For Plato, `` imitation '' is twice removed from reality, being a poor copy of physical appearance, which in itself is a poor copy of ideal essence.

For many years, the Swedish Academy interpreted " ideal " as " idealistic " () and used it as a reason not to give the prize to important but less Romantic authors, such as Henrik Ibsen and Leo Tolstoy.

For a monatomic ideal gas,, and for a diatomic gas ( such as nitrogen and oxygen, the main components of air ).

For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum.

For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case.

For an ideal gas the internal energy is given by

For an ideal I, if a ∉ I and-a ∉ I, then I ∪

For example, they might interpret a tomato as a vegetable — according to the English definition of tomato — even though the ideal Blissymbol of vegetable was restricted by Bliss to just vegetables growing underground.

For example, the same techniques used to model ideal gases can be applied to model the behavior of a hard sphere colloidal suspension.

For a twentieth century thinker committed to the same ideal, government was an essential tool to the same end ... odern liberalism is instead the logical and sociological outcome of classical liberalism.

For an ideal gas the partial pressure is related to the molar concentration by the relation

For example, to synthesize iso-octane ( the 8-carbon ideal gasoline ) from the unfunctionalized alkane isobutane ( a 4-carbon gas ), isobutane is first dehydrogenated into isobutene.

For young Scarlett, the ideal southern belle is represented by her mother, Ellen O ' Hara.

The Southern belle was bred to conform to a subspecies of the nineteenth-century " lady "... For Scarlett, the ideal is embodied in her adored mother, the saintly Ellen, whose back is never seen to rest against the back of any chair on which she sits, whose broken spirit everywhere is mistaken for righteous calm ...

For him " In opposition to the ascetic ideal advocated by the dominant school of thought, hedonism suggests identifying the highest good with your own pleasure and that of others ; the one must never be indulged at the expense of sacrificing the other.

For the imaging scan, I-123 is considered an almost ideal isotope of iodine for imaging thyroid tissue and thyroid cancer metastasis.

For instance, the thymus provides an ideal environment for thymocytes to differentiate into a variety of different functional T cells.

" For me ," he said, " the important distinction is between a stylistic approach to the design ; and an analytical approach giving the process of due consideration to time, place, and purpose ... My analytical approach requires a full understanding of the three essential elements ... to arrive at an ideal balance among them.

For an HDTV set with slightly more than twice the number of rows of pixels, the ideal viewing distance is about half what it is for a standard definition set.

For ideal positional accuracy, it is desirable to operate on a navigational grid where the grid lines are closer to right angles ( orthogonal ) to each other.

For this reason, muonium is an ideal system for studying bound-state QED and also for searching for physics beyond the standard model.

For instance, the ring ( in fact field ) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial.

For example, the direct sum of the R < sub > i </ sub > form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.

For gases, the volume ratio is ideally the same by the ideal gas law, but the mass ratio of a single reaction has to be calculated from the molecular masses of the reactants and products.

For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2. 405.

For a Gaussian intensity ( i. e. power density, W / m < sup > 2 </ sup >) distribution in a single-mode optical fiber, the mode field diameter is that at which the electric and magnetic field strengths are reduced to 1 / e of their maximum values, i. e., the diameter at which power density is reduced to 1 / e < sup > 2 </ sup > of the maximum power density, because the power density is proportional to the square of the field strength.

For these reasons, single-mode fibers can have a higher bandwidth than multi-mode fibers.

For power-law profiles, single-mode operation occurs for a normalized frequency, V, less than approximately

For a Gaussian response system ( or a simple RC roll off ), the rise time is approximated by:

For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.

For example, if one assumes that data arise from a univariate Gaussian distribution, then one has assumed a Gaussian model:.

For example, a mixture of Gaussians with one Gaussian at each data point is dense is the space of distributions.

For example, a common choice for A < sub > o </ sub > is a Gaussian wave packet:

For example, a Gaussian wavefunction ψ might take the form:

For both families, the lowest-order solution describes a Gaussian beam, while higher-order solutions describe higher-order transverse modes in an optical resonator.

For a Gaussian beam propagating in free space, the spot size ( radius ) w ( z ) will be at a minimum value w < sub > 0 </ sub > at one place along the beam axis, known as the beam waist.

For a Gaussian beam, the BPP is the product of the beam's divergence and waist size.

For large numbers the Poisson distribution approaches a normal distribution, typically making shot noise in actual observations indistinguishable from true Gaussian noise except when the elementary events ( photons, electrons, etc.

For example, the inverse of cumulative Gaussian distribution

For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials and the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph Leo Doob in 1948, namely if the random variable in consideration has a finite probability space.

For any affine transformation of the Gaussian plane, z mapping to a z + b, a ≠ 0, a triangle is transformed but does not change its shape.

For a Gaussian distribution this is the best unbiased estimator ( that is, it has the lowest MSE among all unbiased estimators ), but not, say, for a uniform distribution.

For a Gaussian distribution, where, this means the MSE is minimized when dividing the sum by, whereas for a Bernoulli distribution with p = 1 / 2 ( a coin flip ),, the MSE is minimized for.

For example, for a laser producing pulses with a Gaussian temporal shape, the minimum possible pulse duration Δt is given by

For the HeNe laser with a 1. 5 GHz spectral width, the shortest Gaussian pulse consistent with this spectral width would be around 300 picoseconds ; for the 128 THz bandwidth Ti: sapphire laser, this spectral width would be only 3. 4 femtoseconds.

For large α, the shape of the Kaiser window ( in both time and frequency domain ) tends to a Gaussian curve.

For the Gaussian random approximation, the structure function of Tatarski ( 1961 ) can be described in terms

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the " unrolled " tube ( which is flat ).

* For an orthogonal parametrization ( i. e., ), Gaussian curvature is:

* For a surface described as graph of a function, Gaussian curvature is: