### My discoveries in Mathematics

Andy Blunden

1. As a civil engineering student, my undergraduate research project was reviewing a paper which analysed the vibrations of a beam-column caused by a vibration taken to be white noise, using Fourier analysis. I showed that even though the displacement of the column derived from the Fourier series converged, the second derivative (curvature) was divergent, establishing the inappropriateness of white noise as an input and the limitations of Fourier analysis in this context.

2. In the first part of my PhD, I showed that the stationarity of a stochastic process was reflected in the statistical independence of the component harmonics of the Fourier transform, and conversely, the development of the process was reflected in the cross-correlation matrix of the Fourier series. This demonstrated the fallacy of the usual application of Fourier transforms to phenomena which are inherently nonstationary including earthquakes and economic history.

3. In the second part of my PhD, I showed that the accumulated effects of an extended natural process should be analysed in terms of a time-series of events rather than by means of representation by an analytical function of time. I also uncovered a fallacy in the application of Bayles Theorem to hypothesis testing.

4. In the third part of my PhD, I found that the great discovery of Ernst Gumble’s statistics of extremes (double logarithm liner curve-fitting) was simply based on a first order truncation of infinite series, without reference to the distribution of the base variable, calling into question its application in predicting extremes, even in in stationary conditions.

5. In a research project at NELP, using the calculus of variations, I demonstrated that least square fitting of a curve representing a composite spectrum, achieves a least square fit to the composition of the spectrum and derived a method for real time analysis of such spectra which minimised computational time.

6. I reduced by two rounds of computation, the bit-reversal method of implementing a Fourier transform on a digital series.