For the past decade, Wi-Fi signal fingerprinting algorithm has been considered as a prevalent solution for indoor positioning systems. Fingerprinting based methods require a massive database of Wi-Fi signal samples to calibrate the indoor positioning system and to achieve a high location accuracy. Traditionally the calibration procedure requires human intervention and is very time-consuming, which makes a large-scale deployments of indoor positioning systems non-trivial. Objective of this research to minimise the manual workload by combining the conventional sampling algorithm with signal prediction. In contrast to traditional algorithms, proposed method requires only few signal samples to be collected and rest of the data are approximated using Discrete Fourier Transforms. The main objective of our research is to reduce the calibration effort while maintaining an acceptable location accuracy of the indoor positioning systems.

This paper studies the method for establishing an approximate solution of nonlinear two dimensional Volterra integral equations (NLTD-VIE). The Newton-Kantorovich (NK) suppositions are employed to modify NLTD-VIE to the sequence of linear two dimensional Volterra integral equation (LTDVIE). The proper-ties of the two dimensional Gauss-Legenre (GL) quadrature fromula are used to abridge the sequence of LTD-VIE to the solution of the linear algebraic system. The existence and uniqueness of the approximate solution is demonstrated, and an illustrative example is provided to show the precision and authenticity of the method.

Finding a perfect triangle was stated as an open problem by Guy in [6]. Numerous researches have been done in the past to find such a triangle, unfortunately, to date, no one has ever found one, nor has proved its non-existence. However, on the bright side, there are partial results which show that there exist triangles that satisfy five or even six of the seven parameters to be rational. In this paper, we perform an extensive search to investigate if we can extract any perfect triangles from the curve C4 based on the final unsolved case in [9], which will then complete the proof of existence or non-existence of perfect triangle on the curve. Multiple conjectures were tested to eliminate the possibilities of finding a perfect triangle from the last unsolved case of n ≡ 3024 (mod 6052) in [9]. Finally, a theorem was proved, which was subtle enough to eliminate this case, proving that there does not exist any perfect triangle arising from the curve C4.