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An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
[3] [5] A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space. The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, [ note 4 ] matches the speed of light ; indeed, light is one form of electromagnetic radiation (as are X-rays ...
In fact, all the finite-difference formulae are ill-conditioned [4] and due to cancellation will produce a value of zero if h is small enough. [5] If too large, the calculation of the slope of the secant line will be more accurately calculated, but the estimate of the slope of the tangent by using the secant could be worse. [6]
The Chebyshev polynomials can also be defined as the solutions to the Pell equation: () = in a ring R[x]. [6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution: T n ( x ) + U n − 1 ( x ) x 2 − 1 = ( x + x 2 − 1 ) n . {\displaystyle T_{n}(x)+U_{n-1}(x)\,{\sqrt {x^{2}-1 ...
[4] In phenomenological applications, it is often not clear whether the stretched exponential function should be used to describe the differential or the integral distribution function—or neither. In each case, one gets the same asymptotic decay, but a different power law prefactor, which makes fits more ambiguous than for simple exponentials.
2H 2 O → O 2 + 4H + + 4e − Oxidation (generation of dioxygen) 4H + + 4e − → 2H 2 Reduction (generation of dihydrogen) 2H 2 O → 2H 2 + O 2 Total Reaction Of the two half reactions, the oxidation step is the most demanding because it requires the coupling of 4 electron and proton transfers and the formation of an oxygen-oxygen bond.
Og, 118, oganesson : 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 10 6p 6 7s 2 5f 14 6d 10 7p 6 Note that these electron configurations are given for neutral atoms in the gas phase, which are not the same as the electron configurations for the same atoms in chemical environments.