If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

If each of the features makes an independent contribution to the output, then algorithms based on linear functions ( e. g., linear regression, logistic regression, Support Vector Machines, naive Bayes ) and distance functions ( e. g., nearest neighbor methods, support vector machines with Gaussian kernels ) generally perform well.

If the source has a Gaussian spectrum with FWHM spectral width, then a path offset of ± will reduce the fringe visibility to 50 %.

If jitter has a Gaussian distribution, it is usually quantified using the standard deviation of this distribution ( aka.

The Gaussian integers form a principal ideal domain with units 1, − 1, i, and − i. If x is a Gaussian integer, the four numbers x, ix, − x, and − ix are called the associates of x.

If is a Gaussian integer whose norm is a prime number, then is a Gaussian prime, because the norm is multiplicative.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C ( r ) of one complete trip around P. If the surface were flat, she would find C ( r ) = 2πr.

If the kernel used is a Gaussian radial basis function, the corresponding feature space is a Hilbert space of infinite dimensions.

If one is dealing with continuous data, a typical assumption is that the continuous values associated with each class are distributed according to a Gaussian distribution.

If the torus carries the ordinary Riemannian metric from its embedding in R < sup > 3 </ sup >, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0.

If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas.

If f ( x ) does not have many derivatives at all points, or if the derivatives become large, then Gaussian quadrature is often insufficient.

If is given as a Gaussian distribution

If we assume that P < sub > X </ sub >( x ) is Gaussian with variance σ < sup > 2 </ sup >, and if we assume that successive samples of the signal X are stochastically independent ( or, if you like, the source is memoryless, or the signal is uncorrelated ), we find the following analytical expression for the rate – distortion function:

If there is a static spherically symmetric Gaussian charge density

If n is sufficiently large, the above binomial distribution may be approximated by a Gaussian ( normal ) distribution and thus the Pearson test statistic approximates a chi-squared distribution,

If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw – Curtis quadrature are probably more suitable.

If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.

* If both principal curvatures are the same sign: κ < sub > 1 </ sub > κ < sub > 2 </ sub >> 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point.

* If the principal curvatures have different signs: κ < sub > 1 </ sub > κ < sub > 2 </ sub >< 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic point.

If the assembly broke the law, the only thing that might happen is that it would punish those who had made the proposal that it had agreed to.

If the proposal is rejected, the debtor is returned to his prior insolvent state and may have no alternative but to declare personal bankruptcy.

If debts are greater than $ 250, 000, the proposal must be filed under Division 1 of Part III of the Bankruptcy and Insolvency Act.

If a Committee manages to adopt a joint text, it then has to be approved in a third reading by both the Council and Parliament or the proposal is abandoned.

If an RFC is part of a proposal that is on the Standard Track, then at the first stage, the standard is proposed and subsequently organizations decide whether to implement this Proposed Standard.

If the proposal is approved by the Big-8 Management Board, the group is created.

If the proposal reaches consensus, the SIG Chair reports the decision at the APNIC Member Meeting ( AMM ) at the end of the week.

If any changes were made to the proposal during the APNIC meeting, this eight-week comment period gives the community the opportunity to comment on the modified proposal.

If the proposal is deemed to have reached consensus during the eight-week comment period, the SIG Chair will ask the APNIC Executive Council ( EC ) to endorse the proposal.

When President Bush's 2003 budget proposal threatened to cut education grants, she responded, " If he can run deficits for the military, then he can run deficits to educate our children.

If the proposal was accepted, then the elder hand can make another proposal, if desired, and can go on making proposals as long as the dealer accepts them.

If no initial proposal is made, the elder hand becomes a vulnerable player, leaving the dealer with a chance of scoring an extra point.

* If, after the trump card is turned up, and before proposing, or, if there is no proposal, before playing, it be discovered that the non-dealer has more than five cards, he may claim a fresh deal.

* If, after the trump card is turned up, and before proposing, or, if there is no proposal, before playing, it be discovered that the non-dealer has fewer than five cards, he may have his hand completed from the stock, or may claim a fresh deal.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that he has dealt himself more than five cards, the non-dealer may claim a fresh deal.

* If, after the trump card is turned up, and before the dealer accepts or refuses, or, if there is no proposal, before he plays, it be discovered that the dealer has fewer than five cards, the non-dealer may permit the dealer to complete his hand from the stock, or may claim a fresh deal.

If the offender win the point he marks nothing ; if he win the vole he marks only one ; if he win the point when his adversary has played without proposing, or has refused the first proposal, he marks only one.

If the dealer is not content with his own hand, he will give cards, but after the first proposal, it is entirely at his own option whether or not to do so.

If the offender win the point he marks nothing ; if he win the vole, he marks only one ; if he win the point when his adversary has played without proposing, or has refused the first proposal, he marks only one.

If the dealer refuse the first proposal, and fail to win the point, the non-dealer is entitled to mark two.

In 1970, at a meeting about a proposal for affordable housing, held at an all black church in Mount Laurel, Mayor Bill Haines summed up the newcomers perspectives by saying " If you people can't afford to live in our town, then you'll just have to leave.