Abstract: Regularity has often been present in the form of regular
polyhedra or tessellations; classical examples are the nine regular
polyhedra consisting of the five Platonic solids (regular convex
polyhedra) and the four Kleper-Poinsot polyhedra. These polytopes
can be seen as regular maps. Maps are cellular embeddings of
graphs (with possibly multiple edges, loops or dangling edges) on
compact connected (closed) surfaces with or without boundary. The
n-dimensional abstract polytopes, particularly the regular ones, have
gained popularity over recent years. The main focus of research
has been their symmetries and regularity. Planification of polyhedra
helps its spatial construction, yet it destroys its symmetries. To our
knowledge there is no “planification” for n-dimensional polytopes.
However we show that it is possible to make a “surfacification”
of the n-dimensional polytope, that is, it is possible to construct a
restrictedly-marked map representation of the abstract polytope on
some surface that describes its combinatorial structures as well as
all of its symmetries. We also show that there are infinitely many
ways to do this; yet there is one that is more natural that describes
reflections on the sides ((n−1)-faces) of n-simplices with reflections
on the sides of n-polygons. We illustrate this construction with the
4-tetrahedron (a regular 4-polytope with automorphism group of size
120) and the 4-cube (a regular 4-polytope with automorphism group
of size 384).
Abstract: Vertex Enumeration Algorithms explore the methods and procedures of generating the vertices of general polyhedra formed by system of equations or inequalities. These problems of enumerating the extreme points (vertices) of general polyhedra are shown to be NP-Hard. This lead to exploring how to count the vertices of general polyhedra without listing them. This is also shown to be #P-Complete. Some fully polynomial randomized approximation schemes (fpras) of counting the vertices of some special classes of polyhedra associated with Down-Sets, Independent Sets, 2-Knapsack problems and 2 x n transportation problems are presented together with some discovered open problems.
Abstract: In this paper, a different architecture of a collision detection neural network (DCNN) is developed. This network, which has been particularly reviewed, has enabled us to solve with a new approach the problem of collision detection between two convex polyhedra in a fixed time (O (1) time). We used two types of neurons, linear and threshold logic, which simplified the actual implementation of all the networks proposed. The study of the collision detection is divided into two sections, the collision between a point and a polyhedron and then the collision between two convex polyhedra. The aim of this research is to determine through the AMAXNET network a mini maximum point in a fixed time, which allows us to detect the presence of a potential collision.
Abstract: Vertex configuration for a vertex in an orthogonal
pseudo-polyhedron is an identity of a vertex that is determined by the
number of edges, dihedral angles, and non-manifold properties
meeting at the vertex. There are up to sixteen vertex configurations
for any orthogonal pseudo-polyhedron (OPP). Understanding the
relationship between these vertex configurations will give us insight
into the structure of an OPP and help us design better algorithms for
many 3-dimensional geometric problems. In this paper, 16 vertex
configurations for OPP are described first. This is followed by a
number of formulas giving insight into the relationship between
different vertex configurations in an OPP. These formulas
will be useful as an extension of orthogonal polyhedra usefulness on
pattern analysis in 3D-digital images.
Abstract: This paper presents an interactive modeling system of
polyhedra using the isomorphic graphs. Especially, Conway
polyhedron notation is implemented. The notation can be observed as
interactive animation.
Abstract: This paper presents an interactive modeling system of
uniform polyhedra using the isomorphic graphs. Especially,
Kepler-Poinsot solids are formed by modifications of dodecahedron
and icosahedron.
Abstract: We present new finite element methods for Helmholtz and Maxwell equations on general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes.
Abstract: A rare phenomenon of SDS-induced activation of a latent protease activity associated with the purified silkworm excretory red fluorescent protein (SE-RFP) was noticed. SE-RFP aliquots incubated with SDS for different time intervals indicated that the protein undergoes an obligatory breakdown into a number of subunits which exhibit autoproteolytic (acting upon themselves) and/or heteroproteolytic (acting on other proteins) activities. A strong serine protease activity of SE-RFP subunits on Bombyx mori nucleopolyhedrovirus (BmNPV) polyhedral protein was detected by zymography technique. A complete inhibition of BmNPV infection to silkworms was observed by the oral administration assay of the SE-RFP. Here, it is proposed that the SE-RFP prevents the initial infection of BmNPV to silkworms by obliterating the polyhedral protein. This is the first report on a silkworm red fluorescent protein that exhibits a protease activity on exposure to SDS. The present studies would help in understanding the antiviral mechanism of silkworm red fluorescent proteins.
Abstract: The Chinese Postman Problem (CPP) is one of the
classical problems in graph theory and is applicable in a wide range
of fields. With the rapid development of hybrid systems and model
based testing, Chinese Postman Problem with Time Dependent Travel
Times (CPPTDT) becomes more realistic than the classical problems.
In the literature, we have proposed the first integer programming
formulation for the CPPTDT problem, namely, circuit formulation,
based on which some polyhedral results are investigated and a cutting
plane algorithm is also designed. However, there exists a main drawback:
the circuit formulation is only available for solving the special
instances with all circuits passing through the origin. Therefore, this
paper proposes a new integer programming formulation for solving
all the general instances of CPPTDT. Moreover, the size of the circuit
formulation is too large, which is reduced dramatically here. Thus, it
is possible to design more efficient algorithm for solving the CPPTDT
in the future research.
Abstract: This paper presents a novel algorithm for path planning of mobile robots in known 3D environments using Binary Integer Programming (BIP). In this approach the problem of path planning is formulated as a BIP with variables taken from 3D Delaunay Triangulation of the Free Configuration Space and solved to obtain an optimal channel made of connected tetrahedrons. The 3D channel is then partitioned into convex fragments which are used to build safe and short paths within from Start to Goal. The algorithm is simple, complete, does not suffer from local minima, and is applicable to different workspaces with convex and concave polyhedral obstacles. The noticeable feature of this algorithm is that it is simply extendable to n-D Configuration spaces.
Abstract: The symmetric solution set Σ sym is the set of all solutions to the linear systems Ax = b, where A is symmetric and lies between some given bounds A and A, and b lies between b and b. We present a contractor for Σ sym, which is an iterative method that starts with some initial enclosure of Σ sym (by means of a cartesian product of intervals) and sequentially makes the enclosure tighter. Our contractor is based on polyhedral approximation and solving a series of linear programs. Even though it does not converge to the optimal bounds in general, it may significantly reduce the overestimation. The efficiency is discussed by a number of numerical experiments.
Abstract: Imprecision is a long-standing problem in CAD design
and high accuracy image-based reconstruction applications. The visual
hull which is the closed silhouette equivalent shape of the objects
of interest is an important concept in image-based reconstruction.
We extend the domain-theoretic framework, which is a robust and
imprecision capturing geometric model, to analyze the imprecision in
the output shape when the input vertices are given with imprecision.
Under this framework, we show an efficient algorithm to generate the
2D partial visual hull which represents the exact information of the
visual hull with only basic imprecision assumptions. We also show
how the visual hull from polyhedra problem can be efficiently solved
in the context of imprecise input.