A short proof of the Buchstaber-Rees theorem

We give a short proof of the Buchstaber-Rees theorem concerning symmetric powers. The proof is based on the notion of a formal characteristic function of a linear map of algebras.Comment: 11 pages. LaTeX2

Geometric constructions on the algebra of densities

The algebra of densities \Den(M) is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra \Den(M) is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra \Den(M). It allows a natural definition of bracket operations on vector densities of various weights on a (super)manifold $M$, similar to how the classical Fr\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from "vector fields" (derivations) on \Den(M) to "multivector fields". This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.Comment: LaTeX, 23 p

On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory

The classical Kolmogorov-Gelfand theorem gives an embedding of a (compact Hausdorff) topological space X into the linear space of all linear functionals C(X)^* on the algebra of continuous functions C(X). The image is specified by algebraic equations: f(ab)=f(a)f(b) for all functions a, b on X; that is, the image consists of all algebra homomorphisms of C(X) to numbers. Buchstaber and Rees have found that not only X, but all symmetric powers of X can be embedded into the space C(X)^*. The embedding is again given by algebraic equations, but more complicated. Algebra homomorphisms are replaced by the so-called "n-homomorphisms", the notion that can be traced back to Frobenius, but which explicitly appeared in Buchstaber and Rees's works on multivalued groups. We give a further natural generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory. Symmetric powers of a space X or of an algebra A are replaced by certain "generalized symmetric powers" Sym^{p|q}(X) and S^{p|q}A, which we introduce, and n-homomorphisms, by the new notion of "p|q-homomorphisms". Important tool of our study is a certain "characteristic function" R(f,a,z), which we introduce for an arbitrary linear map of algebras f, and whose functional properties with respect to the variable z reflect algebraic properties of the map f.Comment: LaTeX, 7 pages (3+4). In this new version we slightly edited the main text, and added to it an Appendix giving details of some constructions and a short direct proof of Buchstaber--Rees's main theore

On the Jacobian of minimal graphs in R^4

We provide a characterization for complex analytic curves among two-dimensional minimal graphs in $\mathbb{R}^{4}$ via the Jacobia

A Non-Crossing Approximation for the Study of Intersite Correlations

We develop a Non-Crossing Approximation (NCA) for the effective cluster problem of the recently developed Dynamical Cluster Approximation (DCA). The DCA technique includes short-ranged correlations by mapping the lattice problem onto a self-consistently embedded periodic cluster of size $N_c$. It is a fully causal and systematic approximation to the full lattice problem, with corrections ${\cal{O}}(1/N_c)$ in two dimensions. The NCA we develop is a systematic approximation with corrections ${\cal{O}}(1/N_c^3)$. The method will be discussed in detail and results for the one-particle properties of the Hubbard model are shown. Near half filling, the spectra display pronounced features including a pseudogap and non-Fermi-liquid behavior due to short-ranged antiferromagnetic correlations.Comment: 12 pages, 11 figures, EPJB styl

Carbonates in space - The challenge of low temperature data

Carbonates have repeatedly been discussed as possible carriers of stardust emission bands. However, the band assignments proposed so far were mainly based on room temperature powder transmission spectra of the respective minerals. Since very cold calcite grains have been claimed to be present in protostars and in Planetary Nebulae such as NGC 6302, the changes of their dielectric functions at low temperatures are relevant from an astronomical point of view. We have derived the IR optical constants of calcite and dolomite from reflectance spectra - measured at 300, 200, 100 and 10K - and calculated small particle spectra for different grain shapes, with the following results: i) The absorption efficiency factors both of calcite and dolomite are extremely dependent on the particle shapes. This is due to the high peak values of the optical constants of CaCO3 and CaMg[CO3]2. ii) The far infrared properties of calcite and dolomite depend also very significantly on the temperature. Below 200K, a pronounced sharpening and increase in the band strengths of the FIR resonances occurs. iii) In view of the intrinsic strength and sharpening of the 44 mum band of calcite at 200-100K, the absence of this band -- inferred from Infrared Space Observatory data -- in PNe requires dust temperatures below 45K. iv) Calcite grains at such low temperatures can account for the '92' mum band, while our data rule out dolomite as the carrier of the 60-65 mum band. The optical constants here presented are publicly available in the electronic database http://www.astro.uni-jena.de/Laboratory/OCDBComment: 20 pages, 10 figures, accepted by ApJ, corrected typo