Application of Fibonacci Sequence and Lucas Sequence on the Design of the Toilet Siphon Pipe Shape

The purpose of this study was to explore the method for designing the toilet siphon pipe shape to improve flushing performance. The Fibonacci sequence and the Lucas sequence were used to design the structural parameters of the siphon pipe. The flushing processes of the toilet were simulated using the computational fluid dynamics (CFD) method to analyze the flushing performance under different siphon pipe shapes. Experimental studies were conducted to verify the reliability of the simulation results. The results indicated that when the Lucas numbers and the Fibonacci numbers were utilized to regulate the curvature of the siphon pipe in the Xi direction and the Yj direction respectively, the flushing performance of the toilet was optimal. In order to obtain better flushing performance, the curvature of the siphon pipe should be smooth and have obvious transitions at the connections of different sections. When the overall size of the siphon pipe is kept constant, a short siphon pipe length is helpful for the improvement of toilet flushing performance

Golden section, Fibonacci sequence and the time invariant Kalman and Lainiotis filters

We consider the discrete time Kalman and Lainiotis filters for multidimensional stochastic dynamic systems and investigate the relation between the golden section, the Fibonacci sequence and the parameters of the filters. Necessary and sufficient conditions for the existence of this relation are obtained through the associated Riccati equations. A conditional relation between the golden section and the steady state Kalman and Lainiotis filters is derived. A Finite Impulse Response (FIR) implementation of the steady state filters is proposed, where the coefficients of the steady state filter are related to the golden section. Finally, the relation between the Fibonacci numbers and the discrete time Lainiotis filter for multidimensional models is investigated. (C) 2014 Elsevier Inc. All rights reserved