a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

lesche stability of κ entropy abstract the lesche stability condition for the shannon entropy j stat phys 27 1982 419 represents a fundamental test for its experimental robustness for systems obeying the maxwell boltzmann statistical mechanics of course this stability condition must be satisfied by any entropic functional candidate to generate non conventional statistical mechanics in the present effort we show that the entropy recently introduced in literature phys rev e 66 2002 056125 satisfies the lesche stability condition

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

an integrable model with a non reducible three particle r matrix we define an integrable lattice model which in the notation of yang in addition to the conventional two particle r matrices also contains non reducible three particle r matrices the corresponding modified yang baxter equations are solved and an expression for the transfer matrix is found as a normal ordered exponential of a non local hamiltonian

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

a general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids the overlapping covolume case we present a general framework for constructing and analyzing finite volume methods applied to the mixed formulation of second order elliptic problems on quadrilateral grids the control volumes or covolumes in the grids overlap an overlapping finite volume method of this type was first introduced by russell in t f russell tech report 3 reservoir simulation research corp tulsa ok 1995 and was tested for a variety of problems on rectangular and quadrilateral grids in z cai et al comput geosci 1 1997 pp 289 315 later in s h chou and d y kwak siam j numer anal 37 2000 pp 758 771 chou and kwak reformulated it as their mixed covolume method and proved optimal order error estimates using the covolume methodology from s h chou math comp 66 1997 pp 85 104 and s h chou and d y kwak siam j numer anal 35 1998 pp 494 507 however their treatment was restricted to the case of diagonal coefficient tensor and rectangular grids since a different approach was needed for the quadrilateral distorted rectangular case in this paper we give a new framework which can handle not only the rectangular anisotropic case but also the anisotropic and irregular grid cases in which the locally supported test functions are images of the natural unit coordinate vectors under the piola transformation our theory sheds light on how to create new test functions using quadratures and now covers russell s quadrilateral case

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

deriving partition functions and entropic functionals from thermodynamics abstract given an arbitrary probability distribution and a nondegenerate energy spectrum so that a mean energy u can be computed we derive the partition function z and the entropic functional s that satisfy the basic relation d u t d s the procedure is illustrated by considering examples of typical distributions found currently in nature in particular the power law spectrum is shown to correspond to a critical state associated with tsallis entropy

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

deformed exponentials and logarithms in generalized thermostatistics abstract criteria are given that deformed logarithmic and exponential functions should satisfy with a pair of such functions one can associate another function called the deduced logarithmic function it is shown that generalized thermostatistics can be formulated in terms of deformed exponential functions together with the associated deduced logarithmic functions

supraconvergence and supercloseness in quasilinear coupled problems the aim of this paper is to study a finite difference method for quasilinear coupled problems of partial differential equations that presents numerically an unexpected second order convergence rate the error analysis presented allows us to conclude that the finite difference method is supraconvergent as the method studied in this paper can be seen as a fully discrete piecewise linear finite element method we conclude the supercloseness of our approximations

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

generalized entropies and the transformation group of superstatistics superstatistics describes statistical systems that behave like superpositions of different inverse temperatures so that the probability distribution is where the kernel f is nonnegative and normalized f d 1 we discuss the relation between this distribution and the generalized entropic form the first three shannon khinchin axioms are assumed to hold it then turns out that for a given distribution there are two different ways to construct the entropy one approach uses escort probabilities and the other does not the question of which to use must be decided empirically the two approaches are related by a duality the thermodynamic properties of the system can be quite different for the two approaches in that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes the transformation group is the euclidean group in one dimension

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a theorem on the existence of trace form generalized entropies an analytic technique is proposed which allows to generate many new examples of entropic functionals generalizing the standard boltzmanngibbs entropy our approach is based on the existence of a g

microcanonical foundation for systems with power law distributions starting from a microcanonical basis with the principle of equal a priori probability it is shown using the method of steepest descents that besides ordinary boltzmann gibbs theory with the exponential distribution a theory describing systems with power law distributions can also be derived

a higher order formulation of the mimetic finite difference method a new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes this approach provides a higher order accurate approximation to the flux of the exact solution in analogy with the original formulation a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability the scalar solution field is approximated by a piecewise constant function a computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable finally optimal convergence rates and accuracy improvement with respect to the lower order formulation are shown by numerical examples

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

a multipoint flux mixed finite element method on hexahedra we develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell centered finite differences the paper is an extension of our earlier paper for quadrilateral and simplicial grids m f wheeler and i yotov siam j numer anal 44 2006 pp 2082 2106 the construction is motivated by the multipoint flux approximation method and it is based on an enhancement of the lowest order brezzi douglas duran fortin bddf mixed finite element spaces on hexahedra in particular there are four fluxes per face one associated with each vertex a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate first order convergence for pressures and subface fluxes on sufficiently regular grids as well as second order convergence for pressures at the cell centers second order convergence for face fluxes is also observed computationally

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

an extension of the discrete variational method to nonuniform grids the discrete variational method is a method used to derive finite difference schemes that inherit the conservation dissipation property of the original equations although this method has mainly been developed for uniform grids we extend this method to multidimensional nonuniform meshes

the discrete variational derivative method based on discrete differential forms as is well known for pdes that enjoy a conservation or dissipation property numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice lately furihata and matsuo have developed the so called discrete variational derivative method that automatically constructs energy preserving or dissipative finite difference schemes although this method was originally developed on uniform meshes the use of non uniform meshes is of importance for multi dimensional problems on the other hand the theories of discrete differential forms have received much attention recently these theories provide a discrete analogue of the vector calculus on general meshes in this paper we show that the discrete variational derivative method and the discrete differential forms by bochev and hyman can be combined applications to the cahn hilliard equation and the klein gordon equation on triangular meshes are provided as demonstrations we also show that the schemes for these equations are h1 stable under some assumptions in particular one for the nonlinear klein gordon equation is obtained by combination of the energy conservation property and the discrete poincare inequality which are the temporal and spacial structures that are preserved by the above methods

a mimetic discretization of the reissner mindlin plate bending problem we present a mimetic approximation of the reissner mindlin plate bending problem which uses deflections and rotations as discrete variables the method applies to very general polygonal meshes even with non matching or non convex elements we prove linear convergence for the method uniformly in the plate thickness

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

on the stability of analytic entropic forms abstract the stability against small perturbations on the probability distributions also called experimental robustness of analytic entropic forms is analyzed entropies s p associated with a given set of probabilities p i that can be written in the simple form s p i 1 w r p i are shown to be robust if r p i is an analytic function of the p i s the same property holds for entropies s p that are monotonic and analytic functions of s p the tsallis entropy s q p falls in the first class of entropies whenever the entropic index q is an integer greater than 1 a new kind of entropy that follows such requirements is discussed

superconvergence for control volume mixed finite element methods on rectangular grids we consider control volume mixed finite element methods for the approximation of second order elliptic problems on rectangular grids these methods associate control volumes covolumes with the vector variable as well as the scalar obtaining local algebraic representation of the vector equation e g darcy s law as well as the scalar equation e g conservation of mass we establish o h 2 superconvergence for both the scalar variable in a discrete l 2 norm and the vector variable in a discrete h rm div norm the analysis exploits a relationship between control volume mixed finite element methods and the lowest order raviart thomas mixed finite element methods

generalized entropy optimized by a given arbitrary distribution an ultimate generalization of the maximum entropy principle is presented an entropic measure which is optimized by a given arbitrary distribution with the finite linear expectation value of a physical random quantity of interest is constructed it is concave irrespective of the properties of the distribution and satisfies the h theorem for the master equation combined with the principle of microscopic reversibility this offers a unified basis for a great variety of distributions observed in nature as examples the entropies associated with the stretched exponential distribution postulated by anteneodo and plastino 1999 j phys a math gen 32 1089 and the deformed exponential distribution by kaniadaki 2002 phys rev e 66 056125 and naudts 2002 physica a 316 323 are derived to include distributions with divergent moments e g the levy stable distributions it is necessary to modify the definition of the expectation value

stabilities of generalized entropies the generalized entropic measure which is maximized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity is considered to examine if it can be of physical relevance its experimental robustness is discussed in particular lesche s criterion is analysed which states that an entropic measure is stable if its change under an arbitrary weak deformation of the distribution representing fluctuations of experimental data remains small it is essential to note the difference between this criterion and thermodynamic stability a general condition under which the generalized entropy becomes stable is derived examples known in the literature including the entropy for the stretched exponential distribution the quantum group entropy and the entropy are discussed

analysis of stochastic mimetic finite difference methods and their applications in single phase stochastic flows abstract stochastic modeling has become a widely accepted approach to quantify uncertainty in applications where diffusion plays a central role in many of these applications the geometry is complex and important properties of the underlying deterministic continuum model need to be captured accurately to address these problems we present a stochastic mimetic finite difference mfd method for diffusion equations with random input data specifically we use the mfd methodology for the spatial approximation to ensure the necessary accuracy and robustness is achieved to treat the high dimensionality of the stochastic approximation efficiently we use a stochastic collocation method we consider the stochastic mfd approximation in hybrid form and perform a rigorous analysis of its semi discretization and full discretization for the pressure flux and lagrange multipliers convergence rates are developed for statistical moments of the quantities of interest numerical results are presented for single phase flow in random porous media and support the efficiency of the stochastic mfd

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems we present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems under a saturation assumption the global reliability and efficiency of the proposed a posteriori estimator are proved several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies finally we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

remarks about the tsallis formalism in the present paper the conditions for the validity of the tsallis statistics are analyzed the same has been done following the analogy with the traditional case starting from the microcanonical description of the systems and taking into account their self similarity scaling properties in the thermodynamic limit it is analyzed the necessary conditions for the equivalence of microcanonical ensemble with the tsallis generalization of the canonical ensemble it is shown that the tsallis statistics is appropriate for the macroscopic description of systems with potential scaling laws of the asymptotic accessible states density of the microcanonical ensemble our analysis shows many details of the tsallis s formalism the q expectation values the generalized legendre transformations between the thermodynamic potentials as well as the conditions for its validity having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment on the other hand the definition of physical temperature received a modification that differs from the toral result for the case of finite systems we have generalized the microcanonical thermostatistics of gross with the generalization of the curvature tensor for this kind of description

a preconditioned multilevel mimetic finite difference method for solving the pressure velocity system abstract a mimetic finite difference mfd implementation is proposed to solve the pressure velocity system iteratively the recursive procedure only solves for the velocity variable while the pressure is calculated by direct substitution simplifying the problem the symmetric positive definite system obtained is solved with an innovative implementation of a pre conditioned multilevel conjugate gradient algorithm where coarse algebraic objects are easily computed from the corresponding fine level ones based on fundamental geometric properties of the mfd technology a succinct eigen analysis shows second order convergence on the velocity variable for cartesian grids which is corroborated numerically also results consider different topologies and different medium properties showing the applicability of the method and optimal convergence rates for the velocity on cartesian and h 2 uniform curved grids

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

tetrahedral zamolodchikov algebras corresponding to baxter s l operators tetrahedral zamolodchikov algebras are structures that occupy an intermediate place between the solutions of the yang baxter equation and its generalization onto 3 dimensional mathematical physics the tetrahedron equation these algebras produce solutions to the tetrahedron equation and besides specific two layer solutions to the yang baxter equation here the tetrahedral zamolodchikov algebras are studied that arise froml operators of the free fermion case of baxter s eight vertex model

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

maximum entropy approach to stretched exponential probability distributions we introduce a nonextensive entropy functionals whose optimization under simple constraints mean values of some standard quantities yields stretched exponential probability distributions which occur in many complex systems the new entropy functional is characterized by a parameter the stretching exponent such that for d 1 the standard logarithmic entropy is recovered we study its mathematical properties showing that the basic requirements for a well behaved entropy functional are verified i e s possesses the usual properties of positivity equiprobability concavity and irreversibility and verifies khinchin axioms except the one related to additivity since s is nonextensive the entropys is shown to be superadditive for 1

rehabilitation of the lowest order raviart thomas element on quadrilateral grids a recent study d n arnold d boffi and r s falk siam j numer anal 42 2005 pp 2429 2451 reveals that convergence of finite element methods using h mathrm div omega compatible finite element spaces deteriorates on nonaffine quadrilateral grids this phenomena is particularly troublesome for the lowest order raviart thomas elements because it implies loss of convergence in some norms for finite element solutions of mixed and least squares methods in this paper we propose reformulation of finite element methods based on the natural mimetic divergence operator m shashkov conservative finite difference methods on general grids crc press boca raton fl 1996 which restores the order of convergence reformulations of mixed galerkin and least squares methods for the darcy equation illustrate our approach we prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator furthermore we prove that standard and reformulated versions of the mixed galerkin method lead to identical linear systems but the two versions of the least squares method are veritably different the surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial and that the lowest order raviart thomas elements are safe to use in this method however the breakdown in the least squares method is real and there one should use our proposed reformulation

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators