracks in the corresponding directed graphs were identified

local information transfer as a spatiotemporal filter for complex systems we present a measure of local information transfer derived from an existing averaged information theoretical measure namely transfer entropy local transfer entropy is used to produce profiles of the information transfer into each spatiotemporal point in a complex system these spatiotemporal profiles are useful not only as an analytical tool but also allow explicit investigation of different parameter settings and forms of the transfer entropy metric itself as an example local transfer entropy is applied to cellular automata where it is demonstrated to be a useful method of filtering for coherent structure more importantly local transfer entropy provides the first quantitative evidence for the long held conjecture that the emergent traveling coherent structures known as particles both gliders and domain walls which have analogs in many physical processes are the dominant information transfer agents in cellular automata

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

congruences between derivatives of geometric l functions we prove a natural refinement of a theorem of lichtenbaum describing the leading terms of zeta functions of curves over finite fields in terms of weil etale cohomology we then use this result to prove the validity of chinburg s 3 conjecture for all abelian extensions of global function fields to prove natural refinements and generalisations of the refined stark conjectures formulated by amongst others gross tate rubin and popescu to prove a variety of explicit restrictions on the galois module structure of unit groups and divisor class groups and to describe explicitly the fitting ideals of certain weil etale cohomology groups in an appendix coauthored with k f lai and k s tan we also show that the main conjectures of geometric iwasawa theory can be proved without using either crystalline cohomology or drinfeld modules

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

on the equivariant tamagawa number conjecture for tate motives and unconditional annihilation results dfg collaborative research center 701 spectral structures and topological methods in mathematics

a coupled molecular dynamics and extended finite element method for dynamic crack propagation a multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain after recalling the basic equations of molecular dynamics and continuum mechanics the discretization is discussed for the continuum subdomain where the partition of unity property of finite element shape functions is used since in this fashion the crack in the wake of its tip is naturally modelled as a traction free discontinuity next the zonal coupling method between the atomistic and continuum models is recapitulated finally it is discussed how the stress has been computed in the atomic subdomain and a two dimensional computation is presented of dynamic fracture using the coupled model the result shows multiple branching which is reminiscent of recent results from simulations on dynamic fracture using cohesive zone models

on the rubin stark conjecture for a special class of cm extensions of totally real number fields we use greither s recent results on brumer s conjecture to prove rubin s integral version of stark s conjecture up to a power of 2 for an infinite class of cm extensions of totally real number fields called nice extensions under the assumption that a certain iwasawa invariant vanishes as a consequence and under the same assumption we show that the brumer stark conjecture is true for nice extensions up to a power of 2

a feti dp method for the parallel iterative solution of indefinite and complex valued solid and shell vibration problems the dual primal finite element tearing and interconnecting feti dp domain decomposition method ddm is extended to address the iterative solution of a class of indefinite problems of the form k 2 m u f and a class of complex problems of the form k 2 m i d u f where k m and d are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems i is the imaginary complex number and a is a real positive number a key component of this extension is a new coarse problem based on the free space solutions of navier s equations of motion these solutions are waves and therefore the resulting ddm is reminiscent of the feti h method for this reason it is named here the feti dph method for a practically large a range feti dph is shown numerically to be scalable with respect to all of the problem size substructure size and number of substructures the cpu performance of this iterative solver is illustrated on a 40 processor computing system with the parallel solution for various a ranges of several large scale indefinite or complex valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems

recursion theory on the reals and continuous time computation abstract we define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers corresponding to a conceptual analog computer that operates in continuous time this class turns out to be surprisingly large and includes many functions which are uncomputable in the traditional sense we stratify this class of functions into a hierarchy according to the number of uses of the zero finding operator at the lowest level are continuous functions that are differentially algebraic and computable by shannon s general purpose analog computer at higher levels are increasingly discontinuous and complex functions we relate this hierarchy to the arithmetical and analytical hierarchies of classical recursion theory

non reflecting boundary conditions for atomistic continuum and coupled atomistic continuum simulations summary we present a method to numerically calculate a non reflecting boundary condition which is applicable to atomistic continuum and coupled multiscale atomistic continuum simulations the method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements and utilizes standard laplace and fourier transform techniques to eliminate the unnecessary degrees of freedom the eliminated degrees of freedom are accounted for in a time history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials or regular finite element meshes using an automated numerical procedure the new theoretical developments presented in this work allow the application of the method to non nearest neighbour atomic interactions it is also demonstrated that the identical procedure can be used for finite element and mesh free simulations we illustrate the effectiveness of the method on a one dimensional model problem and calculate the time history kernel for fcc gold using the embedded atom method eam copyright 2005 john wiley sons ltd

on special elements in higher algebraic k theory and the lichtenbaum gross conjecture abstract we conjecture the existence of special elements in odd degree higher algebraic k groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of artin l functions of finite dimensional irreducible complex representations we prove this conjecture for an important family of examples and also provide other evidence both theoretical and numerical in its support

on the main conjecture of equivariant iwasawa theory we prove the main conjecture of equivariant iwasawa theory when 0 a word of caution theorems 1 2 3 4 5 here are theorems 1 2 5 3 4 respectively in jams the main conjecture of equivariant iwasawa theory concerns the situation where l is a xed odd prime number k k is a galois extension of totally real number elds with k q and k k nite where k k is the cyclotomic zl extension we set g g k k and k g k k s a xed nite set which will normally be suppressed in the notation of primes of k containing all primes which ramify in k and all archimedean primes and m is the maximal abelian l extension of k unrami ed outside s we set x g m k it asserts that a canonical re nement f fs of the iwasawa module x is determined by the iwasawa l function lk k lk k s of k k the data f and lk k and hom rl g qc k below have been de ned in rw2 p 562 563 1 p 558 we will not repeat the de nitions now only in 1 but brie y explain how lk k should determine f denote by g qg the completed group ring zl g of g over zl and its total ring of fractions respectively the localization sequence of k theory k1 g k1 qg k0t g k0 qg has f in k0t g the reduced norms of the wedderburn components of the semi simple algebra qg induce the map det k1 qg hom rl g qc k the iwasawa l function lk k of k k is derived from the s truncated l adic artin l functions and belongs to the above group hom the equivariant main conjecture asserts that there is a unique element k1 qg satisfying det lk k and moreover that this has f in other words is the nonabelian generalization to k k of the pseudomeasure of an abelian eld extension we acknowledge nancial support provided by dfg and nserc 1where f is denoted f

conservation properties of the bridging domain method for coupled molecular continuum dynamics an overlaid domain decomposition method called the bridging domain method for the coupled simulation of molecular continuum systems is analyzed in this method compatibility between the atomistic and the continuum subdomains is enforced by lagrange multipliers two forms are considered 1 a consistent constraint form which employs the exact non diagonal lagrange multiplier equations and thus enforces compatibility exactly but is computationally demanding and 2 a diagonalized constraint form which is computationally more efficient it is shown that the consistent constraint form conserves linear momentum angular momentum and energy the diagonalized constraint form dissipates energy the diagonalized form is shown to be more effective in suppressing spurious reflections at the interface copyright 2008 john wiley sons ltd

upper bound on the products of particle interactions in cellular automata abstract particle like objects are observed to propagate and interact in many spatially extended dynamical systems for one of the simplest classes of such systems one dimensional cellular automata we establish a rigorous upper bound on the number of distinct products that these interactions can generate the upper bound is controlled by the structural complexity of the interacting particles a quantity which is defined here and which measures the amount of spatio temporal information that a particle stores along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata thereby elucidating why that approach is of general utility the upper bound is tested against several relatively complex domain particle cellular automata and found to be tight

on the cyclotomic main conjecture for the prime 2 we complete the proof of the equivariant tamagawa number conjecture for tate motives over absolutely abelian fields by proving a refined cyclotomic main conjecture at the prime 2

integrality of stickelberger elements and the equivariant tamagawa number conjecture let l k be a nite galois cm extension of number elds with galois group g in an earlier paper the author has de ned a module sku l k over the center of the group ring zg which coincides with the sinnott kurihara ideal if g is abelian and in particular contains many stickelberger elements it was shown that a certain conjecture on the integrality of sku l k implies the minus part of the equivariant tamagawa number conjecture at an odd prime p for an in nite class of non abelian galois cm extensions of number elds which are at most tamely rami ed above p provided that iwasawa s invariant vanishes here we prove a relevant part of this integrality conjecture which enables us to deduce the equivariant tamagawa number conjecture from the vanishing of for the same class of extensions

a generalized space time mathematical homogenization theory for bridging atomistic and continuum scales a generalized space time mathematical homogenization theory which constructs an equivalent continuum description directly from molecular dynamics md equations is developed the noteworthy theoretical findings of this work are i the coarse scale continuum equations pdes obtained from the homogenization of md equations are identical to those obtained from the classical homogenization of the fine scale continuum ii the lower order effective continuum properties such as the fourth order elasticity tensor have similar characteristics to those resulting from the homogenization of fine scale continuum iii the higher order effective properties such as polarization and dispersion tensors substantially differ from those of continuum due to the discreteness effect both the heterogeneous continuum and discrete media possess the size effect but only atomistic medium has the discreteness effect some demonstrative numerical examples are presented to verify the formulation copyright 2005 john wiley sons ltd

bridging domain methods for coupled atomistic continuum models with l2 or h1 couplings a bridging domain method for coupled atomistic continuum models is proposed that enables to compare various coupling terms the approach does not require the finite element mesh to match the lattice spacing of the atomic model it is based on an overlapping domain decomposition method that makes use of lagrange multipliers and weight functions in the coupling zone in order to distribute the energy between the two competing models two couplings are investigated the l 2 coupling enforces the continuity of displacements between the two models directly the h 1 coupling involves the definition of a strain measure for this purpose a moving least square interpolant of the atomic displacement is defined the choice of the weight functions is studied patch tests and a graphene sheet with a crack are studied it is shown that both continuous and discontinuous weight functions can be used with the h 1 coupling whereas the l 2 coupling requires continuous weight functions for the examples developed herein the l 2 coupling produces less error in the zone of interest the flexibility of the h 1 coupling with constant weight function may be beneficial but the results may be affected depending on the topology of the bridging zone

automatic filters for the detection of coherent structure in spatiotemporal systems most current methods for identifying coherent structures in spatially extended systems rely on prior information about the form which those structures take here we present two approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures one local sensitivity filtering is a modification of the local lyapunov exponent approach suitable to cellular automata and other discrete spatial systems the other local statistical complexity filtering calculates the amount of information needed for optimal prediction of the system s behavior in the vicinity of a given point by examining the changing spatiotemporal distributions of these quantities we can find the coherent structures in a variety of pattern forming cellular automata without needing to guess or postulate the form of that structure we apply both filters to elementary and cyclical cellular automata eca and cca and find that they readily identify particles domains and other more complicated structures we compare the results from eca with earlier ones based upon the theory of formal languages and the results from cca with a more traditional approach based on an order parameter and free energy while sensitivity and statistical complexity are equally adept at uncovering structure they are based on different system properties dynamical and probabilistic respectively and provide complementary information

three state potts model in combination with the rock scissors paper game we study a three state potts model extended by allowing cyclic dominance between the states as exemplified in the rock scissors paper game monte carlo simulations are performed on a square lattice while varying the temperature and the strength of cyclic dominance it is shown that the critical phase transition from the disordered state to the ordered one is destroyed by the cyclic dominance which yields a self organizing pattern even at low temperatures the differences and similarities are discussed between the present model and half filled driven lattice gases with repulsive interaction

defect particle kinematics in one dimensional cellular automata let a z be the cantor space of bi infinite sequences in a finite alphabet a and let s be the shift map on a z a cellular automaton is a continuous s commuting self map f of a z and a f invariant subshift is a closed f s invariant subset s a z suppose a a z is s admissible everywhere except for some small region we call a defect it has been empirically observed that such defects persist under iteration of f and often propagate like particles we characterize the motion of these particles and show that it falls into several regimes ranging from simple deterministic motion to generalized random walks to complex motion emulating turing machines or pushdown automata one consequence is that some questions about defect behaviour are formally undecidable

on main conjectures in non commutative iwasawa theory and related conjectures the main conjecture of non commutative iwasawa theory of coates fukaya kato sujatha and venjakob is shown to be equivalent modulo an appropriate torsion hypothesis to a family of main conjectures over cyclotomic zp extensions together with suitable compatibility relations between the associated abelian p adic l functions this result combines with techniques of ritter and weiss and of kakde to give a concrete strategy for deducing main conjectures for non abelian compact p adic lie extensions of arbitrarily large rank from main conjectures in the classical sense as an explicit example of this approach we combine it with a recent result of ritter and weiss to prove modulo an appropriate vanishing hypothesis on invariants the main conjecture of non commutative iwasawa theory for tate motives over compact p adic lie extensions of totally real fields of arbitrarily large rank we then deduce several interesting consequences of this result concerning a range of related conjectures including the equivariant tamagawa number conjecture 1991 mathematics subject classification primary 11s40 secondary 11r29 20c05

computational mechanics of cellular automata an example abstract we illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the space time behavior of elementary one dimensional cellular automaton rule 54 the dominant regular domain of the cellular automation is identified and a domain filter is constructed to locate and classify defects in the domain the primary particles are identified and a range of interparticle interactions is studied the deterministic equation of motion of the filtered space time behavior is derived filters of increasing sophistication are constructed for the efficient gathering of particle statistics and for the identification of higher level defects particle interactions and secondary domains we define the emergence time at which the space time behavior condenses into configurations consisting only of domains particles and particle interactions taken together these techniques serve as the basis for the investigation of pattern evolution and self organization in this representative system

set of measures to analyze the dynamics of nonequilibrium structures we present a class of statistical measures that can be used to quantify nonequilibrium surface growth they are used to deduce information about spatiotemporal dynamics of model systems for spinodal decomposition and surface deposition pattern growth in the cahn hilliard equation used to model spinodal decomposition are shown to exhibit three distinct stages two models of surface growth namely the continuous kardar parisi zhang model and the discrete restricted solid on solid model are shown to have different saturation exponents

generalized mathematical homogenization from theory to practice a generalized mathematical homogenization gmh theory for coupling atomistic and continuum descriptions at finite temperature has been extended to account for many body potentials gmh gives rise to the constitutive law free coarse scale equations where coupled continuum thermo mechanical equations are directly derived from molecular dynamics equations an integrated computational approach which seamlessly integrates abaqus for coarse scale finite element method computations with a molecular dynamics code has been developed the accuracy and efficiency of the method has been studied for crystal silicon structures including failure predictions of nanowires

non commutative fitting invariants and annihilation of class groups abstract one can associate to each finitely presented module m over a commutative ring r an r ideal fitt r m which is called the zeroth fitting ideal of m over r and which is an important natural invariant of m we generalize this notion to o orders in separable algebras where o is a complete commutative noetherian local ring as an application we construct annihilators of class groups assuming the validity of the equivariant tamagawa number conjecture for a certain motive attached to a galois cm extension of number fields

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

on the non abelian brumer stark conjecture and the equivariant iwasawa main conjecture we show that for an odd prime p the p primary parts of refinements of the imprimitive non abelian brumer and brumer stark conjectures are implied by the equivariant iwasawa main conjecture eimc for totally real fields crucially this result does not depend on the vanishing of the relevant iwasawa mu invariant in combination with the authors previous work on the eimc this leads to unconditional proofs of the non abelian brumer and brumer stark conjectures in many new cases

on derivatives of artin l series we present a universal theory of refined stark conjectures we give evidence in support of these explicit and very general conjectures including a full proof in several important cases and explain the connection to previous conjectures of bloch and kato of lichtenbaum and of serre and tate we also deduce a wide range of unconditional consequences of these results concerning the annihilation as galois modules of ideal class groups by explicit elements constructed from the values of higher order derivatives of non abelian artin l series

comparison of several staggered atomistic to continuum concurrent coupling strategies abstract in this contribution several staggered schemes used to couple continuum mechanics cm and molecular mechanics mm are proposed the described approaches are based on the atomistic to continuum correspondence obtained by spatial averaging in the spirit of irving and kirkwood and noll similarities between this and other concurrent coupling schemes are indicated thus providing a broad overview of different approaches in the field the schemes considered here are decomposed into the surface type displacement or traction boundary conditions and the volume type the latter restricts the continuum displacement field and possibly its gradient in some sense to the atomistic discrete displacements using lagrange multipliers a large strain cm formulation incorporating lagrange multipliers and a strategy to solve the resulting coupled linear system using an iterative solver is presented finally the described coupling methods are numerically examined using two examples uniaxial deformation and a plate with a hole relaxed under surface tension accuracy and convergence rates of each method are reported it was found that the displacement surface coupling scheme and the lagrangian volume scheme based on either discrete displacements or the h 1 norm derived from continuous displacement fields provide the best performance

concurrently coupled atomistic and xfem models for dislocations and cracks a method for the modeling of dislocations and cracks by atomistic continuum models is described the methodology combines the extended finite element method with the bridging domain method bdm the former is used to model crack surfaces and slip planes in the continuum whereas the bdm is used to link the atomistic models with the continuum the bdm is an overlapping domain decomposition method in which the atomistic and continuum energies are blended so that their contributions decay to their boundaries on the overlapping subdomain compatibility between the continua and atomistic domains is enforced by a continuous lagrange multiplier field the methodology allows for simulations with atomistic resolution near crack fronts and dislocation cores while retaining a continuum model in the remaining part of the domain and so a large reduction in the number of atoms is possible it is applied to the modeling of cracks and dislocations in graphene sheets energies and energy distributions compare very well with direct numerical simulations by strictly atomistic models copyright 2008 john wiley sons ltd

complexity of two dimensional patterns in dynamical systems such as cellular automata and iterated maps it is often useful to look at a language or set of symbol sequences produced by the system there are well established classification schemes such as the chomsky hierarchy with which we can measure the complexity of these sets of sequences and thus the complexity of the systems which produce them in this paper we look at the first few levels of a hierarchy of complexity for two or more dimensional patterns we show that several definitions of regular language or local rule that are equivalent in d 1 lead to distinct classes in d 2 we explore the closure properties and computational complexity of these classes including undecidability and l nl and np completeness results we apply these classes to cellular automata in particular to their sets of fixed and periodic points finite time images and limit sets we show that it is undecidable whether a ca in d 2 has a periodic point of a given period and that certain local lattice languages are not finite time images or limit sets of any ca we also show that the entropy of a d dimensional ca s finite time image cannot decrease faster than t d unless it maps every initial condition to a single homogeneous state

on the equivariant tamagawa number conjecture in tame cm extensions let l k be a finite galois cm extension with galois group g the equivariant tamagawa number conjecture etnc for the pair h 0 rm spec l 0 mathbb z g naturally decomposes into p parts where p runs over all rational primes if p is odd these p parts in turn decompose into a plus and a minus part let l k be tame above p we show that a certain ray class group of l defines an element in k 0 mathbb z p g mathbb q p which is determined by a corresponding stickelberger element if and only if the minus part of the etnc at p holds for this we use the lifted root number conjecture for small sets of places which is equivalent to the etnc in the number field case for abelian g we show that the minus part of the etnc at p implies the strong brumer stark conjecture at p we prove the minus part of the etnc at p for almost all primes p

equivariant epsilon constants discriminants and etale cohomology let l k be a finite galois extension of number fields we formulate and study a conjectural equality between an element of the relative algebraic k group k 0 mathbb z mathrm gal l k mathbb r which is constructed from the equivariant artin epsilon constant of l k and a sum of structural invariants associated to l k the precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the tamagawa number conjecture of bloch and kato as formulated by flach and the second named author should be compatible with the functional equation of the associated l function we show that more concretely our conjecture also suggests a completely systematic refinement of the central approach and results of classical galois module theory in particular the evidence for our conjecture that we present here already strongly refines many of the main results of galois module theory

the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

ghost forces and spurious effects in atomic to continuum coupling methods by the arlequin approach the main objective of this work is to identify spurious effects and propose a corrective method when coupling a particle model involving long range interaction potentials with a first gradient homogenized model using the arlequin framework the most significant spurious effects are generally created by the so called ghost forces that arise in coupling methods based on the minimization of a global energy functional they depend as well on the coupling formulation itself on the notion of representative volume element and on the discretization of the continuum model the proposed corrective technique is based on post processing of the approximate solution by introducing dead forces that can be systematically evaluated and consistently inserted within the arlequin formulation efficiency of the corrective procedure is demonstrated on 1d and 2d examples copyright 2010 john wiley sons ltd

embedding computation in one dimensional automata by phase coding solitons it is shown that some kind of meaningful computation can be embedded in very simple microscopically homogeneous one dimensional automata and in particular filter automata with a parity next state rule a systematic procedure is given for generating moving periodic structures particles these particles exhibit soliton like properties that is they often pass through one another with phase shifts ways to encode information in the phase of these particles are discussed the search for useful logical operations is reduced to a search for paths in certain graphs as a demonstration of principle the details of implementing a carry ripple adder are given

hybrid iwasawa algebras and the equivariant iwasawa main conjecture let p be an odd prime we give an unconditional proof of the equivariant iwasawa main conjecture for totally real fields for an infinite class of one dimensional non abelian p adic lie extensions crucially this result does not depend on the vanishing of the relevant iwasawa mu invariant