KMS, etc

A general form of the ``Wick rotation'', starting from imaginary-time Green functions of quantum-mechanical systems in thermal equilibrium at positive temperature, is established. Extending work of H. Araki, the role of the KMS condition and of an associated anti-unitary symmetry operation, the ``modular conjugation'', in constructing analytic continuations of Green functions from real- to imaginary times, and back, is clarified. The relationship between the KMS condition for the vacuum with respect to Lorentz boosts, on one hand, and the spin-statistics connection and the PCT theorem, on the other hand, in local, relativistic quantum field theory is recalled. General results on the reconstruction of local quantum theories in various non-trivial gravitational backgrounds from ``Euclidian amplitudes'' are presented. In particular, a general form of the KMS condition is proposed and applied, e.g., to the Unruh- and the Hawking effects. This paper is dedicated to Huzihiro Araki on the occasion of his seventieth birthday, with admiration, affection and best wishes.Comment: 56 pages, submitted to J. Math. Phy

Warm, Trembling, Etc.

Poetry by Fiona Schicho

Using theoretical-computational conflicts to enrich the concept name of derivative

Recent literature has pointed out pedagogical obstacles associated with the use of computational environments in the learning of mathematics. In this paper, we focus on the pedagogical role of the computer's inherent limitations in the development of learners' concept images of derivative. In particular, we intend to discuss how the approach to this concept can be designed to prompt a positive conversion of those limitations for the enrichment of concept images. We present results of a case study with six undergraduate students in Brazil, dealing with situation of theoretical-computational conflict

Enumeration of points, lines, planes, etc

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points $E$ in a projective space determines at least $|E|$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of a $d$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(d-k)$-dimensional subspaces as there are $k$-dimensional subspaces, for every $k$ at most $d/2$. This confirms the "top-heavy" conjecture of Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.Comment: 18 pages, major revisio

Differential Uptake of Gold Nanoparticles by 2 Species of Tadpole, the Wood Frog (Lithobates Sylvaticus) and the Bullfrog (Lithobates Catesbeianus)

Engineered nanoparticles are aquatic contaminants of emerging concern that exert ecotoxicological effects on a wide variety of organisms. We exposed cetyltrimethylammonium bromide–capped spherical gold nanoparticles to wood frog and bullfrog tadpoles with conspecifics and in combination with the other species continuously for 21 d, then measured uptake and localization of gold. Wood frog tadpoles alone and in combination with bullfrog tadpoles took up significantly more gold than bullfrogs. Bullfrog tadpoles in combination with wood frogs took up significantly more gold than controls. The rank order of weight-normalized gold uptake was wood frogs in combination \u3e wood frogs alone \u3e bullfrogs in combination \u3e bullfrogs alone \u3e controls. In all gold-exposed groups of tadpoles, gold was concentrated in the anterior region compared with the posterior region of the body. The concentration of gold nanoparticles in the anterior region of wood frogs both alone and in combination with bullfrogs was significantly higher than the corresponding posterior regions. We also measured depuration time of gold in wood frogs. After 21 d in a solution of gold nanoparticles, tadpoles lost \u3e83% of internalized gold when placed in gold-free water for 5 d. After 10 d in gold-free water, tadpoles lost 94% of their gold. After 15 d, gold concentrations were below the level of detection. Our finding of differential uptake between closely related species living in similar habitats with overlapping geographical distributions argues against generalizing toxicological effects of nanoparticles for a large group of organisms based on measurements in only one species

Manifolds obtained by soldering together points, lines, etc

This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. We explore the notion of solder (or soldering form) in differential geometry and propose an alternative interpretation of it, motivated by the search of an accurate mathematical description of the General Relativity. This new interpretation leads naturally to imagine a rich family of new geometries which has not yet a satisfactory definition in general. We try however to communicate to the reader an intuition of such geometries through some examples and review quickly some possible applications in physics. The basic objects in this geometry are not points (i.e. 0-dimensional), but (p-1)-dimensional.Comment: in Geometry, Topology, QFT and Cosmology, Paris : France (2008