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Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by ...
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
In bioinformatics, the root mean square deviation of atomic positions is the measure of the average distance between the atoms of superimposed proteins. In structure based drug design , the RMSD is a measure of the difference between a crystal conformation of the ligand conformation and a docking prediction.
where is the root mean square width of the beam, is the Courant-Snyder , and is the fraction of the beam to be enclosed in the ellipse, given as a number between 0 and 1. Here the factor of π {\displaystyle \pi } is shown on the right of the equation, and would often be included in the units of emittance, rather than being multiplied in to the ...
The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. [1] The window function means that the signal near the time being analyzed will have higher weight.
Ruze's equation is an equation relating the gain of an antenna to the root mean square (RMS) of the antenna's random surface errors. The equation was originally developed for parabolic reflector antennas, and later extended to phased arrays. The equation is named after John Ruze, who introduced the equation in a paper he wrote in 1952. [1]
It, too, gives a Gaussian distribution. However, specific cases can also be precisely calculated. The average end-to-end distance for freely-rotating (not freely-jointed) polymethylene (polyethylene with each -C-C- considered as a subunit) is l times the square root of 2N, an increase by a factor of about 1.4.