A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres

We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on \zeta behaves like an ``analytic function'' defined on the set of roots of unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q=1.Comment: 66 pages, 8 figure

The critical loads and levels approach for nitrogen

This chapter reports the findings of a Working Group to review the critical loads (CLs) and levels approach for nitrogen (N). The three main approaches to estimating CLs are empirical, mass balance and dynamic modelling. Examples are given of recent developments in Europe, North America and Asia and it is concluded that other countries should be encouraged to develop basic assessments using soil, land cover, and deposition map overlays in order to determine what regions might exceed nitrogen CLs. There is a need for increasing the certainty of critical load (CL) estimates by focusing on empirical data needs, especially for understudied ecosystems such as tropical or Mediterranean, high elevation environments, and aquatic systems. There is also a need to improve steady-state mass balance parameters, especially soil solution terms, such as nitrate leaching, used to determine the CL, and denitrification, which is an equation parameter. Improved dynamic models are needed for predicting plant community changes, and work should continue on existing models to determine CL values. Dynamic models require more data and are more complex than simple calculated CLs but offer more information and allow the development of ‘what if?’ scenarios. Optimal use of CLs requires expert knowledge of ecosystem values to provide reference states so that safe deposition amounts can be determined. Increased interaction between CL and biodiversity specialists to identify critical biodiversity limits would help provide better CL assessments