Minimum distance between a canal surface and a simple surface

The computation of the minimum distance between two objects is an important problem in the applications such as haptic rendering, CAD/CAM, NC verification, robotics and computer graphics. This paper presents a method to compute the minimum distance between a canal surface and a simple surface (i.e. a plane, a natural quadric, or a torus) by finding roots of a function of a single parameter. We utilize the fact that the normals at the closest points between two surfaces are collinear. Given the spine curve C(t), t_min≤t≤t_max, and the radius function r(t) for a canal surface, a point on the spine curve uniquely determines a characteristic circle on the surface. Normals to the canal surface at points on form a cone with a vertex and an axis which is parallel to Then we construct a function of t which expresses the condition that the perpendicular from C(t) to a given simple surface is embedded in the cone of normals to the canal surface at points on K(t). By solving this equation, we find characteristic circles which contain the points of locally minimum distance from the simple surface. Based on these circles, we can compute the minimum distance between given surfaces.     

The extension principle and a decomposition of fuzzy sets

We give an algorithm to decompose a fuzzy interval u. Using this decomposition and the multilinearization of a univariate function f, we obtain an approximation of the fuzzy interval , where is obtained from f by applying the extension principle. We provide approximation bounds. Some numeric illustration is provided.     

On the Euler sequence spaces which include the spaces ?p and ?∞ I

In the present paper, we introduce the Euler sequence space consisting of all sequences whose Euler transforms of order r are in the space ?_p of non-absolute type which is the BK-space including the space ?_p and prove that the spaces and ?_p are linearly isomorphic for 1 ? p ? ∞. Furthermore, we give some inclusion relations concerning the space . Finally, we determine the α-, β- and γ-duals of the space for 1 ? p ? ∞ and construct the basis for the space , where 1 ? p < ∞.     

List edge and list total colorings of planar graphs without short cycles

Let G be a planar graph with maximum degree Δ(G). We use and to denote the list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that and if Δ(G)?6 and G has neither C₄ nor C₆, or Δ(G)?7 and G has neither C₅ nor C₆, where Ck is a cycle of length k.     

Enhanced algorithms for Local Search

Let G=(V,E) be a finite graph, and be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity , so that we obtain a deterministic query complexity of , where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of . Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.