Using Discrete Geometric Models in an Automated Layout

The application of discrete (voxel) geometric models in computer-aided design problems is shown. In this case, the most difficult formalized task of computer-aided design is considered—computer-aided layout. The solution to this problem is most relevant when designing products with a high density of layout (primarily transport equipment). From a mathematical point of view, these are placement problems; therefore, their solution is based on the use of a geometric modeling apparatus. The basic provisions and features of discrete modeling of geometric objects, their place in the system of geometric modeling, the advantages and disadvantages of discrete geometric models, and their primary use are described. Their practical use in solving some of the practical problems of automated layout is shown. This is the definition of the embeddability of the placed objects and the task of tracing and evaluating the shading. Algorithms and features of their practical implementation are described. A numerical assessment of the accuracy and performance of the developed geometric modeling algorithms shows the possibility of their implementation even on modern computers of medium power. This allows us to hope for the integration of the developed layout algorithms into modern systems of solid-state geometric modeling in the form of plug-ins

On the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Given the abstract evolution equation $$y'(t)=Ay(t),\ t\in \mathbb{R},$$ with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on $\mathbb{R}$. The important case of the equation with a normal operator $A$ in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at $0$, then all of them are strongly infinite differentiable on $\mathbb{R}$.Comment: A correction in Remarks 3.1, a few minor readability improvements. arXiv admin note: substantial text overlap with arXiv:1707.09359, arXiv:1706.08014, arXiv:1708.0506