×

Dimensional and scaling analysis. (English) Zbl 1456.00105

Summary: A complete theory of dimensional and scaling analysis is presented and its power is demonstrated through a series of examples. A vector-matrix exponentiation is introduced to simplify notation and calculus.

MSC:

00A71 General theory of mathematical modeling
00A79 Physics
15A03 Vector spaces, linear dependence, rank, lineability
15A04 Linear transformations, semilinear transformations
15A16 Matrix exponential and similar functions of matrices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Alexander, {\it Estimates of speeds of dinosaurs}, Nature, 261 (1976), pp. 129-130.
[2] R. Alexander, {\it Optimization and gaits in the locomotion of vertebrates}, Physiol. Rev., 69 (1989), pp. 1199-1227.
[3] F. Benford, {\it The law of anomalous numbers}, Proc. Amer. Philos. Soc., 78 (1938), pp. 551-572. · Zbl 0018.26502
[4] J. Bertrand, {\it Sur l’homogénéité dans les formules de physique}, Comptes Rendus, 86 (1878), pp. 916-920. · JFM 10.0041.01
[5] E. Buckingham, {\it On physically similar systems: Illustrations of the use of dimensional equations}, Phys. Rev., 4 (1914), pp. 345-376.
[6] E. Buckingham, {\it The principle of similitude}, Nature, 96 (1915), pp. 396-397, . · JFM 45.0949.03
[7] E. Carvallo, {\it Sur une similitude dans les fonctions des machines}, J. Phys. Theoret. Appl., 1 (1892), pp. 209-212. · JFM 24.1075.02
[8] M. Deakin, {\it G.I. Taylor and the Trinity test}, Internat. J. Math. Ed. Sci. Tech., 42 (2011), pp. 1069-1079.
[9] A. Federman, {\it On some general methods of integration of first-order partial differential equations}, Ann. Inst. Polytech. Pierre le Grand à St. Pétersbourg, 16 (1911), pp. 1-59.
[10] I. Helbing, D. Farkas and T. Vicsek, {\it Simulating dynamical features of escape panic}, Nature, 407 (2000), pp. 487-490.
[11] S. Mahajan, {\it Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving}, MIT Press, Cambridge, MA, London, 2010.
[12] M. Nigrini, {\it I’ve got your number – how a mathematical phenomenon can help CPAs uncover fraud and other irregularities}, J. Accountancy, 187 (1999), pp. 79-83.
[13] R. Pinkham, {\it On the distribution of first significant digits}, Ann. Math. Statist., 32 (1961), pp. 1223-1230. · Zbl 0102.14205
[14] B. Rauch, M. Göttsche, G. Brähler, and S. Engel, {\it Fact and fiction in EU-Governmental economic data}, German Econom. Rev., 12 (2011), pp. 243-255.
[15] D. Riabouchinsky, {\it Méthode des variables de dimension zéro et son application en aérodynamique}, l’Aérophile, (1911), pp. 407-408.
[16] C. Shannon, {\it A mathematical theory of communication}, Bell Syst. Tech. J., 27 (1948), pp. 379-423. · Zbl 1154.94303
[17] G. Taylor, {\it The formation of a blast wave by a very intense explosion. I. Theoretical discussion}, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Sci., 201 (1950), pp. 159-174. · Zbl 0036.26404
[18] G. Taylor, {\it The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945}, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Sci., 201 (1950), pp. 175-186, . · Zbl 0036.26404
[19] A. Vachy, {\it Sur les lois de similitude en physique}, Ann. Télégraphiques, 19 (1892), pp. 25-28.
[20] E. van Groesen and J. Molenaar, {\it Continuum Modeling in the Physical Sciences}, Math. Model. Comput. 13, SIAM, Philadelphia, 2007. · Zbl 1124.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.