martin lof random quantum states we extend the key notion of martin lof randomness for infinite bit sequences to the quantum setting where the sequences become states of an infinite dimensional system we prove that our definition naturally extends the classical case in analogy with the levin schnorr theorem we work toward characterizing quantum ml randomness of states by incompressibility in the sense of quantum turing machines of all initial segments we extend the key notion of martin lof randomness for infinite bit sequences to the quantum setting where the sequences become states of an infinite dimensional system we prove that our definition naturally extends the classical case in analogy with the levin schnorr theorem we work toward characterizing quantum ml randomness of states by incompressibility in the sense of quantum turing machines of all initial segments

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

on the convergence properties of the quadrilateral elements of sander and beckers a class of nonconforming quadrilateral membrane elements introduced by sander and beckers is discussed in all details it is proved that two elements that do not pass the patch test may still yield convergence under a suitable condition on mesh subdivisions whereas one such element is found to be divergent for all mesh subdivisions all other finite elements in this class that pass the patch test are convergent the mathematical analysis provides a clear explanation of the convergence behavior appearing in the numerical examples of sander and beckers 1 introduction quadrilateral finite elements are attractive for discretizations of domains of arbitrary shapes but these approximations are hardly conforming therefore such elements have to be tested for convergence in connection with a variational interpretation of the patch test 2 sander and beckers 3 have recently introduced a class of nonconforming quadrilateral membrane elements having 8 to 16 degrees of freedom and have shown by numerical computations for a trapezoidal membrane problem that two elements that do not pass the patch test still exhibit convergence and yield good approximations whereas one such element diverges this result is of great interest and leads to the question whether the patch test is necessary for convergence of these nonconforming elements as stated in 3 additional investigations are certainly required to understand their behavior the present paper establishes a detailed mathematical analysis of these elements we formulate a condition on mesh subdivisions under which two elements that do not pass the patch test indeed yield convergence whereas another element is proved to be divergent whatever the mesh subdivision may be all other quadrilateral elements in this class that pass the patch test are convergent the theoretical analysis clearly explains the convergence behavior discussed in the numerical computations of sander and beckers in their examples a bisection scheme of mesh subdivisions is used for the discretization of the trapezoidal membrane in such cases the proposed condition on the mesh subdivisions is automatically fulfilled and thus convergence occurs for two elements it is seen therefore that the range of applicability of the patch test as a necessary condition for convergence is limited at this point it is worth noting that stummel 5 has presented basic examples which show that spaces of nonconforming trial functions

convergence properties of two nonconforming finite elements abstract two nonconforming triangular elements for second order problems are analyzed one element is found to be convergent the other does not yield approximate solutions converging to the solution of the given boundary value problem but to a wrong limit which is the solution of another variational equation this strange convergence behaviour may be typical for a wider class of nonconforming elements violating the basic precondition for convergence the problem of patch subdivisions is also considered

an explicit analysis of stummel s patch test examples a constructive approach for the treatment of stummel s examples to the patch test is presented together with generalizations of these examples the strange convergence phenomenon of nonconforming approximations is discussed in detail and a kind of modified variational equation for remedying these defects is introduced our variational equation uses a combination of hybrid and penalty methods that define a convergent procedure with optimal order of convergence to the exact solution with respect to a mesh dependent norm

universality in sandpile models a classification of sandpile models into universality classes is presented on the basis of extensive numerical simulations in which we measure an extended set of exponents the manna two state model s s manna j phys a 24 l363 1991 is found to belong to a universality class of random neighbor models which is distinct from the universality class of the original model of bak tang and wiesenfeld p bak c tang and k wiesenfeld phys rev lett 59 381 1987 directed models are found to belong to a universality class which includes the directed model introduced and solved by dhar and ramaswamy d dhar and r ramaswamy phys rev lett 63 1659 1989

upcrossing inequalities for stationary sequences and applications for arrays s i j 1 i j of random variables that are stationary in an appropriate sense we show that the fluctuations of the process s 1 n n 1 can be bounded in terms of a measure of the mean subadditivity of the process s i j 1 i j we derive universal upcrossing inequalities with exponential decay for kingman s subadditive ergodic theorem the shannon macmillan breiman theorem and for the convergence of the kolmogorov complexity of a stationary sample

a new class of zienkiewicz type non conforming element in any dimensions in this paper a new class of zienkiewicz type non conforming finite element in n spatial dimensions with n 2 is proposed the new finite element is proved to be convergent for the biharmonic equation

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

oslo model hyperuniformity and the quenched edwards wilkinson model we present simulations of the one dimensional oslo rice pile model in which the critical height at each site is randomly reset after each toppling we use the fact that the stationary state of this sand pile model is hyperuniform to reach system of sizes 10 7 most previous simulations were seriously flawed by important finite size corrections we find that all critical exponents have values consistent with simple rationals 4 3 for the correlation length exponent d 9 4 for the fractal dimension of avalanche clusters and z 10 7 for the dynamical exponent in addition we relate the hyperuniformity exponent to the correlation length exponent finally we discuss the relationship with the quenched edwards wilkinson model where we find in particular that the local roughness exponent is loc 1

kolmogorov s contributions to information theory and algorithmic complexity briefly information theory says a random object x p x has complexity entropy h xp x log p x with the attendant interpretation that h bits are sufficient to describe x on the average algorithmic complexity says an object x has a complexity k x equal to the length of the shortest binary program that describes x it is a beautiful fact that these ideas are much the same in fact it is roughly true that ek x h moreover if we let pu x pr u prints x be the probability that a given computer u prints x when given a random program it can be shown that log 1 pu x k x for all x thus establishing a vital link between the universal probability measure pu and the universal complexity k more on this later the relationship of these ideas to probability theory was summed up in kolmogorov s 1983 paper which was based on his 1970 talk in nice perhaps only

nonconforming tetrahedral finite elements for fourth order elliptic equations this paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions the newly constructed elements include two nonconforming tetrahedral finite elements and one quasi conforming tetrahedral element these elements are proved to be convergent for a model bi harmonic equation in three dimensions in particular the quasi conforming tetrahedron element is a modified zienkiewicz element while the nonmodified zienkiewicz element a tetrahedral element of hermite type is proved to be divergent on a special grid

a convergence condition for the quadrilateral wilson element the paper deals with the convergence properties of the nonconforming quadrilateral wilson element which violates the patch test the convergence of the element is proved under a certain condition on mesh subdivisions without any modifications of the variational formulation this result extends the range of applicability of wilson s element the necessity of the proposed condition is also discussed

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

the conforming virtual element method for polyharmonic problems abstract in this work we exploit the capability of virtual element methods in accommodating approximation spaces featuring high order continuity to numerically approximate differential problems of the form p u f p 1 more specifically we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems and prove an abstract result that states the convergence of the method in suitable norms

steady state of stochastic sandpile models we study the steady state of the abelian sandpile models with stochastic toppling rules the particle addition operators commute with each other but in general these operators need not be diagonalizable we use their abelian algebra to determine their eigenvalues and the jordan block structure these are then used to determine the probability of different configurations in the steady state we illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example for systems of size 12 and also study the density profile in the steady state

steady state and relaxation spectrum of the oslo rice pile model we show that the one dimensional oslo rice pile model is a special case of the abelian distributed processors model the exact steady state of the model is determined we show that the time evolution operator w for the system satisfies the equation w n 1 w n where n l l 1 2 for a pile with l sites this is used to prove that w has only one eigenvalue 1 corresponding to the steady state and all other eigenvalues are exactly zero also all connected time dependent correlation functions in the steady state of the pile are exactly zero for time difference greater that n generalization to other abelian critical height models where the critical thresholds are randomly reset after each toppling is briefly discussed

sampling rare fluctuations of height in the oslo ricepile model we describe a new monte carlo algorithm for studying large deviations of the height of the pile from its mean value in the oslo ricepile model we have used it to generate events which have probabilities of order 10 1000 these simulations check our qualitative argument pradhan p and dhar d 2006 phys rev e 73 021303 that in the steady state of the oslo ricepile model the probability of large negative height fluctuations h l about the mean varies as exp 4l3 as l with held fixed and 0

nonconforming finite element methods for the equations of linear elasticity in the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions the main difficulty in the analysis is to prove that an appropriate discrete version of korn s second inequality is valid such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears optimal order error estimates uniform for poisson ratio v e 0 1 2 are then derived for the corresponding p2 and p3 methods this contrasts with the use of c finite elements where there is a deterioration in the convergence rate as v 1 2 for piecewise polynomials of degree 3 modifications of the continuous methods and the nonconforming linear method which also give uniform optimal order error estimates are discussed

oslo rice pile model is a quenched edwards wilkinson equation the oslo rice pile model is a sandpile like paradigmatic model of self organized criticality soc in this paper it is shown that the oslo model is in fact exactly a discrete realization of the much studied quenched edwards wilkinson equation qew nattermann et al j phys ii france 2 1483 1992 this is possible by choosing the correct dynamical variable and identifying its equation of motion it establishes for the first time an exact link between soc models and the field of interface growth with quenched disorder this connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of soc

on the necessity and sufficiency of the patch test for convergence of nonconforming finite elements this paper proposes a weak patch test that improves both the patch test and the generalized patch test under a weak superapproximation assumption which can be obtained from certain approximation and weak continuity property it is shown that the weak patch test is equivalent to the generalized patch test thus if a nonconforming finite element passes the patch test and satisfies some approximation and weak continuity property its convergence is then guaranteed furthermore the consistency term of such a nonconforming element can be proved to be of order o h it is shown that if a nonconforming element does not pass the patch test and a condition relative to subdividing a domain by the element is true then one can find a family of triangulations for which the element is divergent

the lower bound property of the morley element eigenvalues in this paper we prove that the morley element eigenvalues approximate the exact ones from below on regular meshes including adaptive local refined meshes for the fourth order elliptic eigenvalue problems with the clamped boundary condition in any dimension and we implement the adaptive computation with the morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth order elliptic eigenvalue problem with variable coefficients

a quadratic finite element method for solving biharmonic problems in ir n a family of simplicial finite element methods having the simplest possible structure is introduced to solve biharmonic problems in n n 3 using the primal variable the family is inspired in the morley triangle for the two dimensional case and in some sense this element can be viewed as its member corresponding to the valuen 2

a c1 virtual element method on polyhedral meshes abstract the purpose of the present paper is to develop c 1 virtual elements in three dimensions for linear elliptic fourth order problems motivated by the difficulties that standard conforming finite elements encounter in this framework we focus the presentation on the lowest order case the generalization to higher orders being briefly provided in the appendix the degrees of freedom of the proposed scheme are only 4 per mesh vertex representing function values and gradient values interpolation error estimates for the proposed space are provided together with a set of numerical tests to validate the method at the practical level

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

the f e m test for convergence of nonconforming finite elements on etablit un nouveau critere de convergence pour les elements finis non conformes qui est facilement verifiable le long de chaque interface

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems