# MATHEMATICAL SETI

## Statistics, Signal Processing, Space Missions

### Claudio Maccone

This book introduces the Statistical Drake Equation where, from a simple product of seven positive numbers, the Drake Equation is turned into the product of seven positive random variables. The mathematical consequences of this transformation are demonstrated and it is proven that the new random variable

*N* for the number of communicating civilizations in the Galaxy must follow the lognormal probability distribution when the number of factors in the Drake Equation is allowed to increase at will.

*Mathematical SETI* also studies the proposal FOCAL (Fast Outgoing Cyclopean Astronomical Lens) space mission to the nearest Sun Focal Sphere at 550 AU and describes its consequences for future interstellar precursor missions and truly interstellar missions. In addition the author shows how SETI signal processing may be dramatically improved by use of the Karhumen-Loève Transform (KLT) rather than Fast Fourier Transform (FFT). Finally, he describes the efforts made to persuade the United Nations to make the central part of the Moon Far Side a UN-protected zone, in order to preserve the unique radio-noise-free environment for future scientific use.

**Table of Contents**
Preface

Acknowledgments and dedication

Foreword by Frank Drake

Foreword by Giovanni Bignami

List of figures

List of tables

List of abbreviations and acronyms

**PART I SETI STATISTICS **

1. The statistical Drake equation

2. Letting Maxima do the calculations

3. How many planets for man and aliens?

4. Statistical Fermi paradox and Galactic travels

5. How long does a civilization live?

6. Life-span modeling by finite *b*-lognormals

7. Civilizations as finite *b*-lognormals: Mathematical history

8. Darwinian exponential growth and cladistics

9. Societal statistics by the statistical Drake equation

10. Cubics of historical recovery

11. Exponential evolution in time as a geometric Brownian motion

**PART II SPACE MISSIONS TO EXPLOIT GRAVITATIONAL LENSING**

12. So much to gain at 550 AU

13. FOCAL mission to 1,000 AU as an interstellar precursor

14. Belt of focal spheres between 550 and 17,000 AU

15. Galactic Internet by star gravitational lensing

16. Extragalactic Internet by black hole gravitational lensing

**PART III KLT FOR OPTIMAL SIGNAL PROCESSING**

17. A simple introduction to the KLT and BAM-KLT

18. KLT of radio signals from relativistic spaceships in uniform and decelerated motion

19. KLT of radio signals from relativistic spaceships in hyperbolic motion

20. KLT of radio signals from relativistic spaceships in arbitrary motion

21. Brownian motion and its time rescaling

22. Maccone first KLT theorem: KLT of all time-rescaled Brownian motions

23. KLT of the *B*(*t*^{2H})

24. Maccone second KLT theorem: KLT of all time-rescaled square Brownian motions

25. KLT of the *B*^{2}(*t*^{2H}) time-rescaled quare Brownian motion

26. Maccone third KLT theorem: Asymptotic KLT of GBM

27. A Matlab code for KLT simulations

28. KLT applications and a Fortran code (*Stephane Dumas*)

**PART IV THE UNITED NATIONS AND PROTECTION OF THE MOON'S FARSIDE**

29. The United Nations and protection of the Moon's farside

**PART V EPILOGUE**

30. Epilogue: Evolution, progress, and SETI

Index

**Extent: **760 pages, Black and white/colour images integrated with text

**Binding:** Hardback

**Published:** 2012

**ISBN: **978-3-642-27436-7

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