A Kato's second type representation theorem for solvable sesquilinear forms

Kato's second representation theorem is generalized to solvable sesquilinear forms. These forms need not be non-negative nor symmetric. The representation considered holds for a subclass of solvable forms (called hyper-solvable), precisely for those whose domain is exactly the domain of the square root of the modulus of the associated operator. This condition always holds for closed semibounded forms, and it is also considered by several authors for symmetric sign-indefinite forms. As a consequence, a one-to-one correspondence between hyper-solvable forms and operators, which generalizes those already known, is established.Comment: 20 page

A Survey on Solvable Sesquilinear Forms

The aim of this paper is to present a unified theory of many Kato type representation theorems in terms of solvable forms on Hilbert spaces. In particular, for some sesquilinear forms $\Omega$ on a dense domain $\mathcal{D}$ one looks for an expression $$\Omega(\xi,\eta)=\langle T\xi , \eta\rangle, \qquad \forall \xi\in D(T),\eta \in \mathcal{D},$$ where $T$ is a densely defined closed operator with domain $D(T)\subseteq \mathcal{D}$. There are two characteristic aspects of solvable forms. Namely, one is that the domain of the form can be turned into a reflexive Banach space need not be a Hilbert space. The second one is the existence of a perturbation with a bounded form which is not necessarily a multiple of the inner product.Comment: 11 page

Finitely generated abelian groups of units

In 1960 Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper we address Fuchs' question for {\it finitely generated abelian} groups and we consider the problem of characterizing those groups which arise in some fixed classes of rings $\mathcal C$, namely the integral domains, the torsion free rings and the reduced rings. To determine the realizable groups we have to establish what finite abelian groups $T$ (up to isomorphism) occur as torsion subgroup of $A^*$ when $A$ varies in $\mathcal C$, and on the other hand, we have to determine what are the possible values of the rank of $A^*$ when $(A^*)_{tors}\cong T$. Most of the paper is devoted to the study of the class of torsion-free rings, which needs a substantially deeper study.Comment: 28 page

Links of prime ideals and their Rees algebras

In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number $1$. This led to the construction of large families of Cohen--Macaulay Rees algebras. The first goal of this paper is to extend this result to arbitrary Cohen--Macaulay rings. The means of the proof are changed since one cannot depend so heavily on linkage theory. We then study the structure of the Rees algebra of these links, more specifically we describe their canonical module in sufficient detail to be able to characterize self--linked prime ideals. In the last section multiplicity estimates for classes of such ideals are established

The use of tethered satellites for the collection of cosmic dust and the sampling of man made orbital debris far from the space station

The use of a tethered subsatellite employed downward into the earth's upper atmosphere to an altitude of about 110 km above the earth would eliminate the orbital contamination problem while at the same time affording a measure of atmospheric braking to reduce the velocities of many particles to where they may be captured intact or nearly so with properly designed collectors. The same technique could also be used to monitor the flux of all types of man-made orbital debris out to a distance of more than a hundred kilometers in any direction from the space station. In this way the build up of any debris belt orbiting earth could be determined. The actual collecting elements used for both purposes could be of several different materials and designs so as to optimize the collection of different types of particles with different densities. Stacks of foils, films, plastics, and foams, as well as simple capture cells would be mounted in clusters around the outside of a tethered satellite and protected by iris covers until the tethered had been fully deployed. If the orientation history of the satellite were known the direction of the incoming material could be infered. A chief advantage in deploying such tethered collectors from the Space Station instead of from the shuttle is the ability to maintain deployment of the tether for days instead of hours resulting in much greater yields of intact particles and impact debris