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Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry ...
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional ...
It was found that an easily oxidized alcohol was an essential ingredient. A thin layer of about 10–15 μm was applied, which turned from yellow to dark green when it was cured. There is still a question as to the correct mechanism. Chrome green is a mixture of Prussian blue and chrome yellow, while the chrome oxide green is chromium(III ...
For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations. Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in ...
Lamport's bakery algorithm is a computer algorithm devised by computer scientist Leslie Lamport, as part of his long study of the formal correctness of concurrent systems, which is intended to improve the safety in the usage of shared resources among multiple threads by means of mutual exclusion.
The fine structure energy corrections can be obtained by using perturbation theory.To perform this calculation one must add three corrective terms to the Hamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to the spin–orbit coupling, and the Darwin term coming from the quantum fluctuating motion or zitterbewegung of the electron.
The solution would then be obtained by truncating the expansion to basis functions, and finding a solution for the (). In general, this is done by numerical methods, such as Runge–Kutta methods. For the numerical solutions, the right-hand side of the ordinary differential equation has to be evaluated repeatedly at different time steps.
Infrared (IR) spectroscopy by ATR is applicable to the same chemical or biological systems as the transmission method. One advantage of ATR-IR over transmission-IR is the limited path length into the sample. This avoids the problem of strong attenuation of the IR signal in highly absorbing media such as aqueous solutions.