On the n-th row of the graded Betti table of an n-dimensional toric variety

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of projective space where we prove a special case of a conjecture of Ein and Lazarsfeld. We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.Comment: 20 pages, 9 figure

Crystallography and Chemistry of Perovskites

Despite the simplicity of the original perovskite crystal structure, this family of compounds shows an enormous variety of structural modifications and variants. In the following, we will describe several examples of perovskites, their structural variants and discuss the implications of distortions and non-stoichiometry on their electronic and magnetic properties.Comment: 11 pages, 8 figures, further information http://www.peter-lemmens.d

On the dynamics of sup-norm non-expansive maps

We present several results for the periods of periodic points of sup-norm non-expansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map $f\colon M\to M$, where $M\subset \mathbb{R}^n$, is at most $\max_k\, 2^k \big(\begin{smallmatrix}n\\ k\end{smallmatrix}\big)$. This upper bound is smaller than 3n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions $f\colon\mathbb{R}^n\to\mathbb{R}^n$ Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is $\big(\begin{smallmatrix}n\\ \lfloor n/2\rfloor\end{smallmatrix}\big)$. We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets

Unique geodesics for Thompson's metric

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting $x$ and $y$ in the cone of positive self-adjoint elements in a unital $C^*$-algebra if, and only if, the spectrum of $x^{-1/2}yx^{-1/2}$ is contained in $\{1/\beta,\beta\}$ for some $\beta\geq 1$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $C^\circ$ is the interior of a finite-dimensional closed cone $C$, then the Thompson's metric space $(C^\circ,d_C)$ can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, $C$ is a polyhedral cone. Moreover, $(C^\circ,d_C)$ is isometric to a finite-dimensional normed space if, and only if, $C$ is a simplicial cone. It will also be shown that if $C^\circ$ is the interior of a strictly convex cone $C$ with $3\leq \dim C<\infty$, then every Thompson's metric isometry is projectively linear.Comment: 30 page

Probability assignment in a quantum statistical model

The evolution of a quantum system, appropriate to describe nano-magnets, can be mapped on a Markov process, continuous in $\beta$. The mapping implies a probability assignment that can be used to study the probability density (PDF) of the magnetization. This procedure is not the common way to assign probabilities, usually an assignment that is compatible with the von Neumann entropy is made. Making these two assignments for the same system and comparing both PDFs, we see that they differ numerically. In other words the assignments lead to different PDFs for the same observable within the same model for the dynamics of the system. Using the maximum entropy principle we show that the assignment resulting from the mapping on the Markov process makes less assumptions than the other one. Using a stochastic queue model that can be mapped on a quantum statistical model, we control both assignments on compatibility with the Gibbs procedure for systems in thermal equilibrium and argue that the assignment resulting from the mapping on the Markov process satisfies the compatibility requirements.Comment: 8 pages, 2 eps figures, presented at the 26-th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 200