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We consider resolvents (A ϵ + 1) − 1 of elliptic second-order differential operators A ϵ = − div a (x / ϵ) ∇ in R d with ε-periodic measurable matrix a (x / ϵ) and study the asymptotic behaviour of (A ϵ + 1) − 1 , as the period ε goes to zero. We provide a construction for the leading terms of the 'operator asymptotics' of (A ϵ + 1) − 1 in the sense of L 2 -operator-norm convergence and prove order ϵ 2 remainder estimates. We apply the modified method of the first approximation with the usage of Steklov's smoothing. The class of operators covered by our analysis includes uniformly elliptic families with bounded coefficients and also with unbounded coefficients from the John–Nirenberg space BMO (bounded mean oscillation). [ABSTRACT FROM AUTHOR]

As the nineteenth book of his Decretum, Bishop Burchard of Worms (d. 1025) included a penitential titled the Corrector or Medicus. Although the Corrector has traditionally been described as a manual of private confession, this article argues that it was primarily used to judge public penances at church courts and synods. Focusing on the manuscript transmission of the Decretum in the eleventh and twelfth centuries, we find that the Corrector was almost never copied apart from the rest of the Decretum and that it was almost always copied alongside other legal texts. Turning to individual abbreviations, we also find that Burchard's collection was frequently reworked and rearranged so as to integrate canonical and penitential texts. These findings call into question current definitions of canon law collections and penitentials which presume functional exclusivity. [ABSTRACT FROM AUTHOR]

A divergent-type elliptic operatorof arbitrary even order 2mis studied. Coefficients of the operator are-periodic,is a small parameter. The resolvent equationis solvable in the Sobolev spaceof ordermfor any, provided the parameteris sufficiently large,, where the bounddepends only on constants from ellipticity condition. The limit equation is of the same type but with constant coefficients, that is,. The limit operatorcan be considered here, for instance, in the sense ofG-convergence. We prove that the resolventapproximatesin operator-norm with the estimate, as. We find also the approximation of the resolventin operator-norm. This is the sum, whereis a correcting operator whose structure is given. We prove the estimate, as. [ABSTRACT FROM PUBLISHER]

In, we consider a selfadjoint operator,, given by the differential expression, whereis the first-order differential operator, andare matrix-valued functions inperiodic with respect to some lattice. It is assumed thatgis bounded and positive definite, whileandQare, in general, unbounded. We study the generalized resolvent, whereis a-periodic, bounded and positive definite matrix-valued function, andis a complex-valued parameter. Approximations for the generalized resolvent in the- and-norms with two-parametric error estimates (with respect to the parametersand) are obtained. [ABSTRACT FROM PUBLISHER]

The postbuckling response of structures with multi winding loops is characterized by the appearance of multi adjacent equilibrium paths, which often makes the iterations difficult to converge to the desired path. In this paper, some key issues for tracing the postbuckling response of a structure using an incremental-iterative approach are discussed. Concerning the finite element equations used, it is essential that the corrector used for recovering the element forces from the element displacements be made as accurate as possible, and that the predictor for computing the structural displacements under given load increments, which are approximate by nature due to linearization involved, be accurate to the level not to misguide the iterations. As for the incremental-iterative scheme, it is required to be: (1) numerically stable when encountering the limit points, (2) adjustable in load increments to reflect the stiffness variation, and (3) self-adaptive in changing the loading direction. To demonstrate the ideas involved, some examples with highly complicated postbuckling responses will be solved in this paper. [ABSTRACT FROM AUTHOR]

A P-stable linear multistep method for solving general third order initial value problems of ordinary differential equations without first reducing the problems into a system of first order equations is considered. The approach for the development of this method is essentially based on collocation of the differential system generated from a basis function. A predictor for the evaluation of $y_{n+k}$ for an odd $k\ge 3$ in the main method is also proposed. The two resulting methods, the corrector and the predictor are P-stable for $k = 3$ . These as a block are tested on a number of problems to show their efficiency. When the methods (the corrector and the predictor) are evaluated at $x = x_{n+3}$ identical schemes are obtained as special cases of the methods, while the set of first and second derivatives obtained from the corrector are different from those obtained from the predictor. [ABSTRACT FROM AUTHOR]