• 한국수학사학회지
• /
• 제30권5호
• /
• pp.261-287
• /
• 2017

De Zolt's axiom which is a precise formulation of Euclid's Common Notion 5, "the whole is greater than the part", for the notion of 'content' holds in any Hilbert plane. In this article, we study the history of de Zolt's axiom which has its origin in Euclid's Common Notions, and introduce an example of a plane geometry in which de Zolt's axiom does not hold. We show that there is no area function in this geometry and every square is equidecomposable with a square which is properly contained in the first one. From this we also show that there are two equidecomposable rectangles which have the same base and do not have the same altitude, and there is a rectangle which is equicomplementable with an emptyset.

• 대한기계학회논문집A
• /
• 제31권9호
• /
• pp.967-974
• /
• 2007

A general form solution is considered for a piezoelectric material containing impermeable cracks subjected to a combined mechanical and in-plane electrical loading. The analysis is based upon the Hilbert problem formulation. Using this solution, typically for a central crack in transverse isotropic piezoelectric material, a closed form solution is obtained, where one concentrated mechanical and electrical load is subjected to the crack surface. This problem could be used as a Green's function to generate the solutions of other problems with the same geometry but of different loading conditions.