1,330 research outputs found
Realization spaces of 4-polytopes are universal
Let be a -dimensional polytope. The {\em realization space}
of~ is the space of all polytopes that are combinatorially
equivalent to~, 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~ defined over~, there is a -polytope whose realization
space is ``stably equivalent'' to~. This implies that the realization space
of a -polytope can have the homotopy type of an arbitrary finite simplicial
complex, and that all algebraic numbers are needed to realize all -
polytopes. The proof is constructive. These results sharply contrast the
-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
Resolution of signal components of a resultant electromagnetic wave received in a multipath field of microwave frequencies
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 sites as well as by the
high-temperature expansion (HTE) data. We find that due to pronounced
finite-size effects the FTLM data for are not representative for the
infinite system below . A similar restriction to
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 and for a single maximum in at . For the
susceptibility we find indications of a monotonous increase of
upon decreasing of reaching at . Moreover, the EM
allows to estimate the ground-state energy to .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
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