On the completeness of factor rings

Let be a complete local domain containing the integers with maximal ideal such that is at least the cardinality of the real numbers. Let be a nonmaximal prime ideal of such that is a regular local ring. We construct an excellent local ring such that the completion of is , the generic formal fiber of is local with maximal ideal and if is a nonzero ideal of , then is complete.

     

Characterization of completions of reduced local rings

We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring with maximal ideal are (i) or , and (ii) for all , if is an integer of , then .

     

Completion of Semiuniform Convergence Spaces

Semiuniform convergence spaces form a common generalization of filter spaces (including symmetric convergence spaces [and thus symmetric topological spaces] as well as Cauchy spaces) and uniform limit spaces (including uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. Here, for each semiuniform convergence space a completion is constructed, called the simple completion. This one generalizes Császár's -completion of filter spaces. Thus, filter spaces are characterized as subspaces of convergence spaces. Furthermore, Wyler's completion of separated uniform limit spaces can be easily derived from the simple completion.     

Exponential iterated integrals and the relative solvable completion of the fundamental group of a manifold

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K.T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of π₁(M,x) can be expressed via these new integrals. The ring of exponential iterated integrals contains the coordinate rings for a class of universal representations, called the relative solvable completions of π₁(M,x). We consider exponential iterated integrals in the particular case of fibered knot complements, where the fundamental group always has a faithful relative solvable completion.     

Positive extension and completion problems for a class of structured matrices

A class of matrices, defined by a displacement rank property, is introduced. Completion and extension problems are studied for matrices in this class, under certain positivity constraints. The extension problem is reduced to a standard interpolation problem for Schur matrix valued functions.     

Cauchy Complete Nearness Spaces

A nearness space is Cauchy complete if every regular Cauchy filter on the space is convergent. We show that the category CCNear ₂ of Cauchy complete N ₂ spaces is reflective in the category Near ₂ C of N ₂-spaces and Cauchy maps and that the reflection of an N ₂-space is given by the strict extension associated with regular Cauchy filters on the space.     

Completions of Goldschmidt Amalgams of Type G4 in Dimension 3

The subgroups of GL₃(k) which are completions of the Goldschmidt G ₄-amalgam are determined. We also draw attention to five related graphs which are remarkable in that they have large girth and few vertices.     

Characterizing and computing minimal cograph completions

A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a characterization that allows us to check in linear time whether a given cograph completion is minimal. The second result is a vertex incremental algorithm to compute a minimal cograph completion H of an arbitrary input graph G in O(|V(H)|+|E(H)|) time. An extended abstract of the result has been already presented at FAW 2008 [D. Lokshtanov, F. Mancini, C. Papadopoulos, Characterizing and computing minimal cograph completions, in: Proceedings of FAW’08-2nd International Frontiers of Algorithmics Workshop, in: LNCS, vol. 5059, 2008, pp. 147158. [1]].     

Solution of the quadratically hyponormal completion problem

For , let be a collection of () positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on to guarantee the existence of a quadratically hyponormal unilateral weighted shift with as the initial segment of weights. We prove that admits a quadratically hyponormal completion if and only if the self-adjoint matrix

is positive and invertible, where , , , , , and, for notational convenience, . As a particular case, this result shows that a collection of four positive numbers always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.

     

Cotorsion theories and splitters

Let be a subring of the rationals. We want to investigate self splitting -modules (that is . Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism -algebras with a free -module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.