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  Encyclopedia of Keywords > Science > Mathematics > Algebra > Polynomials > Approximation   Michael Charnine

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    This Review contains major "Approximation"- related terms, short phrases and links grouped together in the form of Encyclopedia article.


  1. An approximation is an inexact representation of something that is still close enough to be useful.
  2. An approximation (represented by the symbol ≈) is an inexact representation of something that is still close enough to be useful.
  3. The approximation is far more accurate for relatively small values of the parallax error when compared to the parallax. (Web site)
  4. The approximation is close to the true variance only in large samples.
  5. The approximation is known as first-order raytracing or Gaussian optics.


  1. Spaces between words offer an easy first approximation, but this approximation is not good enough for machine translation (MT), where many word sequences are not translated word-for-word. (Web site)


  1. Note that this is only an approximation that holds true when the orbiting body is of considerably lesser mass than the central one.
  2. This holds true to any (loop) order in perturbation theory, and thus it also holds true to trea level (zero loops), i.e., in the classical approximation. (Web site)
  3. Our figures on admission rates are therefore a good approximation of true, diagnosed incidence. (Web site)


  1. The approximation involves replacing the true distribution of the data (unknown) with the empirical distribution of the data. (Web site)
  2. Derivation of principal components as the best approximation to the data in a linear subspace. (Web site)
  3. Apr 01, 2006; Abstract: Here we were concerned with least square approximation by exponential functions for given data.


  1. We show that the pathwise uniqueness of the solution implies the convergence in the strong sense of the approximation to the solution.
  2. The most basic principle in approximation theory is this: the smoother the function, the faster the convergence as N - infty. (Web site)


  1. E = M + e * sin(M) * (1.0 + e * cos(M)) If e, the eccentricity, is less than about 0.05-0.06, this approximation is sufficiently accurate. (Web site)


  1. In the best case, this assumption is only an approximation. (Web site)
  2. To achieve this, an assumption known as the paraxial approximation is made.
  3. This assumption is mathematically exactly correct for a spherically symmetrical object (such as, to a reasonable approximation, a planet). (Web site)

Objective Lens

  1. As a first approximation (but see next paragraph), this is a good way of steering the incoming beam onto the optic axis of the objective lens. (Web site)
  2. Use the lateral magnification formula for the objective lens, and two rough equalities from the diagram to get an approximation for m ob. (Web site)


  1. Going beyond the Gross–Pitaevskii (GP) mean-field approximation, we show that bosons can localize and form polygonal-ring-like crystalline patterns.


  1. A cognitive model is an approximation to animal cognitive processes (predominantly human) for the purposes of comprehension and prediction. (Web site)
  2. Hawking radiation is a prediction of this semiclassical approximation.
  3. Linear regression is optimal linear (mean-square) prediction; we do this because we hope a linear approximation will work well enough over a small range. (Web site)

Best Linear Approximation

  1. The importance of the Jacobian lies in the fact that it represents the best linear approximation to a differentiable function near a given point.


  1. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once.


  1. For scattering by particles similar to or larger than a wavelength, see Mie theory or Discrete dipole approximation (they apply to Rayleigh regime as well). (Web site)


  1. The gradient of a function f from the Euclidean space to at any particular point x 0 in characterizes the best linear approximation to f at x 0. (Web site)


  1. If the model is, in fact, AVTD, the approximation in the symplectic numerical scheme should become more accurate as the singularity is approached.
  2. In all cases of durable state communism, there was some approximation to the Soviet "model".
  3. In these cases, geometric similitude and the same value of the Froude number in model and prototype produce a good approximation to dynamic similitude. (Web site)


  1. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation.


  1. As in all cases in the physical world, the trajectory is always an approximation of a parabola. (Web site)
  2. A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a parabola. (Web site)
  3. A second order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a parabola.


  1. Figure 2 shows the graph of an approximation to the antiderivative G(x), also truncated to 8 terms. (Web site)
  2. Using an approximation with fewer than n 0 terms may result in a very bad approximation. (Web site)
  3. A series expansion up to the 5th power in eccentricity gives an approximation to the true anomaly in terms of the mean anomaly. (Web site)

Linear Algebra

  1. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. (Web site)


  1. One expects that the functions S N ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. (Web site)
  2. Cutoffs have a physical motivation in zero energy ontology but it could be an excellent approximation to take them to infinity. (Web site)


  1. Many methods have been developed for calculating added mass, typically using the potential flow approximation. (Web site)
  2. An algorithm for calculating the Maximum Likelihood estimator is presented, based on data augmentation and stochastic approximation.


  1. The results indicate that the approximation based on limiting distribution are unsatisfactory unless number of treatments is very large.
  2. Results are not much better for the second-order approximation to the Euler equation.


  1. An approximation for the bias of the ML estimator of the AR parameters is also investigated.
  2. For example, the bias on the error calculated for log x increases as x increases since the expansion to 1+x is a good approximation only when x is small.


  1. The Hill sphere is but an approximation, and other forces (such as radiation pressure) can make an object deviate from within the sphere.
  2. To a very rough approximation, the nucleus can be considered a sphere of uniform charge density.
  3. Forty slices and twenty stacks give quite a good approximation for a sphere. (Web site)


  1. In Fig. 5, we display as a function of the parameters R δ the total energies for N = 6 bosons calculated at several levels of approximation.
  2. Stochastic approximation methods can be used to estimate the parameters.
  3. We examine the utility of the mean field approximation for the NNBM, and describe how Monte Carlo sampling techniques can be used to learn its parameters. (Web site)


  1. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation. (Web site)
  2. It means that the zero P values were replaced by 0.005 for the SSOWH and SDNB test when used with the chi-square approximation procedure.
  3. If, however, y and z are small, their values will not to a first approximation be altered if the electric and magnetic deflections occur simultaneously. (Web site)


  1. It is a good approximation, however, when the wavelength is very small compared with the size of structures with which the light interacts.
  2. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N, using a second order approximation.
  3. The linear approximation of the first order differential can be fitted to a trend of an O(2) polynomial trend in the BGP table size.


  1. The absolute error is the magnitude of the difference between the exact value and the approximation.
  2. The difference between reality and simulation includes at least two aspects: approximation and discretization. (Web site)
  3. To a first approximation, the modulation rate of a given filter is the difference between the adjacent frequencies processed by that filter. (Web site)

Quantum Mechanics

  1. However, it obscures the fact that all this talk of virtual states is just an approximation to quantum mechanics, in which energy is conserved at all times. (Web site)
  2. Even though these semiclassical models were only a first approximation, they were accurate for simple systems and they led to quantum mechanics. (Web site)
  3. In quantum mechanics, perturbation theory is a set of approximation schemes for describing a complicated quantum system in terms of a simpler one.


  1. My research interests are in information theory, signal processing, mathematical statistics, approximation theory, optimization and distributed algorithms.
  2. Additional topics include boundary value problems, evolution equations, optimization, and approximation.1979 edition. (Web site)
  3. Sadegh P., Constrained optimization via stochastic approximation with simultaneous perturbation gradient approximation, Automatica, 33, 1997, 889-892.


  1. Furthermore, the values in the point table 302 and slope table 304 are defined over a limited range for which the approximation is valid. (Web site)
  2. A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a straight line with a slope. (Web site)
  3. A first order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a straight line with a slope.


  1. By assuming that the boundary layer is thin, we can make the approximation that the static pressure is constant across the thickness of the boundary layer.

Mean Value Theorem

  1. Each rectangle, by virtue of the Mean Value Theorem, describes an approximation of the curve section it is drawn over.
  2. One can use the mean value theorem (for real-valued functions of many variables) to see that this does not rely on taking first order approximation. (Web site)


  1. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis.
  2. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables.

Best Approximation

  1. Proof. It's clear that the sequence of polynomials of best approximation converges to f(x) uniformly (due to Weierstrass approximation theorem). (Web site)
  2. But this is true due to a special property of polynomials of best approximation known from Chebyshev alternance theorem.
  3. The higher order rule is used to compute the best approximation to an integral over a small range.


  1. Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. (Web site)
  2. The Icosahedron is the most nearly spherical of all the Platonic Solids, but it is not really all that close an approximation of a sphere. (Web site)
  3. This may be the source of the common assumption that the icosahedron is the Platonic solid that gives the closest approximation to the sphere.


  1. First, they are solving a set of ordinary differential equations that are themselves an approximation to the underlying stochastic kinetics of the system. (Web site)
  2. The approach was verified by solving a test problem and its approximation error was analytically estimated for periodized wavelets.
  3. The flow field is computed by solving a finite volume approximation of the Navier–Stokes equations on moving grids augmented by turbulence models.


  1. As can be seen, as the number of terms rises, the error of the approximation is reduced in width and energy, but converges to a fixed height.
  2. To a rough approximation, factoring a number of a certain size and calculating the discrete logarithm of numbers the same size takes the same amount of work. (Web site)
  3. A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. (Web site)


  1. Other Alpha max plus beta min algorithm: an approximation of the square-root of the sum of two squares.
  2. To sum up, on first approximation, in order to predict the case of a subject in Georgian, one must know the Class and Series of the governing verb form. (Web site)
  3. If we minimize this sum, we will obtain a good approximation to the data.


  1. An approximate distribution of this statistic does not seem to be available, so there is no approximation or probability reported.
  2. However, if the observed statistic had been less extreme, the chi-square approximation would not have worked as well. (Web site)


  1. While direct computation of is rather complex, this technique provides a simple approximation by sampling (e.g.,) matrices from. (Web site)
  2. In this Electric Dipole approximation, we can make general progress on computation of the matrix element. (Web site)


  1. The concept of an ideal gas is an approximation often used in physics and chemistry in order to simplify calculations. (Web site)
  2. The calculations are based on the Born-Oppenheimer approximation, and incorporate the effects of the strong interaction through a scattering length approach.
  3. Newton's law of universal gravitation provide an excellent approximation for most calculations.


  1. Isaac Newton 's law of Universal Gravitation (1687) was a good approximation of the behaviour of gravity. (Web site)
  2. The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation.
  3. Although this approximation is crude, it allowed him to calculate the deflection of light by gravity, and show that it is nonzero. (Web site)

Quantum Theory

  1. General principles of quantum theory; approximation methods; spectra; symmetry laws; theory of scattering.
  2. As already mentioned, in quantum theory a molecule might be ascribed a definite shape by using the Born-Oppenheimer approximation. (Web site)


  1. The approximation to the chi-square distribution breaks down if expected frequencies are too low. (Web site)
  2. As long as the expected frequencies are sufficiently large, the chi-square approximation should be adequate for practical purposes. (Web site)


  1. Light and Optics The Greeks had applied the methods of geometry to the study of optics, and Ptolemy had a crude approximation to the law of refraction.


  1. These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods. (Web site)
  2. One of the many things this is useful for is expressing the point-wise error of a numerical approximation in terms of the mesh size.
  3. The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse).


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  Originally created: March 20, 2008.
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