Universal equations for Pythagorean and Sauter-type formulas of mean value calculation and classification of the Extended Pythagorean Means
1
Faculty of Geoengineering, Mining and Geology
Wroclaw University of Science and Technology, Wroclaw, POLAND
Corresponding author
Jan Drzymała
Faculty of Geoengineering, Mining and Geology Wroclaw University of Science and Technology ul. Na Grobli 15, 50-421 Wroclaw, POLAND
Faculty of Geoengineering, Mining and Geology Wroclaw University of Science and Technology ul. Na Grobli 15, 50-421 Wroclaw, POLAND
Mining Science 2017;24:227-235
KEYWORDS
TOPICS
ABSTRACT
A general equation, combining the Pythagorean and Sauter-type formulas, which can be used for calculation of a mean of a set of values is given and discussed in the paper. A classification of means, forming the Extended Pythagorean Means family, including power (arithmetic, geometric, quadratic, cubic etc.), inverted power (harmonic, inverted quadratic, inverted cubic etc.), and Sauter-type (volume-surface, volume-length etc.) is presented. The calculated means, depending on their type, are different. Formulas for calculating mean values for a set of values treated individually, in groups and as a fraction (weighted mean) are also given.
REFERENCES (17)
1.
mathematics, information theory and to the natural and social sciences, Encyclopedia of Mathematics.
4.
ANONIMUS, 2017, Generalized f-mean, Project Gutenberg Self-Publishing eBooks, World Heritage,.
5.
BULLEN P.S., 1990, Averages still on the move, Mathematics Magazine, 63 (4), 250–255.
6.
BULLEN S., 2003, Handbook of means and their inequalities, Second Edition, Kluwer Academic Publishers,Dordrecht, Mathematics and its Applications, Vol. 560.
7.
KOLMOGOROV A., 1930, On the notion of mean. In: Mathematics and Mechanics, Kluwer, 1991, 144–146.
8.
KOWALCZUK P.B., DRZYMALA J., 2016, Physical meaning of the Sauter mean diameter of spherical particulate matter, Part. Sci. Technol., 34(6), 645–647.
9.
NEUMANN E., PALES A.S., 2003, On comparison of Stolarsky and Gini means, J. Math. Anal. Appl. 278, 274–284.
10.
PACEK A.W., MAN C.C., NIENOW A.W., 1998, On the Sauter mean diameter and size distribution in turbulent liquid/liquid dispersion in a stirred vessel, Chemical Eng.Sci., 53, 2005–2011.
11.
SAUTER J., 1926, Die Grössenbestimmung der im Gemischnebel von Verbrennungskraft-maschinen Vorhandenen Brennstoffteilchen, Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, Hefte 279, 1–34, VDI-Verlag, GMBH, Berlin SW19.
13.
SOKOLOWSKI M., 1996, Energy consumed in comminution. A new of a general law of comminution. New test stands and testing results, Réc. progrés en génie des procédés, 10, Ed. Tec & Doc, Lavoisier,Paris.
14.
STOLARSKY K.B., 1996, Hölder means, Lehmer means, and x-1 log cosh x, J. Math. Anal. Appl., 202,810–818.
16.
ACZEL J., DHOMBRES J.G., 1989, Functional equations in several variables with applications to mathematics, information theory and to the natural and social sciences, Encyclopedia of Mathematics and its Applications, 31, Cambridge Univ. Press, Cambridge. Universal equations for Pythagorean and Sauter-type formulas... 235.
17.
ANONIMUS, 2017, Generalized f-mean, Project Gutenberg Self Publishing eBooks, World Heritage, self.gutenberg.org/artic/generalized_f-mean.
| eISSN: | 2353-5423 |
| ISSN: | 2300-9586 |
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