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In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians ; the solutions to which are given by eigenvalues corresponding to their modes of vibration.
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
The series converges extremely slowly. Baillie [4] remarks that after summing 10 24 terms the remainder is still larger than 1. [9]The upper bound of 80 is very crude. In 1916, Irwin [10] showed that the value of the Kempner series is between 22.4 and 23.3, since refined to the value above, 22.92067...
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rates such as speeds, [1] [2] ...
Voisin, Claire (2002), "A counterexample to the Hodge conjecture extended to Kähler varieties", International Mathematics Research Notices, 2002 (20): 1057– 1075, doi: 10.1155/S1073792802111135, MR 1902630, S2CID 55572794. Weil, André (1977), "Abelian varieties and the Hodge ring", Collected papers, vol. III, pp. 421– 429
The theorem is a corollary of Harnack's inequality. If u n (y) is a Cauchy sequence for any particular value of y, then the Harnack inequality applied to the harmonic function u m − u n implies, for an arbitrary compact set D containing y, that sup D |u m − u n | is arbitrarily small for sufficiently large m and n. This is exactly the ...