Abstract
In this paper, we define \(\alpha \)-power sums of two or more elements on symmetric cones. For two elements, \(\alpha \)-power sums, which are generalized parallel sums, are defined on our previous paper. We mention interpolation for \(\alpha \)-power sums, which is not defined on our previous paper. It is shown that the synthesized resistances of \(\alpha \)-series parallel circuits naturally correspond to \(\alpha \)-power sums. We also mention relations with power sums and arithmetic, geometric, harmonic and \(\alpha \)-power means, where \(\alpha \) is a parameter of dualistic structure on information geometry.
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References
Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)
Bernstein, D.S.: Matrix Mathematics; Theory, Facts, and Formulas. Princeton University Press, New Jersey (2009)
Bhatia, R.: Positive Definite Matrices. Princeton University Press, New Jersey (2007)
Uhlmann, A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 80, 21–32 (1977)
Fujii, J., Kamei, E.: Uhlmann’s interpolational method for operator means. Math. Jap. 34, 541–547 (1989)
Kamei, E.: Paths of operators parametrized by operator means. Math. Jap. 39, 395–400 (1994)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Ohara, A.: Geodesics for dual connections and means on symmetric cones. Integr. Eq. Oper. Theory 50, 537–548 (2004)
Amari, S.: Information Geometry and Its Applications. Springer, Tokyo, Japan (2016)
Morley, T.D.: An alternative approach to the parallel sum. Adv. Appl. Math. 10, 358–369 (1989)
Berkics, P.: On parallel sum of matrices. Linear Multilinear A. 65, 2114–2123 (2017)
Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendon Press, Oxford (1994)
Uohashi, K., Ohara, A.: Jordan algebras and dual affine connections on symmetric cones. Positivity 8, 369–378 (2004)
Uohashi, K.: Generalized parallel sums on symmetric cones and series parallel circuits (2019, submitted)
Hambley, A.R.: Electrical Engineering: Principles and Applications, 4th edn. Prentice Hall, New Jersey (2007)
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Uohashi, K. (2019). \(\alpha \)-power Sums on Symmetric Cones. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_14
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DOI: https://doi.org/10.1007/978-3-030-26980-7_14
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