Fusion Rings Related to Affine Weyl Groups

The construction of the fusion ring of a quasi-rational CFT based on $\hat{sl}(3)_k$ at generic level $k\not \in {\Bbb Q}$ is reviewed. It is a commutative ring generated by formal characters, elements in the group ring ${\Bbb Z}[\tilde{W}]$ of the extended affine Weyl group $\tilde{W}$ of $\hat{sl}(3)_k$. Some partial results towards the $\hat{sl}(4)_k$ generalisation of this character ring are presented.Comment: 13 pages; two figures. Talk at ``Lie Theory and Its Applications in Physics III'', Clausthal, 11-14 July, 1999, to appear in the Proceedings, eds. H.-D. Doebner et a

An Extension of the Character Ring of sl(3) and Its Quantisation

We construct a commutative ring with identity which extends the ring of characters of finite dimensional representations of sl(3). It is generated by characters with values in the group ring $Z[\tilde{W}]$ of the extended affine Weyl group of $\hat{sl}(3)_k$ at $k\not \in Q$. The `quantised' version at rational level $k+3=3/p$ realises the fusion rules of a WZW conformal field theory based on admissible representations of $\hat{sl}(3)_k$.Comment: contains two TeX files: main file using harvmac.tex, amssym.def, amssym.tex, 35p.; file with figures using XY-pic package, 4p; v2: minor corrections, Note adde

Temperature and Mortality in New York City: Past, Present and Future

The complex interplay between climate change, demographics and socioeconomic conditions is transforming the global environmental health landscape. In the aftermath of recent heat waves around the world, especially the 2003 heat wave in Europe, heat is being recognized as an emerging public health issue worldwide, particularly in urban areas. This work explores the historical and future heat-related mortality in New York City, from the beginning of the 20th until the end of the 21st century. New York City is among the largest cities in the world and has been a thriving metropolis over the entire period covered by this study. The unique makeup of the city makes it particularly suitable for studying the impacts of heat over an extended period of time. The presented work encompasses multiple domains of knowledge and illustrates the necessity for applying highly interdisciplinary approaches in addressing the emerging challenges of our time. The background chapter provides an overview of methodological approaches and findings from previous studies with direct relevance to the specific aims of this work. Chapter I is focused on characterizing the impacts of heat on daily mortality since 1900. Here, heat effects are presented in a historical context and changes over time are analyzed and discussed. Chapter II provides a comparative assessment of recent historical and heat impacts until 2100 in New York City, Boston and Philadelphia. This analysis illustrates the differences and similarities between heat impacts in New York City and the other two major urban areas in the U.S. Northeast. Chapter III provides a more comprehensive assessment of future heat-related mortality in New York City under a number of adaptation, climate change and demographic scenarios. The concluding chapter presents a summary of findings and recommendations for future research

Non-critical string pentagon equations and their solutions

We derive pentagon type relations for the 3-point boundary tachyon correlation functions in the non-critical open string theory with generic c_{matter} < 1 and study their solutions in the case of FZZ branes. A new general formula for the Liouville 3-point factor is derived.Comment: 18 pages, harvmac; misprints corrected, section 3.2 extended, a new general formula for the Liouville 3-point factor adde

Conformal Field Theories, Graphs and Quantum Algebras

This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions --open, twisted-- are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract ``quantum'' algebra, whose $6j$- and $3j$-symbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of ``Weak $C^*$- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001 --Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in Math., Birkhauser. References and comments adde