An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give several proofs and extensions of this beautiful formula to hexagon etc. and consider special cases of affine–regular polygons. The Gauss pentagon formula is, in...

A quick proof, by construction, of the first part of this result goes as follows: consider a rotation of π/3 radians about the point C that maps point A to point B, and M to M . Clearly, △MCM ′ is equilateral and MM ′ = MC. Since MA = M B, we conclude that △MBM ′ has sides equal to MA,MB, and MC (see Figure 1; all marked angles are equal to π/3 radians). The theorem is not classical in that not...

For emerging safety-critical systems, novel design methodologies are becoming necessary to cope with early stage design validation, performance and timing prediction, and design space exploration. In this paper, we propose a tool integration technique for architectural exploration of an aircraft electric power system (EPS) controller using Ptolemy II and Metro II to satisfy requirements imposed...

This paper intends to clarify recent efforts proposed by the Berkeley school in order to give a formal semantics to the Ptolemy toolbox. We show that the proposed semantics is indeed a generalisation of a Kahn semantics based on tag systems. We make this proposal a precise one by showing for several Ptolemy domains what is the corresponding tag system and how operators can be defined. We first ...

We present a solution to a complex multi-tone transient detection problem to illustrate the integrated use of symbolic and numeric processing techniques which are supported by well-established underlying models. Examples of such models include synchronous dataflow for numeric processing and the blackboard paradigm for symbolic heuristic search. Our transient detection solution serves to emphasi...

Recently, Takahashi has introduced the James and von Neumann-Jordan type constants. In this paper, we present some sufficient conditions for uniform normal structure and therefore the fixed point property of a Banach space in terms of the James and von Neumann-Jordan type constants and the Ptolemy constant. Our main results of the paper significantly generalize and improve many known results in...