In this report, we will look at the connection between the Fibonacci and Euler numbers. By using a combinatorial argument including the Fibonacci and Euler numbers, we will prove our main theorem:  Fn·En ≥ n! From the main theorem and the asymptotics of these numbers, we will conclude that π ≤ 2ϕ. We follow the proof in the article of Alejandro H. Morales, Igor Pak & Greta Panova, but we will give a more detailed proof and some extra facts about the Golden Ratio, ϕ, and the Fibonacci and Euler numbers. Finally, the article discusses the number of linear extensions of certain partially ordered sets, or posets. We see that there exist a two-dimensional poset Un and complement poset Ûn, both with n elements, such that the number of linear extensions are respectively En and Fn. We conclude that the Fibonacci and Euler numbers are related to each other.    

The whitening transformation transforms a random matrix into a whitened matrix with expectation 0 and covariance matrix I. By removing the first and second order statistical structures, higher order structures can be looked at for better classification. This is why Stage Gate 11 B.V. has employed whitening in the preprocessing of their hyperspectral data. The aim of this work is to gain insight into the whitening transformation and how it influences hyperspectral data. To gain this insight, synthetic data was created and used to make synthetic scans. The signal-to-noise ratio of a target spectrum was calculated, and Monte Carlo simulations were used to reveal hidden patterns in the data. In case of a high contrast scenario, multi-area whitening was employed and the cosine similarity between the target spectrum and its signature was determined. It was observed that the shape and intensity of the whitened target spectrum differs, depending on if pixels were used as observations or wavelengths. However, both are subject to the ‘bleeding’ effect. Further, it was found that if the number of pixels in the scan is greater than the number of spectral bands (548), then the signal-to-noise ratio becomes better as the number of whitened pixels in the scan increases. In case of a high contrast scenario, multi-area whitening guarantees the uniformity of the spectra, resulting in a higher cosine similarity between the target spectrum and its signature. But as multi-area whitening uses a smaller number of pixels in the scan, it cannot be concluded if multi-area whitening is better than global whitening, as it is not known how the increase in cosine similarity and the decrease in signal-to-noise ratio relate to the classification process. Finally, it is concluded that when working with real and unknown data, using pixels as observations is much more feasible.