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In reality, values of a specimen (its strength, etc.) have certain probability distributions. Deterministic values in reality are also uncertain and have error margins. Thus, averages and standard deviations are used for estimates in engineering and nature, and major factors in estimations are based on the nominal value and the upper and lower allowed (survival, design, standards) limits. Observations beyond those limits indicate mostly a failure. However, probability distributions in nature and engineering show a relatively long “tail” beyond the limits of several standard deviations. The values of deviations from the mean (or target) can be very large, which can endanger the whole specimen. The phenomena at the “tail” domain (and not in the domain up to the limits) are the main reason for the fact that larger specimens are generally more vulnerable than smaller ones, having the same mean and probability distribution. The present paper extends this concept to relevant phenomena in nature and in technology.
distribution tails, probability distributions, size effect, statistics, strength of specimen, upper standard limit
[1] Gould, S.J., Full House, Crown Publishers: New York, 1996.
[2] Rosenhouse, G., A universal law that explains why larger samples are more vulnerable than smaller ones. 5th Int. Conf. on Quality, Reliability, and Maintenance – QRM, ed. G.J. McNulty, Oxford: UK, pp. 315–318, 2004.
[3] da Vinci, Leonardo, References: McCurdy, E., The Notebooks of Leonardo da Vinci, 1500s, p. 546, 1945; or Ravaisson-Mollien, C., Les Manuscrits de Leonard de Vinci, translated to French by Institut de France (1881–1891), Vol. 3.
[4] Lund, J.R. & Byrne, J.P., Leonardo da Vinci’s tensile strength tests: implications for the discovery of engineering mechanics. Civil Eng. and Env. Syst., 18(3), pp. 243–250, 2001. (Permission of Gordon and Breach Sci. Publishers.)
[5] Rossmanith, H.P., The importance of engineering fracture mechanics in structural integrity: a short history of fracture mechanics. Technology, Law and Insurance, 2(4), pp. 195–229, 1997.
[6] Todhunter, I. & Pearson, K., A History of the Theory of Elasticity and the Strength of Materials from Galilei to the Present Time, Cambridge, 1886.
[7] Parsons, W.B., Engineers and Engineering in the Renaissance, MIT Press: Cambridge, MA, 1939.
[8] Beke, B.B., Principles of Comminution, Akademiai Kiado: Budapest, 1964.
[9] Fisher, R.A. & Tippett, L.H.C., Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proc. Cambridge Philosophical Society, 24, pp. 180–190, 1928.
[10] Weibull, W., The phenomenon of rupture in solids. Proc. Roy. Swedish Inst. of Engng Research, 153, pp. 1–55, 1939.
[11] Frécht, M., Sur la loi de probabilité de l’écart maximum. Ann. Soc. Plon. Math., 6, p. 93, 1927.
[12] Novák, D, Vorechovský, M., Pukl, R. & Cˇervenka, V., Statistical nonlinear fracture analysis: size effect of concrete beams. Int. Conference on Fracture Mechanics of Concrete Structures FraMCos 4, eds. De Borst et al. Balkema: Cachan, France, and Rotterdam, 2001.
[13] Galilei, Galileo, Limceo. Diuscorsi e Dimonstrrazioni Matematiche, intorno á due Nuove Scienze, Elsevirii: Leiden, 1638; or Galilei, Galileo, Dialogues Concerning Two New Sciences, translated by Henry Crew & Alfonso de Slavio, 1954,
[14] Kim, H.C., Elvin, D.E.C. & Rosenhouse, G., The source-strength specimen size effects on acoustic emission energy release during work-hardening of NaCl single crystals. Phil. Mag. A, 40(1), pp. 49–64, 1979.
[15] Ollendorff, F., Schaudinischky, H. & Rosenhouse, G., Noise level threshold for wakening and wake-up dependence on noise level. Acustica, 32(2), pp. 100– 03, 1975.
[16] Ficon, Federal Agency Review Selected Airport Noise Analysis Issues, Federal Interagency Committee on Noise: Washington, DC, 1992.
[17] Mariotte, E., Traité du Mouvement des Eaux, posthumously edited by M. de la Hire, English trans. by J.T. Desvaguliers, p. 249, 1718; or Maroitte Collected Works, 2nd edn., p. 1686, 1740.
[18] Hooke, R., De Potentia Restitutiva; or Of Spring Explaining the Power of Springing Bodies, p. 23, 1678 (1686).
[19] Griffi th, A.A., The phenomena of rupture and fl ow of solids. Philosophical Trans. Roy. Soc. of London, Series A, 221, pp. 163–198, 1921.
[20] Bernard, A.B. & Busse, M.R., Who wins the Olympic Games: economic resources and medal totals. Review of Economics and Statistics, 86(1), pp. 413–417, 2004.
[21] Ball, D.W., Olympic Games competition: structural correlates of national success. Journal of Comparative Sociology, 15, pp. 186–200, 1972.
[22] Johnson, D.K.N. & Ali, A., Coming to play or coming to win: participating and success at the Olympic Games, Wellesley College Working paper 2000-10, 2000.
[23] Hoffmann, R., Ging, L.C. & Ramasamy, B., The socio-economic determinants of international soccer performance. J. Appl. Economics, V(2), pp. 253–272, 2002.
[24] Turchin, P., War and Peace and War: The life Cycle of Imperial Nations, Pi Press: New York, 2006.
[25] Feigenbaum, A.V., Quality Control: Principles, Practice, and Administration, McGraw-Hill, New York, 1951; re-issue: Total Quality Control, McGraw-Hill Professional, 2004.
[26] Mikel, H. & Schroeder, R., Six Sigma, Random House: New York, 2002.
[27] Shirky, C., Power laws, weblogs and inequality (Chapter 3). Extreme Democracy, eds. J. Lebkowsky & M. Radcliffe, Lulu Marketplace: US, 2005. Brynjolfsson, E., Hu, J.Y. & Smith, M.D., Consumer surplus in the digital economy: estimating the value of increased product variety at online book sellers. Management Science, 49(11), pp. 1580–1596, 2003.
[28] Free, D., The original “long tail” entrepreneur. http://www.seedsofgrowth.com/the-originallong- tail-entrepreneur, September 21, 2006, accessed January 2008. Also: http://en.wikipedia.org/wiki/Interchangeable_parts, accessed January 2008.
[29] Li, W., Zipf’s law. http://www.nslij-genetics.org/wli/zipf, accessed January 2008. See also: http://planemath.org/encyclopedia/Zipfs Law.html, accessed January 2008.
[30] Anderson, C., The long tail. Wired, Issue 12.10, October 2004. http://www.wired.com/wired/archive/12.10/tail.html, accessed January 2008.
[31] Anderson, C. The Long Tail, Random House: London, 2006.
[32] Kilkki, K., A practical model for analyzing long tails. First Monday, 12(5), 2007. http://www.fi rstmonday.org/issues/issue12_54/kilkki, accessed January 2008.
[33] Nielsen, J., Data visualization of web states: logarithmic charts and the dropping tail. http://www.useit.com/alertbox/visualizing-traffi c-analysis.html, August 14, 2006, accessed January 2008.
[34] Evans, M., Hastings, N. & Peacock, B., Statistical Distributions, 3rd edn., Wiley: New York, 2000.