Power theorem

Two-Secant Power Theorem— f given a circle C and an exterior point Q, let L be a secant line through Q, intersecting C at points R and S, and let be another secant line through Q, intersecting C at U and T, then... 

Two-Chord Power Theorem—RS and TU are chords of the same circle, intersecting at Q then... 

In its more general form, we have the power theorem... 

The most powerful theorem in group theory, for our purposes, is the great orthogonality theorem (GOT) which states that for irreps D and D, of respective dimensions na and n, ... 

Many methods for the fitting of data obtained from pulsed NMR have been described in the literature. The methods may, for example, be classified as either time domain (TD) or FD methods. Alternatively, they may be described as black-box methods or as interactive methods. An excellent review is given by de Beer and van Ormondt.20 There is now a consensus21 that FD and TD fitting methods are equivalent in terms of y2 parameter estimation if potential artifacts introduced by Fourier transformation are handled properly. TD and FD fitting can truly be equivalent (i.e. same 2 minima) if 2 is determined over the whole data range (a consequence of the power theorem on the Fourier transform) and if the model used to fit the experimental spectrum is correct. Very often, however, the model used is only an approximation. 

The Cayley-Hamilton theorem is one of the most powerful theorems of matrix theory. It states A matrix satisfies its own characteristic equation. That is, if the characteristic equation of an m X m matrix [A] is... 

Example 10.6.1 illustrates a powerful theorem due to Metropolis et al. (1973), They considered all unimodal maps of the form x , =rflx,), where f(x) also satisfies /(0) = /(l) = 0. (Forthe precise conditions, see their original paper.) Metropolis et al. proved that as r is varied, the order in which stable periodic solutions appear is independent of the unimodal map being iterated. That is, the periodic attractors always occur in the same sequence, now called the universal or V-sequence. This amazing result implies that the algebraic form of /(%) is irrelevant only its overall shape matters. 

In these more complicated examples, we are able to demonstrate ergodicity without finding an explicit solution (as in the Ornstein-Uhlenbeck example) or study of the dynamics generator (as in Brownian dynamics), given some assumptions on the behavior of solutions. We state (without proof) a powerful theorem on the ergodicity of degenerate stochastic diffusions, whose proof is essentially contained in [257, Theorem 2.5] (see also [44, 160, 161, 253, 266]). We denote by Hfix) the open ball in D centered on the points of radius while B(D) is the Borel a-algebra on T) (see Sect. 5.2.1). 

In recent years a very considerable bod of theory has been developed which makes it possible to understand and utilize the infrared and Raman spectra of a large number of polyatomic molecules. The purpose of this book is to develop essential elements of this theory, starting from its simplest form and advancing to fairly elaborate and powerful theorems useful in more complicated applications. In order to hold the volume lo a reasonable size, only vibrational spectra are treated in detail. However, vibrational (as opposed to rotational) spectra constitute the great bulk of published infrared and Raman data. The theory herein presented applies to gases and is a useful approximation for liquids but requires some extension for crystals. 

The power theorem is based upon Presvel s equality ... 

The claims that he underestimated the historical achievements of Russian scientists did hurt Frumkin, however, and briefly he turned his attention to the history of electrochemistry in Russia in order to reevaluate Russian contributions. He rated the studies of Moritz Hermann von Jacobi (21 September 1801-10 March 1874) most highly. (The Russian version of his name is Boris Semyonovich Yakobi. ) Jacobi had discovered the maximum power theorem, and his name is also associated with the development of galvanic cells for testing electric motors. In addition, Frumkin noted the priority of Pyotr Romanovich Bagration (24 September 1818-17 January 1876), who had created the first galvanic dry cell in 1843. Finally, Frumkin drew attention to the work of Kazan professor Robert Andreyevich Colley (Kolli) (25 June 1845-2 August 1891) back in 1878. Colley was the first person to use the shift of the electrode potential in a certain period of time as a measure of the interfacial capacitance and found a value of 150 pF cm for platinum. 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

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