Cost Problems for Parametric Time Petri Nets*

We investigate the problem of parameter synthesis for time Petri nets with a cost variable that evolves both continuously with time, and discretely when firing transitions. More precisely, parameters are rational symbolic constants used for time constraints on the firing of transitions and we want to synthesise all their values such that some marking is reachable, with a cost that is either minimal or simply less than a given bound. We first prove that the mere existence of values for the parameters such that the latter property holds is undecidable. We nonetheless provide symbolic semi-algorithms for the two synthesis problems and we prove them both sound and complete when they terminate. We also show how to modify them for the case when parameter values are integers. Finally, we prove that these modified versions terminate if parameters are bounded. While this is to be expected since there are now only a finite number of possible parameter values, our algorithms are symbolic and thus avoid an explicit enumeration of all those values. Furthermore, the results are symbolic constraints representing finite unions of convex polyhedra that are easily amenable to further analysis through linear programming. We finally report on the implementation of the approach in Romeo, a software tool for the analysis of time Petri nets.

Spatial Geometric Cells — Quasipolyhedra

Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.

On Polyhedral Realization with Isosceles Triangles

Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer i such that all convex i-iterated Kleetopes have a scalene face. However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles. We answer another question of Malkevitch by observing that a spherical tiling of Dawson (Renaissance Banff, Bridges Conference, pp. 489–496, 2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.

Continuous flattening of the 2-dimensional skeleton of a regular 24-cell

Bellow theorem says that any polyhedron with rigid faces cannot change its volume even if it is flexible. The problem on continuous flattenig of polyhedra with non-rigid faces proposed by Demaine et al. was solved for all convex polyhedra by using the notion of moving creases to change some of the faces. This problem was extended to a problem on continuous flattening of the 2-dimensional skeleton of higher dimensional polytopes. This problem was solved for all regular polytopes except three types, the 24-cell, the 120-cell, and the 600-cell. This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton, which is related to the Jitterbug by Buckminster Fuller.

Semi-infinite programming for trajectory optimization with non-convex obstacles

This article presents a novel optimization method that handles collision constraints with complex, non-convex 3D geometries. The optimization problem is cast as a semi-infinite program in which each collision constraint is implicitly treated as an infinite number of numeric constraints. The approach progressively generates some of these constraints for inclusion in a finite nonlinear program. Constraint generation uses an oracle to detect points of deepest penetration, and this oracle is implemented efficiently via signed distance field (SDF) versus point cloud collision detection. This approach is applied to pose optimization and trajectory optimization for both free-flying rigid bodies and articulated robots. Experiments demonstrate performance improvements compared with optimizers that handle only convex polyhedra, and demonstrate efficient collision avoidance between non-convex CAD models and point clouds in a variety of pose and trajectory optimization settings.

Data-based polyhedron model for optimization of engineering structures involving uncertainties

This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

Geometric model of microscopic raphide crystals in plant cells

In the present study, we showed that the microscopic structures of some plant crystals have the geometric model and mathematical formulas. Plant crystals are the storage of many mineral acid salts in many plants, such as chloride, phosphate, carbonate, silicate anhydrides and sulfates, formed due to metabolism. The crystals formed take different shapes. The shaping of plant crystals is not a simple structure. They are created in specific shapes and sizes by this biomineralisation process. Seventy-five per cent of flowering plants make one or more kinds of crystals. One of these is called a raphide crystal. Our study determined that the microscopic structures of some raphide crystals show the elongated triangular bipyramid that is a mathematics definition. In geometry, the elongated triangular bipyramid is one of the Johnson solids (J14), convex polyhedra, whose faces are regular polygons. At the same time, it was determined that the crystals show a minimal surface feature. The feature takes an essential place in geometry. The minimal surface feature provides the advantages of resistance and minimal space occupation to the crystals