Definition 0.2. For any two sets X and Y, let Map (X, Y) denote the set of all maps from X to Y. Define a K-vector space M by M = ⊕ n≥0 Map (Rn, K) (where each Map (Rn, K) becomes a K-vector space by pointwise addition and multiplication with scalars). We make M into a K-algebra, whose multiplication is defined as follows: For any n ∈ N, any m ∈ N, any f ∈ Map (Rn, K) and g ∈ Map (Rm, K), we de...

There is a long history of constructions of non-shellable triangulations of -dimensional (topological) balls. This paper gives a survey of these constructions, including Furch’s 1924 construction using knotted curves, which also appears in Bing’s 1962 survey of combinatorial approaches to the Poincaré conjecture, Newman’s 1926 explicit example, and M. E. Rudin’s 1958 non-shellable triangulation...

The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function f : R → R and a convex polyhedron P ⊂ R, whether, for some initial point x0 in P, the trajectory of the unique solution to the differential equation ẋ(t) = f(x(t)),x(0) = x0, t ≥ 0, is entirely contained in P. We show that this problem is decidable, by reducing it in polynomial t...

We prove that [n/2J vertex guards are always sufficient and sometimes necessary to guard the surface of an n-vertex polyhedral terrain. We also show that l(4n 4)/13J edge guards are sometimes necessary to guard the surface of an n-vertex polyhedral terrain. The upper bound on the number of edge guards is ln/3J (Everett and Rivera-Campo, 1994). Since both upper bounds are based on the four color...

The serpentine minerals lizardite, polyhedral serpentine, chrysotile, antigorite and 15-sector 30-sector polygonal have been studied by micro-Raman spectroscopy, using selected samples, that had previously characterized. appropriate crystal orientations were determined optical microscopy of petrographic sections. Oriented spectra, obtained Nd-YAG green laser radiation (532 nm), deconvolved, ext...

Parallel and heterogeneous computing are growing in audience thanks to the increased performance brought by ubiquitous manycores and GPUs. However, available programming models, like OPENCL or CUDA, are far from being straightforward to use. As a consequence, several automated or semi-automated approaches have been proposed to automatically generate hardware-level codes from high-level sequenti...

The recent discovery that there is a tight polyhedral immersion of the projective plane with one handle, while there is no smooth tight immersion of the same surface, provides a rare example in low dimensions of a significant difference between smooth and polyhedral surfaces. In this paper the author shows that the obstruction to smoothing the polyhedral model is not local in nature, and descri...

Consider a convex polyhedral set represented by a system of linear inequalities. A prime representation of the polyhedron is one that contains no redundant constraints. We present a sharp upper bound on the difference between the cardinalities of any two primes.

In undergoing this life, many people always try to do and get the best. New knowledge, experience, lesson, and everything that can improve the life will be done. However, many people sometimes feel confused to get those things. Feeling the limited of experience and sources to be better is one of the lacks to own. However, there is a very simple thing that can be done. This is what your teacher ...

We propose a method to model and control topological changes by a smooth deformation of a polyhedral mesh using curves and loops. As changing the genus of a surface is not a continuous transformation, the topological change is made when an intermediate shape between the two topologies has been obtained. The creation and the deletion of holes are studied. The deletion of a hole uses non nullhomo...