Engineering design of hydraulic fracturing proppants typically focuses on maximizing permeability retention under stress, resistance to high temperatures, and controlling properties such as specific gravity and particle size; less attention is paid to the proppant's wetting characteristics and their potential impact on the deliverability of a hydraulic fracture. The purpose of this work is to investigate the effect of proppant wettability on the conductivity of hydraulic fractures through laboratory measurements. Results demonstrate a competing effect between permeability and wettability, the latter having a diminished impact on displacement efficiency at high permeability values.

The Gröbner bases of the ideal generated by the symmetric functions are presented. Moreover we show how they can be fruitfully applied to a problem arising from coding theory.

Duality was introduced in Computer Algebra in 1982 by MoIler and since that has been widely used. We give a survey of Moller algorithm and its applications, presenting a new one to the computation of canonical modules. "Its dual application" allow us to give answer to a question posed to us by Stetter.

A generalization of the FGLM technique is given to compute Gröbner bases for two-sided ideals of free finitely generated algebras. Specializations of this algorithm are presented for the cases in which the ideal is determined by either functionals or monoid (group) presentations. Generalizations are discussed in order to compute Gröbner bases on (twisted) semigroup rings.

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.

In 1965, Buchberger introduced the notion of Grobner bases for a polynomial ideal and an algorithm (Buchberger algorithm) for their computation; since the end of the seventies, Grobner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in Algebraic Geometry and Commutative Algebra; moreover, Grobner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras, of interest in Differential Algebra (e.g. Weyl algebras, enveloping algebras of Lie algebras).The aim of this paper is to give an introduction, as elementary as I was able to make it, to both commutative and noncommutative algebras: Grobner bases are in a sense a finite model of an infinite linear Gauss-reduced basis of an ideal viewed as a vector space and Buchberger algorithm is the corresponding generalization of the Gaussian elimination algorithm.Moreover the paper contains a survey of some applications of Buchberger theory to noncommutative algebras; together with these results surveyed, this paper contains some minor new points: e.g. the ''useless pair criteria'' in the noncommutative case and the final result on the existence and ''computability'' of Grobner bases for two-sided ideals in any finitely presented algebra.

We present an efficient algorithm for the transformation of a Gröbner basis of a zero-dimensional ideal with respect to any given ordering into a Gröbner basis with respect to any other ordering. This algorithm is polynomial in the degree of the ideal. In particular the lexicographical Gröbner basis can be obtained by applying this algorithm after a total degree Gröbner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchberger's algorithm.

In this paper we study 0-dimensional polynomial ideals defined by a dual basis, i.e. as the set of polynomials which are in the kernel of a set of linear morphisms from the polynomial ring to the base field. For such ideals, we give polynomial complexity algorithms to compute a Gröbner basis, generalizing the Buchberger-Möller algorithm for computing a basis of an ideal vanishing at a set of points and the FGLM basis conversion algorithm.

Up to now, no computational model is known to perform effective commutative algebra for ideals in a computable ring of formal power series, while the theory for this is quite developed at least since [7]. The particular case of algebraic formal power series comes out naturally, when studying singular points of algebraic varieties, for instance, in the NewtonPuiseux algorithm for determining the analytic branches of a curve at a singular point and, more generally, when studying analytic components of a complex algebraic variety. We propose here to develop a computational model for algebraic formal power series, already introduced in [l], based on a symbolic codification of the series by means of the Implicit Function Theorem, i.e., we will consider algebraic series as the unique solutions of suitable functional equations, which we call Locally Smooth Systems. We then reduce the problem of handling a finite set of algebraic series to some corresponding problem involving suitable polynomial rings. In this model we will show that most of the usual local commutative algebra can be effectively performed on algebraic series, since we can reduce to the polyno-

This paper discusses a general lifting technique for solving polynomial equations in gradedstructures A, where the solution is understood to lie in the completion A^@^ of A. It shows that the classical Hensel lifting and the main constructions related to the Buchberger algorithm for Grobner bases are both instances of this technique. So, while the setting of it is too general to allow for an effective solution of equations, this technique stresses a theoretical relation between two basic algorithms in computer algebra and could be used as a theoretical model to attack computation problems under the same viewpoint.

To every ideal I in the polynomial ring A:=k[X"1,..., X"n] new invariants are attached, such as the Grobner Fan F(I), the Grobner region G(I) and the set ATO(I) of the almost term-orderings of I, i.e. orderings which ''behave like'' term-orderings with respect to I. These invariants arise by considering the reduced Grobner bases of I with respect to all the term-orderings. Moreover F(I), G(I), ATO(I) can be got in a constructive way, as we show by producing a suitable algorithm which computes them.