first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

the solutions of linear fuzzy stochastic differential systems as a completion of previous work of feng fuzzy sets and systems 115 2000 351 on the general theory of fuzzy stochastic differential systems the solutions of linear fuzzy stochastic differential systems for general coefficient matrix are discussed

mixed covolume methods for elliptic problems on triangular grids we consider a covolume or finite volume method for a system of first order pdes resulting from the mixed formulation of the variable coefficient matrix poisson equation with the neumann boundary condition the system may represent either the darcy law and the mass conservation law in anisotropic porous media flow or fourier law and energy conservation the velocity and pressure are approximated by the lowest order raviart thomas space on triangles we prove its first order optimal rate of convergence for the approximate velocities in the l 2 and h mbox div omega norms as well as for the approximate pressures in the l 2 norm numerical experiments are included

a new iterative method in waring s problem on introduit une nouvelle methode iterative dans le probleme de waring on ameliore toutes les bornes superieures anterieures pour g k lorsque k est superieur ou egal a 5

mean square integral and differential of fuzzy stochastic processes abstract this paper defines the concepts of integral and differential in mean square associated with a class of fuzzy stochastic processes and studies their integrability and differentiability properties the definition and the mean square calculus properties of a gaussian fuzzy stochastic process are also discussed

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

the hasse principle for systems of diagonal cubic forms we establish the hasse principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r times r submatrix thereby achieving the theoretical limit of the circle method for such systems

the cubic case of the main conjecture in vinogradov s mean value theorem abstract we apply a variant of the multigrade efficient congruencing method to estimate vinogradov s integral of degree 3 for moments of order 2 s establishing strongly diagonal behaviour for 1 s 6 consequently the main conjecture is now known to hold for the first time in a case of degree exceeding 2

multigrid algorithms for a vertex centered covolume method for elliptic problems we analyze v cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem as an application we study the convergence of multigrid algorithms for a covolume method or a vertex centered finite volume element method for variable coefficient elliptic problems on polygonal domains as in standard finite element methods the v cycle algorithm with one pre smoothing converges with a rate independent of the number of levels various types of smoothers including point or line jacobi and gauss seidel relaxation are considered

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

symmetric positive definite flux continuous full tensor finite volume schemes on unstructured cell centered triangular grids novel cell centered full tensor finite volume methods are presented for general unstructured grids in two spatial dimensions the numerical schemes are flux continuous and based on computing the transmissibilities in a local subcell transform space ensuring that local flux matrices are symmetric as a result the global discretization matrix is shown to be symmetric positive definite for any grid type a symmetric physical space method is also introduced and the symmetric methods are shown to be closely related discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme the schemes are compared with the so called multipoint flux approximation mpfa o method i aavatsmark t barkve o boe and t mannseth siam j sci comput 19 1998 pp 1700 1716 good agreement between the methods is obtained but the new scheme shows improved behavior in challenging cases

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

a mass conservative generalized multiscale finite element method applied to two phase flow in heterogeneous porous media in this paper we propose a method for the construction of locally conservative flux fields through a variation of the generalized multiscale finite element method gmsfem the flux values are obtained through the use of a ritz formulation in which we augment the resulting linear system of the continuous galerkin cg formulation in the higher order gmsfem approximation space in particular we impose the finite volume based restrictions through incorporating a scalar lagrange multiplier for each mass conservation constraint this approach can be equivalently viewed as a constraint minimization problem where we minimize the energy functional of the equation restricted to the subspace of functions that satisfy the desired conservation properties to test the performance of the method we consider equations with heterogeneous permeability coefficients that have high variation and discontinuities and couple the resulting fluxes to a two phase flow model the increase in accuracy associated with the computation of the gmsfem pressure solutions is inherited by the flux fields and saturation solutions and is closely correlated to the size of the reduced order systems in particular the addition of more basis functions to the enriched multiscale space produces solutions that more accurately capture the behavior of the fine scale model a variety of numerical examples are offered to validate the performance of the method

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

on high order conservative finite element methods abstract we describe and analyze a volumetric and residual based lagrange multipliers saddle point reformulation of the standard high order finite method to impose conservation of mass constraints for simulating the pressure equation on two dimensional convex polygons with sufficiently smooth solution and mobility phase we establish high order a priori error estimates with locally conservative fluxes and numerical results are presented that confirm the theoretical results

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

on embedding problems of fuzzy number spaces iv in this paper we are going to follow the preceding papers to investigate the embedding problem of fuzzy number space ne 1 by means of a concrete embedding operator using the embedding operator some subclasses of ne 1 and metrices on ne 1 are also investigated

gaussian fuzzy random variables in this paper we define the concepts of a gaussian fuzzy random variable and a gaussian fuzzy random vector and discuss their properties the covariance operator the characteristic functional and the linear transformation of a gaussian fuzzy random variable are investigated

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

unified analysis of finite volume methods for second order elliptic problems we establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions under the framework optimal order convergence in the h 1 and l 2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finite volume methods applied to second order elliptic problems

subconvexity for additive equations pairs of undenary cubic forms we investigate pairs of diagonal cubic equations with integral coefficients for a class of such diophantine systems with 11 or more variables we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

fuzzy stochastic differential systems abstract basing on the work of feng fuzzy sets and systems 102 1999 271 280 on mean square calculus for fuzzy stochastic processes we discuss in this paper the general theory of fuzzy stochastic differential systems including the existence and uniqueness of a solution the continuity of the solution with respect to the initial value and the stability of systems when there are perturbations of the coefficients and the initial conditions the explicit representation of solutions for the first order linear fuzzy stochastic systems are established

on waring s problem for cubes and smooth weyl sums non trivial estimates for fractional moments of smooth cubic weyl sums are developed complemented by bounds for such sums of use on both the major and minor arcs in a hardy littlewood dissection these estimates are applied to derive an upper bound for the s th moment of the smooth cubic weyl sum of the expected order of magnitude as soon as s ge 7 691 related arguments demonstrate that all large integers n are represented as the sum of eight cubes of natural numbers all of whose prime divisors are at most exp c log n log log n 1 2 for a suitable positive number c this conclusion improves a previous result of g harcos in which nine cubes are required 1991 mathematics subject classification 11p05 11l15 11p55

a new class of high order finite volume methods for second order elliptic equations in the numerical simulation of many practical problems in physics and engineering finite volume methods are an important and popular class of discretization methods due to the local conservation and the capability of discretizing domains with complex geometry however they are limited by low order approximation since most existing finite volume methods use piecewise constant or linear function space to approximate the solution in this paper a new class of high order finite volume methods for second order elliptic equations is developed by combining high order finite element methods and linear finite volume methods optimal convergence rate in h 1 norm for our new quadratic finite volume methods over two dimensional triangular or rectangular grids is obtained

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

the hasse principle for pairs of diagonal cubic forms by means of the hardy littlewood method we apply a new mean value theorem for exponential sums to confirm the truth over the rational numbers of the hasse principle for pairs of diagonal cubic forms in thirteen or more variables

the asymptotic formulae in waring s problem for cubes write rs n for the number of representations of a natural number as the sum of s positive integral cubes we investigate the higher moments of rs n establishing asymptotic formulae for n n rs n h when s 5 and h 3 when s 6 and h 28 5 and when s 7 and h 12 hitherto such an asymptotic formula was available only when s 4 and h 2

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

error estimates for a mixed finite volume method for the p laplacian problem in this work we propose and analyze a mixed finite volume method for the p laplacian problem which is based on the lowest order raviart thomas element for the vector variable and the p1 nonconforming element for the scalar variable it is shown that this method can be reduced to a p1 nonconforming finite element method for the scalar variable only one can then recover the vector approximation from the computed scalar approximation in a virtually cost free manner optimal a priori error estimates are proved for both approximations by the quasi norm techniques we also derive an implicit error estimator of bank weiser type which is based on the local neumann problems

rational solutions of pairs of diagonal equations one cubic and one quadratic we obtain an essentially optimal estimate for the moment of order 32 3 of the exponential sum having argumentx 3 x 2 subject to modest local solubility hypotheses we thereby establish that pairs of diagonal diophantine equations one cubic and one quadratic possess non trivial integral solutions whenever the number of variables exceeds 10

iterative galerkin enriched multiscale finite volume method the multiscale finite volume msfv method was designed to efficiently compute approximate numerical solutions of elliptic and parabolic problems with highly heterogeneous coefficients for a wide range of test cases it has been demonstrated that msfv solutions are in excellent agreement with reference data obtained with a standard fine scale finite volume method however for some problems involving large coherent structures with strong contrasts e g channels and shale layers the msfv method produces unsatisfactory results due to the dwindling validity of the localization assumption recently the iterative msfv i msfv method was introduced to iteratively improve the localization assumption and it was shown that the method converges to the fine scale reference solution in this work it is explained how the convergence rate of the i msfv method can be enhanced by consistent enrichment of the initial coarse space spanned by nodal basis functions a new set of enrichment basis functions associated with additional coarse scale degrees of freedom dof is introduced by construction the sum of all nodal and enrichment basis functions is one in the entire computational domain further a hybrid finite volume galerkin formulation for the coarse scale problem and a generic placement strategy for the additional dof are devised the resulting iterative galerkin enriched msfv i ge msfv method has all features of the i msfv method but is more robust and has improved convergence properties

a triangle based mixed finite element finite volume technique for modeling two phase flow through porous media triangle based discretization techniques offer great advantages relative to standard finite difference methods for the modeling of flow through geometrically complex geological features the purpose of this paper is to develop and apply a triangle based method for the modeling of two phase flow through porous formations the formulation includes the effects of gravity compressibility and capillary pressure the technique entails a triangle based mixed finite element method for solution of the variable coefficient parabolic pressure equation and a second order tvd type total variation diminishing finite volume scheme for solution of the essentially hyperbolic saturation equation the method is applied to a variety of example problems and is shown to perform very well on problems involving geometric complexity coupled with heterogeneous generally anisotropic permeability descriptions

a convergence analysis of generalized multiscale finite element methods abstract in this paper we consider an approximation method and a novel general analysis for second order elliptic differential equations with heterogeneous multiscale coefficients we obtain convergence of the generalized multi scale finite element method gmsfem method that uses local eigenvectors in its construction the analysis presented here can be extended without great difficulty to more sophisticated gmsfems for concreteness the obtained error estimates generalize and simplify the convergence analysis of j comput phys 230 2011 937 955 the gmsfem method construct basis functions that are obtained by multiplication of approximation of local eigenvectors by partition of unity functions only important eigenvectors are used in the construction the error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction the error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space