hamiltonian formulation of surfaces with constant gaussian curvature dirac s method for constrained hamiltonian systems is used to describe surfaces of constant gaussian curvature a geometrical free energy for which these surfaces are equilibrium states is introduced and interpreted as an action an equilibrium surface can then be generated by the evolution of a closed space curve since the underlying action depends on second derivatives the velocity of the curve and its conjugate momentum must be included in the set of phase space variables furthermore the action is linear in the acceleration of the curve and possesses a local symmetry reparametrization invariance which implies primary constraints in the canonical formalism these constraints are incorporated into the hamiltonian through lagrange multiplier functions that are identified as the components of the acceleration of the curve the formulation leads to four first order partial differential equations one for each canonical variable with the appropriate choice of parametrization only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve to illustrate the formalism several evolutions of pseudospherical surfaces are discussed

statistical properties of locally free groups with applications to braid groups and growth of random heaps the main statistical characteristics of locally free groups the growth the drift and the entropy are considered and relations between them are established our results assert that i the statistical properties of random walks markov chains on locally free and braid groups are not the same as the uniform statistics on these groups and ii the stabilization of the statistical characteristics exists when the number of generators of the group grows

a higher order lagrangian discontinuous galerkin hydrodynamic method for solid dynamics abstract we present a new multidimensional high order lagrangian discontinuous galerkin dg hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher order elements we also present new one dimensional test problems that have an analytic solution corresponding to a hyperelastic plastic wave a modal dg approach is used to evolve fields relevant to conservation laws these fields are approximated high order taylor series polynomials the stress fields are represented using nodal quantities the constitutive models used to calculate the deviatoric stress are either a hypoelastic plastic infinitesimal strain hyperelastic plastic or finite strain hyperelastic plastic model these constitutive models require new methods for calculating high order polynomials for the velocity gradient and deformation gradient in an element the plasticity associated with the strength model is determined using a radial return method with a j 2 yield criterion and perfect plasticity the temporal evolution of the governing equations is achieved with the total variation diminishing runge kutta tvd rk time integration method a diverse suite of 1d and 2d test problems are calculated the new 1d piston test problems which have analytic solutions for each elastic plastic model are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new lagrangian dg method 2d test problems are calculated to demonstrate the stability and robustness of the new lagrangian dg method on multidimensional problems with high order elements which have faces that can bend

a free lagrange augmented godunov method for the simulation of elastic plastic solids a lagrangian finite volume godunov scheme is extended to simulate two dimensional solids in planar geometry the scheme employs an elastic perfectly plastic material model implemented using the method of radial return and either the stiffened gas or osborne equation of state to describe the material the problem of mesh entanglement common to conventional two dimensional lagrangian schemes is avoided by utilising the free lagrange method the lagrangian formulation enables features convecting at the local velocity such as material interfaces to be resolved with minimal numerical dissipation the governing equations are split into separate subproblems and solved sequentially in time using a time operator split procedure local riemann problems are solved using a two shock approximate riemann solver and piecewise linear data reconstruction is employed using a muscl based approach to improve spatial accuracy to illustrate the effectiveness of the technique numerical simulations are presented and compared with results from commercial fixed connectivity lagrangian and smooth particle hydrodynamics solvers autodyn 2d the simulations comprise the low velocity impact of an aluminium projectile on a semi infinite target the collapse of a thick walled beryllium cylinder and the high velocity impact of cylindrical aluminium and steel projectiles on a thin aluminium target the analytical solution for the collapse of a thick walled cylinder is also presented for comparison

distortion program size complexity with respect to a fidelity criterion and rate distortion function a novel concept of distortion kolmogorov chaitin complexity is proposed bearing analogy to ordinary kolmogorov chaitin complexity that represents the information content of a single message distortion kolmogorov chaitin complexity can be regarded as the amount of information about a single message that must be conveyed in order to reproduce it with a bounded distortion this explanation is justified by the properties of distortion kolmogorov chaitin complexity and the equivalences between distortion kolmogorov chaitin complexity and the rate distortion function which are analogous to those between ordinary kolmogorov chaitin complexity and the shannon entropy some implications for universal almost sure data compression are also discussed

shape selection in non euclidean plates abstract we investigate isometric immersions of disks with constant negative curvature into r 3 and the minimizers for the bending energy i e the l 2 norm of the principal curvatures over the class of w 2 2 isometric immersions we show the existence of smooth immersions of arbitrarily large geodesic balls in h 2 into r 3 in elucidating the connection between these immersions and the non existence singularity results of hilbert and amsler we obtain a lower bound for the l norm of the principal curvatures for such smooth isometric immersions we also construct piecewise smooth isometric immersions that have a periodic profile are globally w 2 2 and numerically have lower bending energy than their smooth counterparts the number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disk we discuss the implications of our results on recent experiments on the mechanics of non euclidean plates

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

self avoiding walks and polygons on non euclidean lattices self avoiding walks and polygons on four non euclidean lattices are studied by the method of series expansions it appears that the polygons and walks have different critical points from each other and that both are in a different universality class to their cartesian lattice counterparts an exact solution is given for one lattice

random operator approach for word enumeration in braid groups we investigate analytically the problem of enumeration of nonequivalent primitive words in the locally free and braid groups for by analysing the random word statistics and target spaces on these groups we develop a symbolic dynamics method for exact word enumeration in locally free groups and give arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations we touch briefly the connection of these problems with conventional random operator theory localization phenomena and statistics of systems with quenched disorder we also discuss the relation of particular problems of random operator theory to the theory of modular functions

approximating rate distortion graphs of individual data experiments in lossy compression and denoising classical rate distortion theory requires specifying a source distribution instead we analyze rate distortion properties of individual objects using the recently developed algorithmic rate distortion theory the latter is based on the noncomputable notion of kolmogorov complexity to apply the theory we approximate the kolmogorov complexity by standard data compression techniques and perform a number of experiments with lossy compression and denoising of objects from different domains we also introduce a natural generalization to lossy compression with side information to maintain full generality we need to address a difficult searching problem while our solutions are therefore not time efficient we do observe good denoising and compression performance

dimension 1 sequences are close to randoms abstract we show that a sequence has effective hausdorff dimension 1 if and only if it is coarsely similar to a martin lof random sequence more generally a sequence has effective dimension s if and only if it is coarsely similar to a weakly s random sequence further for any s t every sequence of effective dimension s can be changed on density at most h 1 t h 1 s of its bits to produce a sequence of effective dimension t and this bound is optimal

ising models on the lobachevsky plane we consider the ising model on a lattice which is the orbit of a discrete cocompact group acting on the hyperbolic plane for large values of the inverse temperature we construct an uncountable number of mutually singular gibbs states

on the gaussianity of kolmogorov complexity of mixing sequences it has been proved that for all stationary and ergodic processes the average kolmogorov complexity of the first n observations converges almost surely to its shannon s conditional entropy this paper studies the convergence rate of this asymptotic result in particular we show that if the process satisfies certain mixing conditions then a central limit theorem will be respected furthermore we show that under slightly stronger mixing conditions one may obtain non asymptotic concentration bounds for the kolmogorov complexity

rate distortion and denoising of individual data using kolmogorov complexity we examine the structure of families of distortion balls from the perspective of kolmogorov complexity special attention is paid to the canonical rate distortion function of a source word which returns the minimal kolmogorov complexity of all distortion balls containing that word subject to a bound on their cardinality this canonical rate distortion function is related to the more standard algorithmic rate distortion function for the given distortion measure examples are given of list distortion hamming distortion and euclidean distortion the algorithmic rate distortion function can behave differently from shannon s rate distortion function to this end we show that the canonical rate distortion function can and does assume a wide class of shapes unlike shannon s we relate low algorithmic mutual information to low kolmogorov complexity and consequently suggest that certain aspects of the mutual information formulation of shannon s rate distortion function behave differently than would an analogous formulation using algorithmic mutual information we explore the notion that low kolmogorov complexity distortion balls containing a given word capture the interesting properties of that word which is hard to formalize in shannon s theory and this suggests an approach to denoising

random hierarchical matrices spectral properties and relation to polymers on disordered trees we study the statistical and dynamic properties of the systems characterized by an ultrametric space of states and translationary non invariant symmetric transition matrices of the parisi type subjected to locally constant randomization using the explicit expression for eigenvalues of such matrices we compute the spectral density for the gaussian distribution of matrix elements we also compute the averaged survival probability sp having sense of the probability to find a system in the initial state by time t using the similarity between the averaged sp for locally constant randomized parisi matrices and the partition function of directed polymers on disordered trees we show that for times t tcr where tcr is some critical time a lacunary structure of the ultrametric space occurs with the probability 1 const t this means that the escape from some bounded areas of the ultrametric space of states is locked and the kinetics is confined in these areas for infinitely long time the explosion of the interest to the statistical properties of ensembles of random matrices around 1950s has been motivated in first turn by a series of physical problems in nuclear physics later the physical applications of the random matrix theory rmt were extended to mesoscopics and statistical physics one can mention recent works 1 2 which deal with physical applications of the eigenvalue statistics of ensembles of random matrices the typical problems of rmt concern the evaluation of the averaged spectral density as well as distributions of level spacings under the supposition that all matrix elements are independent randomly distributed entries which take values in some specific ensemble 3 despite the standard rmt can be applied to a wide range of physical phenomena rmt in its basic form does not cover a particular class of hierarchical complex systems important for glasses real networks and proteins the description of such complex systems deals with the concept of hierarchical organization of energy landscapes 4 5 a complex system is assumed to have a large number of metastable states corresponding to local minima in the potential energy landscape with respect to the transition rates the minima are suggested to be clustered in hierarchically nested basins of minima i e larger basins consist of smaller basins each of these consists of even smaller ones etc the basins of local energy minima are separated by a hierarchically arranged set of barriers large basins are separated by high barriers and smaller basins within each larger one are separated by lower barriers since the transitions between the basins are determined by the passages over the highest barriers separating them the transitions between any two local minima obey the strong triangle inequality for ultrametric distances hence the hierarchy of basins possesses the ultrametric geometry which has the natural visualization by the uniform p adic i e p 1 branching cayley tree

an unfinished tale of nonlinear pdes do solutions of 3d incompressible euler equations blow up in finite time abstract the solutions of time dependent pdes may show a bewildering variety of behaviors the example of the kuramoto sivashinsky ks equation illustrates how simple an equation can be with extremely complex solutions the equations for incompressible fluids are also notorious for the extremely complex behavior of their solutions but in the latter case the situation is in a sense far less satisfactory than for the ks equations since one still ignores if their solution remains smooth as time goes on with smooth initial data in 3d this leaves in a quite uncertain state all theories of 3d turbulence which may be either a way of coping with our ignorance of the true dynamics of real fluids or a way of analyzing the behavior of the smooth solutions of the navier stokes equations below in this homage to professor kuramoto we present some remarks on this question limited to the 3d euler equations without viscosity the existence of solutions of the 3d incompressible euler equations for fluids blowing up in finite time remains an open question if the initial energy is finite the consensus seems to be that the singularity if there is any is local in space and time although it is difficult to find a straight statement supporting such a claim in the literature based on the properties of self similar equations we claim that this is not possible at least if one assumes such a self similar blow up including some refinements in the time dependence that extend the class of self similar solutions actually the possible blow up solutions are very much constrained by the conservation of energy and of circulation based on scaling transformations we argue that the singularity cannot be a simple self similar blow up but requires some sort of oscillations on a logarithmic time scale the space dependence cannot be simple either the various constraints lead to a self similar collapse towards a line at least for an axisymmetric flow with swirl with a wavelength along the axis decreasing by steps our final result is a set of explicit equations such that if they have a smooth solution satisfying certain conditions the original euler equations have a finite time singularity

discretization of hyperelasticity on unstructured mesh with a cell centered lagrangian scheme we present a new cell centered lagrangian scheme on unstructured mesh for hyperelasticity it is based on the recently proposed glace scheme 11 for compressible gas dynamics we show how to use the multiplicative decomposition of the gradient of deformation and the entropy property to derive the new scheme we also prove the compatibility of this discretization with usual calculations of mass our motivation is to use hyperelasticity models for the study of finite plasticity which is an extension of hypoelasticity to finite deformations hyperelasticity is a natural choice for extended models in solid mechanics because of its mathematical structure which is a system of conservation laws with full rotational invariance we study these properties for the lagrangian system and detail the various eulerian formulations we present several test problems in 1d and 2d planar cases which shows the capability of the scheme to capture complex shock waves and to simulate solid fluid problems in this article we use a special equation of state 21 its interest is twofold we can calculate multi dimensional plastic phenomenon such as split shock in 1d uniaxial cases or particular shapes in taylor test case and it gives interesting multi dimensional test cases for hyperelastic planar schemes

on the plant leaf s boundary jupe a godets and conformal embeddings the stable profile of the boundary of a plant s leaf fluctuating in the direction transverse to the leaf s surface is described in the framework of a model called a surface a godets sg it is shown that the information on the profile is encoded in the jacobian of a conformal mapping the coefficient of deformation corresponding to an isometric embedding of a uniform cayley tree into the 3d euclidean space the geometric characteristics of the leaf s boundary such as the perimeter and the height are calculated in addition a symbolic language allowing us to investigate the statistical properties of a sg with annealed random defects of the curvature of density q is developed it is found that at q 1 the surface exhibits a phase transition with the critical exponent 1 from

the similarity metric a new class of distances appropriate for measuring similarity relations between sequences say one type of similarity per distance is studied we propose a new normalized information distance based on the noncomputable notion of kolmogorov complexity and show that it is in this class and it minorizes every computable distance in the class that is it is universal in that it discovers all computable similarities we demonstrate that it is a metric and call it the similarity metric this theory forms the foundation for a new practical tool to evidence generality and robustness we give two distinctive applications in widely divergent areas using standard compression programs like gzip and gencompress first we compare whole mitochondrial genomes and infer their evolutionary history this results in a first completely automatic computed whole mitochondrial phylogeny tree secondly we fully automatically compute the language tree of 52 different languages

kolmogorov s structure functions and model selection in 1974 kolmogorov proposed a nonprobabilistic approach to statistics and model selection let data be finite binary strings and models be finite sets of binary strings consider model classes consisting of models of given maximal kolmogorov complexity the structure function of the given data expresses the relation between the complexity level constraint on a model class and the least log cardinality of a model in the class containing the data we show that the structure function determines all stochastic properties of the data for every constrained model class it determines the individual best fitting model in the class irrespective of whether the true model is in the model class considered or not in this setting this happens with certainty rather than with high probability as is in the classical case we precisely quantify the goodness of fit of an individual model with respect to individual data we show that within the obvious constraints every graph is realized by the structure function of some data we determine the un computability properties of the various functions contemplated and of the algorithmic minimal sufficient statistic

first passage quantities of brownian motion in a bounded domain with multiple targets a unified approach in this paper we introduce a general computation method to systematically determine the mean first passage time mfpt the global mean first passage time gmfpt and splitting probabilities for a continuous brownian motion in a confined 2d or 3d domain with multiple absorbing targets in the bulk or on the boundary this method is applied to spherically symmetric domains in the limit of small sized targets and asymptotic expansions of the mpft gmfpt and splitting probabilities are obtained in four distinct cases 3d domains with targets in the bulk 3d domains with targets on the boundary 2d domains with targets on the bulk and 2d domains with targets on the boundary this approach gives a unified description of existing exact results which were obtained using specific technics and also yields new results in particular for n targets splitting probabilities

close or connected distance and connectivity effects on transport in networks laboratoire de physique th eorique de la mati ere condens ee umr cnrs upmc universit e pierre et marie curie 4 place jussieu 75255 paris cedex dated february 25 2011 we develop an analytical approach which provides the dependence of the mean rst passage time mfpt for random walks on complex networks both on the target connectivity and on the source target distance our approach puts forward two strongly di erent behaviors depending on the type compact or non compact of the random walk in the case of non compact exploration we showthat the mfpt scales linearly with the inverse connectivity of the target and is largely independentof the starting point on the contrary in the compact case the mfpt is controlled by the source target distance and we nd that unexpectedly the target connectivity becomes irrelevant for remotetargets

minimum complexity pursuit for universal compressed sensing the nascent field of compressed sensing is founded on the fact that high dimensional signals with simple structure can be recovered accurately from just a small number of randomized samples several specific kinds of structures have been explored in the literature from sparsity and group sparsity to low rankness however two fundamental questions have been left unanswered what are the general abstract meanings of structure and simplicity do there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension in this paper we address these two questions using algorithmic information theory tools such as the kolmogorov complexity we provide a unified definition of structure and simplicity leveraging this new definition we develop and analyze an abstract algorithm for signal recovery motivated by occam s razor minimum complexity pursuit mcp requires approximately 2 randomized samples to recover a signal of complexity and ambient dimension n we also discuss the performance of the mcp in the presence of measurement noise and with approximately simple signals

filtering random noise from deterministic signals via data compression we present a novel technique for the design of filters for random noise leading to a class of filters called occam filters the essence of the technique is that when a lossy data compression algorithm is applied to a noisy signal with the allowed loss set equal to the noise strength the loss and the noise tend to cancel rather than add we give two illustrative applications of the technique to univariate signals we also prove asymptotic convergence bounds on the effectiveness of occam filters

a nominally second order cell centered lagrangian scheme for simulating elastic plastic flows on two dimensional unstructured grids in this paper we describe a cell centered lagrangian scheme devoted to the numerical simulation of solid dynamics on two dimensional unstructured grids in planar geometry this numerical method utilizes the classical elastic perfectly plastic material model initially proposed by wilkins m l wilkins calculation of elastic plastic flow meth comput phys 1964 in this model the cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state regarding the deviatoric stress its time evolution is governed by a classical constitutive law for isotropic material the plasticity model employs the von mises yield criterion and is implemented by means of the radial return algorithm the numerical scheme relies on a finite volume cell centered method wherein numerical fluxes are expressed in terms of sub cell force the generic form of the sub cell force is obtained by requiring the scheme to satisfy a semi discrete dissipation inequality sub cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node centered solver which results from total energy conservation the nominally second order extension is achieved by developing a two dimensional extension in the lagrangian framework of the generalized riemann problem methodology introduced by ben artzi and falcovitz m ben artzi j falcovitz generalized riemann problems in computational fluid dynamics cambridge monogr appl comput math 2003 finally the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases

high order curvilinear finite elements for elastic plastic lagrangian dynamics this paper presents a high order finite element method for calculating elastic plastic flow on moving curvilinear meshes and is an extension of our general high order curvilinear finite element approach for solving the euler equations of gas dynamics in a lagrangian frame 1 2 in order to handle transition to plastic flow we formulate the stress strain relation in rate or incremental form and augment our semi discrete equations for lagrangian hydrodynamics with an additional evolution equation for the deviatoric stress which is valid for arbitrary order spatial discretizations of the kinematic and thermodynamic variables the semi discrete equation for the deviatoric stress rate is developed for 2d planar 2d axisymmetric and full 3d geometries for each case the strain rate is approximated via a collocation method at zone quadrature points while the deviatoric stress is approximated using an l 2 projection onto the thermodynamic basis we apply high order energy conserving explicit time stepping methods to the semi discrete equations to develop the fully discrete method we conclude with numerical results from an extensive series of verification tests that demonstrate several practical advantages of using high order finite elements for elastic plastic flow

the ising model on hyperlattices the authors describe the construction of regular lattices in two dimensional hyperbolic space by means of the action of a discrete subgroups of su 1 1 they consider an ising model on such lattices and show how the thermodynamic limit can be handled they give high and low temperature expansions of the free energy magnetic susceptibility and magnetization and find that these quantities diverge at a critical temperature with mean field exponents beta 1 2 gamma 1 they also conjecture the long distance behaviour of correlation functions at the critical temperature

high order curvilinear finite element methods for lagrangian hydrodynamics the numerical approximation of the euler equations of gas dynamics in a movinglagrangian frame is at the heart of many multiphysics simulation algorithms in this paper we present a general framework for high order lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements this method is an extension of the approach outlined in dobrev et al internat j numer methods fluids 65 2010 pp 1295 1310 and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields including generic finite elements on two and three dimensional meshes with triangular quadrilateral tetrahedral or hexahedral zones we discretize the kinematic variables of position and velocity using a continuous high order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high order parametric mapping from a standard reference element this enables the use of curvilinear zone geometry higher ord