Finite domination and Novikov rings. Iterative approach

Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x -1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ? L R((x)) and C ? L R((x -1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology 34(3) (1995), 619–632). Here R((x)) = R[[x]][x -1] and R((x -1)) = R[[x -1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

Finite domination and Novikov rings: Laurent polynomial rings in two variables

Let C be a bounded cochain complex of finitely generatedfree modules over the Laurent polynomial ring L = R[x, x−1, y, y−1].The complex C is called R-finitely dominated if it is homotopy equivalentover R to a bounded complex of finitely generated projective Rmodules.Our main result characterises R-finitely dominated complexesin terms of Novikov cohomology: C is R-finitely dominated if andonly if eight complexes derived from C are acyclic; these complexes areC ⊗L R[[x, y]][(xy)−1] and C ⊗L R[x, x−1][[y]][y−1], and their variants obtainedby swapping x and y, and replacing either indeterminate by its inverse.

Finite domination and Novikov rings. Laurent polynomial rings in several variables

We present a homological characterisation of those chain complexes of modules over a Laurent polynomial ring in several indeterminates which are finitely dominated over the ground ring (that is, are a retract up to homotopy of a bounded complex of finitely generated free modules). The main tools, which we develop in the paper, are a non-standard totalisation construction for multi-complexes based on truncated products, and a high-dimensional mapping torus construction employing a theory of cubical diagrams that commute up to specified coherent homotopies.

Finite domination and Novikov homology over strongly Z-graded rings

Let L be a unital Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent to a bounded complex of finitely generated projective L0-modules, generalising known results for twisted Laurent polynomial rings. The crucial hypothesis is that L is a strongly graded ring. 

The conjugacy problem for two-by-two matrices over polynomial rings

We give an effective solution of the conjugacy problem for two-by-two matrices over the polynomial ring in one variable over a finite field.

A dynamic mechanical method for determining the silicone elastomer solubility of drugs and pharmaceutical excipients in silicone intravaginal drug delivery rings

The silicone elastomer solubilities of a range of drugs and pharmaceutical excipients employed in the development of silicone intravaginal drug delivery rings (polyethylene glycols, norethisterone acetate, estradiol, triclosan, oleyl alcohol, oxybutynin) have been determined using dynamic mechanical analysis. The method involves measuring the concentration-dependent decrease in the storage modulus associated with the melting of the incorporated drug/excipient, and extrapolation to zero change in storage modulus. The study also demonstrates the effect of drug/excipient concentrations on the mechanical stiffness of the silicone devices at 37°C.

In vitro release of nonoxynol-9 from silicone matrix intravaginal rings

The controlled-release characteristics of matrix silicone intravaginal rings loaded with between 100 and 971 mg of nonoxynol-9 have been investigated with a view to developing a ring that may offer a new female-controlled method for the prevention of transmission of sexually transmitted diseases, particularly HIV. Intravaginal rings containing 253, 487 and 971 mg of nonoxynol-9 provided a daily release of 2 mg or more over the 8-day release period, the minimal amount of nonoxynol-9 considered to provide an effective vaginal concentration for the prevention of HIV. Furthermore, the maximum daily release of N9 was about 6 mg, an amount significantly smaller than that observed for other nonoxynol-9 products whose large daily doses may in fact increase the occurrence of HIV by causing epithelial damage to the vaginal tissue. The release mechanism of the liquid nonoxynol-9 from the intravaginal rings has also been investigated and compared to models describing the release of solid drugs from the rings. It has been demonstrated through release studies and surface microscopy that a drug depletion zone is not established in such liquid-loaded intravaginal ring systems, with implications for the release kinetics. (C) 2003 Elsevier B.V. All rights reserved.

Influence of silicone elastomer solubility and diffusivity on the in vitro release of drugs from intravaginal rings

The in vitro release characteristics of eight low-molecular-weight drugs (clindamycin, 17beta-estradiol, 17beta-estradiol-3-acetate, 17beta-estradiol diacetate, metronidazole, norethisterone, norethisterone acetate and oxybutynin) from silicone matrixtype intravaginal rings of various drug loadings have been evaluated under sink conditions. Through modelling of the release data using the Higuchi equation, and determination of the silicone solubility of the drugs, the apparent silicone elastomer diffusion coefficients of the drugs have been calculated. Furthermore, in an attempt to develop a quantitative model for predicting release rates of new drug substances from these vaginal ring devices, it has been observed that linear relationships exist between the log of the silicone solubility of the drug (mg ml(-1)) and the reciprocal of its melting point (K-1) (y = 3.558x - 9.620, R = 0.77), and also between the log of the diffusion coefficient (cm(2) s(-1)) and the molecular weight of the drug molecule (g mol(-1)) (y = - 0.0068x - 4.0738, R = 0.95). Given that the silicone solubility and silicone diffusion coefficient are the major parameters influencing the permeation of drugs through silicone elastomers, it is now possible to predict through use of the appropriate mathematical equations both matrix-type and reservoir-type intravaginal ring release rates simply from a knowledge of drug melting temperature and molecular weight. (C) 2003 Elsevier Science B.V. All rights reserved.