Finite Polygons Research Articles

Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $\mathbb R^d$

For any norm on \mathbb{R}^d with countably many extreme points, we prove that there is a set E \subset \mathbb{R}^d of Hausdorff dimension d whose distance set with respect to this norm has zero linear measure. This was previously known only for norms associated to certain finite polygons in \mathbb{R}^2 . Similar examples exist for norms that are very well approximated by polyhedral norms, including some examples where the unit ball is strictly convex and has C^1 boundary.

Polytopal composite finite elements for modeling concrete fracture based on nonlocal damage models

The paper presents an assumed strain formulation over polygonal meshes to accurately evaluate the strain fields in nonlocal damage models. An assume strained technique based on the Hu-Washizu variational principle is employed to generate a new strain approximation instead of direct derivation from the basis functions and the displacement fields. The underlying idea embedded in arbitrary finite polygons is named as Polytopal composite finite elements (PCFEM). The PCFEM is accordingly applied within the framework of the nonlocal model of continuum damage mechanics to enhance the description of damage behaviours in which highly localized deformations must be captured accurately. This application is helpful to reduce the mesh-sensitivity and elaborate the process-zone of damage models. Several numerical examples are designed for various cases of fracture to discuss and validate the computational capability of the present method through comparison with published numerical results and experimental data from the literature.

Division point measures from primitive substitutions

In this note we extend results of Olli concerning limits of point measures arising from substitutions. We consider a general primitive substitution on a finite polygon set in R2 and show that limits of certain atomic measures each converge to Lebesgue measure.

THE SHORTEST PATH THROUGH THE INTERIOR OF A POLYGON

As part or an expert system dealing with water penetration through window frames, the authors have developed an algorithm for finding the shortest path between two points within a compact, simply connected area in the 2D plane with a finite polygon boundary. The algorithm is based on A∗, a heuristic search algorithm, with improvements taking advantage of the knowledge that only certain points on the polygon need be considered, and a heuristic using the remaining distance to the destination. The algorithm illustrates the use of a generic artificial intelligence technique along with the importance of problem-specific knowledge for obtaining efficiency.

An Automated Finite Polygon Division Method for 3-D Objects

The automatic generation of color-shading data for CAD/CAM objects can produce mass-property calculations in less time with fewer errors.

Statistical Mechanics of LiquidHe4

The partition function proposed by Feynman for liquid ${\mathrm{He}}^{4}$, based on his path integral method, is evaluated for a simple cubic lattice considering long-range permutations as well as nearest-neighbor permutations (to which the previous analysis by one of the authors was restricted). The result indicates a second-order phase transition at the $\ensuremath{\lambda}$ point. The marked improvements over the previous treatment are: (1) the specific heat behaves as ${T}^{3}$ near absolute zero, (2) the specific heat peak is more pronounced at the $\ensuremath{\lambda}$ point, and (3) when triangles are added as possible finite polygons above ${T}_{\ensuremath{\lambda}}$ the specific heat just above ${T}_{\ensuremath{\lambda}}$ increases over the previous result, showing an improvement. Equating the theoretical $\ensuremath{\lambda}$ point with the experimental, a value for the effective mass of a helium atom about 1.6 times the normal mass is obtained.

Theorems on the Simple Finite Polygon and Polyhedron

Theorems on the Simple Finite Polygon and Polyhedron