Realization spaces of 4-polytopes are universal

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all $4$- polytopes. The proof is constructive. These results sharply contrast the $3$-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.Comment: 10 page

On the Finding of Final Polynomials

Final polynomials have been used to prove non-representability for oriented matroids, i.e. to decide whether geometric embeddings of combinatorial structures exist. They received more attention when Dress and Sturmfels, independently, pointed out that non-representable oriented matroids always possess a final polynomial as a consequence of an appropriate real version of Hilbert's Nullstellensatz. We discuss the more difficult problem of determining such final polynomials algorithmically. We introduce the notion of bi-quadratic final polynomials, and we show that finding them is equivalent to solving an LP-Problem. We apply a new theorem about symmetric oriented matroids to a series of cases of geometrical interest

Cayley-Bacharach Formulas

The Cayley-Bacharach Theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.Comment: 13 pages, 4 figure

The Complexity of Finding Small Triangulations of Convex 3-Polytopes

The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is proved to be NP-hard. We discuss other related complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof appeared at the proceedings of SODA 200

Adapting Planck's route to investigate the thermodynamics of the spin-half pyrochlore Heisenberg antiferromagnet

The spin-half pyrochlore Heisenberg antiferromagnet (PHAF) is one of the most challenging problems in the field of highly frustrated quantum magnetism. Stimulated by the seminal paper of M.~Planck [M.~Planck, Verhandl. Dtsch. phys. Ges. {\bf 2}, 202-204 (1900)] we calculate thermodynamic properties of this model by interpolating between the low- and high-temperature behavior. For that we follow ideas developed in detail by B.~Bernu and G.~Misguich and use for the interpolation the entropy exploiting sum rules [the ``entropy method'' (EM)]. We complement the EM results for the specific heat, the entropy, and the susceptibility by corresponding results obtained by the finite-temperature Lanczos method (FTLM) for a finite lattice of $N=32$ sites as well as by the high-temperature expansion (HTE) data. We find that due to pronounced finite-size effects the FTLM data for $N=32$ are not representative for the infinite system below $T \approx 0.7$. A similar restriction to $T \gtrsim 0.7$ holds for the HTE designed for the infinite PHAF. By contrast, the EM provides reliable data for the whole temperature region for the infinite PHAF. We find evidence for a gapless spectrum leading to a power-law behavior of the specific heat at low $T$ and for a single maximum in $c(T)$ at $T\approx 0.25$. For the susceptibility $\chi(T)$ we find indications of a monotonous increase of $\chi$ upon decreasing of $T$ reaching $\chi_0 \approx 0.1$ at $T=0$. Moreover, the EM allows to estimate the ground-state energy to $e_0\approx -0.52$.Comment: 17 pages, 24 figure

Towards many-class classification of materials based on their spectral fingerprints

Hyperspectral sensors are becoming cheaper and more available to the public. It is reasonable to assume that in the near future they will become more and more ubiquitous. This gives rise to many interesting applications, for example identification of pharmaceutical products and classification of food stuffs. Such applications require a precise models of the underlying classes, but hand-crafting these models is not feasible. In this paper, we propose to instead learn the model from the data using machine learning techniques. We investigate the use of two popular methods: support vector machines and random forest classifiers. In contrast to similar approaches, we restrict ourselves to linear support vector machines. Furthermore, we train the classifiers by solving the primal, instead of dual optimization problem. Our experiments on a large dataset show that the support vector machine approach is superior to random forest in classification accuracy as well as training time