a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

conformal invariance in 2 1 dimensional stochastic systems stochastic partial differential equations can be used to model second order thermodynamical phase transitions as well as a number of critical out of equilibrium phenomena in 2 1 dimensions many of these systems are conjectured and some are indeed proved to be described by conformal field theories we advance in the framework of the martin siggia rose field theoretical formalism of stochastic dynamics a general solution of the translation ward identities which yields a putative conformal energy momentum tensor even though the computation of energy momentum correlators is obstructed in principle by dimensional reduction issues these are bypassed by the addition of replicated fields to the original 2 1 dimensional model the method is illustrated with an application to the kardar parisi zhang kpz model of surface growth the consistency of the approach is checked by means of a straightforward perturbative analysis of the kpz ultraviolet region leading as expected to its c 1 conformal fixed point

conformal invariance of isoheight lines in a two dimensional kardar parisi zhang surface the statistics of isoheight lines in the 2 1 dimensional kardar parisi zhang kpz model is shown to be conformally invariant and equivalent to those of self avoiding random walks this leads to a rich variety of exact analytical results for the kpz dynamics we present direct evidence that the isoheight lines can be described by the family of conformally invariant curves called schramm loewner evolution or sle with diffusivity 8 3 it is shown that the absence of the nonlinear term in the kpz equation will change the diffusivity from 8 3 to 4 indicating that the isoheight lines of the edwards wilkinson surface are also conformally invariant and belong to the universality class of domain walls in the o2 spin model

extensions of states on mv quantales abstract we consider the lattice of the ordered right continuous families of elements of a dual quantale with the index set 0 1 called its rc extension and the lattice of the ordered n 1 tuples with n 1 of elements of this dual quantale its n extension equipped with a dual quantale structure that generalizes the canonical extension proposed by hohle and weber of particular interest is the existence of states on n extensions of finite mv chains we generalize the recent results of weber by incorporating effectible girard quantales which are introduced by the author

self consistent expansion for the kardar parisi zhang equation with correlated noise a minor modification of the self consistent expansion sce for the kardar parisi zhang kpz system with uncorrelated noise is used to obtain the exponents in systems where the noise has spatial long range correlations for d dimensional systems with correlations of the form d stackrel ensuremath rightarrow r ensuremath r ensuremath t ensuremath t ensuremath 2d 0 stackrel ensuremath rightarrow r ensuremath r ensuremath 2 ensuremath rho ensuremath d ensuremath delta t ensuremath t ensuremath ensuremath rho g0 we find a lower critical dimension d 0 ensuremath rho 2 2 ensuremath rho above which a perturbative edwards wilkinson ew solution appears below the lower critical dimension two solutions exist each in a different distinct region of ensuremath rho for small ensuremath rho s the solution of kpz with uncorrelated noise is recovered for large ensuremath rho s a ensuremath rho dependent solution is found the existence of only one solution in each region of ensuremath rho is not a result of a competition between two solutions but a direct outcome of the sce equation

foundations for an algebraic theory of conditioning abstract we investigate some mathematical results in ring theory in order to lay down an algebraic framework for a theory of measure free conditional objects

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

finite element methods for semilinear elliptic stochastic partial differential equations we study finite element methods for semilinear stochastic partial differential equations error estimates are established numerical examples are also presented to examine our theoretical results

classification of 2 1 dimensional growing surfaces using schramm loewner evolution statistical behavior and scaling properties of isoheight lines in three different saturated two dimensional grown surfaces with controversial universality classes are investigated using ideas from schramm loewner evolution text sle ensuremath kappa we present some evidence that the isoheight lines in the ballistic deposition bd eden and restricted solid on solid rsos models have conformally invariant properties all in the same universality class as the self avoiding random walk saw equivalently text sle 8 3 this leads to the conclusion that all these discrete growth models fall into the same universality class as the kardar parisi zhang kpz equation in two dimensions

surface scaling properties of decagonal quasicrystals under nonequilibrium vertical growth restricted solid on solid rsos growth models are studied on two different decagonal quasicrystal lattices namely the penrose tiling lattice and the random tiling lattice there exist two types of growth blocks fat and skinny tiles which may have different sticking probabilities we found that the rsos growths on both lattices belong to the kardar parisi zhang universality class when they have the same sticking probabilities in spite of the lack of periodicity in the substrates however when they have tile type dependent sticking probabilities the rsos models on two lattices may produce different scaling behaviors growth on penrose tiling shows that the roughness exponent is around 0 4 while that on random tiling is around 0 49 our observation may provide an effective way to investigate the bulk structures of decagonal quasicrystals

geometrical exponents of contour loops on synthetic multifractal rough surfaces multiplicative hierarchical cascade p model in this paper we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces which are generated by multiplicative hierarchical cascading processes to this end two different classes of multifractal rough surfaces are numerically simulated as the first group singular measure multifractal rough surfaces are generated by using the p model the smoothened multifractal rough surface then is simulated by convolving the first group with a so called hurst exponent h the generalized multifractal dimension of isoheight lines contours d q correlation exponent of contours x l cumulative distributions of areas and perimeters are calculated for both synthetic multifractal rough surfaces our results show that for both mentioned classes hyperscaling relations for contour loops are the same as that of monofractal systems in contrast to singular measure multifractal rough surfaces h plays a leading role in smoothened multifractal rough surfaces all computed geometrical exponents for the first class depend not only on its hurst exponent but also on the set of p values but in spite of multifractal nature of smoothened surfaces second class the corresponding geometrical exponents are controlled by h the same as what happens for monofractal rough surfaces

measure free conditioning and extensions of additive measures on finite mv algebras using the well known representation of any finite mv algebra as a product of finite mv chains as factors we obtain a representation of its canonical extension as a girard algebra product of the canonical extensions of the mv chain factors based on this representation and using the results from our last paper we characterize the additive measures on any finite mv algebra resp the weakly and the strongly additive measures on its canonical girard algebra extension and that as convex combinations of the corresponding measures on the respective factors after that we apply the results to measure free defined conditional events which for this reason are considered as elements of the canonical extension of a finite mv algebra of events this definition for conditional events is naturally equivalent to the well known definition as lattice intervals of events here the weak additivity is considered as a reasonable notion for a global additivity property of a measure extension these extensions are essentially different from the measure extension which has a form analogous to the classical conditional probability but which is only levelwise additive moreover the measure free approach of this paper is completely different from the classical approach in which conditional measures of events given a fixed condition event are considered based on additive measures states of events without having defined conditional events

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

universality and corrections to scaling in the ballistic deposition model in order to analyze some controversies on the equivalence between ballistic deposition bd and the kardar parisi zhang kpz theory we simulated the bd model in one and two dimensions effective exponents betal were obtained in the growth regions which were rigorously determined for various lengths l effective exponents alphal were obtained from saturation widths in the steady state regimes in d 1 we found betal beta al lambda and alphal alpha bl delta with asymptotic exponents consistent with the kpz values beta 1 3 and alpha 1 2 and correction to scaling exponents 0 2 or approximately lambda or approximately 0 4 and 0 6 or approximately delta or approximately 0 8 these strong finite size corrections explain the previous discrepancies between numerical estimates for bd and the exact kpz results in d 2 we could only obtain reliable estimates of alphal which are consistent with kpz values if finite size corrections with delta approximately 0 4 are considered

l 2 projected least squares finite element methods for the stokes equations two new l2 least squares ls finite element methods are developed for the velocity pressure vorticity first order system of the stokes problem with dirichlet velocity boundary condition a key feature of these new methods is that a local or almost local l2 projector is applied to the residual of the momentum equation such l2 projection is always defined onto the linear finite element space no matter which finite element spaces are used for velocity pressure vorticity variables consequently the implementation of this l2 projected ls method is almost as easy as that of the standard l2 ls method more importantly the former has optimal error estimates in l2 norm with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal order interpolations and for all three variables velocity pressure and vorticity using unequal order interpolations numerical experiments are given to demonstrate the theoretical results

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

the existence and asymptotic behaviour of energy solutions to stochastic age dependent population equations driven by levy processes in this paper we introduce a class of stochastic age dependent population equations driven by levy processes existence and uniqueness of energy solutions for stochastic age dependent population dynamic system are proved under lipschitz condition in hilbert space the moment boundedness of the approximate solution by the galerkin method is considered we discuss by using the energy equality the exponential stability theorems of the energy solution to stochastic age dependent population equations

scaling of ballistic deposition from a langevin equation an exact lattice langevin equation is derived for the ballistic deposition model of surface growth the continuum limit of this equation is dominated by the kardar parisi zhang kpz equation at all length and time scales for a one dimensional substrate the solution of the exact lattice langevin equation yields the kpz scaling exponents without any extrapolation for a two dimensional substrate the scaling exponents are different from those found from computer simulations this discrepancy is discussed in relation to analytic approaches to the kpz equation in higher dimensions

numerical study of the kardar parisi zhang equation we integrate numerically the kardar parisi zhang kpz equation in 1 1 and 2 1 dimensions using a euler discretization scheme and the replacement of nablah 2 by exponentially decreasing functions of that quantity to suppress instabilities when applied to the equation in 1 1 dimensions the method of instability control provides values of scaling amplitudes consistent with exactly known results in contrast to the deviations generated by the original scheme in 2 1 dimensions we spanned a range of the model parameters where transients with edwards wilkinson or random growth are not observed in box sizes 8 or l or 128 we obtain a roughness exponent of 0 37 or alpha or 0 40 and steady state height distributions with skewness s 0 25 0 01 and kurtosis q 0 15 0 1 these estimates are obtained after extrapolations to the large l limit which is necessary due to significant finite size effects in the estimates of effective exponents and height distributions on the other hand the steady state roughness distributions show weak scaling corrections and evidence of stretched exponential tails these results confirm previous estimates from lattice models showing their reliability as representatives of the kpz class

strong and weak convergence order of finite element methods for stochastic pdes with spatial white noise we investigate the strong and weak convergence order of piecewise linear finite element methods for a class of semilinear elliptic equations with additive spatial white noise using a spectral truncation of white noise taking the number of modes for spectral approximation of the white noise at the order of 1 h d 1 hd we show that the strong convergence order of the finite element approximation is h 2 d 2 h2 d 2 where h is the element size and d le 3 d 3 is the dimension moreover we show that the weak convergence order is h min 4 d 2 hmin 4 d 2 we then consider a fourth order equation and an advection reaction equation and show that a spectral approximation of the white noise can lead to higher convergence order in both strong and weak sense when the solutions are smooth numerical results confirm our prediction for one and two dimensional elliptic problems

white noise driven spdes with reflection we study reflected solutions of a nonlinear heat equation on the spatial interval 0 1 with dirichlet boundary conditions driven by space time white noise the nonlinearity appears both in the drift and in the diffusion coefficient roughly speaking at any point t x where the solutionu t x is strictly positive it obeys the equation and at a point t x whereu t x is zero we add a force in order to prevent it from becoming negative this can be viewed as an extension both of one dimensional sdes reflected at 0 and of deterministic variational inequalities existence of a minimal solution is proved the construction uses a penalization argument a new existence theorem for spdes whose coefficients depend on the past of the solution and a comparison theorem for solutions of white noise driven spdes

what is the connection between ballistic deposition and the kardar parisi zhang equation ballistic deposition bd is believed to belong to the kardar parisi zhang kpz universality class in this paper we study the validity of this belief by rigorously deriving a continuum equation from the bd microscopic rules which deviates from the kpz equation we show that in one dimension and in the presence of noise the deviation is not important this is not the case in the absence of noise in more than one dimension and in the presence of noise we obtain an equation that superficially seems to be a continuum equation but in which the symmetry under rotations around the growth direction is broken

universality in two dimensional kardar parisi zhang growth we analyze simulation results of a model proposed for etching of a crystalline solid and results of other discrete models in the 2 1 dimensional kardar parisi zhang kpz class in the steady states the moments w n of orders n 2 3 4 of the height distribution are estimated results for the etching model the ballistic deposition model and the temperature dependent body centered restricted solid on solid model suggest the universality of the absolute value of the skewness s ensuremath equiv w 3 w 2 3 2 and of the value of the kurtosis q ensuremath equiv w 4 w 2 2 ensuremath 3 the sign of the skewness is the same as of the parameter ensuremath lambda of the kpz equation which represents the process in the continuum limit the best numerical estimates obtained from the etching model are s 0 26 ifmmode pm else textpm fi 0 01 and q 0 134 ifmmode pm else textpm fi 0 015 for this model the roughness exponent ensuremath alpha 0 383 ifmmode pm else textpm fi 0 008 is obtained accounting for a constant correction term intrinsic width in the scaling of the squared interface width this value is slightly below previous estimates of extensive simulations and rules out the proposal of the exact value ensuremath alpha 2 5 the conclusion is supported by results for the ballistic deposition model independent estimates of the dynamical exponent and of the growth exponent are 1 605l zl 1 64 and ensuremath beta 0 229 ifmmode pm else textpm fi 0 005 respectively which are consistent with the relations ensuremath alpha z 2 and z ensuremath alpha ensuremath beta

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

a virtual element method for stochastic stokes equations abstract in this paper a virtual element method for the stochastic stokes equations driven by an additive white noise is proposed and analyzed the velocity is approximated by the lowest order virtual element which is originally designed for the poisson equation and the projection is also taken as the one originally for the poisson equation while the pressure is approximated by the traditional discontinuous piecewise constant element for stable approximations we adopt a stabilization associating with the pressure jumps we show the inf sup condition and derive the stability we moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the green function numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained

stochastic galerkin methods for the steady state navier stokes equations we study the steady state navier stokes equations in the context of stochastic finite element discretizations specifically we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion for the resulting stochastic problem we formulate the model and linearization schemes using picard and newton iterations in the framework of the stochastic galerkin method and we explore properties of the resulting stochastic solutions we also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic galerkin nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems

conformal invariance of isoheight lines of the 2 1 dimensional wolf villain surfaces in order to learn more about the statistical properties of the surface fluctuations of the wolf villain wv model and confirm the applicability of the schramm loewner evolution theory sle to wv surfaces the contour lines of the 2 1 dimensional saturated surfaces of the wv model are investigated numerically the results obtained show that the isoheight lines of the wv surfaces are conformally invariant and can also be described in the frame of the sle theory with the diffusivity 4 the calculated fractal dimension of the contours is df 1 31 0 02

uncertainty measures problems concerning additivity additivity of an uncertainty measure on an mv algebra has a clear meaning if the divisibility is dropped we come up to a so called girard algebra there we discuss strong resp weak additivity based on so called divisible disjoint unions resp on additivity for all sub mv algebras we obtain a description of those extensions from additive measures on an mv algebra to the canonical girard algebra extension of pairs which are strongly additive and valuation measures finally we prove the non existence of strongly additive measure extensions if the underlying mv algebra is a finite chain with more than two non trivial elements

conditional measures and their applications to fuzzy sets abstract given a decomposable measure with respect to a continuous t conorm as introduced by the author in an earlier paper see section 1 we can construct conditional measures as implications these fulfil a generalized product law replacing the product in the classical law by any other strict t norm and turn out to be decomposable with respect to an operation v depending on and the condition set v section 2 more general conditional measures are introduced axiomatically and are shown to be conditional measures with respect to some decomposable measure section 3 bayesian like models are given which are alternatives to that presented by the author in a recent paper section 4 finally some applications to fuzzy sets are sketched section 5

conditioning on mv algebras and additive measures i abstract we present a lattice ordered semigroup approach for the foundation of conditional events which covers the special situations where the underlying unconditional events are boolean or fuzzy respectively our proposal is quite different from other ring theoretical approaches the problem of extending additivity of uncertainty measures from unconditional to conditional events will be discussed

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

conditioning a state by a lukasiewicz event a probabilistic approach to ulam games abstract the notion of state in an mv algebra generalizes the notion of finitely additive measure on a boolean algebra in this paper the conditioning of a state by a ukasiewicz event is examined with particular attention to ulam game

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

dynamic scaling and phase transitions in interface growth abstract the dynamic scaling approach is an effective tool for understanding the temporal evolution of fluctuating interfaces the surface width w obeys the scaling form w l t l t l where and are exponents which characterize how the surface width grows with the length scale l and the time t applications of dynamic scaling to a number of surface growth models are discussed the results of a large scale simulation of the ballistic deposition model in three dimensions are presented and compared with recent conjectures the question of universality in interface growth is addressed and a finite temperature generalization of the restricted solid on solid model is studied this model appears to undergo a phase transition from the usual rough phase at high temperatures to a new phase at low temperatures numerical solutions of the generalized langevin equation indicate that there is no phase transition in 2 1 dimensions thus the relation between the continuum equation and the discrete models is still an open question

scaling of clusters and winding angle statistics of isoheight lines in two dimensional kardar parisi zhang surfaces we investigate the statistics of iso height lines of 2 1 dimensional kardar parisi zhang model at different level sets around the mean height in the saturation regime we find that the exponent describing the distribution of the height cluster size behaves differently for level cuts above and below the mean height while the fractal dimensions of the height clusters and their perimeters remain unchanged the winding angle statistics also confirms again the conformal invariance of these contour lines in the same universality class of self avoiding random walks saws the study of surfaces their static statistical properties as well as growth and evolution dynamics and their morphological properties has been attracting an ever increasing amount of interest over the last two decades a method of characterizing surfaces is by looking at their iso height contour lines and the islands that are generated by a cut through the surface at a certain constant height such a situation can be found in the patterns exhibited by topographical islands and continents these islands and their coastlines have some fractal features with a fractal dimension related to the roughness exponent given by the structure function h h x h x r 2 i r 2 where h x is the height of interface as a function of its position theoretical modeling of the growth processes started with the work by edwards and wilkinson ew 1 who suggested that one might describe the dynamics of the height fluctuations by a simple linear diffusion equation kardar parisi and zhang kpz 2 realized that there is a relevant term proportional to the square of the height gradient which represents a correction for lateral growth the kpz equation is given by

a complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite mv chain we consider extensions of the unique additive measure on a finite mv chain to uncertainty measures on its canonical girard algebra extension if the underlying mv chain has more than two non trivial elements in a previous paper we have proved the non existence of strongly additive measure extensions where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so called divisible disjoint unions this negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all mv subalgebras of the canonical girard algebra extension we obtain a characterization of all such mv subalgebras which are in fact mv chains and using this result we come up to a characterization of all weakly additive measure extensions moreover we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions except for underlying mv chains with one or two or four non trivial elements

schramm loewner evolution theory of the asymptotic behaviors of 2 1 dimensional wolf villain model abstract although extensive analytical and numerical work has focus on investigating the 2 1 dimensional wolf villain wv model some problems concerning its asymptotical behaviors such as the universality class to which it belongs remain controversial the schramm loewner evolution sle theory is an attractive approach to describe the fluctuation phenomena and random processes and has also been applied to the analysis of the stochastic growth of surfaces in this work we applied sle theory to the analysis of the saturated surface to conduct in depth research with a new perspective on the asymptotical behaviors of the 2 1 dimensional wv model on the basis of the analysis of the saturated surface contour lines we determine that the diffusion coefficient calculated for 2 1 dimensional wv model is 2 91 0 01 and that for family model is 2 88 0 01 accordingly we can conclude from the view of sle theory that the 2 1 dimensional wv model similar to the family model also belongs to the edwards wilkinson universality class

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

topoi and categories of fuzzy sets abstract let h be a completely distributive lattice and hence a heyting algebra goguen s category of fuzzy sets set h eytan s logos fuz h and the topos of sheaves on h sh h are interconnected by pairs of adjoint functors between them each of the categories has a predicate calculus these predicate calculi are related through the functors between the categories change of base lattice gives rise to several functors which preserve or reflect specific kinds of statements in the predicate calculi this paper gives details of the categories the predicate calculi attached to them the functors between the categories and the preservation properties of those functors