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It is convenient to denote cavity frequencies with a complex number ~ = /, where = (~) is the angular resonant frequency and = (~) is the inverse of the mode lifetime. Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics [1], and Waldron in the radio frequency domain. [2]
A crystal may have zero, one, or multiple axes of symmetry but, by the crystallographic restriction theorem, the order of rotation may only be 2-fold, 3-fold, 4-fold, or 6-fold for each axis. An exception is made for quasicrystals which may have other orders of rotation, for example 5-fold. An axis of symmetry is also known as a proper rotation.
Contributing structures of the carbonate ion. In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or polyatomic ions by the combination of several contributing structures (or forms, [1] also variously known as resonance structures or canonical structures) into a resonance hybrid (or hybrid structure) in valence bond theory.
The steric number of a central atom in a molecule is the number of atoms bonded to that central atom, called its coordination number, plus the number of lone pairs of valence electrons on the central atom. [11] In the molecule SF 4, for example, the central sulfur atom has four ligands; the coordination number of sulfur is four. In addition to ...
Haboush's theorem (algebraic groups, representation theory, invariant theory) Harnack's curve theorem (real algebraic geometry) Hasse's theorem on elliptic curves (number theory) Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry) Hironaka theorem (algebraic geometry) Hodge index theorem (algebraic surfaces)
Pushing a person in a swing is a common example of resonance. The loaded swing, a pendulum, has a natural frequency of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate. A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the ...
Thus, the lifetime of a particle is the direct inverse of the particle's resonance width. For example, the charged pion has the second-longest lifetime of any meson, at 2.6033 × 10 −8 s. [2] Therefore, its resonance width is very small, about 2.528 × 10 −8 eV or about 6.11 MHz. Pions are generally not considered as "resonances".
In such a scheme, the negative constitutive parameters are designed to appear around the Mie resonances of the inclusions: the negative effective permittivity is designed around the resonance of the Mie electric dipole scattering coefficient, whereas negative effective permeability is designed around the resonance of the Mie magnetic dipole ...