[permalink] [id link]
The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.
from
Wikipedia
Some Related Sentences
Gaussian and theory
-- The theory of elasticity of Gaussian networks has been developed on a more general basis and the equations of state relating variables of pressure, volume, temperature, stress and strain have been precisely formulated.
Consequently the Gaussian theory only supplies a convenient method of approximating to reality ; and no constructor would attempt to realize this unattainable ideal.
This ray, named by Abbe a principal ray ( not to be confused with the principal rays of the Gaussian theory ), passes through the center of the entrance pupil before the first refraction, and the center of the exit pupil after the last refraction.
Gaussian molecular orbital methods were described in the 1986 book Ab initio molecular orbital theory by Warren Hehre, Leo Radom, Paul v. R.
Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers.
This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.
In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution, is a generalization of the one-dimensional ( univariate ) normal distribution to higher dimensions.
In applied mathematics, the Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instrument errors in filtering theory and unknown forces in control theory.
* Gaussian adaptation-Based on information theory.
They are also used in systems theory in connection with nonlinear operations on Gaussian noise.
In probability theory and statistics, a Gaussian process is a stochastic process whose realizations consist of random values associated with every point in a range of times ( or of space ) such that each such random variable has a normal distribution.
* Scale space for theory of Gaussian image smoothing and multi-scale feature detection
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity.
* A field of Gaussian rationals in number theory
Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae.
Generators for each such field can be written down by Gaussian period theory.
Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
The Gaussian integral is encountered very often in physics and numerous generalizations of the integral are encountered in quantum field theory.
As in quantum field theory the " fat tails " can only be obtained by complicated " nonperturbative " methods, mainly by numerical ones, since they contain the deviations from the usual Gaussian approximations, e. g. the Black-Scholes theory.
Often, in popular culture, an endangering huge wave is loosely denoted as a rogue wave, while it has not been ( and most often cannot be ) established that the reported event is a rogue wave in the scientific sense — i. e. of a very different nature in characteristics as the surrounding waves in that sea state and with very low probability of occurrence ( according to a Gaussian process description as valid for linear wave theory ).
In the language of the renormalization group, this theory is known as the Gaussian fixed point.
Gaussian and however
with an ideal uniform PRNG with range ( 0, 1 ) as input would produce a sequence of ( positive only ) values with a Gaussian distribution ; however
The desired acceptance rate depends on the target distribution, however it has been shown theoretically that the ideal acceptance rate for a one dimensional Gaussian distribution is approx 50 %, decreasing to approx 23 % for an-dimensional Gaussian target distribution.
Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface ; intuitively, this means that ants living on the surface could determine the Gaussian curvature.
The overall size of the mode is determined by the Gaussian beam radius w, and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation.
For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw – Curtis quadrature are generally far more accurate ; Clenshaw – Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.
Effects can be applied to a picture including Smart Blur -- a type of Gaussian blur effect which however retains sharpness around sharper edges -- Mesh Warp, Camera Lens Flare, Trace Contour and others.
In the second subcase, however, it is either impossible to absorb all of the torsion, or there is some ambiguity ( as is often the case in Gaussian elimination, for example ).
It is similar to standard minimum-shift keying ( MSK ); however the digital data stream is first shaped with a Gaussian filter before being applied to a frequency modulator.
Gaussian and is
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.
For a Gaussian response system ( or a simple RC roll off ), the rise time is approximated by:
This is in P, since an XOR-SAT formula is a system of linear equations mod 2, and can be solved by Gaussian elimination.
* A Gaussian function, a specific kind of function whose graph is a bell-shaped curve
* Normal ( Gaussian ) distribution, whose density function is a Gaussian function
" Four normal distribution | Gaussian distributions in statisticsThis unproved statement put a strain on his relationship with János Bolyai ( who thought that Gauss was " stealing " his idea ), but it is now generally taken at face value.
Among these choices, Gaussian units are the most common today, and in fact the phrase " CGS units " is often used to refer specifically to CGS-Gaussian units.
** Normal dynamics, is a stochastic motion having a Gaussian probability density function in position with variance MSD that follows, MSD ~ t, where MSD is the mean squared displacement of the process, and t is the time the process is seen ( normal dynamics and Brownian dynamics are very similar ; the term used depends on the field )
This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers.
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.
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations.
Gaussian elimination alone is sufficient for many applications, and requires fewer calculations than the Gauss – Jordan version.
The notes were widely imitated, which made ( what is now called ) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century.
0.429 seconds.