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Typically data is discretized into partitions of K equal lengths/width (equal intervals) or K% of the total data (equal frequencies). [1] Mechanisms for discretizing continuous data include Fayyad & Irani's MDL method, [2] which uses mutual information to recursively define the best bins, CAIM, CACC, Ameva, and many others [3]
Functions which are not smooth can be made smooth using a mollifier prior to discretization. As an example, discretization of the function that is constantly yields the sequence [..,,,,..] which, interpreted as the coefficients of a linear combination of Dirac delta functions, forms a Dirac comb.
A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures. Examples of the variational formulation are the Galerkin method, the discontinuous Galerkin method, mixed methods, etc.
In applied mathematics, the non-uniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both).
This means that in order to be accurate, the integration, for most discretization methods, must be done with a time step, , small enough such that a fluid particle moves only a fraction of the mesh spacing in each step. That is, = ′ <
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
For example, the Cauchy–Kowalevski theorem for Cauchy initial value problems essentially states that if the terms in a partial differential equation are all made up of analytic functions and a certain transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be non ...
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). [1] That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval.