No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

Tangram is one of the oldest puzzle games in China. It can make up volatile and wonderful patterns. To master the game, it would be ideal if the players can go through several complexity levels from the easiest to the most complicated. However, tangram today does not have the difficulty grade. Only a similar puzzle game—Japanese jigsaw puzzle has the difficulty grade which is not very reasonable. And the grading process is not provided. Therefore, we try to provide the theoretical basis of dividing the difficulty grades by means of probability of the patterns. We hope we can make some complements to this ancient game. Among the complicated influential factors, we sum up three key elements: the number of answers，the length of side, the basic figures. Based on these points, we have devised three plans, the procedural thinking of computer, the length-of-side principle, division principle, to calculate the probability. Then, we made comprehensive analysis on the probability. Finally, we got the difficulty grade of some classical tangram patterns. Moreover, we summarize some skills of piecing together the patterns.

It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. In this work, we present stronger and more general versions of such a trace theorem in a new nonlocal function space $\cS(\Omega)$ satisfying $H^1(\Omega)\subset \cS(\Omega)\subset L^2(\Omega)$. The new space $\cS(\Omega)$ is associated with a nonlocal norm characterized by a nonlocal interaction kernel that is defined heterogeneously with a special localization feature on the boundary. Through the heterogeneous localization, we are able to show that the $H^{1/2}$ norm of the trace on the boundary can be controlled by the nonlocal norm that are weaker than the classical $H^1$ norm. In fact, the trace theorems can be essentially shown without imposing any extra regularity of the function in the interior of the domain other than being square integrable. Implications of the new trace theorems to the coupling of local and nonlocal equations and possible further generalizations are also discussed.