In optics , a Gaussianbeam is a Light beambeam of electromagnetic radiation whose transverse electric ... . When refraction refracted by a lens optics lens , a Gaussianbeam is transformed into another Gaussianbeam characterized by a different set of parameters , which explains why it is a convenient, widespread mathematical models in physics model in laser optics. Image Gaussianbeam w40mm lambda30mm.png thumb 250px Instantaneous intensity of a Gaussianbeam. Image Laser gaussian profile.svg .... The mathematical function that describes the Gaussianbeam is a solution to the paraxial approximation ... of a Gaussian function, represents the complex number complex amplitude of the beam s electric ... form For a Gaussianbeam, the complex electric field amplitude is given by math E r,z E 0 frac ... Omega math . Beam parameters The geometry and behavior of a Gaussianbeam are governed by a set ... Beam diameter Image GaussianBeamWaist.svg thumb 350px right Gaussianbeam width w z as a function of the axial ... math total angular spread For a Gaussianbeam propagating in free space, the spot size w z will be at a minimum ... Because of this property, a Gaussian laser beam that is focused to a small spot spreads out rapidly ... . Non Gaussian beams also exhibit this effect, but a Gaussianbeam is a special case where the product of width and divergence is the smallest possible. Since the gaussianbeam model uses the paraxial ... the Gaussianbeam model is valid only for beams with waists larger than about 2 . Laser beam quality is quantified by the beam parameter product BPP . For a Gaussianbeam, the BPP is the product of the beam ... of the real beam to that of an ideal Gaussianbeam at the same wavelength is known as M sup 2 sup M squared . The M sup 2 sup for a Gaussianbeam is one. All real laser beams have M sup 2 sup values ... a key role in the analysis of gaussianbeam propagation, and especially in the analysis of optical ... of the complex beam parameter math q math , a Gaussian field with one transverse dimension is proportional ... more details
wiktionary beamBeam may refer to Beam structure , a construction element Beam nautical , the most extreme width or breadth of a nautical vessel, or a point alongside the ship at the mid point of its length A narrow, propagating stream of particles or energy Bessel beamGaussianbeam Light beam Laser beam Particle beam Charged particle beam Cathode ray , or electron beamBeam Inc. , a NYSE listed spirits company that produces Jim Beam , a popular brand of Bourbon whiskey , and a variety of other brands Beam music , a connection line in musical notation Balance beam , a piece of gymnastics equipment BEAM robotics , an automatically moving machine based on analog electronics Beam search , a search algorithm Beam Software , a computer game developer BEAM.TV , a global distribution network Blaster Beam , a musical instrument BEAM, an aggressive chemotherapy regimen involving the use of Carmustine BCNU Carmustine , Etoposide , Cytarabine Ara C Cytarabine , and Melphalan BEAM, neuroimaging brain electrical activity mapping BEAM, the virtual machine at the core of the Erlang programming language Erlang programming language Beam, operate the Transporter Star Trek Transporter in the fictional Star Trek series See also Beam theory , a means of calculating the load carrying and deflection characteristics of beams Battle of the Beams , radio countermeasures used against navigational system used by the Luftwaffe during WWII The Beam disambiguation disambiguation ang B am da Trave de Beam ja pt Trave simple Beam tl Beam th ... more details
The Beam may refer to The Beam instrument , a musical instrument created by Michael Stearns The Beam tale , Grimm s Fairy Tale s 149th story See also Beam disambiguation disambig ... more details
A T beam , used in construction , is a Structural engineering load bearing structure of reinforced concrete , wood or metal , with a t shaped Cross section geometry cross section . The top of the t shaped cross section serves as a flange or compression member in resisting compression physical compressive Stress physics stresses . The web of the Beam structure beam below the compression flange serves to resist shear stress and to provide greater separation for the coupled forces of bending . ref Cite book first Francis D.K. last Ching year 1995 title A Visual Dictionary of Architecture edition publisher John Wiley and Sons location New York pages 203 id ISBN 0 471 82451 3 ref The T beam has a big disadvantage compared to an i beam ibeam beam because it has no bottom flange with which to deal with tensile force s. One way to make a T beam more efficient structurally is to use an inverted T beam with a floor slab or bridge deck joining the tops of the beams. Done properly, the slab acts as the compression flange. Footnotes Reflist Concrete beams are often poured integrally with the slab, forming a much stronger T shaped beam. These beams are very efficient because the slab portion carries the compressive loads and the reinforcing bars placed at the bottom of the stem carry the tension. A T beam typically has a narrower stem than an ordinary rectangular beam. These stems are typically spaced from 4 0 apart to more than 12 0 . The slab portion above the stem is designed as a one way slab spanning between stems see Lecture 6 . External links http www.efunda.com math areas TBeam.cfm Diagram of a tapered T beam http www.efunda.com math areas SquareTbeam.cfm Diagram of a square T beam http www.roymech.co.uk Useful Tables Sections Tee Prop.htm BS 4 Standard steel T section beam tables, 1993 http sketchup.engineeringtoolbox.com bs 4 tee sections c 45.html BS 4 Standard steel T section beam tables, 1971 Category Structural system Element arch stub ... more details
Gaussian field may refer to A field of Gaussian rational s in number theory Gaussian free field , a concept in statistical mechanics A Gaussian random field , a field of Gaussian distributed random variables Disambig ... more details
For Gaussianbeam optics GaussianbeamGaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation , in which only rays which make small angles with the optical axis of the system are considered. ref A.Lipson, S.G.Lipson, H.Lipson, http books.google.it books?id aow3o0dhyjYC&pg PA51 v onepage&q&f false Optical Physics , IV edition, 2010, University Press, Cambridge, UK, p.51 ref In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere . In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements. Gaussian optics is named after Carl Friedrich Gauss , who showed that an optical system can be characterized by a series of Cardinal point optics cardinal points , which allow one to calculate its optical properties. ref W.J.Smith, http books.google.it books?id DrtM bAnf YC&pg PA22 v onepage&q&f false Modern optical engineering the design of optical systems , McGraw Hill 2007, p.22 ref References reflist Category Geometrical optics optics stub ... more details
astronomy stellar photometry , Gaussianbeam characterization, and emission spectrum emission spectroscopy ..., Gaussian functions are also associated with the vacuum state in quantum field theory . Gaussianbeam ...refimprove date August 2009 Image Normal Distribution PDF.svg thumb 360px right Normalized Gaussian curves ... &radic 2&pi , b &mu , c &sigma In mathematics , a Gaussian function named after Carl Friedrich ... of a function graph of a Gaussian is a characteristic symmetric bell curve shape that quickly ... of the centre of the peak, and c controls the width of the bell . Gaussian functions are widely ... to define Gaussian filter s, in image processing where two dimensional Gaussians are used for Gaussian ... s and to define the Weierstrass transform . Properties Gaussian functions arise by applying the exponential function to a general quadratic function . The Gaussian functions are thus those functions ... of maximum FWTM for a Gaussian could be of interest and is math mathrm FWTM 2 sqrt 2 ln 10 c math . Gaussian ... &infin is 0. Gaussian functions are among those functions that are Elementary function differential algebra elementary but lack elementary antiderivative s the integral of the Gaussian ... exactly, using the Gaussian integral math int infty infty e x 2 ,dx sqrt pi math and one ... if a 1 c &radic 2&pi , and in this case the Gaussian is the probability density ... figure. Gaussian functions centered at zero minimize the Fourier Fourier transform Uncertainty principle uncertainty principle . The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is again a Gaussian, with math c sqrt c 1 2 c 2 2 math . Taking ... of a Gaussian function with parameters a , b 0 and c yields another Gaussian function, with parameters ac , b 0 and 1 c . So in particular the Gaussian functions with b ... article with pi in the exponent, also the way that I prefer , the Gaussian must also have a pi in its ... more details
Unreferenced date July 2009 Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution , which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian distributed. A special case is white Gaussian noise , in which the values at any pairs of times are statistically independent and uncorrelated . In applications, Gaussian noise is most commonly used as additive white noise to yield additive white Gaussian noise . Explanation Gaussian noise is properly defined as the noise with a Gaussian amplitude distribution. This says nothing of the correlation of the noise in time or of the spectral density of the noise. Labeling Gaussian noise as white describes the correlation of the noise. It is necessary to use the term white Gaussian noise to be precise. Gaussian noise is sometimes misunderstood to be white Gaussian noise, but this is not the case. See also Gaussian process References Reflist physics stub probability stub DEFAULTSORT Gaussian Noise Category Noise Category Stochastic processes Category Time series analysis ar es Ruido gaussiano it Rumore gaussiano pt Ru do gaussiano su Gaussian noise zh ... more details
Orphan date February 2009 Contaminated Gaussian noise is a non Gaussian noise , drawn from a Gaussian mixture process. This process has the following probability density function math f x 1 epsilon phi x sigma 1 epsilon phi x sigma 2 math physics stub Category Noise ... more details
Unreferenced date September 2008 Gaussian broadening refers to broadening effects in spectral line s, these can be produced by Doppler broadening and natural broadening . This effect is similar to Gaussian blur effect in image processing produced by convolution with the Gaussian function . The term is named after Carl Friedrich Gauss . Category Emission spectroscopy Category Scattering Category Spectroscopy Category Article Feedback 5 physics stub ... more details
In probability theory and statistics , a Gaussian process is a stochastic process whose realizations ... random variables has a multivariate normal distribution . Gaussian processes are important in statistical ... is modelled as a Gaussian process, the distributions of various derived quantities can be obtained ... in estimating the average using sample values at a small set of times. Definition A Gaussian ... the Gaussian distribution , cn date April 2012 although Gauss was not the first to study that distribution see normal distribution History history of the normal Gaussian distribution . Alternative definitions Alternatively, a process is Gaussian if and only if for every finite set of indexed family ... , ldots, mathbf X t k math is a multivariate normal distribution multivariate Gaussian random variable ..., the Gaussian property can be formulated as follows X sub t sub t T is Gaussian ... Important Gaussian processes The Wiener process is perhaps the most widely studied Gaussian process ... process is a stationary process stationary Gaussian process. The Brownian bridge is a Gaussian ... is a Gaussian process whose covariance function is a generalisation of Wiener process. Applications A Gaussian process can be used as a prior probability distribution over Function mathematics functions ... Gaussian Processes for Machine Learning url http www.gaussianprocess.org gpml year 2006 publisher ... set of math N math points in the desired domain of your functions, take a multivariate Gaussian whose ... stochastic kernel kernel , and sampling mathematics sample from that Gaussian. Inference of continuous values with a Gaussian process prior is known as Gaussian process regression , or Kriging . ref ... publisher Springer Science Business Media Springer ref Gaussian processes are also useful as a powerful non linear interpolation tool. Gaussian processes can also be used for probabilistic classification. ref cite book last Rasmussen first C.E. coauthors Williams, C.K.I title Gaussian Processes ... more details
other uses GaussianGaussian is a computational chemistry software program initially released in 1970 ... year 2004 ref ref cite web url http www.gaussian.com g people pople.htm title John Pople and Gaussian ref and his research group at Carnegie Mellon University as Gaussian 70. ref W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, Gaussian 70 Quantum Chemistry Program Exchange ... A. A.2.4 Gaussian page 336 ref The name originates from Pople s use of Gaussian orbitals to speed up ... version of the program is Gaussian 09. ref cite web url http www.gaussian.com g tech g ur m citation.htm title Gaussian Citation ref Originally available through the Quantum Chemistry Program ... and licensed by Gaussian, Inc. Gaussian quickly became a popular and widely used electronic structure ... According to the most recent Gaussian manual, the package can do ref cite web url http www.gaussian.com g tech g09ur.htm title Gaussian online manual ref Molecular mechanics . AMBER . Universal ..., Gaussian86, Gaussian88, Gaussian90, Gaussian 92, Gaussian93, Gaussian 94, Gaussian95, Gaussian96, Gaussian 98, Gaussian 03, Gaussian 09 License controversy In the past, Gaussian, Inc. has attracted ... Banned By Gaussian ref has published a list of scientists who it claims are not permitted to use GAUSSIAN ... Chemical and Engineering News title Grumblings about Gaussian url http pubs.acs.org isubscribe ... on Computational Software ref Gaussian corporation disputes the accuracy of these descriptions ... on the Banned by Gaussian Website ref noting that all of the listed institutions do in fact have licenses ... Spartan TeraChem TURBOMOLE References reflist External links http www.gaussian.com Gaussian Homepage http www.bannedbygaussian.org Gaussian detractors http www.gaussian.com g misc silly.htm Gaussian s Response DEFAULTSORT Gaussian Category Computational chemistry software de GAUSSIAN es Gaussian fr Gaussian id GAUSSIAN it GAUSSIAN ja GAUSSIAN pl GAUSSIAN ru Gaussian zh GAUSSIAN ... more details
In number theory , a Gaussian integer is a complex number whose real and imaginary part are both integer s. The Gaussian integers, with ordinary addition and multiplication of complex numbers , form an integral domain , usually written as Z i . The Gaussian integers are a special case of the quadratic integer s. This domain does not have a total order ing that respects arithmetic. Image Gaussian integer lattice.png thumb 217px Gaussian integers as lattice point s in the complex plane Formally, Gaussian ... within the complex plane the Gaussian integers may be seen to constitute the 2 dimensional integer lattice . The Field norm norm of a Gaussian integer is the natural number defined as math ... with norm 1, i.e. the elements 1, &minus 1, i and &minus i . As a unique factorization domain The Gaussian ... i . If x is a Gaussian integer, the four numbers x , ix , &minus x , and &minus ix are called the associates of x . The prime element s of Z i are also known as Gaussian primes . An associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes. The positive integer Gaussian primes are the prime numbers Congruence class congruent to 3 modulo 4, OEIS A002145 . One should not refer to only these numbers as the Gaussian primes , which term refers to all the Gaussian primes, many of which do not lie in Z . ref name Gaussian Primes naming error http oeis.org A002145 COMMENT , OEIS sequence A002145 COMMENT section ref Image gauss primes 768x768.png 170px thumb Some of the Gaussian primes A Gaussian integer math a bi math is a Gaussian prime if and only if either one of a , b is zero and the other is a prime number ... 2 factors as math 2 1 i 1 i i 1 i 2 math as a Gaussian integer, the second factorisation in which i is a unit showing that 2 is divisible by the square of a Gaussian prime it is the unique prime number with this property. The necessary conditions can be stated as following if a Gaussian integer is a Gaussian ... more details
Linear analog electronic filter filter2 hide filter3 hide File Gaussian Filter.svg thumb Shape of a typical Gaussian filter In electronics and signal processing , a Gaussian filter is a filter signal processing filter whose impulse response is a Gaussian function . Gaussian filters are designed to give ... connected to the fact that the Gaussian filter has the minimum possible group delay . Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function this transformation is also known as the Weierstrass transform . Definition The one dimensional Gaussian filter ... of Fast Gaussian Binomial Filters for Speech and Image Processing, IEEE Transactions on Acoustics ... from the origin in the vertical axis, and is the standard deviation of the Gaussian distribution. Digital implementation Cleanup section date November 2008 The Gaussian function is non zero for math .... The filter function is said to be the kernel of an integral transform. The Gaussian kernel is continuous. Most commonly, the discrete equivalent is the sampled Gaussian kernel that is produced by sampling points from the continuous Gaussian. An alternate method is to use the discrete Gaussian kernel which has superior characteristics for some purposes. Unlike the sampled Gaussian kernel, the discrete Gaussian kernel is the solution to the discrete diffusion equation . Since the Fourier transform of the Gaussian function yields a Gaussian function, the signal preferably after being divided into overlapping windowed blocks can be transformed with a Fast Fourier transform , multiplied with a Gaussian ... is explicitly known. Due to the central limit theorem , the Gaussian can be approximated by several ... a cubic B spline as filter window which approximates the Gaussian quite well. Borrowing the terms ... s do. A gaussian kernel requires math 6 sigma 1 math values, eg. for a math sigma math of 3 ... is over using a gaussian rather than a poor approximation. When applied in two dimensions ... more details
merge to cyclotomic field date January 2012 discuss Talk cyclotomic field Gaussian rational In mathematics , a Gaussian rational number is a complex number of the form p qi , where p and q are both rational number s. The set of all Gaussian rationals forms the Gaussian rational field mathematics field , denoted Q i , obtained by adjoining the imaginary number i to the field of rationals. It thus provides an example of an algebraic number field , which is both a quadratic field and a cyclotomic field since i is a 4th root of unity . Like all quadratic fields it is a Galois extension of Q with Galois group cyclic group cyclic of order two, in this case generated by complex conjugation , and is thus an abelian extension of Q , with conductor algebraic number theory conductor 4. The field of Gaussian rationals is neither ordered field ordered nor topologically complete space complete . The Gaussian integer s Z i form the ring of integers of Q i . References Ian Stewart mathematician Ian Stewart , David O. Tall , Algebraic Number Theory , Chapman and Hall , 1979, ISBN 0412138409. Chap.3. Category Cyclotomic fields Numtheory stub fr Rationnel de Gauss it Intero di Gauss Campo dei quozienti ... more details
Image H3224020.GIF 200px The blurred image with 2 br Image blurred using Gaussian blur with 2 A Gaussian blur also known as Gaussian smoothing is the result of blurring an image by a Gaussian ... of focus lens or the shadow of an object under usual illumination. Gaussian smoothing is also used ... a Gaussian blur to an image is the same as convolution convolving the image with a Gaussian function ... of a Gaussian is another Gaussian, applying a Gaussian blur has the effect of reducing the image s high frequency components a Gaussian blur is thus a low pass filter . Mechanics File Halftone, Gaussian Blur.jpg thumb right Gaussian blur can be used in order to obtain a smooth grayscale digital image of a halftone print The Gaussian blur is a type of image blurring filter that uses a Gaussian ... mathematics transformation to apply to each pixel in the image. The equation of a Gaussian function ..., and is the standard deviation of the Gaussian distribution. When applied in two dimensions, this formula produces a surface whose contour s are concentric circles with a Gaussian distribution from .... The original pixel s value receives the heaviest weight having the highest Gaussian value and neighboring ... see also scale space implementation . In theory, the Gaussian function at every point on the image ... pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels ... close to that obtained by the entire gaussian distribution. In addition to being circularly symmetric, the Gaussian blur can be applied to a two dimensional image as two independent one dimensional ... matrix can also be achieved by applying a series of single dimensional Gaussian matrices in the horizontal ... image h text image right math for a non separable kernel. Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square ... successive gaussian blurs with radii of 6 and 8 gives the same results as applying a single gaussian ... more details
unsolved mathematics In the complex plane, is it possible to walk to infinity in the Gaussian integers using the Gaussian primes as stepping stones and taking bounded length steps? File Gaussian moat 15x15.svg thumb 300px The Gaussian primes with real and imaginary part at most seven, showing portions of a Gaussian moat of width two separating the origin from infinity In number theory , the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. More colorfully, if one imagines the Gaussian primes to be stepping stones in a sea of complex numbers, the question is whether one can walk from the origin to infinity with steps of bounded size, without getting wet. The problem was first posed in 1962 by Basil Gordon although it has sometimes been erroneously attributed to Paul Erd s and it remains unsolved. ref name gww citation last1 Gethner first1 Ellen last2 Wagon first2 Stan author2 link Stan Wagon last3 Wick first3 Brian doi 10.2307 2589708 issue 4 journal The American Mathematical Monthly mr 1614871 pages 327 337 title A stroll through the Gaussian primes volume 105 year 1998 . ref With the usual prime number s, such a sequence is impossible the prime number theorem implies that there are arbitrarily large prime gap gaps in the sequence of prime numbers, and this can also be proved directly for any n , the n &minus 1 consecutive numbers n 2, n 3, ..., n n are all composite. ref name gww The problem of finding a path between two Gaussian primes that minimizes the maximum hop size is an instance ... in each subset. Thus, the Gaussian moat problem may be phrased in a different but equivalent form ... results for Gaussian moat problem volume 88 year 2005 . ref It is known that, for any positive number k , there exist Gaussian primes whose nearest neighbor is at distance k or larger. In fact ... more details
Unreferenced stub auto yes date December 2009 A Gaussian year is defined as 365.2568983 day s. It was adopted by Carl Friedrich Gauss as the length of the sidereal year in his studies of the dynamics of the solar system. A slightly different value is now accepted as the length of the sidereal year, and the value accepted by Gauss is given a special name. A particle of negligible mass, that orbits a body of 1 solar mass in this period, has a mean axis for its orbit of 1 astronomical unit by definition. The value is derived from Kepler s laws of planetary motion Kepler s third law as math mbox 1 Gaussian year frac 2 pi k , math where k is the Gaussian gravitational constant . See also Astronomical unit DEFAULTSORT Gaussian Year Category Astronomy too general Category Units of time Category Celestial coordinate system Astronomy stub es A o gaussiano fr Ann e gaussienne it Anno gaussiano sk Gaussov rok sl Gaussovo leto zh ... more details
Unreferenced stub auto yes date December 2009 A Gaussian pyramid is a technique used in image processing, especially in texture synthesis . The technique involves creating a series of images which are weighted down using a Gaussian average Gaussian blur and scaled down. When this technique is used multiple times, it creates a stack of successively smaller images, with each pixel containing a local average that corresponds to a pixel neighborhood on a lower level of the pyramid. See also Pyramid image processing Laplacian pyramid Category Image processing Compu graphics stub de Gau Laplace Pyramide ... more details
Unreferenced date December 2009 In mathematics , Gaussian measure is a Borel measure on finite dimensional Euclidean space R sup n sup , closely related to the normal distribution in statistics . There is also a generalization to infinite dimensional spaces. Gaussian measures are named after the Germany German mathematician Carl Friedrich Gauss . Definitions Let n N and let B sub 0 sub R sup n sup denote the complete measure completion of the Borel sigma algebra Borel &sigma algebra on R sup n sup . Let sup n sup B sub 0 sub R sup n sup 0, denote the usual n dimensional Lebesgue measure . Then the standard Gaussian measure sup n sup B sub 0 sub R sup n sup 0, is defined by math gamma n A frac 1 sqrt 2 pi n int A exp left frac 1 2 x mathbb R n 2 right , mathrm d lambda n x math for any measurable ... generally, the Gaussian measure with mean R sup n sup and variance sup 2 sup > 0 is given by math ... n 2 right , mathrm d lambda n x . math Gaussian measures with mean 0 are known as centred Gaussian ... mu, sigma 2 n math as 0, and is considered to be a degenerate Gaussian measure in contrast, Gaussian measures with finite, non zero variance are called non degenerate Gaussian measures . Properties of Gaussian measure The standard Gaussian measure sup n sup on R sup n sup is a Borel measure ... so Gaussian measure is a Radon measure is not Translation geometry translation Invariant mathematics ... measure push forward of standard Gaussian measure by the translation map T sub h sub R ... in A gamma mu, sigma 2 n A . math Gaussian measures on infinite dimensional spaces It can be shown that There is no infinite ... dimensional vector space . Even so, it is possible to define Gaussian measures on infinite ... on a separable space separable Banach space E is said to be a non degenerate centered Gaussian measure ... degenerate centered Gaussian measure on R in the sense defined above. For example, Classical ... is a Gaussian measure. See also Cameron Martin theorem DEFAULTSORT Gaussian Measure Category Measures ... more details
Image Gaussian curvature.PNG thumb From left to right a surface of negative Gaussian curvature hyperboloid , a surface of zero Gaussian curvature cylinder geometry cylinder , and a surface of positive Gaussian curvature sphere . In differential geometry , the Gaussian curvature or Gauss curvature of a point ... is the content of Carl Friedrich Gauss Gauss s Theorema egregium . Symbolically, the Gaussian ... p on a regular surface in R sup 3 sup , the Gaussian curvature is also given by math K mathbf p det S mathbf p , math where S is the shape operator . A useful formula for the Gaussian curvature ... of f vanishes this can always be attained by a suitable rigid motion . Then the Gaussian curvature ... curvature is less than that of a plane triangle. The surface integral of the Gaussian curvature ... states that Gaussian curvature of a surface can be determined from the measurements of length on the surface ... of the Gaussian curvature of a surface S in R sup 3 sup certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the inner ... mathematics invariant . In particular, the Gaussian curvature is invariant under isometry isometric ... its image. Theorema Egregium is then stated as follows The Gaussian curvature of an embedded smooth surface in R sup 3 sup is invariant under the local isometries. For example, the Gaussian curvature ... regular of class C sup 2 sup closed surfaces in R sup 3 sup with constant positive Gaussian curvature ... C sup &omega sup regular surface in R sup 3 sup of constant negative Gaussian curvature. In fact, the conclusion ... for C sup 1 sup surfaces. The pseudosphere has constant negative Gaussian curvature except at its singular ... Reference Works. ref Alternative Formulas Gaussian curvature of a surface in R sup 3 sup can be expressed ... gives Gaussian curvature solely in terms of the first fundamental form math K left det begin vmatrix ... i.e., math F 0 math , Gaussian curvature is math K frac 1 2 sqrt EG left frac partial ... more details
Image NCEP T62 gaussian grid.png thumb NCEP T62 Gaussian grid points 400px right NCEP T62 Gaussian grid points A Gaussian grid is used in the earth science s as a gridded horizontal coordinate system for scientific modeling on a sphere i.e., the approximate shape of the Earth . The grid is rectangular, with a set number of orthogonal coordinates usually latitude and longitude . The gridpoints along the longitudes are equally spaced, while they are unequally spaced along the latitudes, where they are defined by their Gaussian quadrature . There are no grid points at the geographical pole poles . In a regular Gaussian grid, the number of gridpoints along the longitudes is constant, usually double the number along the latitudes. In a reduced or thinned Gaussian grid, the number of gridpoints in the rows decreases towards the poles, which keeps the gridpoint separation approximately constant across the sphere. Examples of Gaussian grids CCCma global climate model s of climate change http www.cccma.ec.gc.ca data grids geom llg 96x48.shtml 96 48 http www.cccma.ec.gc.ca data grids geom llg 128x64.shtml 128 64 European Centre for Medium Range Weather Forecasts http www.ecmwf.int publications manuals libraries interpolation gaussianGridsFIS.html http www.ecmwf.int publications manuals libraries interpolation n48FIS.html N48 192 96 http www.ecmwf.int publications manuals libraries interpolation n80FIS.html N80 320 160 http www.ecmwf.int publications manuals libraries interpolation n128FIS.html N128 512 256 http www.ecmwf.int publications manuals libraries interpolation n160FIS.html N160 640 320 http www.ecmwf.int publications manuals libraries interpolation n200FIS.html N200 800 400 http www.ecmwf.int publications manuals libraries interpolation n256FIS.html N256 1024 512 http www.ecmwf.int publications manuals libraries interpolation n400FIS.html N400 1600 800 http www.ecmwf.int publications manuals libraries interpolation n512FIS.html N512 2048 1024 http www.ecmwf.int publications ... more details
In computational chemistry and molecular physics , Gaussian orbitals also known as Gaussian type orbitals ... Peter M.W. title Molecular integrals Over Gaussian Basis Functions journal Advances in Quantum Chemistry ... The principal reason for the use of Gaussian Basis set chemistry basis functions in molecular quantum chemical calculations is the Gaussian Product Theorem , which guarantees that the product of two ... number of basis functions generally required in a Gaussian calculation. For reasons of convenience, many Gaussian integral evaluation programs work in a basis of Cartesian Gaussians even when spherical ... R r A r l e alpha r math Gaussian primitives for radial orbitals read math R r A r l e alpha r 2 math , where math A math is normalisation constant. Using tables of Gaussian orbitals There is a large database of Gaussian orbitals https bse.pnl.gov bse portal EMSL portal . The database determines ... define Gaussian primitives, which needs to be contracted summed over to determine a radial .... Radial orbital math R l r math is a sum of Gaussian primitives with an angular momentum ... p math are normalisation factors to ensure the norm of Gaussian primitives math r l A l, alpha exp ... form expression for normalisation factors math A l, alpha math can be obtained using Gaussian integrals ... over Cartesian Gaussian functions were first proposed by Boys ref S.F. Boys, http www.jstor.org ... calculations. McMurchie and Davidson 1978 introduced Hermite Gaussian functions to take advantage ... R. title One and two electron integrals over Cartesian Gaussian functions journal Journal of Computational ... involving contracted Gaussian basis functions. journal J. Comp. Phys. year 1978 volume 27 issue ..., A. title Efficient recursive computation of molecular integrals over Cartesian Gaussian functions ... on organic molecules with gaussian type functions. Introductory review and mathematical formalism , Theoretica Chimica Acta, 6, 191, 1966. ref was the first package for ab initio calculations using Gaussian ... more details
Image Gaussian2.jpg thumb A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, ideal wire. A Gaussian surface is a closed surface in three dimensional ... for. Gaussian surfaces are usually carefully chosen to exploit symmetry symmetries of a situation to simplify the calculation of the surface integral . If the Gaussian surface is chosen such that for every ... out of the integral. Common Gaussian surfaces see also charge distribution File SurfacesWithAndWithoutBoundary.svg right thumb 200px Examples of valid left and invalid right Gaussian surfaces. Left Some valid Gaussian surfaces include the surface of a sphere, surface of a torus, and surface ... as Gaussian surfaces, such as the disk mathematics disk surface , square surface, or hemisphere ... surface integral, the Gaussian surface commonly abbreviated G.S. or g.s. does not necessarily encompass ... the interior contribution. Furthermore it is not necessary to choose a Gaussian surface that utilises ... less laborious if an appropriate surface is chosen. Most calculations using Gaussian surfaces begin ... theorem and Coulomb s law . Spherical surface A sphere spherical Gaussian surface is used when finding ... with Circular symmetry spherical symmetry The spherical Gaussian surface is chosen so that it is concentric .... It is immediately apparent that for a spherical Gaussian surface of radius r R the enclosed charge is zero hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 by letting Q sub A sub 0 in Gauss s law, where Q sub A sub is the charge enclosed by the Gaussian surface . With the same example, using a larger Gaussian surface outside the shell where r R ... cylindrical Gaussian surface is used when finding the electric field or the flux produced by any ... lambda h math where, q is the charge enclosed in Gaussian surface. There are three surfaces a , b and c ... a , b and c . Image Gaussian surface.jpg right frame Closed surface in the form of cylinder having ... more details
File Carl Friedrich Gauss.jpg right thumb 250px Carl Gauss Gaussian units comprise a metric system of units ... called the Gaussian unit system , Gaussian cgs units , or often just cgs units . ref One of many examples of using the term cgs units to refer to Gaussian units is http nlpc.stanford.edu nleht ... units, not just Gaussian units, as described below. The most common alternative to Gaussian ..., and continue to increase in popularity at the expense of Gaussian units. ref name Rowlett ref name JacksonEditions Other alternative unit systems also exist, as discussed below. Conversions between Gaussian ... dimensions in another. History Gaussian units existed before the CGS system. The British Association report of 1873 that proposed the CGS contains gaussian units derived from the foot grain second and metre gram second as well. There are also references to foot pound second gaussian units. Alternative ... in electromagnetism l1 Alternative CGS units in electromagnetism The main alternative to the Gaussian ... for metre kilogram second. ref name Rowlett The Gaussian unit system is just one of several electromagnetic ..., scientific literature such as theoretical physics and astronomy , Gaussian units were predominant ... by J.D. Jackson. The second edition, published in 1975, used Gaussian units exclusively, but the third ... differences between Gaussian and SI units Rationalized unit systems One difference between Gaussian ... title Gaussian, SI and Other Systems of Units in Electromagnetic Theory work Physics 221A, University ... law , do have factors of 4 in them. In Gaussian units, which are not rationalized , the situation ... between Gaussian and SI units is in the definition of the unit of charge. In SI, a separate base ... units kilogram, metre, second . On the other hand, in Gaussian units, the unit of electrical charge ... s law in Gaussian units is particularly simple math F frac Q 1Q 2 r 2 math where F is the repulsive ... does not even exist in Gaussian units. Since the unit of charge is built out of mechanical units ... more details
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